IMPERIAL COLLEGE LONDON

Counterparty Risk, Funding Risk and Central Clearing

Stephen Yang Zhang

Submitted in partial fulfilment of the requirements for the degree of Doctor of Philosophy in for Imperial College London and the Diploma of Imperial College London Counterparty , Funding Risk and Central Clearing

To Kate, Mum and Dad, for your support and encouragement throughout my life.

To Alicia, who continually makes me proud to be a father.

* I hereby certify that this thesis constitutes my own work and that all material, which is not my own work, has been properly acknowledged.

* The copyright of this thesis rests with the author and is made available under a Creative Commons Attribution Non- Commercial No Derivatives licence. Researchers are free to copy, distribute or transmit the thesis on the condition that they attribute it, that they do not use it for commercial purposes and that they do not alter, transform or build upon it. For any reuse or redistribution, researchers must make clear to others the licence terms of this work.

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Counterparty Credit Risk, Funding Risk and Central Clearing

Contents

Acknowledgement 3 List of Tables 4 List of Figures 6 1. Introduction 8 2. Counterparty Credit Risk and CVA/DVA 16 2.1 Introduction 16 2.2 General Pricing Formula of Unilateral CVA 24 2.3 No Arbitrage Pricing in Real Market 26 2.4 No Arbitrage Pricing across Different Asset Classes 39 2.5 No Arbitrage Pricing for Unilateral CVA 65 2.6 Bilateral CVA without First to Default Features 72 2.7 Approximations for CVA/DVA 76 2.8 General Pricing Formula ofof Bilateral CVA 80 2.9 CVA/DVA Pricing 83 3. Funding Risk and FVA 113 3.1 Background: Widening LIBOR-OIS Spread 113 3.2 FVA Pricing without Counterparty Credit Risk 116 3.3 Economic Drivers of FVA 124 3.4 FVA Pricing with Counterparty Credit Risk 135 3.5 CVA, DVA and FVA at Trade Level 145 3.6 CVA, DVA and FVA at Portfolio Level 152 3.7 xVA 159 3.8 FCA/FBA Accounting (FTP=ΔCA-ΔCL) 161 3.9 FVA/FDA Accounting (FTP=ΔCA) 164 3.10 DVA Hedging 168 4. Central Clearing and Initial Margins 189 4.1 Introduction to Central Clearing 189 4.2 A Brief Review of Clearing Mechanisms of Exchanges 191 4.3 Clearing for OTC Derivatives 193 4.4 Advantages and Disadvantages of Central Clearing 199 4.5 Regulatory Arbitrages Created by Central Clearing 206 4.6 xVA and Central Clearing 209 4.7 Risk Conversion by Requirements 213 4.8 CVA, DVA, FVA and MVA Pricing 224 5. Conclusion 237 Appendix 238 Reference 253

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Counterparty Credit Risk, Funding Risk and Central Clearing

Acknowledgement

I would like to thank my supervisor, Prof. Enrico Biffis, for his support, encouragement and patience over my 4 years PhD studies at Imperial College. And I am grateful for many helpful discussions with Prof. Damiano Brigo, Dr. Laura Ballotta, Mr. Jonathan Durden, Mr. Tianyu Wang, Dr. Nicholas Burgess, and Dr. Warwick Palmer.

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Counterparty Credit Risk, Funding Risk and Central Clearing

List of Tables

Table 2.1: Asymmetric Treatments of the Surviving Party in the Default State of Its Counterparty 67 Table 2.2: Term Structure of Dealer’s CDS Curve 83 Table 2.3: CVA and DVA for Uncollateralised 10 Years Spot Starting IRS Fixed Payer 86 Table 2.4: CVA and DVA for Uncollateralised 10 Years Spot Starting IRS Fixed Receiver 86 Table 2.5: CVA and DVA for Uncollateralised 5x15 Forward Starting IRS Fixed Payer 88 Table 2.6: CVA and DVA for Uncollateralised 5x15 Forward Starting IRS Fixed Receiver 88 Table 2.7: CVA and DVA for Collateralised 5x15 Forward Starting IRS Fixed Payer 90 Table 2.8: CVA and DVA for Collateralised 5x15 Forward Starting IRS Fixed Receiver 90 Table 2.9: CVA and DVA for Uncollateralised 5 into 10 Settled European Payer 92 Table 2.10: CVA and DVA for Uncollateralised 5 into 10 Cash Settled European Receiver Swaption 92 Table 2.11: CVA and DVA for Uncollateralised 5 into 10 Physically Settled European Payer Swaption 94 Table 2.12: CVA and DVA for Uncollateralised 5 into 10 Physically Settled European Receiver Swaption 94 Table 2.13: CVA and DVA for Uncollateralised 5 into 10 Physically Settled Bermudan Payer Swaption 96 Table 2.14: CVA and DVA for Uncollateralised 5 into 10 Physically Settled Bermudan Receiver Swaption 96 Table 2.15: CVA and DVA for Collateralised 5 into 10 Physically Settled European Payer Swaption 99 Table 2.16: CVA and DVA for Collateralised 5 into 10 Physically Settled European Receiver Swaption 99 Table 2.17: CVA and DVA for Collateralised 5 into 10 Physically Settled Bermudan Payer Swaption 99 Table 2.18: CVA and DVA for Collateralised 5 into 10 Physically Settled Bermudan Payer Swaption 99 Table 2.19: CVA and DVA for Uncollateralised 15 Years FX (EURUSD) Forward Long EUR at Maturity 101 Table 2.20: CVA and DVA for Uncollateralised 15 Years FX (EURUSD) Forward Short EUR at Maturity 101 Table 2.21: CVA and DVA for Uncollateralised 15 Years FX (EURUSD) Long EUR at Maturity 103 Table 2.22: CVA and DVA for Uncollateralised 15 Years FX (EURUSD) Swap Short EUR at Maturity 103 Table 2.23: CVA and DVA for Collateralised 15 Years FX (EURUSD) Forward Long EUR at Maturity 106 Table 2.24: CVA and DVA for Collateralised 15 Years FX (EURUSD) Forward Short EUR at Maturity 106 Table 2.25: CVA and DVA for Collateralised 15 Years FX (EURUSD) Swap Long EUR at Maturity 107 Table 2.26: CVA and DVA for Collateralised 15 Years FX (EURUSD) Swap Short EUR at Maturity 107 Table 2.27: CVA and DVA for Uncollateralised 15 Years EURUSD CCS Long EUR at Maturity 109 Table 2.28: CVA and DVA for Uncollateralised 15 Years EURUSD CCS Short EUR at Maturity 109 Table 2.29: CVA and DVA for Collateralised 15 Years EURUSD CCS Long EUR at Maturity 110 Table 2.30: CVA and DVA for Collateralised 15 Years EURUSD CCS Short EUR at Maturity 110 Table 2.31: Counterparty Credit Risk Reduction by Variation Margin for IRDs 111 Table 2.32: Counterparty Credit Risk Reduction by Variation Margin for FXDs 112 Table 3.1: Explanatory Power of Principal Components of GS, MS and JPM 175 Table 3.2: Summary of R^2 of CDS as Explanatory Variable for GS, MS and JPM 180 Table 3.3: Summary of R^2 of 1 Lag in CDS as Explanatory Variable for GS, MS and JPM 182 Table 3.4: Summary of R^2 of 1st Order Changes in CDS as Explanatory Variable for GS, MS and JPM 184 Table 3.5: Summary of R^2 of three different explanatory variables as Explanatory Variable for GS, MS and JPM 184 Table 3.6: Summary of R^2 of 1st Order Changes in CDS as Explanatory Variable for DVAs GS, MS and JPM 186 Table 3.7: DVA Book Sensitivity to CDS changes for GS, MS and JPM 187 Table 3.8: DVA Hedging Equity Portfolio Holdings 188 Table 3.9: Risk/Return Profile of DVA Hedging Equity Portfolio 188 Table 4.1: BCBS-ISOCO (2013) Standardised Initial Margin Schedule 222

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Counterparty Credit Risk, Funding Risk and Central Clearing

Table 4.2: the Term Structure of Derivative Dealer’s Funding Spread 224 Table 4.3: xVA for 5x15 Froward Starting Fixed Payer under Different Margin Requirements 226 Table 4.4: xVA for 5x15 Froward Starting Fixed Receiver under Different Margin Requirements 227 Table 4.5: xVA for 5 into 10 European Payer Swaption under Different Margin Requirements 228 Table 4.6: xVA for 5 into 10 European Receiver Swaption under Different Margin Requirements 229 Table 4.7: xVA for 5 into 10 Bermudan Payer Swaption under Different Margin Requirements 230 Table 4.8: xVA for 5 into 10 Bermudan Receiver Swaption under Different Margin Requirements 231 Table 4.9: xVA for 15 Years EURUSD Cross Long EUR at Maturity under Different Margin Requirements 231 Table 4.10: xVA for 15 Years EURUSD Cross Currency Swap Short EUR at Maturity under Different Margin Requirements 233

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Counterparty Credit Risk, Funding Risk and Central Clearing

List of Figures

Figure 2.1: Uncollateralised 10 Years Spot Starting IRS Fixed Payer Exposure Profile 85 Figure 2.2: Uncollateralised 10 Years Spot Starting IRS Fixed Receiver Exposure Profile 85 Figure 2.3: Uncollateralised 5x15 Forward Starting IRS Fixed Payer Exposure Profile 87 Figure 2.4: Uncollateralised 5x15 Forward Starting IRS Fixed Receiver Exposure Profile 87 Figure 2.5: Collateralised 5x15 Forward Starting IRS Fixed Payer Exposure Profile 89 Figure 2.6: Collateralised 5x15 Forward Starting IRS Fixed Receiver Exposure Profile 89 Figure 2.7: Uncollateralised 5 into 10 Cash Settled European Payer Swaption Exposure Profile 91 Figure 2.8: Uncollateralised 5 into 10 Cash Settled European Receiver Swaption Exposure Profile 91 Figure 2.9: Uncollateralised 5 into 10 Physically Settled European Payer Swaption Exposure Profile 93 Figure 2.10: Uncollateralised 5 into 10 Physically Settled European Receiver Swaption Exposure Profile 93 Figure 2.11: Uncollateralised 5 into 10 Physically Settled Bermudan Payer Swaption Exposure Profile 95 Figure 2.12: Uncollateralised 5 into 10 Physically Settled Bermudan Receiver Swaption Exposure Profile 95 Figure 2.13: Collateralised 5 into 10 Physically Settled European Payer Swaption Exposure Profile 97 Figure 2.14: Collateralised 5 into 10 Physically Settled European Receiver Swaption Exposure Profile 97 Figure 2.15: Collateralised 5 into 10 Physically Settled Bermudan Payer Swaption Exposure Profile 98 Figure 2.16: Collateralised 5 into 10 Physically Settled Bermudan Payer Swaption Exposure Profile 98 Figure 2.17: Uncollateralised 15 Years FX (EURUSD) Forward Long EUR at Maturity Exposure Profile 100 Figure 2.18: Uncollateralised 15 Years FX (EURUSD) Forward Short EUR at Maturity Exposure Profile 101 Figure 2.19: Uncollateralised 15 Years FX (EURUSD) Swap Long EUR at Maturity Exposure Profile 102 Figure 2.20: Uncollateralised 15 Years FX (EURUSD) Swap Short EUR at Maturity Exposure Profile 103 Figure 2.21: Collateralised 15 Years FX (EURUSD) Forward Long EUR at Maturity Exposure Profile 104 Figure 2.22: Collateralised 15 Years FX (EURUSD) Forward Short EUR at Maturity Exposure Profile 105 Figure 2.23: Collateralised 15 Years FX (EURUSD) Swap Long EUR at Maturity Exposure Profile 105 Figure 2.24: Collateralised 15 Years FX (EURUSD) Swap Short EUR at Maturity Exposure Profile 106 Figure 2.25: Uncollateralised 15 Years EURUSD CCS Long EUR at Maturity Exposure Profile 108 Figure 2.26: Uncollateralised 15 Years EURUSD CCS Short EUR at Maturity Exposure Profile 108 Figure 2.27: Collateralised 15 Years EURUSD CCS Long EUR at Maturity Exposure Profile 109 Figure 2.28: Collateralised 15 Years EURUSD CCS Short EUR at Maturity Exposure Profile 110 Figure 3.1: GS DVAs 170 Figure 3.2: MS DVAs 171 Figure 3.3: JPM DVAs 172 Figure 3.4: GS DVAs PCA Variance Analysis 174 Figure 3.5: MS DVAs PCA Variance Analysis 174 Figure 3.6: JPM DVAs PCA Variance Analysis 175 Figure 3.7: GS DVA Analysis 176 Figure 3.8: MS DVA Analysis 177 Figure 3.9: JPM DVA Analysis 177 Figure 3.10: Prediction Test of CDS as Explanatory variable for GS 178 Figure 3.11: Prediction Test of CDS as Explanatory variable for MS 179 Figure 3.12: Prediction Test of CDS as Explanatory variable for JPM 179 Figure 3.13: Prediction Test of 1 Lag in CDS as Explanatory variable for GS 180 Figure 3.14: Prediction Test of 1 Lag in CDS as Explanatory variable for MS 181 Figure 3.15: Prediction Test of 1 Lag in CDS as Explanatory variable for JPM 181 Figure 3.16: Prediction Test of 1st Order Changes in CDS as Explanatory variable for GS 182

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Counterparty Credit Risk, Funding Risk and Central Clearing

Figure 3.17: Prediction Test of 1st Order Changes in CDS as Explanatory variable for MS 183 Figure 3.18: Prediction Test of 1st Order Changes in CDS as Explanatory variable for JPM 183 Figure 3.19: Prediction Test of 1st Order Changes in CDS as Explanatory variable for GS DVAs 185 Figure 3.20: Prediction Test of 1st Order Changes in CDS as Explanatory variable for MS DVAs 185 Figure 3.21: Prediction Test of 1st Order Changes in CDS as Explanatory variable for JPM DVAs 186 Figure 4.1: Exposure Profile Evolvements of 5 x 15 Forward Starting IRS Fixed Payer 226 Figure 4.2: Exposure Profile Evolvements of 5 x 15 Forward Starting IRS Fixed Receiver 227 Figure 4.3: Exposure Profile Evolvements of 5 into 10 European Payer Swaption 228 Figure 4.4: Exposure Profile Evolvements of 5 into 10 European Receiver Swaption 229 Figure 4.5: Exposure Profile Evolvements of 5 into 10 Bermudan Payer Swaption 230 Figure 4.6: Exposure Profile Evolvements of 5 into 10 Bermudan Receiver Swaption 231 Figure 4.7: Exposure Profile Evolvements of 15 Years EURUSD Cross Currency Swap Long EUR at Maturity 232 Figure 4.8: Exposure Profile Evolvements of 15 Years EURUSD Cross Currency Swap Short EUR at Maturity 233

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Counterparty Credit Risk, Funding Risk and Central Clearing

Counterparty Credit Risk, Funding Risk and Central Clearing

Stephen Yang ZHANG

School of Business, Imperial College London

Room 349, ACEX, South Kensington Campus, Exhibition Road, SW7 2AZ

Abstract

In this thesis we have a review of the critical issues of CVA/DVA/FVA pricing framework, provide detailed economic interpretations of these xVA terms and present empirical studies on DVA hedging practice in the marketplace and a new approach to hedge DVAs. The economic drivers and implications of central clearing and initial margins on derivatives are addressed as well.

1. Introduction Counterparty credit risk is the risk that a counterparty cannot honour its obligations specified in the contract before the of the contract. Credit valuation adjustment, or CVA, is the market price of counterparty credit risk. Pre Great Financial Crisis (GFC), it is customary for derivative dealers to run market risk neutral trading books and charge their counterparties CVAs on standalone basis to reflect the possible losses due to defaults of their counterparties. CVA represents the difference of value of positions with a default free counterparty and the same counterparty with default possibility. Counterparty credit risk is identified as one of the main contributors to the Great Financial Crisis (GFC). Nathanael (2010) and BIS (2011) reported that two thirds of the losses had been caused by CVA Mark to Market during this crisis (only one third due to the actual defaults of counterparties). Derivative dealers, therefore, have to manage their counterparty credit risk. The proper valuation and hedging strategies of counterparty credit risk is therefore extremely important within this context. Brigo, Morini and Pallavicini (2013) provided an excellent discussion of CVA and its implications for current financial markets.

It has been widely discussed regarding the credit risk component embedded in various underlying assets. Such kind of credit risk had been standardised and traded in many years in the form of risky bonds and credit derivatives. The pricing and trading of counterparty credit risk is a relatively new concept and becomes popular after the financial crisis. Suppose a enters into an OTC derivative transaction with its client. Implicitly the bank sells an to default to its client. The client could this option at its default time to reduce the net liabilities. This option is CVA. Another way to interpret CVA is that CVA is or credit protection sold by CVA desk to every trading desk in a bank. If the trading desk passes this CVA to its client, CVA becomes a credit protection product that the bank sells to its client to cover the default risk of the client. In essence, the pricing of

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Counterparty Credit Risk, Funding Risk and Central Clearing this counterparty credit risk is identical to pricing default risk premium of a defaultable and derivatives. Now a simple rate swap is actually a defaultable and a standard CDS is a defaultable CDS as the default risk premium of the protection seller is implicitly taken into account in transactions. As the market price of credit risk measures the difference between a defaultable bond and default free bond, CVA measures the market price of reduction of net receivables in OTC transactions due to counterparty credit risk. And therefore, the market value of a bank’s overall PnL is the net PnL of all trading desks marked down by CVA or the market price of counterparty the bank takes on its .

Derivative dealers provide liquidity to markets and profit by making bid ask spread. The underlying mathematical models have to be calibrated to market prices on regular basis such that the derivative dealers’ positions will not be arbitraged by other market participants. However, the derivative dealing business is not counterparty credit risk free. Since CVA is a real component on a bank’s income statement, CVA, or the market price of counterparty credit risk, should be hedged and risk managed as other components of risks in derivatives transactions. As we could observe CVAs of major bank’s quarterly income statements, the PV01 of CVA of a top derivative dealer could be amounted to the millions of dollars.

Here we could make a brief comparison of CDS transactions and CVA transactions. The CDS premium is market price of default loss of a bond, whereas CVA is market price of default loss of a derivative (or a portfolio of derivatives). From the perspective of CDS protection buyer, the CDS protection buyer is exposed to changes before default happens and recovery/liquidity risk (different closeout amounts in different market scenarios) at the default time. CVA protection buyer (like trading desks in an investment bank) faces the same risks as they have in CDS transactions. Furthermore, CVA protection buyer is exposed to exposure dynamics and Wong Way Risk (WWR). For example, the difference between a cash settled and physical settled swaption and model dependent early exercise decisions of Bermudan have a significant impact on CVA desks’ trading and hedging activities. Wrong Way Risk, i.e. the positive dependence between size of exposure profile and counterparty credit riskiness, creates a significant challenge to derivative dealing business. The trading and hedging of wrong way risk and right way risk emerged from dependencies between counterparty credit spreads and exposure dynamics on the portfolio level is subject to top management decisions.

In fact, CVA is more complex and more difficult to value than any of the transactions between these two parties, especially with more realistic features, netting and management under ISDA Master Agreement (Hull and White (2012)). The economics and business models of derivatives dealing changed significantly due to CVA. The cost of trading derivatives with uncollateralised

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Counterparty Credit Risk, Funding Risk and Central Clearing counterparties and hedging with collateralised counterparties raised up significantly post GFC and made old business model unsustainable. Therefore, derivative dealers now have to seek for different approaches to actively manage their CVA.

Debit valuation adjustment, or DVA, is the market price of credit risk of the market participant itself. Suppose two parties enter into a derivative transaction. If one of the two parties defaults before the expiry of this transaction, ‘survivor’ has to pay the entire amount of its net liabilities but ‘defaulter’ only to pay the recovery of its net liabilities. Therefore, each party implicitly sold a default option to each other at inception such that the defaulted party only has to pay recovery in its default state. The market prices of these default options are CVA and DVA. The fair market price of this derivative, therefore, should be default free market price adjusted by credit risk of counterparty (i.e. CVA) and credit risk of itself (i.e. DVA).

Several arbitrage strategies could be used to monetise counterparty credit risk. An interesting topic would be how to convert or monetise DVA or a bank’s default gain before it actually defaults. Here DVA/default gain is defined as the net benefits the bank could obtain if it defaults. The bank only has to pay the recovery of Mark to Market instead of full MtM in normal market conditions of the portfolio if the bank is actually out of money at its default time. The market price of credit risk of bank itself is DVA. Details regarding DVA hedging would be presented in Funding Risk Chapter of this thesis. Regarding CVA, normally a bank would charge its clients a premium to cover client’s default risk. Appropriate strategies will help the bank reduce its PnL and reduce the regulatory capital and hence improve the long term prospect of the bank and boost its return on capital.

Proper pricing and management of CVA is crucial in running derivatives portfolios. It would be short of appropriate protection over default risk and credit spread volatilities if counterparty credit risk is under-priced or too aggressively priced, and the dealer would lose business to other market dealers if counterparty credit risk of over-priced or too conservatively priced. It is obvious that the decisions of CVA desk would be on the opposite side of major trading desks in many occasions.

Suppose a bank enters into an OTC derivative transaction with counterparty. In essence, the bank becomes the credit protection seller of its counterparty and the credit protection buyer of itself. It could be interpreted as the bank will only receive the recovery of net receivables (net of collateral) if the counterparty defaults and have to pay the recovery of net payables (net of collateral) if it defaults. So a plain vanilla interest rate swap transaction is actually an interest rate swap with a long position in bank’s own credit and a short position in counterparty’s credit. So in order to recover the old paradigm of derivatives trading and make underlying asset price a pure revelation of market risk, each

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Counterparty Credit Risk, Funding Risk and Central Clearing party in derivatives transaction should be compensated by the market price of counterparty credit risk (i.e. CVA, DVA) that they undertake.

The protection the bank buys and sells have the same life-cycle as the underlying derivatives. If the initiates additional derivatives transactions with its counterparty, the bank enters into additional corresponding long/short positions in bank’s/counterparty’s protection. And if the underlying trades are unwound/terminated, the bank has to unwind/terminate the corresponding long/short protection positions in bank’s/counterparty’s credit.

Sorensen and Bollier (1994) and Duffie and Huang (1996) initiated counterparty credit risk research in Finance. Duffie and Huang (1996) proposed a simple switch discount rate model to incorporate asymmetric default risk of the two counterparties in a swap trading and examined this impact on the swap rates. Further models on counterparty credit risk analysis following Duffie and Huang (1996) extends to compute the valuation adjustment the same way as a credit spread. UCVA with netting agreements is discussed in Brigo and Masetti (2005). Brigo and Pallavicini (2007) discussed UCVA on interest rate swaps. UCVA with wrong way risk on credit derivatives under no arbitrage arguments was covered in Brigo and Chourdakis (2012). Inclusion of bilateral defaults becomes desirable after several default events of the leading financial institutions after crisis 2008. An arbitrage-free valuation framework for bilateral counterparty credit risk (BCVA) was formalized and developed by Brigo, Capponi and Pallavicini (2014).

The general hedging strategies for CVA/DVA work like any other strategies and they could create value or positive PnL if the net future receivables are greater than net future payables. The emergence of CVA/DVA terms is due to asymmetric default treatments and the hedging strategies of CVA/ DVA is actually to structure a trade to achieve symmetry in default scenario. Let’s set aside these technical discussions for a while. Generally we have two choices to hedge CVAs in transactions.

One choice is to buy a CCDS. CCDS stands for contingent CDS and works similar to CDS. The protection buyer of CCDS receives a notional equal to the loss given default (LGD) fraction of Mark to Market (if positive) of a portfolio at the default time of reference credit. The valuation of CCDS and counterparty risk under correlation is discussed in Brigo and Pallavicini (2008). If the counterparty defaults during the life of the trade and CCDS is bought to protect the bank from suffering losses, CCDS is triggered and provides protection by replacing the underlying trade with an identical instrument upon the default of the counterparty. Hence CCDS is a knock in OTC product covers the loss of a transaction or a portfolio triggered by default of the counterparty. However, CCDS is not the ultimate solution to counterparty risk and its liquidity isn’t sufficient enough to

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Counterparty Credit Risk, Funding Risk and Central Clearing establish a well-functioned market for trading counterparty risk (Brigo, Morini and Pallavicini (2013)). ISDA tried but failed to create a liquid standardised market for single name .

The other choice is to structure credit-linked . This security is triggered upon default of counterparty and enters into the offsetting positions with respect to the underlying instrument. It works like structuring a cancellable payer swap by a payer swap and receiver swaption. The only difference is the swaption is triggered not by the swap rate but by the default of counterparty. So this credit-linked security, or credit extinguisher, is a knock out derivative of the underlying trade. The idea of credit extinguisher is similar to the set-off provision of ISDA (Tang and Li (2007)). Details could be found in Counterparty Credit Risk chapter.

Derivative dealers become aware of funding risk in current market environments. This is due to the negative carry they experienced in derivative dealing. Suppose derivatives dealer has a portfolio of trades with positive Mark to Market (MtM) with an uncollateralised counterparty and hedged the portfolio with a fully collateralised counterparty. That is, underlying portfolio has a positive MtM and the corresponding hedging portfolio has a negative MtM. The dealer, therefore, has to borrow collateral at his/her own funding curve in the market and post to the hedging party and hedging party reimburse the collateral with OIS rate. Since the return on the collateral (i.e. OIS rate) is significantly lower than the cost of collateral (i.e. funding rate), this negative carry creates a serious problem.

Someone may argue that the underlying portfolio might have an equal probability of having a negative Mark to Market and the hedging portfolio therefore has a positive Mark to Market. Since the return on collateral (i.e. funding rate) is significantly higher than the cost of collateral (i.e. OIS rate), the positive carry gives dealer a funding benefit that could offset with his/her funding cost. The reality is, it is very hard to use a short term cash (i.e. the funding benefit) to retire/buy back long term debt of a dealer. Given the short term nature of funding benefits, it is more realistically to lend cash in repo market. Then the effective positive carry becomes return on collateral (i.e. repo rate) minus cost of collateral (i.e. OIS rate). That leaves almost no funding benefits for the derivative dealer.

Funding valuation adjustment (FVA) is the price of costs (FCA) and benefits (FBA) of funding derivatives trading due to a simple fact that (fully) collateralised derivative positions cannot be repoed or cannot be funded at the repo rate.

Suppose a bank has net receivables (net of collateral) from a derivative transaction, the bondholders of the bank actually sells protection to the bank to cover expected loss triggered by counterparty’s default from this transaction. Bondholders are interested in extra return offered by bank’s bond and therefore provide protection on bank’s operations. The market price of this protection is Funding

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Counterparty Credit Risk, Funding Risk and Central Clearing

Cost Adjustment (FCA). If the bank has net payables (net of collateral) from a derivatives transaction, the bondholders of the bank implicitly buy protection from the bank to cover the expected loss triggered by bank’s default in this transaction. The bank only has to pay the recovery of net payables if it defaults and hence leaves more assets to recover the value of the bond . The market price of this protection is Funding Benefit Adjustment (FBA). The Details could be found in Funding Risk Chapter.

Some arbitrage opportunities emerge if a bank intends to hold FBA and pass FCA charged by bondholders to its counterparty. The bank’s counterparty could simply unwind this trade and enter into the same position with another bank that has better funding conditions. The CVA changes will not be significant as it is dominated by the counterparty’s credit quality but FCA will drop to a significant lower level. If the bank hedges the portfolio with an exact offsetting portfolio under identical CSA terms, FBA that the bank has on hedging portfolio is FCA in the original portfolio. So FCA is actually a term that accounts for the net borrowing cost a bank has to finance its receivables of an OTC transaction or the equivalent funding cost throughout the life of OTC trade. As suggested by Albanese et al. (2014), a bank will not have any FCA if the underlying asset in the transaction could be repoed. In other words, if the net receivables (net of collateral) could be repoed in the capital market, the trade is funded at repo rate and the bank does not have to borrow from bondholders to finance the receivables. An OTC trade, like plain vanilla interest rate swap, could not be repoed and therefore there is always a FCA attached to the final price.

Piterbarg (2010) presented a general pricing framework for FVA without counterparty credit risk. Burgard and Kjaer (2012) extended to FVA with counterparty credit risk and symmetric funding curves. Pallavicini, Perini, and Brigo (2012) provided a robust FVA framework with counterparty credit risk and realistic asymmetric funding curves. Hull and White (2014) examined the overlaps between funding benefits and DVA and provides a different interpretation of funding costs.

Collateral becomes the centre of CVA, DVA and FVA pricing. As we will discuss in final chapter of this thesis, initial margins and related regulations push the market from uncollateralisation to overcollateralization, implicitly converting counterparty credit risk to funding risk.

Albanese and Andersen (2014) laid out a general framework for XVA accounting. In their analysis, if a derivative dealer recognises DVA or funding benefits adjustment (FBA) in pricing, their CET1 (Common Tier 1 Equity) is going to be marked down by changes in DVA or changes in FBA. If dealer doesn’t recognise DVA or funding benefits, their CET1 will not change. It is logical for dealers to hedge/monetise DVA or funding benefits, but there is a lack of literatures covering this part. The empirical analysis on DVA/FBA hedging in Funding Risk chapter provides evidence on the DVA

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Counterparty Credit Risk, Funding Risk and Central Clearing hedging practice in the industry and proposes a new DVA hedging strategy via trading equities. The economic interpretation is discussed in detail in Funding Risk chapter.

The Great Financial Crisis (GFC) made Basel Committee modify the guidelines for OTC markets with capturing Wrong Way Risk (i.e. positive dependence between size of exposure and counterparty credit riskiness) as well as other amendments regarding netting rules and collateralization, e.g. extension of margin period, use of central counterparty (CCP), and refinement of rules on re- hypothecation and closeout conventions. Brigo, Capponi, Pallavicini and Papatheodorou (2011), Brigo and Morini (2010) and Brigo and Pallavicini (2014) presented in depth analysis of impact of re- hypothecation, closeout and CCP on CVA/DVA/FVA.

Regulations push the market moving from bilateral clearing to central clearing. The key idea supports central clearing is that central clearing makes ‘defaulter’ pays for default losses via initial margins and variation margins. In classic bilateral clearing system, ‘survivor’ pays for default losses as ‘defaulter’ only has to pay recovery of its outstanding liabilities. Duffie and Zhu (2011) showed that we need an unrealistically high number of participants in CCPs to achieve a significant reduction of exposures Rama et al. (2012) rectified assumptions used in Duffie and Zhu (2011) and found greater netting benefits than Duffie and Zhu (2011). Brigo and Pallavicini (2014) provided a unified valuation framework of XVAs pricing on CCP cleared derivatives with initial margins. The advantages and disadvantages of central clearing and pricing examples of CVA, DVA and FVA with initial margins are provided and discussed in the last chapter.

This thesis is structured as follows.

In section 2, economic interpretations and mathematical modelling of CVA/DVA are introduced in detail. Several strategies to make a more competitive price by reducing CVA and key aspects of CVA/DVA between reality and theory are discussed as well. Pricing examples of CVA/DVA on interest rate swaps and other related products are provided at the end of the section.

In section 3, different pricing theories of FVA and their economic interpretations are reviewed in detail. The rationale behind using Treasury curve, repo curve and funding curve for discounting is further explored in this section. Evidence on DVA/FBA hedging practice in current marketplace is presented and a new hedging strategy is proposed at the end of this section.

In section 4, economic motivations of initial margins and central clearing have on derivative trading activities are discussed in detail. The comparison between central clearing system and bilateral

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Counterparty Credit Risk, Funding Risk and Central Clearing clearing system is included as well. A detailed pricing example of CVA/DVA/FVA with initial margins is provided at the end of this section.

Section 5 concludes this thesis.

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Counterparty Credit Risk, Funding Risk and Central Clearing

2. Counterparty Credit Risk and CVA/DVA

2.1 Introduction Let’s use an example to explain credit valuation adjustment (CVA) and debit valuation adjustment (DVA).

What is CVA/DVA?

Suppose party A sells an option to party B and buys the option with identical terms as a hedge from party C. Now party A does default. Then party C has to pay full MtM to party A but party A only has to pay recovery amount (REC) to party B. It is clear that asymmetric treatment of default leads to a net default gain for party A and these default gains would be distributed to bondholders of party A and enhance the recovery rate of party A’s outstanding debt. Thus, party A actually bought a default option triggered by A’s credit event with a positive payoff equal to loss given default of underlying trade from party B. Party A calls this default option DVA. Party B names this default option CVA. Given the fact that party A could receive a positive payoff if party A defaults, party A has to pay for this option to make its CVA book flat with party B.

In theory, if party A holds a significant DVA on its book, the DVA economically converts into higher recovery rate of debt and we should expect credit spread/funding spread of party A tightens in the process. However, the market may not have full access to this information and the market could not have full information regarding the underlying trade dynamics, party A in this case, has much better information and hence could apply certain strategy monetise this DVA term. The market will price in this information after party A executes certain strategies.

Before we move on to how to monetise DVA, let’s have a brief review of CVA practice before Great Financial Crisis (GFC) and credit risk modelling methodologies.

CVA Practice before GFC Let’s continue with above example to explain CVA practice before GFC. The party A sells an option to party B and buys the option with identical terms as a hedge from party C. Before GFC, the party A, as a derivative dealer in options market, intends to catch some bid/ask spread in the trading activities and doesn't take counterparty credit risk seriously. Party A runs a flat book which makes party A market risk neutral. However, the counterparty credit risk is out there and party A does not have a flat CVA book, or equivalently, party A is not counterparty credit risk neutral.

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Hull (2000) laid out a simple classic example on adjusting derivatives pricing for counterparty credit risk on standalone trade level. Here we assume there does not exist any dependency between size of exposure and counterparty credit riskiness. Therefore, Wrong Way Risk, i.e. the positive dependency between size of exposure and counterparty riskiness, is not under consideration in our example.

Party A just had a derivative transaction (i.e. sold an option in above example) with Party B and the time to maturity of a derivative transaction is 푇. The default risk free price of this derivative is 푓0.

Party A could default at times 푡1, 푡2, 푡3, … , 푡푛 and 푡푛 = 푇. The Mark to Market (MtM) of that trade from party B’s perspective at time 푡푖 is 푓푖. The risk neutral probability of default of Party A at time 푡푖 observed at time 0 is 푞푖 = ℚ0(푡푖−1 < 휏 < 푡푖) and the recovery rate of Party A is 푅. Instantaneous interest rate is assumed to be constant 푟. Hence the exposure or the maximum loss of party B in this + + transaction at time 푡푖 is 푓푖 , where 푓푖 = max(푓푖, 0). And the market price of expected loss at time 푡푖 is ℚ + (1 − 푅) ∙ 푞푖 ∙ 피푖 (푓푖 ).

And its market price at time 푡0 is ℚ ( ) ℚ( +) ( ) ℚ( +) (2.1.1) 피0 ( 1 − 푅 ∙ 퐷퐹푖 ∙ 1푡푖−1<휏<푡푖 ∙ 피푖 푓푖 ) = 1 − 푅 ∙ 퐷퐹(0, 푖) ∙ 푞푖 ∙ 피0 푓푖 , where 퐷퐹 denotes the discounting factor and 퐷퐹(0, 푖) = 푒−푟∙푡푖.

If we sum up the entire spectrum of cash flows, the market price of the total expected loss at time 푡0 is 푛 (2.1.2) ℚ + (1 − 푅) ∙ ∑ 퐷퐹(0, 푖) ∙ 푞푖 ∙ 피0 (푓푖 ). 푖=1 From the perspective of accounting, the derivatives transactions could be grouped into three categories: 1. The trade is always an asset to the institution (for example, long an option) 2. The trade is always a liability to the institution (for example, short an option) 3. The trade could be an asset or a liability to the institution (for example, an interest rate swap)

Since party A sold an option to Party B, that option is always an asset to Party B in Party B’s book + which means 푓푖, i.e. the of this option, is always positive and hence 푓푖 = 푓푖. And if we assume this option is an European option, there is only one contingent cash flow in this trade and it pays out at maturity, hence the discounted exposure is equal to its current price of the option 푓0 and ℚ + 푓0 = 퐷퐹(0, 푖) ∙ 피푖 (푓푖 ) = 퐷퐹(0, 푖) ∙ 푓푖. Then the market price of the cost of default is

푛 (2.1.3) 푓0 ∙ (1 − 푅) ∙ ∑ 푞푖. 푖=1 And the fair price of the option after taking into account the default risk of party A is

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푛 푛 (2.1.4) ∗ 푓0 = 푓0 − 푓0 ∙ (1 − 푅) ∙ ∑ 푞푖 = 푓0 ∙ (1 − (1 − 푅) ∙ ∑ 푞푖). 푖=1 푖=1 The zero coupon bond issued by party A shared some similarities in this case. Suppose the risk free zero coupon bond price is 퐵0. If we have the same simplifying assumptions regarding credit risk, the price of zero coupon bond bearing credit risk of party A is

푛 (2.1.5) ∗ 퐵0 = 퐵0 ∙ (1 − (1 − 푅) ∙ ∑ 푞푖). 푖=1 If we compare the fair price of the option and market price of party A’s zero coupon bond,

∗ ∗ 푛 (2.1.6) 푓0 퐵0 = = 1 − (1 − 푅) ∙ ∑ 푞푖. 푓0 퐵0 푖=1 If we further assume the of a risk free zero coupon bond with maturity 푇 is 푟 and the yield to maturity of zero coupon bond with maturity T issued by party A is 푟 + 푠, where 푠 denotes the credit spread of party A,

∗ −((푟+푠)−푟)∙푇 −푠∙푇 (2.1.7) 푓0 = 푓0 ∙ 푒 = 푓0 ∙ 푒 . This indicates the derivative with a single positive cash flow at time T could be priced by further discounting by the counterparty’s credit spread. This could be applied to options, caplets (or floorlets) and cash settled swaptions.

Credit Risk Models Traditional credit risk modelling could be divided into two categories: structural models and reduced form models.

Structural Models Merton (1974) initiated the structural models on credit risk modelling. The value of the firm is assumed to follow a random process similar to prices. The outstanding debt of the firm matures at time T. The firm defaults if and only if the value of its assets is below its debt level at time T. Black and Cox (1976) introduced a default barrier into the model and made it possible for the firm to default before the maturity of outstanding debt. In Black and Cox (1976), the firm defaults at the first moment that the value of assets falls below the barrier. This default barrier made the model closer to reality but the model still cannot be calibrated exactly to the term structure of CDS.

Brigo, Morini and Tarenghi (2011) extended Black and Cox (1976) to two analytically tractable models (AT1P and SBTV) and these two enhanced structural models could be calibrated exactly to

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CDS term structure. Hull, Predescu and White (2010) extended Black and Cox (1976) to capture default dependency among a large number of credits. Ballotta, Fusai and Marazzina (2015) used correlated Levy process based structutral model to price counterparty credit risk.

Reduced From Models In reduced form models, default process is an exogenous default process and the default event is defined as the first jump of a Poisson process with given intensity (deterministic or stochastic intensity). Therefore, market observable variables do not trigger default in reduced form models and default free market information do not provide any information on default process. This is suited to calibrate the credit spreads and hence widely used in credit derivatives pricing. The main disadvantage of reduced form model is that it does not provide any economic interpretation of default mechanism.

Generally, reduced form or intensity model provides the best description of the relationship between default probability and credit spread. O’Kane (2008) gave a comprehensive review of how to extract the default probability from the corporate bonds credit spread (liquidity component of credit spread is well addressed in Brigo, Predescu and Capponi (2010)). Here the credit spread is treated as an elegant measure of a bond credit risk.

This thesis uses reduced form model.

How to monetise DVA?

So how to monetise this DVA term? Party A could sell a credit contingent option to party C such that party C only has to pay REC (recovery of party A) to party A if the trade is out of money from party C’s perspective upon default of party A. This contract actually gives symmetric treatment of default to party A which means party A receives recovery amount from party C and pays recovery amount to party B at default time. Since party C receives REC from party A if the trade is in the money for party C, and now party C only has to pay REC to party A of the trade is out of money for party A, party C actually enters into a credit extinguisher position with party A and the trade will cancelled at REC level upon default of party A regardless of of underlying position. So what should party C do in this scenario? Party C could sell party A’s CDS to monetise this credit contingent option.

Another way to review this credit contingent option is that this option is protection party A sold on itself. Party A will be incentivised to unwind this credit contingent option with party C to receive full amount if the trade is in the money for party A prior to A’s default. Party C will therefore charge party A for the market price of credit contingent option and unwind its hedging positions of this

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Counterparty Credit Risk, Funding Risk and Central Clearing credit contingent option. These activities will be expected to cause the credit spread of party A to widen and market will price in these information into views of party A’s credit quality.

How to reduce CVA or cheapen trade price?

Now suppose party A and party B enter into ISDA Credit Support Annex (CSA) (let’s assume it is a two way CSA agreement). Since party A sells an option to party B, party A has to post collateral to party B to cover party B’s default loss. And party B could re-hypothecate the collateral to its counterparties to obtain a better funding curve and reduce its funding cost. So it is important to recognise that funding should be calculated on funding sets level.

In above discussions, the credit contingency option or credit extinguisher between party A and party C is one way party A to use to sell protection on itself to obtain symmetric default treatments for party A. Here if party C is the market marker, party C buys credit contingent option from party actually help party A monetise its DVA or Funding Benefits Adjustment (FBA) and this becomes one way party C could use to cheapen the trade price to party A (and retain the business) and reduce party C’s contingent liability when A defaults.

Another way to cheapen trade price and hedge counterparty credit risk is to create fully funded structured notes like credit linked notes (CLNs) with CVA packaged into coupons. This structures notes could provide a good amount of coupon and yield to maturity to potential investors in current low yield environment. The advantage of this CVA hedging strategy is that the hedging strategy per se does not expose the hedger to further counterparty credit risk from investors. The investors of structured notes have posted notional of the notes to the hedger as initial margin in OTC transaction and a high initial margin will reduce the CVA to zero. We will come back to the design of this structured notes in the hedging chapter. Other potential candidates for hedging instruments for CVA are equity derivatives like out of money put options and equity default swaps.

CVA has the same origin with counterparty’s bond credit spread. It could be shown in the following example. Implicitly the bondholders sold a (knocked in) triggered by default of the bond to the bond issuer. The bond issuer should pay the principal to bondholders at maturity but they only have to pay recovery if the bond defaults. The put option gives the bondholders payoff equal to loss given default of the principal at default time. And because the bondholders sold this put option to the bond issuer, the price bondholders paid to obtain the was lower compared to a Treasury bond investment. The price was lower by an amount equal to present value of that put option.

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Although CVA shares the same origin with bond credit spread, the magnitude of net CVA of a derivative transaction in general is much smaller than the credit spread of the bond. As in bond transaction, the credit spread is determined by the expected loss of underlying bond. The net CVA for a derivative like interest rate swap is determined by the net cash flows and no principal is at risk in an interest rate swap transaction. However, the impact of net CVA could be significant and greater than bid/ask spread, especially for interest rate products with longer maturity and tighter bid/ask spread and commodities products with lower credit quality counterparties and longer maturities.

For counterparty risk management, netting agreements, margin (collateral) agreements, credit contingency agreements and mutual termination clauses (break clauses) could be used to enhance counterparty credit quality and reduce counterparty risk exposure. Netting agreements and margin agreements are often in the form of a master agreement and hence they normally cover trades across different trading desks or different asset classes of the same netting set or legal entity. The trades under the same netting set are netted against each other and reduce the overall exposure. If collateral is posted/received during the transaction process, in the event of counterparty default, the accumulated collateral could be liquidated to offset default loss. The details of netting and collateral agreements and associated gap risk would be discussed in the Counterparty Credit Risk Chapter. The credit linked collateral agreement and credit contingency agreement could also be used to mitigate counterparty credit risk. The margin requirements would be increased if the counterparty’s credit rating is lowered. This type of credit linked collateral agreement share the same spirit of implicit margin requirements imposed by CCPs. The credit contingency agreements would be treated differently at trade level and portfolio level. This agreement allows both parties to reduce or terminates transactions upon some pre-specified credit events. If both parties sign break clauses or mutual termination clauses, both parties would have the right to terminate the trades at market price at pre-specified dates. This essentially shortens the life of the underlying transaction and reduces the volatility of MtM, and hence reduces the credit exposure.

Other strategies could be employed to reduce counterparty credit risk like restructuring/re-striking the existing portfolio periodically such that brings the MtM back to zero and converts the receivables into upfront cash flow such that significantly reduce the exposure profile. Structuring plays a vital role in CVA/counterparty risk mitigation. The counterparty risk linked cash flows could be built into coupons of notes and the issuer of notes (such as the bank) could release the capital allocated to specific counterparty risk. This type of transaction could completely eliminate the counterparty risk as the principal is received upfront to cover the potential loss from default of the portfolio. The restructuring of timing of cash flows which enables the bank the collect receivables sooner and pay the payables later would change the exposure profile in favour of the bank and help reduce the counterparty risk. The placements of funded CDOs transferring counterparty credit risk to the

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Counterparty Credit Risk, Funding Risk and Central Clearing market follow the same logic. The fully funded structure borrows the risk transferring idea from corporate will be discussed in details in hedging chapter.

Similar to market risk practice like Value at Risk (VaR), counterparty credit risk metrics like potential exposures (PE) of a given portfolio are introduced to measure counterparty risk under historical measure. The potential exposure (PE) at 95% level means the MtM of the portfolio will not exceed the PE in 95% of all possible scenarios. PE measures the potential loss from positive MtM (in the money trades) due to counterparty risk and VaR measures the potential loss from negative loss from negative MtM (out of money trades) due to market risk. VaR is introduced to measure potential loss over a relative short time horizon but PE is concerned with potential loss over the life of the transaction.

The complexities in reality

The realities are far more complex than what we have discussed here. One classic example is the dependency between counterparty credit quality and exposure. The bank has set strict internal credit control to avoid the scenario that the bank has more exposure when the default likelihood of counterparty goes up. This is called wrong way risk (WWR) and leads to greater expected default loss. The reverse scenario right way risk reduces the expected default loss as the bank has more exposure when the counterparty is less likely to default. The identification of wrong way risk needs rigorous investigation and is not always straightforward. If the bank have FX transactions with a Russian counterparty and Russian counterparty is obliged to deliver USD at maturity, the depreciation of RUB (Russian Ruble) against USD will significantly diminish the Russian counterparty’s capacity to deliver USD and therefore depreciation of RUB actually drives up the bank’s exposure and the Russian counterparty is more likely to default as Russian counterparty finds it more difficult to obtain funding in foreign currencies like USD. This type of wrong way risk (WWR) is easy to detect as there is a clear link between underlying transaction and counterparty’s credit quality. However, some links are a bit ambiguous and the impact of exposure on credit quality is hard to identify in practice. For example, does the oil driller or oil refiner credit quality improve if the crude oil price goes up? It is very hard to identify as the credit quality of oil producer relies on the outstanding hedging positions of the oil producer. If the oil producer is concerned with the further declines in crude oil positions, the oil producer enters into millions of short forward positions to hedge the downside risk of future oil price. Therefore, the oil producer is more likely to default if the underlying crude oil price rises as the producer has to post more collateral to meet margin calls on hedging positions.

Therefore, CVA/DVA pricing is the market price of credit protections on receivables or payables on an OTC portfolio. The existence of netting/margin agreements and asymmetric treatments of default

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Counterparty Credit Risk, Funding Risk and Central Clearing calls for a portfolio level model with flexibilities to adjust for different pricing systems across different desks.

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2.2 General Pricing Formula of Unilateral CVA

In the pre-crisis context, the condition that only one of the two parties of a transaction defaults and a default risk free always exists is taken into account with the counterparty credit risk pricing. Brigo, Buescu, and Morini (2011) termed this price adjustment to the default free party as unilateral credit valuation adjustment (UCVA). A classic introduction to UCVA is Alavian, Ding, Whitehead and Laudicina (2010) and Gregory (2009). UCVA pricing with netting, with collateral and with several asset classes are presented in a series of pioneering works by Damiano Brigo (details refer to Brigo (2012)).

Here we denote the time 푡 default risk adjusted price of net cash flows of a single transaction of derivative positions with maturity 푇 by 푉̃(푡, 푇). The default risk free price of the net cash flows of the same transaction is denoted by 푉(푡, 푇). The counterparty defaults at time 휏 under this simplified framework and its default process is assumed to follow a Poisson jump process. In such intensity models, the default time is the first jump of the Poisson random variable (Elizalde, 2012). 1(∙) denotes the indicator function and therefore 1휏>푡 is the probability of the counterparty which does not default before time 푡 (survival probability at time t). 퐷퐹(∙) denotes the discounting factor. 푀푡푀푡 denotes Mark to Market of underlying transaction at time 푡.

Then if the counterparty does not default before the expiry, the corresponding payoff becomes

1휏>푇푉(푡, 푇). And if the counterparty defaults before 푇 , the corresponding payoff is constituted with two components: 1) The cash flows received up to the default time 휏 푉(푡, 휏) 2) The residual net cash flows at default time 휏 if the Mark to Market (MtM) is positive, + 푅 ∙ 푀푡푀휏 and the residual net cash flows at default time 휏 if the Mark to Market (MtM) is negative, − 푀푡푀휏 − Where 푅 denotes the recovery fraction of the single transaction value, 푀푡푀휏 = min(푀푡푀휏, 0) and + 푀푡푀휏 = max (푀푡푀휏, 0). Here if the MtM of the single transaction payoff is positive, the institution can still recover a fraction 푅 of the exposure. While if the single transaction payoff is negative, the institution has to settle this outstanding amount immediately although this counterparty has already defaulted.

Therefore, summing all these expressions together yields

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̃ ℚ + − (2.2.1) 푉(푡, 푇) = 피푡 (1휏>푇 ∙ 푉(푡, 푇) + 1휏≤푇 ∙ 푉(푡, 휏) + 1휏≤푇 ∙ (푅 ∙ 푀푡푀휏 + 푀푡푀휏 ) ∙ 퐷퐹(푡, 휏)). And the unilateral CVA at time 푡 is ℚ ̃ ℚ ℚ + ℚ (2.2.2) 피푡 (푉(푡, 푇)) = 피푡 (푉(푡, 푇)) − 피푡 (퐿퐺퐷1푡<휏≤푇퐷퐹(푡, 휏)푀푡푀휏 ) = 피푡 (푉(푡, 푇)) − 푈퐶푉퐴푡

ℚ + with 푈퐶푉퐴푡 = 피푡 (퐿퐺퐷1푡<휏≤푇퐷퐹(푡, 휏)푀푡푀휏 ), where loss given default (LGD) is 퐿퐺퐷 = 1 − 푅. The detailed proof could be found at Brigo, Morini and Pallavicini (2013). Since

푉(푡, 푇) = 1휏>푇푉(푡, 푇) + 1휏≤푇푉(푡, 푇), the term inside the expectation brackets in right hand side of (2.2.2) could be rewritten as + + 푉(푡, 푇) − 퐿퐺퐷1푡<휏≤푇퐷퐹(푡, 휏)푀푡푀휏 = 1휏>푇푉(푡, 푇) + 1휏≤푇푉(푡, 푇) + (푅 − 1)1휏≤푇퐷퐹(푡, 휏)푀푡푀휏 , that is, + 푉(푡, 푇) − 퐿퐺퐷1푡<휏≤푇퐷퐹(푡, 휏)푀푡푀휏 (2.2.3) + = 1휏>푇푉(푡, 푇) + 1휏≤푇푉(푡, 푇) + 푅1휏≤푇퐷퐹(푡, 휏)푀푡푀휏 + − 1휏≤푇퐷퐹(푡, 휏)푀푡푀휏

Since 1휏≤푇푉(푡, 푇) = 1휏≤푇(푉(푡, 휏) + 퐷퐹(푡, 휏)푉(휏, 푇)), conditional expectation of the second and fourth term at time 휏 is ℚ + (2.2.4) 피휏 (1휏≤푇푉(푡, 푇) − 1휏≤푇퐷퐹(푡, 휏)푀푡푀휏 )

ℚ ℚ + = 피휏 (1휏≤푇 (푉(푡, 휏) + 퐷퐹(푡, 휏)푉(휏, 푇) − 퐷퐹(푡, 휏)피휏 (푉(휏, 푇) )))

ℚ ℚ + = 1휏≤푇 (푉(푡, 휏) + 퐷퐹(푡, 휏)피휏 (푉(휏, 푇)) − 퐷퐹(푡, 휏)피휏 (푉(휏, 푇) ))

ℚ − = 1휏≤푇 (푉(푡, 휏) + 퐷퐹(푡, 휏)피휏 (푉(휏, 푇) ))

− = 1휏≤푇(푉(푡, 휏) + 퐷퐹(푡, 휏)푀푡푀휏 ) By law of iterated expectations, we could obtain the exact formula in (2.2.1) after substituting (2.2.4) into (2.2.2).

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2.3 No Arbitrage Pricing in Real Market

The Classic No Arbitrage Framework

In the financial engineering theory, the no arbitrage pricing theorem (Harrison and Pliska (1981)), together with martingale representation theorem and change of measure, is served as the model factory for derivatives across different asset classes.

An arbitrage opportunity is perceived as an opportunity to generate riskless profits without bearing any risk. In theory, the back to back transaction done by brokers and derivative dealers in certain areas are actually their efforts to arbitrage the market. The brokers and derivative dealers buy low and sell high with the bid ask spread as their profits accumulate different sources of risk on their book and provide liquidity to the market in reality. The existence of counterparty credit risk makes the bid ask spread not default risk free and the loss due to default of a trading counterparty is significantly greater than profits from bid ask spread. As we will discuss later, the profits of derivative dealers should be marked down by CVA. It is common to establish an arbitrage trading strategy via static or dynamic trading to replicate or hedge a given security with other securities with liquidity.

In classic finance theory, under a frictionless market structure where there is no liquidity constraints, no transaction cost etc., the no arbitrage price of a given portfolio is the market price of its replicating portfolio which perfectly mimics identical cash flows. The risk preference of different market participants have no impacts on the price of this portfolio as stated in the risk neutral pricing framework. If the portfolio is traded above or below its no arbitrage price, arbitrage opportunities would emerge and these opportunities could be eliminated through arbitrage activities taken by different market participants.

The symptom of less or no arbitrage opportunities indicates the existence of an efficient market. In reality, no arbitrage is defined as the scenario that the transaction price is within the bounds of bid ask spread. The natural question we might ask is why the derivative dealer or dealers could still make profits from bid/ask spread if the price they transacted with different counterparties is no-arbitrage price. The comparative advantage of holding large and diversified portfolios gives the market markers or dealers ability to reduce their hedging costs. For normal size transactions, derivative dealers would not hedge on trades level as the new entries of transactions are normally partially hedged by the existing trades on the trading book.

Therefore the no arbitrage pricing framework is widely applied for trading activities given the nature of market marking is to replicate the market price of an instrument regardless of risk preference of participants in the market. The rest of market participants (other than derivative dealers) could use

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Counterparty Credit Risk, Funding Risk and Central Clearing forecasting models based on econometrics theory to identify the relative value opportunities and take on market risks in their activities.

Generally speaking, derivative dealers use calibrations to make their price consistent with their competitors to avoid arbitrage opportunities. That is, the model price of a given instrument is competitive if and only if it is consistent with market price and it could be replicated/hedged correctly with liquid instruments available in the market. Of course, the classic variables like correlation, correlation skew and volatility of volatility etc. could not be hedged with instruments with enough liquidity in the market. Then the risk preference of market participants and demand and supply would be taken into consideration in pricing of derivatives exposed to these un-hedgable market variables. CVA and Bermudan Swaption embedded in cancellable swaps and callable notes are classic instruments exposed to these un-hedgable variables and would be discussed in detail in the Counterparty Credit Risk Chapter.

Although the derivative dealers should trade risk neutral positions in theory, it is very costly to maintain risk neutral positions in reality. Normally derivative dealers will maintain delta neutral positions or first order market neutral positions of their portfolio. The other hedging like Vega hedging will suffer from high transaction costs. From the past few years market movements and profits generated by trading arm of investment banks, it could be observed that derivative dealers actually have a long position of Vega in their portfolio. That is, their profits would benefit from a surge in market volatility as investors would have more investment opportunities and hedgers would have a greater incentive to hedge against market movements and hence derivative dealers would have more trading opportunities. Under certain circumstances, the derivative dealers would set up partially hedging positions against their trading portfolio to reduce their volatility impact on their PnL. DVA hedging and curve options embedded in non-inversion notes are examples of this category. This kind of hedging is proprietary trading in nature and should be placed within the risk limits set by the bank’s policy. Details of DVA hedging could be found in Funding Risk chapter.

The divergence between reality and theory

The methodology applied in finance literature to obtain a no arbitrage price of any instrument is through dynamic replication or hedging with other instruments via self-financing strategy. A trading strategy would only use the cash flows itself generates without resorting to outside funding is defined as a self-financing strategy. An arbitrage opportunity could be defined as the opportunity to create a portfolio with zero investment upfront but a positive expected payoff via self-financing strategy in the future. Or in other words, the portfolio has zero probability to realise a negative payoff but a non-zero probability to have a positive payoff in the future. In theory, the self-financing strategy

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Counterparty Credit Risk, Funding Risk and Central Clearing gives rise to perfect hedging strategy and the derivative dealers could apply this self-financing strategy in trading and profit from bid/ask spread without any exposure to underlying market dynamics. In reality, self-financing strategy is expensive (due to asymmetric borrowing/lending rates) and perfect replication is impossible (due to counterparty credit risk). Therefore, derivative dealers normally set up an internal risk limits regime like VaR and traders would convert the overall VaR limits into limits on Delta, Gamma, and Vega etc. positions and profit from proprietary positions within the limits of Greeks.

Here we briefly discuss several important factors that lead to the divergence between real world market structure and theoretical market structure.

1. Counterparty Credit Risk

Counterparty credit risk is one of the most important factors and has significant impact on pricing and hedging decisions made by derivative dealers. The details would be addressed in this chapter.

2. Risk Free Rate and Effective Funding Rate

The risk free rate is widely used in different text books and papers without proper definition of risk free rate. US treasury rate is default risk free but not the exact risk free rate due to the exemptions treasury bondholders have from state and local tax. Therefore the US treasury rate is lower than real risk free rate as it gives treasury bondholders additional tax benefits. Under the self-financing strategy structure, the arbitrage free price of derivatives is determined by the derivative dealer’s borrowing and lending curves. Hence the ideal risk free rate is not that important in real life pricing. The effective financing and funding rates of a given derivative dealer for certain instruments come into play here. If the underlying instrument like Treasury bond has a liquid repo market, the effective financing rate for that instrument is the repo rate. If the underlying instrument like interest rate swaps does not have an available repo market, the financing cost of that specific instrument has to be absorbed by the derivative dealer and hence the effective financing rate is the firm’s funding rate set by the bondholders of the derivative dealer. The repo transaction is collateralised with underlying asset and hence the repo transaction is exposed to the credit and liquidity risk and volatility of the underlying asset. The quality of the collateral could be improved by setting strict margining requirements like initial margins and haircuts etc. and all these features should have an impact on the effective repo rate of the repo transaction. Compared to Treasury bonds transactions, repo transactions do not give any parties tax benefits and hence repo rate is free from tax benefits bias. Here if the underlying asset is the Treasury bond in repo transaction, the Treasury bond is free of default risk and has abundant liquidity, the repo rate should be close to yield to maturity of the underlying Treasury bond. (If the

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Counterparty Credit Risk, Funding Risk and Central Clearing time to maturity of repo is equal to the time to maturity of the underlying Treasury bond, we expect the repo rate should be slightly higher than the yield to maturity.) The repo transaction gives rise to a strategy which derivative dealer could employ to fund at a rate close to yield of Treasury. Or a low credit rated derivative dealer could borrow at a similar rate of US Government through this repo transaction. At the same time, the derivative dealer could choose to lend money at US Treasury rate through reverse repo transaction. The collateral of these kind of lending activities is Treasury bond and hence the derivative dealer does not expose itself to counterparty credit risk. For non-Treasury asset repo transactions, the repo rate is not risk free and should contain risk premium of credit/liquidity of the underlying asset and joint default of the underlying asset and borrower.

Given the fact there is no repo markets for derivatives, the derivatives transactions are normally funded by the derivative dealer’s bondholders and the credit quality of the derivative dealer has a huge impact on the funding rate of derivatives. For exchange traded derivatives, like options, the risk free rate could be implied from the put call parity. For OTC derivatives, especially products structured with multiple interest rate curves, the pricing is more intricate. If we want to price an OTC option on Treasury bond, the yield to maturity is used to discount the Treasury bond and the derivative dealer’s own funding curve (normally expressed with LIBOR plus a spread) is used to discount the option payoff.

3. Wrong Way Risk (WWR)

Generally, the derivative dealer has to issue corporate bonds or commercial papers to fund its operations and the funding rate they obtain through this kind of transactions heavily depends on the issuer’s credit quality. Therefore, it is very interesting to explore the potential strategies the derivative dealer could employ to lower borrowing cost and enhance lending benefits. For example, a hedge fund manager is interested in buying a corporate bond but there is no repo market for that specific asset. The manager would approach an investment bank to structure a for that corporate bond. The investment bank would enter the market to buy a corporate bond and delivers the coupon and depreciation/appreciation of the bond to the hedge fund manager. The hedge fund manager is going to pay LIBOR plus a spread to the investment bank. The spread the hedge fund manager going to pay is significantly lower than the spread charged by its bondholders. The design of the structure of the trade is the key to reduce the hedge fund manager’s funding cost. The investment bank retains the ownership of the corporate bond in this transaction and hence could use the corporate bond as the collateral pledged for this total return swap. That’s the main reason why the spread the hedge fund manager going to pay is significantly lower compared to his/her own unsecured funding spread. The synthetic repo structure is introduced using total return swap structure. As discussed in Hull (2009), the spread over LIBOR is a compensation for the Wrong Way

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Risk (WWR) investment bank bears in this transaction. That is, the investment will have a significant loss if the hedge fund defaults and the corporate bond depreciates at the same time. Hence this spread should be priced with credit quality of the hedge fund, credit quality of the corporate bond and the default dependency (or default correlation) between these two entities taken into consideration. This is one of the classic method that hedge fund uses to reduces its funding cost. Another strategy to achieve lower funding cost to issue structured notes with exotic features. The generic idea behind the structured notes is that the bond issuer sells options to the bondholders and built the price of these options into the coupons of the notes. The options here could be swaptions (to convert the notes into a callable notes) or combinations of digital caps and floors (to convert into range accrual notes). The bondholders who bought these options have to give up some upside future scenarios to enhance the coupons they are going to receive. Via structured notes, derivative dealers and other financial institutions could achieve a funding equal to LIBOR while pay out coupon higher than that of par vanilla bond (or par swap rate). Some firms sells put options on itself to reduce its funding cost when buy back their own . The options buyers in this scenario has to be cautious as the firms might not be able to fulfil their obligation when the stock price plummet in the future. This is a classic Wrong Way Risk (WWR) trade.

4. Bid/Ask Spread

Bid/ask spread is another feature of real market compared to the perfect market in theory. Generally it is accepted that the price does not give rise to any arbitrage opportunities if it sits within the boundaries of the bid/ask spread. The bid/ask spread imposed by derivative dealers enables the derivative dealers to take profits after covering hedging cost and other sorts of risks. In principal, the price of counterparty credit risk should be built into bid/ask spread. However, the derivative dealer has to provide a different bid/ask price for every single counterparty and it is very inefficient to operate in current IT structure. As we will discuss in this chapter, the CVA or the price of counterparty credit risk would be significantly greater than bid/ask spread in some scenarios. Derivative dealers normally mark their bid/ask spread to its market competitors. The bid/ask spread sometimes are measured in terms of hedge ratios like Greeks. Bid/ask spreads for linear products like interest rate swaps are expressed in terms of Delta and bid/ask spreads for non-linear products like interest rate swaptions are expressed in terms of Vega. The bid/ask spreads charged by derivative dealers normally are not enough to offset all hedging cost for a standalone derivatives transaction. This actually shows one of the major comparative advantages of derivative dealers is that all the new trades could be partially offset by the existing trades in the portfolio as long as the portfolio is large and well diversified. The remaining positions on Vega, Gamma, and Cross Gamma etc. would not completely hedged away and the money saved would be used to take on some proprietary trading positions within certain risk limits. Although every market participant could follow the self-financing

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Counterparty Credit Risk, Funding Risk and Central Clearing framework to hedge the risk by themselves, the existence of large and well diversified portfolio ensures the price including bid/ask spread charged by derivative dealers competitive and make it cost effective to have derivatives transactions with derivative dealers.

5. The Existence of Un-hedgeable Variables

Un-hedgeable variables like correlations and correlation skews have impact on the price which could not be fully hedged by trading liquid products. Any market participant could offload the trades with great exposure to these un-hedgeable variables in their book by trading back to back contracts with another counterparty but this normally incurs significant costs. The risk preference of the counterparty and demand and supply of that specific contract have a significant impact on pricing. For example, if we observe higher at lower strike and implied volatility is a monotonic function of strikes, we call this structure volatility skew and it could be produced by using CEV (constant elasticity of variance) model. If we observe higher implied volatility at both end of the strikes and volatility behaves like a non-monotonic function of the strikes, we call this structure volatility smile. Suppose the derivative dealer trades a Bermudan swaption. The derivative dealer could use a set of European swaptions to hedge its Vega positions, but the Bermudan swaption position could not be completely hedged by using European swaptions. Or we will have different prices for the same Bermudan swaption priced by different interest rate models calibrating to the same set of European option prices. This indicates that the Bermudan swaptions depends on market variables which could not be hedged away by European swaptions. Forward volatility term structure is the un-hedgeable market variable here. The decision of Bermudan swaption holder is based on the conditional expectation of future option price. And the conditional expectation of future option price relies on the conditional quantities like forward volatility skew/smile conditioned on future times and states. European swaptions, on the other hand, are priced by unconditional expectation of volatility skew/smiles observed today. Therefore, the information extracted from European swaptions could not fully determine the Bermudan swaptions price as the forward volatility term structure is not a pricing factor of European swaption. The swaptions on forward starting swap could ease the problem but there is not enough liquidity for such kind of contracts. Therefore, Bermudan swaption could not be priced using a dynamic linear combination of European swaptions (or a portfolio of European swaptions). In fact, Bermudan swaption is a compound options on a basket of European swaptions. If the option holder decides to exercise the Bermudan swaption, one of the European swaption in that basket activates and cancels the rest of options in the basket. Given the fact it is not possible to price this un-hedgeable forward volatility skew/smile with market products, the fair price (no arbitrage price) of this un-hedgeable variable has to be priced by the risk preference of market participants and demand and supply of Bermudan swaptions. In reality, Bermudan swaptions are often used to cancel swaps/bonds transactions (in the same currency). Bermudan swaptions normally

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Counterparty Credit Risk, Funding Risk and Central Clearing are issued with callable bonds and bond issuers normally buy these Bermudan swaptions from the bondholders (and compensate bond holders with higher coupon) and hedge (sell) these Bermudan swaptions with (to) derivative dealers. The derivative dealers become net buyers of the Bermudan swaptions and the abundant supply of Bermudan swaptions supressed the price and hence implied volatility.

No arbitrage principles requires the prices produced by derivative dealer’s models should be consistent with the prices of hedging instruments available in the market. This procedure is called calibration. Local calibration matches the model parameters with the initial yield curve and selected Vega positions. Global calibration matches the model parameters with the entire volatility matrix. Financial engineers could choose different levels of calibrations to match the model prices to the liquid hedging instruments prices. However, for some exotic products like Bermudan swaptions and total return swaps, their prices relies on the forward volatility term structure and default dependency between underlying credit and counterparty, the liquid hedging instruments prices could not provide an indication of prices of these un-hedgeable variables. These un-hedgeable variables like correlation and correlation skew could be estimated from historical analysis and derivative dealers would ‘assume’ the parameters are stable throughout the entire analysis. Or derivative dealers could reach other brokers and derivative dealers to ask for quotes on underlying transactions or these variables and calibrates their model to these quotes to produce market compatible prices for exotic products. If a derivative dealer finds itself keeps on losing trades to its competitors, or if the derivative dealers finds itself keeps on winning trades from its competitors, these kind of scenarios normally indicate there is something wrong with the models. If the derivative dealer could justify the additional benefit of winning trades comes from the portfolio effect in CVA/DVA pricing, it could be possible but someone should be assigned to investigate these issues. The change of market structure or market regime poses model risk to the calibrations result and therefore the financial engineers or traders should review their un-hedgeable variables calibration results on regular basis to capture these changes in the market structure. And because the unexpected market structure change does have an impact on derivative dealer’s PnL, the derivative dealer should set aside a portion of its PnL as reserves for these possible market events. These reserves could be calculated via hedge ratios or Greeks of the underlying transactions with respect to the un-hedgeable variables.

Due to the existence of un-hedgeable variables like correlations skew and forward volatility term structure, the models for pricing should be different from the models for hedging to some extent. The models for pricing should be calibrated to all liquid hedging instruments available, but they fail to capture the potential impact of un-hedgeable variables have on the future hedging activities. model like Derman-Kani model is able to provide market price of options consistent with liquid European options but fails to produce market consistent volatility skew/smile which is the key

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Counterparty Credit Risk, Funding Risk and Central Clearing in determination of prices of American/Bermudan options with early exercise features. For examples, the volatility skew (higher implied volatility at lower end strikes) is well observed in the market. If the fixed income market rallies, or in other words, the rates drop, the volatility at the lower end strikes should increase in this circumstance. The general for interest rates assumes a log-normal distribution for forward rates and constant percentage volatility across different strike prices. Hence when the market rallies and interest rate drops correspondingly, if the derivative dealer holds short positions in Vega and calculates its Vega using Black’s model, the derivative dealer will lose money and the Vega hedge position should be recognised partially hedged by Delta. In interest rate models family, handling the volatility skew and smile is a difficult task and it needs different sorts of rates models to produce the desired consistent market volatility dynamics. The choice of interest rate models for counterparty credit risk pricing is even more complex and the discussion is included in the Counterparty Credit Risk Chapter. Put this logic in another way, suppose the derivative dealer buys a derivative from a counterparty using pricing model and decides to hedge this transaction by back to back trade, it does not make sense for the derivative dealer to choose the pricing model as it provides the same price when the derivative dealer bought the derivative. In order to capture the bid/ask spread, the derivative dealer should choose a model produces a price greater or equal to the mid-price of that derivative. Black Scholes models have a clear advantage in quoting prices of derivatives but different models would be chosen to produce hedge prices due to the reasons we discussed above.

6. The Tradability of Market Variable

The market is incomplete because there does not exist an insurance market for every single risk factor in the market. Some risk factors, which are un-hedgeable variables as discussed above, cannot be priced using information extracted from hedging instrument with enough liquidity. If the market variable is not tradable, that does not indicates that market variable is un-hedgeable. Interest rate is not a tradable asset but it could be hedged by different sorts of interest rate derivatives. Even if the underlying asset is tradable in the marketplace, this does not indicate the market will complete on this specific market variable. A classic example is the electricity markets. The storage cost for electricity is too high such that the self-financing strategy is not a pragmatic solution to replicate the derivatives contracts written on electricity. In fact, market participants would writes forward or futures contracts on given non-tradable variables to make them tradable. For examples, weather derivatives like HDD (heating degree days) are traded in CME and emerging markets currencies forwards (like non- deliverable forwards) are traded among different derivative dealers. ETFs (exchange traded funds) provides a potential solution to market completeness. In the end of the day, the market is complete if and only if different market participants have different views on market variables and would like to trade products covering these market variables with each other.

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Up to this part, the prices of different contracts are determined by no arbitrage principles. In terms of economics theory, the market clears at its equilibrium price where demand equals supply. This type of pricing model is widely used by exchanges in determination of prices of exchange traded products. Exchanges runs a flat book with different market participants and matches buy and sell orders to find the market prices of products. The no arbitrage theory requires us to construct a self-financing strategy to dynamically hedge the derivatives with underlying assets with abundant liquidity. The price of the dynamical hedging portfolio is the price of the derivative. From this perspective, no arbitrage principle is essentially a relative value analysis method to eliminate the price difference between the derivative and underlying assets. The un-hedgeable variables could be estimated from historical data under the real world measure or calibrated to the quotes from different derivative dealers. It is very hard to derivative dealers to be market neutral to un-hedgeable variables.

The structure

Generally speaking, the derivative dealers are the sell side of the financial industry. They provide liquidity to the market and maintain a market neutral position. The net income will come from bid/ask spread in theory. The discussion on the nature of CVA/DVA/FVA trading will be included in Counterparty Credit Risk and Funding Risk Chapters. The buy side of the industry, includes hedge fund and asset management firms etc., is the supplier of the capital and purchase different categories of financial instruments from the sell side to achieve capital appreciations. The OTC derivatives, which was perceived as a leveraging tool to achieve capital appreciation (as the needs to pay little to initiate a transaction using derivatives), is under heavy regulatory scrutiny at the moment. Both parties of the OTC derivatives transactions are asked to post margins to reduce the leverage embedded in the market. The detailed discussion is included in Central Clearing Chapter.

The capital appreciation target of the buy side could be realised through investment in private firms at a discount price or investment into asset or liabilities of public firms with tailored risk management solutions bought from derivative dealers. The derivative dealers would choose to dynamic hedge the risk management solutions they sold or offload these transactions to other market participants. The risk of unfunded derivatives structure has been recognised and both parties of the transactions have to post collateral under current regulatory regime. However, the derivative dealers could choose to structure the unfunded derivatives into the coupon of a coupon bearing bond and convert the unfunded derivatives transactions into funded structures. The main investors for these kind of structured notes are asset management firms seeking for high coupons and pension funds that are not allowed to directly participate in certain OTC derivatives transactions. The coupon enhancement, or yield enhancements, come from the options the investors sold to the derivative dealer like

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Counterparty Credit Risk, Funding Risk and Central Clearing callability of the bond and range accrual features of the coupon. Investors thereby take on more risks through these transactions. Insurance firms, commercial banks are exposed to asset liability mismatch problem (like commercial banks use short term deposits (liabilities) to fund long term (assets)) and the sensitivities of assets and liabilities to a given market variable like interest rate are different. This serves as the basis for the demand of interest rate swaps to change the sensitivities of asset and liabilities to reduce the commercial bank’s exposure to interest rate volatilities.

The underlying driving factors of derivatives trading

One of the underlying driving factor of derivatives trading is the discrepancy in the views between no arbitrage models (under risk neutral measure) and forecasting/econometrics models (under the real world measure). Under no arbitrage pricing framework, the spot quantity is expected to follow the path of the forward curve. Or in other words, the forward quantity is expected to be realised and becomes the future spot quantity. For example, if the forward curve of 3 month LIBOR is upward sloping, the spot rate realised in the future is exactly equal to the forward curve prediction under no arbitrage framework. The econometrics models generate their predictions though historical analysis and claims that the spot 3m LIBOR rate is expected to be normally distributed around current observation therefore the future spot rate will not locates at the same points as indicated by the forward curve but remains stationary around current spot rate. If an investor believes in the analysis of econometrics, he/she would incline to enter into a fixed receiver interest rate swap as the swap rate is priced higher under no arbitrage framework than econometrics models. As discussed previously, certain types of investors are inclined to trade derivatives or structured notes because of the higher coupon attached to the transaction. The general idea behind this higher coupon is investors sold some options or implied volatilities to the derivative dealers and derivative dealer in return enhance the coupon paid to the investor. The option the investor sold does not have a high probability of getting exercised under real world measure and hence investor are willing to take on this type of risks in exchange for the higher coupon as the extra coupon is worth more than the options or implied volatilities they sold to derivative dealers under risk world measure.

Hence, the driving factors for derivatives trading could be summarised as undervalued assets are created with exotic derivatives through structured notes or other structures under real word measure. On the other side of this transaction, the derivative dealer does not have to lose on this trade but close out the derivatives position via proper hedging strategies to lock in its profit. This is an interesting situation as both side could make money at the same time. Which side of the market is providing profits to both parties?

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Let’s use interest rate swap transaction as an example. The spot rate LIBOR rate is expected to ‘climb up’ the forward curve under risk neutral measure in an upward sloping yield curve environment. The swap rate priced under no-arbitrage framework is going to be much higher than the swap rate priced under real world measure as discussed previously. The investor would enter into a fixed receiver swap. On the other side of the trade, the derivative dealer will enter into a fixed payer swap and hedge by interest rate futures and other interest rate swaps. The market marker could remain neutral to market movements but still earn bid/ask spread. If the markets moves as anticipated by the investor, the investor would make profits and the derivative dealer will earn bid/ask spread. So where is the source of the profits for both sides of the transaction? There are natural fixed payers in the market. The governments and corporates that issue fixed coupon bonds and individuals who make regular mortgage payments are the market participants who have to lock in the regular coupon payments to hedge against surge in long end yields. These transactions co-create the shape of forward curve with fixed receiver transactions in the markets and becomes the natural back to back hedges for the derivative dealers. If the bonds markets sells off and rates rise significantly, the fixed receiver investor will face significant losses and the economic loss from these transactions together with the economic benefits for governments/corporates and individuals who make regular fixed payments determines the shape of the forward curve. Other factors like FOMC (Federal Open Market Committee) decisions on Fed Funds Rate and QE (quantitative easing programme) also have significant impact on determination of interest rate levels.

The common bond structure is fixed coupon bonds. Suppose a firm issues a fixed coupon bond and the treasury of that firm enters into a fixed receiver swap at the same time. The funding spread attached to the floating leg of that interest rate swap could be positive or negative. It depends on the coupon size of the bond, credit quality of the firm and demand and supply of that bond. Essentially this transaction helps the firm convert its fixed rate liability into floating rate liability. Suppose current market forward curve is upward sloping. If the treasury holds the view that the interest rate would stabilise around current level of spot rate, such kind of liability conversion/transformation effectively cut the funding cost of that firm. At the same time, such kind conversion makes the outstanding debt immune to underlying rate changes. The effective duration of a floater is close to zero. Let’s make a simple comparison with/without conversion. If the bond market rallies and the rates level decreases significantly, the outstanding debt with fixed coupon will become more valuable to bondholders but create a loss for the bond issuer. The logic is that the firm could issue a fixed coupon bond with a significant lower coupon in this scenario. The transformation from fixed coupon to floating coupon with a fixed spread essentially reduces the sensitivity of the bond price to the underlying market interest rate volatility. This kind of conversion/cost saving strategy is popular when market expects the interest rate is more likely to decrease or stabilise around current level in the real world measure. Technically speaking, after conversation, the firm actually issued a floating coupon bond with a fixed

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Counterparty Credit Risk, Funding Risk and Central Clearing funding spread attached to the floating coupon. The fixed coupon spread exposes the debt of the firm to volatility of underlying market interest rates but the sensitivity to the interest rate movements of this fixed funding spread is significantly lower than that of fixed coupon. There are scenarios that the firms want to enter into fixed payer positions. Suppose the firm plans to issue debt in 5 years and wants to lock in the financing cost of the debt as the interest rate might rally (or the bond market might sell off) in 5 years. The bond issuer, the firm, therefore would like to enter into a 5 years forward starting fixed payer swap to lock in the coupon rate. This hedging structure could effective reduce the financing cost in a bearish market in real world measure. Normally the forward starting swap is cash settled and hence the credit charge (i.e. CVA) is significant lower than the physically settled interest rate swap. The detailed discussion of difference of impact on counterparty risk pricing between cash settlement and physical settlement is included in the Counterparty Credit Risk Chapter.

The firm could issue a callable fixed coupon bond to reduce its sensitivity of outstanding debt to underlying interest rate volatility. If the market rallies and interest rates go down, the bond issuer has the option to call back the outstanding debt and refinance its operations at a lower coupon rate. And if the market sells off and interest rates go up, the bond issuer would maintain its current coupon rate since it is lower than the market expected fair coupon rate. The callable feature here is similar to prepayment feature of mortgage related products. Essentially the bond issuer bought a Bermudan on the bond (or the corresponding Bermudan put option on the rates) from the bond holders at the inception of this transaction. The options or implied volatilities the firm (or bond issuer) bought will be built into the coupons and bond holders could receive a higher coupon compared to a standard fixed coupon bond without callable features to compensation for the risk the bond will be called back when the interest rate declines in the future. The bond holders therefore might receive the principal when the interest rate is low and have to reinvest the cash in a low interest rate environment. The impact of negative convexity (from the perspective of bond holders) associated with the short position in options reduce the profitability of investment in callable bond. The bond issuer of callable bond has to pay a higher coupon and hence could sell the option bought to the derivative dealer. Normally the callable bond is hedged with a cancellable fixed receiver swap. The cancellable fixed receiver swap is made up with a plain vanilla fixed receiver swap and a fixed payer Bermudan swaption. If the derivative dealer decides to cancel the fixed receiver swap (i.e. exercise the Bermudan swaption at one of the pre-determined dates and enter into a fixed payer swap), the bond issuer should call back their outstanding debt. The Bermudan swaptions market is normally a one-side market and the bond issuers are the main suppliers of Bermudan swaptions. This special market structure makes the implied volatilities of Bermudan swaptions lower compared to their counterparts of European swaptions. The details of comparisons between Bermudan and European swaptions from the perspective of CVA pricing are included in the Counterparty Credit Risk Chapter.

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In reality, several extra factors have to be taken into account (either from perspective of the bond issuer or derivative dealer) to reach the decision of early exercising the Bermudan swaptions. The movement in CVA/FVA and regulatory capital caused by early exercise should be recognised in the first place. And there exists the potential disputes between the bond issuer and derivative dealer on optimal exercise time of the Bermudan swaption. The bond issuer eliminates his/her exposure to underlying market interest rate though the cancellable swap transaction with the derivative dealer. However, the potential movements of the funding spread of the bond issuer are not properly managed in this structure. Suppose the market interest rate remains at the same level in 6 months but there is a significant drop in the funding spread of bond issuer (as its credit quality improves), the bond issuer may find it is optimal to call back the bond and refinance at lower coupon. This phenomenon is common for low credit rated bond issuers. At the same time, the underlying interest rate is stationary around its previous level and the derivative dealer finds it is far from optimal to exercise Bermudan swaption. High credit rated bond issuers would like to enter into cancellable swaps with high credit rated derivative dealers. Several derivative dealers set up certain types of SPV (special purpose vehicles) and DPC (derivative product company) with sufficient capital to get accredited with high credit rating in order to trade with bond issuers with high credit rating. This type of counterparty credit risk management policy got tested in the global financial crisis and the implications of this structure are discussed in Central Clearing Chapter.

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2.4 No Arbitrage Pricing across Different Asset Classes

The emergence of structured notes market is mainly due to the fact the structured notes provide enhanced coupon or yields compared to standard LIBOR quality bond. The bondholders could earn extra yield/coupon from their investment in structured notes without taking on exposure to credit risk in the high yield bonds market. The economic value of the enhanced coupon comes from the fact the notes will pay very low coupons under certain market scenarios of interest rates movements. The investors make their judgements under real world measure and express their opinions on under- valuations of the notes through their investment. Difference in beliefs between no arbitrage model users and econometrics model users gives rise the structured notes market. The bond issuer normally enters into a swap with embedded exotic options with the derivative dealer to eliminate his/her exposure to the underlying market movements. The bond issuers pay the coupon with exotic features to the derivative dealer in exchange for LIBOR plus funding spread. This back to back transaction structure helps the bond issuer to reduce his/her exposure to market volatility. The counterparty credit risk is still there and it needs proper management of the bond issuer. The classic callable bond (replicated with swaptions), knock out bonds (replicated with swaptions and annuities), range accruals (replicated with Caps/Floors) and non-inversion notes (replicated with curve options) etc. are based on replication strategy under no arbitrage framework to provide extra coupon to the investors. The general set up of structured notes is very flexible and derivative dealer could create various kinds of structures to meet different risk preference of investors. For example, suppose investors perceive the yield curve of treasury market is going to steepen in the future. The derivative dealer could create non-inversion notes which pays a high coupon if yield curve steepens and a low coupon otherwise. Essentially the investors are taking positions in the CMS ()/CMT (constant maturity treasury rates) spread or the slope of yield curve. This is equivalent to take positions in second component of yield curve in PCA analysis. The probability of curve inversion is higher under risk neutral measure and hence coupon of notes with net short positions in inversion options are more valuable under risk neutral measure. Other interesting products like target redemption (TAR) notes and their implications on counterparty credit risk management are included in case studies part of the Counterparty Credit Risk Chapter.

The similar strategies of structured notes in the interest rates market could be expanded to FX and credit markets. Under no arbitrage framework, two currencies should be governed by interest rate parity and interest rate differentials lead to an upward sloping forward curve for the currency with low rates. Or in other words, the low rates currency is expected to appreciate against high rates currency under no arbitrage framework. For example, let’s say spot rate for EURUSD is 1.10. 10 year US Treasury yield is 1.84% and 10 year German bund yield is 0.19%. The no arbitrage USD price or forward price for €1 in 10 years is therefore 1.30 (as we could deposit €1 and $ 1.10 for 10 years).

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1 푢푛푖푡 표푓 푙표푤 푟푎푡푒푠 푐푢푟푟푒푛푐푦 Hence if FX spot rate is expressed as , we could observe an upward 1 푢푛푖푡 표푓 ℎ푖푔ℎ 푟푎푡푒푠 푐푢푟푟푒푛푐푦 sloping forward curve or FX spot rate ‘climb’ up the forward curve under no arbitrage framework. Econometrics models under real world measure would suggest the future FX spot rate is expected to stay around the current spot rate. Due to the interest rate differentials among different economies or currencies, the investor seeks for high coupons would like to enhance yield through investment in high rates economies or currencies. A reverse dual currency note (RDC) is a bond that pays out foreign rate coupon in domestic currency. The derivative dealer would price RDC under the risk neutral measure and the extra coupon created by foreign rate would be offset by higher forward rate (as the spot rate ‘climbs’ up the forward curve). The investor would like to buy this notes as the price offered by derivative dealers is undervalued under real world measure. A popular structure based on RDC is called power reverse dual currency note (PRDC) and it gives the opportunity for investors in the low rates economy to receive leveraged high coupon in domestic currency. Clark (2011) gives a nice discussion of this product. Suppose a Japanese investor intends to enhance yield through investment in PRDC USD notes. The coupon rate is

푈푆퐷 퐹푋푖 퐽푃푌 (2.4.1) 푐푖 = 푐푖 − 푐푖 , 퐹푋0 푈푆퐷 퐽푃푌 where 푐푖 is USD coupon rate, 푐푖 is JPY coupon rate and 퐹푋푖 is USDJPY FX rate at time 푖.

퐹푋0 is normally the spot USDJPY rate at inception of the transaction as in classic cross currency swap settings. The issuer of PRDC notes may place a cap and floor to prevent 푐푖 moving outside of the desired territory.

푈푆퐷 퐹푋푖 퐽푃푌 푓푙표표푟 푐푎푝 (2.4.2) 푐푖 = min (푚푎푥 (푐푖 − 푐푖 , 푐푖 ) , 푐푖 ), 퐹푋0 푐푎푝 푓푙표표푟 where 푐푖 denotes the cap rate for the coupon and 푐푖 denotes the floor rate of the coupon. The issuer of PRDC notes normally enters into a PRDC cross currency swap with a derivative dealer to hedge the PRDC exotic coupon. The issuer here would pay JPY notional to the derivative dealer in exchange for USD notion at inception and receive 푐푖 as specified above from the derivative dealer to offload the exotic coupon liability with compensation of TIBOR plus spread to the derivative dealer. From the perspective of investor of PRDC notes, he/she shorts the cap and call option linked to FX rates and longs the floor. If 퐹푋푖 declines relative to 퐹푋0 at time 푖, that is, USD depreciates against JPY, the investor would face loss on this trade. This is similar to the experience of Japanese exporting manufacturers who will lose money if JPY appreciates against foreign currencies. The coupon here could be further enhanced by callable, knockout or early redemption (TAR) features with increments 푓푙표표푟 푐푎푝 in 푐푖 and 푐푖 . The PRDC coupon discussed here contain short positions in FX options and hence is exposed to FX volatility smile/skew and the correlations between FX rates and interest rate dynamics. The call option and cap embedded in coupon are puts on JPY and the floor is call on JPY from the investor’s perspective. The investor in PRDC longs cross gamma term. Normally the

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Counterparty Credit Risk, Funding Risk and Central Clearing knock-out feature is activated when the investor is in the money and it serves as a stop loss strategy for the derivative dealer side. The net short position of cross gamma term accounting for co- movements between FX rates and interest rates posed significant losses for derivative dealers on PRDCs. PRDCs could be created for emerging markets currencies as well. The one important difference for emerging market currency PRDC is that the coupon now is exposed to the credit of emerging market as well due to existence of high correlation between emerging market currency and its credit quality. This extra correlation makes the PRDC written on emerging market becomes a hybrid product on credit, FX and interest rates.

Investors could choose to invest into an written on his/her target assets (e.g. equities, bonds, etc.) to realise the same payoff without significant initial investments. This structure saves investor’s capital but changes the counterparty from asset issuer to swap dealer. For example, Db X- trackers ETFs issued by Deutsche Bank asset management arm used to be backed by asset swaps. The asset swap essentially replicates the cash flows of the underlying asset and gives investor additional flexibility to expand into other investment domains. An investor holding government bonds want to pursue higher coupon but prefer to retain the holding of underlying bond (because of high credit rating, liquidity etc.). The investor in this case could keep holding the government bond and enter into an asset swap to swap the government coupons into structured coupons like PRDC, range accruals etc. For high yield bond investors, the asset swap enables the investor to transform the fixed coupon received into LIBOR plus a fixed funding spread. Thus the investor could retain a pure position on the credit risk component of the bond without exposing to underlying interest rate movements.

All the classic instruments discussed above essentially are credit hybrid products due to counterparty credit risk. Its impact is more pronounced in case of trading with emerging markets sovereigns. Normally emerging markets sovereigns finance by issuing debt in strong currencies like USD and EUR. They would approach a derivative dealer to structure a cross currency swap transaction to transform the outstanding debt denominated in USD or EUR into debt denominated in their domestic currency. The emerging markets sovereign pays USD received from new issued debt to the derivative dealer in exchange for equivalent value of domestic currency at inception. Actually the emerging markets sovereign pays USD to buy a USD denominated coupon bond from the derivative dealer and the derivative dealer pays the equivalent value of domestic currency to the sovereign to buy coupon bearing bond in that currency. The existence of correlation between FX rate and credit quality of emerging markets sovereign poses an interesting question on counterparty credit risk management as it creates the significant dependency of the value of this transaction on the credit condition of the emerging market sovereign. That is, the domestic currency will be heavily depreciated if the emerging market sovereign runs into financial distress. From the perspective of the

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Counterparty Credit Risk, Funding Risk and Central Clearing derivative dealer, the derivative dealer has to close out this transaction at the default time of the sovereign entity. That is, the derivative dealer has to pay the Mark to Market of the foreign coupon bearing bond to the sovereign entity and the sovereign entity has to pay off it's the Mark to Market of domestic coupon bearing bond. Although the sovereign entity is in financial distress, technically speaking, it does not have any difficulty to repay its debt in domestic currency as the sovereign has the right to print more money and convert the credit risk into inflation risk. The derivative dealer in this case does have Right Way Risk (RWR) from a counterparty risk management perspective along with a great exposure to the market risk of exchange rates. Another interesting question is how to structure derivatives transactions to profit by hedging counterparty credit risk from the derivative dealer’s perspective. The derivative dealer could sign a credit extinguishing agreement with the sovereign entity (or buy a credit extinguisher from the sovereign entity) to release the duty to pay full amount of Mark to Market if the sovereign defaults. This transaction removes the asymmetric treatments of default ( that is, the surviving party has to pay off its full liabilities if the portfolio is out of money but claim the recovery of assets if the portfolio is in the money). The market price of this credit extinguisher is equal to CVA in theory. The derivative dealer is ought to pay the price of the extinguisher to the sovereign and profits by selling credit protections on this sovereign to make back the price it pays for credit extinguisher.

The credit events defined here could be default event, debt reprofiling and debt restructuring etc. The scope of credit events could expand into equity space due to the interconnection between CDS and deep out of money puts options (DOOM). The equity default swap, which consists of a portfolio of deep out of money puts, is triggered by the event that the equity falls below the pre-specified level. The equity default swap could be used as DVA hedging instruments and will be discussed in the DVA hedging chapter. The connections between equities and debts on credit risk give rise to some capital structure arbitrage opportunities. Merton (1974) lays out the general capital structure of a firm using option pricing theory. The shareholders of a firm acquired the call option (with implicit down and out barrier) on the firm’s assets. The bondholders, on the other hand, acquired the rest of the firm’s balance sheet. The relative discount the bondholders receive in their investment into corporate debt (compared to risk free bond) is essentially the put option (with implicit down and in barrier) that the bondholders sold to the firm at inception. The put option (with down and in barrier) would become very expensive if the firm defaults and essentially it works like credit protection or CDS. The detailed discussions are included in FVA chapter. The investors could conduct relative value analysis to figure out the relative cheapness of the options embedded in equities and debts with market neutral position to the underlying assets movements. Similar strategies could be constructed on evaluation of the call options on equities embedded in convertible bonds. The relative value analysis on the implied volatilities of calls in convertible bonds over the standard calls could produce some interesting strategies. Derivative dealers could use their on advantage in funding and holding of well-

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Counterparty Credit Risk, Funding Risk and Central Clearing diversified portfolio to securitise a portfolio of illiquid assets into a baskets of tranches with enhanced liquidity. The examples like CDO (collateralised debt obligations), synthetic CDO or CSO (collateralised swap obligations) could create tailored risk return profile to meet the demand of investors and provide liquidity to the underlying assets. The market marking activities for these products call for higher usage of regulatory capital and tighter scrutiny of counterparty risk management.

In reality, the arbitrage strategies are normally ‘quasi’ arbitrage strategies. The replication portfolio created by a self-financing strategy has a positive expected value in the future but with a positive probability to have a negative return. Statistical arbitrage and relative value analysis are examples of this category and the existence of counterparty credit risk further complicates the design of portfolio.

In general, derivatives market gives the opportunities for investors to adjust their risk return profile by their view or preference over certain types of risk and for arbitrageurs or speculators to enhance their expected return by leveraging with significant lower cost (as they do not have to hold the underlying assets). The derivative dealers could provide cost effective derivatives solutions to the investors due to their holding of large and well diversified portfolios that gives comparative advantages in funding and hedging over the investors. Investors would find it cheaper to trade a derivative with derivative dealers than to replicate the derivatives by themselves in most scenarios. Large corporates normally have to hold certain types of securities to maintain long term clients’ relationship and therefore expose themselves of the market movements of valuations of these securities on their balance sheet. Through off balance sheet derivatives transactions like total return swaps (TRS), the corporates could physically keep holding these securities but offload all the risks to the derivative dealers. The un-hedgeable market variables and non-tradable market variables make the tailored risk management derivatives solutions more expensive and derivative dealers have to invest more capital to provide services to their clients on hedging these variables. The Black Monday in 1987 is a classic example. The writers of put options could hedge their Delta position by selling more shares when the underlying stock price declines. However, a sizeable downward jump in stock price makes such kind of delta hedging strategies mission impossible as the derivative dealers or writers of the put options do not have enough time to sell a sufficient number of shares to delta neutral his book. Therefore, the derivative dealers demand higher premiums for out of money put options for bearing this kind of jump risk and lead to the volatility skew as we observe in the equity options market.

The Black Scholes pricing theory is built in a counterparty risk free frictionless market with sufficient liquidity. There is no transaction cost like bid/ask spread and no short selling restrictions in a frictionless market. Market counterparties like Lehman Brothers cannot default and cannot have an

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Counterparty Credit Risk, Funding Risk and Central Clearing impact on the market prices of securities even if counterparty defaults (on the assumption that the surviving party could replace his/her portfolio with another counterparty without any cost and the portfolio he/she trades with the defaulted counterparty could recover the full marketing to market with zero loss given default). The size of any transaction will not have any impact on the execution in the market with sufficient liquidity. Then there is at least one probability measure such that the asset prices relative to a numeraire are martingales. Here we assume the assets and numeraires do not produce intermediate cash flows.

퐶푡 ℕ 퐶푇 (2.4.3) = 피푡 ( ), 푁푡 푁푇 where 0 < 푡 < 푇 and ℙ푁 refers to the equivalent martingale measure with respect to numeraire 푁. The derivatives will be perfectly replicated by underlying assets available in the market if the equivalent martingale measure is unique. And hence we have a complete market. We might have multiple self-financing strategies to replicate a derivative even if the martingale is unique. In reality, the martingale relationship is not unique as there are so many un-hedgeable variables and risk factors that cannot be diversified in the marketplace. For assets and numeraires with intermediate cash flows, we could slice the whole stream of cash flows into portfolios that pays no cash flows before certain time and priced them at corresponding forward measure, 퐶 퐶 (2.4.4) 푠,푇 ℕ푇 푡,푇 = 피푠 ( ), 푁푠,푇 푁푡,푇 where 0 < 푠 < 푡 < 푇 and 퐶푡,푇 and 푁푡,푇 are market prices at time 푡 for all cash flows received by the derivative holder after time 푇. The swaption measure or annuity measure for swaps and swaptions pricing is equivalent to this setting. This technique is useful for pricing derivatives pay out intermediate cash flows during the life of the trade but these cash flows are not path dependent. For intermediate path dependent cash flows like dividends in American/Bermudan option pricing, the intermediate cash flows are re-invested into the assets to form up a self-financing strategy,

퐶푡퐷푡 ℕ 퐶푇퐷푇 (2.4.5) = 피푡 ( ), 푁푡 푁푇 푡 ∫ 푑푢푑푢 where 0 < 푠 < 푡 < 푇 and 퐷푡 denotes the cumulative dividends up to time 푡, i.e. 퐷푡 = 푒 0 . The general martingale relationship could be extended to include pricing under multiple numeraires as cross currency swaps, long dated FX derivatives and multi-currency margining constitutes a large component of CVA/FVA analysis. In some cases, the derivatives pricing with different numeraires could be simplified into Quanto adjustments or convexity adjustments for the forward curve of 퐹푋 underlying asset process. Let’s use plain vanilla European FX call as an example. 푆푡 is the FX spot rate and is the price of $1 (foreign currency) in terms of EUR (domestic currency). Or equivalently 퐹푋 픻 speaking, 푆푡 refers to USDEUR rate in our example. 푃 (푡, 푇) denotes the default free zero coupon bond price at time 푡 in domestic currency, i.e. EUR. And 푃픽(푡, 푇) denotes the default free zero

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퐹푋 픽 coupon bond price at time 푡 in foreign currency, i.e. USD. Therefore, 푆푡 푃 (푡, 푇) is the spot domestic (EUR) price of foreign (USD) zero coupon bond. Hence we have 퐹푋 픽 퐹푋 픽 푆푠 푃 (푠, 푇) 푆푡 푃 (푡, 푇) (2.4.6) = 피픻푇 ( ), 푃픻(푠, 푇) 푠 푃픻(푡, 푇) where 푃픻(푡, 푇) is the numeraire for domestic currency forward measure. The FX forward rate is defined as 퐹푋 픽 푆푡 푃 (푡, 푇) (2.4.7) 퐹퐹푋(푡, 푇) = . 푃픻(푡, 푇) Then the classic interest rate parity following martingale relationship could be expressed as

퐹푋 픻푇 퐹푋 (2.4.8) 퐹 (푠, 푇) = 피푠 (퐹 (푡, 푇)). The European FX call option is traded in domestic currency (EUR) and its current market price is 픻 푉푡 , 픻 픻 푉푠 푉푡 (2.4.9) = 피픻푇 ( ), 푃픻(푠, 푇) 푠 푃픻(푡, 푇) where 0 < 푠 < 푡 < 푇. The European FX call option notional is 푁픽 denominated in the foreign currency (USD) and the FX forward rate is going to converge to FX spot rate at maturity 푇 by no arbitrage principal, 픻 픽 퐹푋 픽 퐹푋 푉 = 푁 max(푆 − 퐾, 0) = 푁 (푆 − 퐾)ퟏ 퐹푋 (2.4.10) 푇 푇 푇 푆푇 −퐾>0 픽 퐹푋 = 푁 (퐹 (푇, 푇) − 퐾)ퟏ퐹퐹푋(푇,푇)−퐾>0, Then the price could be found by using classic Black Scholes framework. For general cross currency swaps and PRDC products, a three factors model is usually applied to calibrate to the FX spot rate, foreign rates term structure and domestic rates term structure at the same time. An excellent introduction to three factors model and its calibration could be found at Clark (2011). Normally the plain vanilla FX derivatives like non-deliverable options (NDO) and no-deliverable forwards (NDF) are cash settled. Some emerging markets currencies are not free floating and under close scrutiny of administration hence the form of NDF and NDO could enhance the liquidity of products. Another important family of martingale with multiple numeraires is Quanto products. The investors are attracted by returns/yields of foreign assets (equities, fixed income, etc.) but do not like to expose to FX rates movements. Quanto products enable the investors to capture the foreign assets payoffs (denominated in foreign currencies) and convert into domestic currency at pre-specified FX rate 푆̅퐹푋 픻 without exposing to FX rates movements. Here 푉푡 denotes the Quanto derivative price denominated in domestic currency at time 푡, then the martingale under domestic forward measure is 픻 픻 푉푠 푉푡 (2.4.11) = 피픻푇 ( ), 푃픻(푠, 푇) 푠 푃픻(푡, 푇) where 0 < 푠 < 푡 < 푇 and the Quanto payoff at maturity 푇 is 픻 퐹푋 픽 푉푇 = 푆̅ 퐶푇 , (2.4.12)

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픽 where 퐶푇 is the payoff of foreign asset (denominated in foreign asset) at time 푇. Then the Quanto product price at time 푠 is 푉픻 푆̅퐹푋퐶픽 (2.4.13) 푉픻 = 푃픻(푠, 푇)피픻푇 ( 푇 ) = 푃픻(푠, 푇)피픻푇 ( 푇 ) 푠 푠 푃픻(푇, 푇) 푠 푃픻(푇, 푇)

픻 픻푇 ̅퐹푋 픽 = 푃 (푠, 푇)피푠 (푆 퐶푇 ). 픽 픽 Given the fact that the forward price will converge to spot price at maturity 푇, i.e. 퐹 (푇, 푇) = 퐶푇 , the price of Quanto product is

픻 픻 픻푇 ̅퐹푋 픽 픻 ̅퐹푋 픻푇 픽 (2.4.14) 푉푠 = 푃 (푠, 푇)피푠 (푆 퐶푇 ) = 푃 (푠, 푇)푆 피푠 (퐹 (푇, 푇)). So the real challenge is, how to price the forward foreign price under forward domestic measure? The answer is to use the classic Quanto adjustments. The natural hedging instruments for Quanto products specified above would be the domestic asset 퐹푋 픽 픽 푆푡 퐶푇 and foreign asset 퐶푇 , and therefore we could construct two martingales under domestic and foreign forward measures for these two assets, respectively. 퐹푋 픽 퐹푋 픽 푆푠 퐶푠 푆푡 퐶푡 (2.4.15) = 피픻푇 ( ), 푃픻(푠, 푇) 푠 푃픻(푡, 푇)

픽 픽 퐶푠 퐶푡 (2.4.16) = 피픽푇 ( ), 푃픽(푠, 푇) 푠 푃픽(푡, 푇) where 0 < 푠 < 푡 < 푇, thus we have 푃픻(푠, 푇) 푆퐹푋퐶픽 퐶픽 (2.4.17) 픽 픻푇 푡 푡 픽 픽푇 푡 퐶푠 = 퐹푋 피푠 ( 픻 ) = 푃 (푠, 푇)피푠 ( 픽 ), 푆푠 푃 (푡, 푇) 푃 (푡, 푇) Hence 퐶픽 퐶픽 푆퐹푋푃픽(푡, 푇) 푃픻(푠, 푇) (2.4.18) 픽푇 푡 픻푇 푡 푡 피푠 ( 픽 ) = 피푠 ( 픽 픻 ) 퐹푋 픽 , 푃 (푡, 푇) 푃 (푡, 푇) 푃 (푡, 푇) 푆푠 푃 (푠, 푇)

푑ℙ픽푇 and the Randon-Nikodym derivative 푀 = 피픻푇 ( ) therefore satisfies the following equation 푡 푡 푑ℙ픻푇 푑ℙ픽푇 푆퐹푋푃픽(푡, 푇) (2.4.19) 피픻푇 ( ) 푡 푡 푑ℙ픻푇 푃픻(푡, 푇) 픽 = 퐹푋 . 픻 푑ℙ 푇 푆 푃픽(푠, 푇) 피 푇 ( ) 푠 푠 푑ℙ픻푇 푃픻(푠, 푇) Therefore, 푑ℙ픽푇 푆퐹푋푃픽(푡, 푇) (2.4.20) 픻푇 푡 퐹푋 푀푡 = 피 ( ) = = 퐹 (푡, 푇) 푡 푑ℙ픻푇 푃픻(푡, 푇) The original two master equations governing the domestic spot price and foreign spot price could be updated using corresponding forward prices. The foreign asset martingale could be updated as 픽 픽 퐶푠 퐶푡 (2.4.21) = 피픽푇 ( ) ⇒ 퐹픽(푠, 푇) = 피픽푇 (퐹픽(푡, 푇)). 푃픽(푠, 푇) 푠 푃픽(푡, 푇) 푠 The domestic asset martingale could be updated as

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푆퐹푋퐶픽 푆퐹푋퐶픽 푆퐹푋푃픽(푠, 푇) 퐶픽 (2.4.22) 푠 푠 = 피픻푇 ( 푡 푡 ) ⇒ 푠 푠 푃픻(푠, 푇) 푠 푃픻(푡, 푇) 푃픻(푠, 푇) 푃픽(푠, 푇)

퐹푋 픽 픽 푆푡 푃 (푡, 푇) 퐶푡 = 피픻푇 ( ) 푠 푃픻(푡, 푇) 푃픽(푡, 푇)

퐹푋 픽 픻푇 퐹푋 픽 ⇒ 퐹 (푠, 푇)퐹 (푠, 푇) = 피푠 (퐹 (푡, 푇)퐹 (푡, 푇)). By the definition of covariance of two variables, we have

픻푇 퐹푋 픽 (2.4.23) 퐶표푣푠 (퐹 (푡, 푇), 퐹 (푡, 푇))

픻푇 퐹푋 픽 픻푇 퐹푋 픻푇 픽 = 피푠 (퐹 (푡, 푇)퐹 (푡, 푇)) − 피푠 (퐹 (푡, 푇))피푠 (퐹 (푡, 푇)). Substitute the classic interest rate parity and domestic asset martingale into above equation,

픻푇 퐹푋 픽 퐹푋 픽 퐹푋 픻푇 픽 (2.4.24) 퐶표푣푠 (퐹 (푡, 푇), 퐹 (푡, 푇)) = 퐹 (푠, 푇)퐹 (푠, 푇) − 퐹 (푠, 푇)피푠 (퐹 (푡, 푇)) Then the foreign asset forward price under domestic forward measure is

픻푇 퐹푋 픽 (2.4.25) 퐶표푣푠 (퐹 (푡, 푇), 퐹 (푡, 푇)) 피픻푇 (퐹픽(푡, 푇)) = 퐹픽(푠, 푇) − . 푠 퐹퐹푋(푠, 푇) So the Quanto adjustment effectively is an approximation of covariance term as in Hull (2009). Or we could use Randon-Nikodym derivative to look for a closed form solution. If we assume the FX forward rate 퐹퐹푋(푡, 푇) and foreign asset forward price 퐹픽(푡, 푇) are log-normally distributed, then we have

퐹푋 퐹퐹푋 퐹푋 픻푇 (2.4.26) 푑퐹 (푡, 푇) = 휎푡 퐹 (푡, 푇)푑푊푡 , as FX forward rate is a martingale under domestic forward measure,

픽 퐹픽 픽 픽푇 (2.4.27) 푑퐹 (푡, 푇) = 휎푡 퐹 (푡, 푇)푑푊푡 , as foreign asset forward price is a martingale under foreign forward measure. 퐹푋 As stated in previous section, the Randon-Nikodym derivative 푀푡 is equal to 퐹 (푡, 푇), we have

퐹푋 1 푡 퐹푋 2 푡 퐹푋 픻 푀푡 퐹 (푡, 푇) − (𝜎퐹 ) 푑푢+ 𝜎퐹 푑푊 푇 (2.4.28) 2 ∫푠 푢 ∫푠 푢 푢 = 퐹푋 = 푒 . 푀푠 퐹 (푠, 푇) By Girsanov’s theorem,

픽푇 ̃ 픻푇 (2.4.29) 푑푊푡 = 휇푡푑푡 + 푑푊푡 . And 푀 픽푇 픽푇 픽푇 픻푇 푡 픽푇 (2.4.30) 푊푠 = 피푠 (푊푡 ) = 피푠 ( 푊푡 ). 푀푠 Thus 푀 (2.4.31) 픽푇 픽푇 픻푇 푡 픽푇 피푠 (푑푊푡 ) = 피푠 (푑 ( 푊푡 )) = 0. 푀푠 By Ito’s Lemma, 푀 푀 푀 푀 푡 픽푇 푡 픽푇 푡 픽푇 푡 픽푇 (2.4.32) 푑 ( 푊푡 ) = 푑 ( ) ∙ 푊푡 + ( ) ∙ 푑푊푡 + 푑 ( ) ∙ 푑푊푡 푀푠 푀푠 푀푠 푀푠

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If the correlation between Brownian motions under different measure is not zero,

픻푇 픻푇 픽푇 픻푇 픻푇 ̃ 픻푇 픻푇 픻푇 ̃ 픻푇 (2.4.33) 피푡 (푑푊푡 푑푊푡 ) = 피푡 (푑푊푡 (휇푡푑푡 + 푑푊푡 )) = 피푡 (푑푊푡 푑푊푡 )

= 휌푡푑푡. Therefore, 푀 푀 푀 (2.4.34) 푡 픽푇 퐹퐹푋 푡 픽푇 픻푇 푡 ̃ 픻푇 푑 ( 푊푡 ) = 휎푡 (( ) 푊푡 ) 푑푊푡 + ( ) 푑푊푡 푀푠 푀푠 푀푠

퐹퐹푋 푀푡 + (휇푡 + 휌푡휎푡 ) ( ) 푑푡. 푀푠

픻푇 푀푡 픽푇 As 피푠 (푑 ( 푊푡 )) = 0, we have 푀푠

퐹퐹푋 (2.4.35) 휇푡 = −휌푡휎푡 . Then the foreign asset price under domestic forward measure is 픽 푑퐹 (푡, 푇) 픽 픽 퐹푋 픽 픽 (2.4.36) = 휎퐹 푑푊픽푇 = 휎퐹 (휇 푑푡 + 푑푊̃ 픻푇 ) = −휌 휎퐹 휎퐹 푑푡 + 휎퐹 푑푊̃ 픻푇 . 퐹픽(푡, 푇) 푡 푡 푡 푡 푡 푡 푡 푡 푡 푡 Hence

푡 퐹퐹푋 퐹픽 픻푇 픽 픽 − ∫ 𝜌푢𝜎푢 𝜎푢 푑푢 (2.4.37) 피푠 (퐹 (푡, 푇)) = 퐹 (푠, 푇)푒 푠 . And the Quanto derivative price is therefore

픻 푇 퐹퐹푋 퐹픽 (2.4.38) 픻 픻 ̅퐹푋 푇 픽 픻 ̅퐹푋 픽 − ∫푠 𝜌푢𝜎푢 𝜎푢 푑푢 푉푠 = 푃 (푠, 푇)푆 피푠 (퐹 (푇, 푇)) = 푃 (푠, 푇)푆 퐹 (푠, 푇)푒 . As discussed before, the hedging models should be slightly different to pricing models in practice. The local volatility models are very successful in terms of delivering consistent pricing but fail to produce meaningful hedging ratios. The existence of un-hedgeable variables like volatility skew and correlation skew creates a substantial divergence between pricing and hedging models. For example, the dependency of volatility on the moneyness of the option (i.e. volatility skew/smile) makes the Delta hedging in Black Scholes framework ineffective as Delta produced by Black Scholes framework ignores the term called Delta due to Vega. If the underlying interest rate or asset price declines, the implied volatility would increase as suggested by volatility skew. Hence the implied volatility is actually a function of underlying market prices. The second order impact of Delta due to Vega would be significant of the portfolio holding by market marker has a substantial Vega position. Typically the derivative dealer do not run a Vega neutral portfolio due to the associated heavy hedging cost. So if the model is not capable of producing market consistent volatility skew/smile, the derivative dealer cannot run a true Delta neutral portfolio. Suppose a derivative dealer expects the bond market to rally and maintains short positions in the interest rate. The Vega of short term interest rate is negative. The FOMC declares to cut Fed Fund Rates to a historical low and the market volatility goes up significantly. The portfolio will lose money although the market moves in the same way as the derivative dealer anticipates. This is due to the net short Vega position of the portfolio. 푉푡 denotes the present value of the portfolio held by the derivative dealer. 푆푡 is the current market price of

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underlying asset and 휎푡 is the Black Scholes implied volatility. The change of market value of the portfolio is therefore

휕푉푡 휕푉푡 퐵푆 퐵푆 (2.4.39) 푑푉푡 = 푑푆푡 + 푑휎푡 + ⋯ = Δ푡 푑푆푡 + νt 푑휎푡 + ⋯ 휕푆푡 휕휎푡

퐵푆 휕푉푡 퐵푆 where Δ푡 is the Black Scholes Delta as an approximation for and νt is the Black Scholes Vega 휕푆푡 휕푉 as an approximation for 푡. However, this set of approximations isolates the impact of underlying 휕𝜎푡 휕푉 휕𝜎 asset price changes on implied volatility 푡 푡 from the Vega position. The impact of Delta due to 휕𝜎푡 휕푆푡 Vega is our concern and the impact of interaction with respect to other terms like Gamma and Theta are not discussed here. An improved approximation with Delta due to Vega taken into account is

휕푉푡 휕푉푡 휕휎푡 휕푉푡 휕푉푡 휕휎푡 (2.4.40) 푑푉푡 = ( + ) 푑푆푡 + ( 푑휎푡 − 푑푆푡) + ⋯ 휕푆푡 휕휎푡 휕푆푡 휕휎푡 휕휎푡 휕푆푡

퐵푆 휕휎푡 퐵푆 휕휎푡 = (Δ푡 + ) 푑푆푡 + νt (푑휎푡 − 푑푆푡) + ⋯ 휕푆푡 휕푆푡 That is, the derivative dealer should take the partial Vega position (the second component in Vega account) into their Delta hedging management to build up an improved Delta neutral portfolio.

Interest rate products are the centre of OTC derivatives market business. Different families of interest rates models have been developed for the past three decades. Short rates, instantaneous forward rates, and full term structure of forward rates could be applied with different processes (like normal, log normal etc.) to produce market consistent volatility skews/smiles, forward volatility term structure and correlation dynamics. A terrific discussion on the theory and practice of interest rates models is Brigo and Mercurio (2006). The main interest rate models used in this thesis are short rate models. The core of interest rates dynamics is mean reversion. The classic mean reversion process used in is Ornstein-Unlenbeck process, ℚ (2.4.41) 푑푋푡 = (휃푡 − 휅푡푋푡)푑푡 + 휎푡푑푊푡 , where 휃푡, 휅푡 and 휎푡 are deterministic processes. The solution to Ornstein-Unlenbeck process is 푡 푡 푡 푠 푠 (2.4.42) − ∫ 휅푠푑푠 ∫ 휅푢푑푢 ∫ 휅푢푑푢 ℚ 푋푡 = 푒 0 (푋0 + ∫ 푒 0 휃푠푑푠 + ∫ 푒 0 휎푠푑푊푠 ). 0 0

The short rate 푟푡 is specified as a function of 푋푡. For Vasciek model (normally distributed short rate models),

푟푡 = 푋푡, (2.4.43) And ℚ (2.4.44) 푑푟푡 = (휃 − 휅푟푡)푑푡 + 휎푑푊푡 . It could be extended to other normally distributed short rates models like Hull-White and Ho-Lee with further adjustments on long term equilibrium rate 휃 and mean reversion speed 휅.

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For Black Karakinski model (log-normally distributed short rate models),

푟푡 = ln 푋푡, (2.4.45) and ℚ (2.4.46) 푑 ln 푟푡 = (휃 − 휅 ln 푟푡)푑푡 + 휎푑푊푡 . For CIR model (constant elasticity of variance short rates models), 1 (2.4.47) 1−훽 푟푡 = ((1 − 훽)푋푡) , 1 where 0 < 훽 < 1 and we could have CIR if 훽 = , 2 ℚ (2.4.48) 푑푟푡 = 푎푡(푏 − 푟푡)푑푡 + 휎√푟푡푑푊푡 . The default free zero coupon bond governs the behaviours of short rate models and the price of zero

푡 ∫ 푟푠푑푠 coupon bond 푃(푡, 푇) is a martingale with account 푁푡 = 푒 0 as numeraire,

푃(푡, 푇) ℚ 푃(푇, 푇) (2.4.49) = 피푡 ( ). 푁푡 푁푇 Since 푃(푇, 푇) = 1,

푇 ℚ − ∫ 푟푠푑푠 (2.4.50) 푃(푡, 푇) = 피푡 (푒 푡 ). Counterparty credit risk is priced on netting set level and funding risk is priced on funding set level. Both of them call for models could deliver consistent market price of derivatives across all asset classes. So which model is the ‘ultimate’ model to price every derivative? Or do we have a model could give consistent price across different products in interest rate space? Generally different models are used to price interest rate products with different features. For example, two or more factors models should be used to price curve options (steepeners or flatteners) and CMS spreads derivatives as the payoff of these derivatives have strong dependencies on the correlation between different components of the entire yield curve. Constant elasticity of variance (CEV) models should be used to price cancellable swaps, knock out swaps and swaps with embedded options as swaptions, caps and floors embedded in these transactions are sensitive to volatility skew. For European cancellable swaps, the European swaptions are ought to be priced at the volatility implied at swap rate. For Bermudan cancellable swaps, forward volatility skew have to be taken into account in pricing and multi-factor models with CEV or have to be used to capture the forward volatility dynamics. Let’s use a simple example to explain why Bermudan swaptions have to be priced with forward volatility skew. Suppose the derivative dealers traded a 3nc1 (i.e. 3 years swap not callable within 1 year) Bermudan cancellable swap in which the derivative dealer has to pay fixed for 3 years and could cancel the swap at 1 year and 2 years. At the end of the first year, the original 3 years swap becomes 2 years swap and the derivative dealer has the right to cancel the swap right now and in 1 year. The right to cancel a fixed payer in 1 year is a 1 into 1 fixed receiver swaption. Therefore the derivative dealer has to revaluate the payoff from immediate cancel and remaining value of not cancelling the swap. The 1 into 1 year fixed receiver swaption has to be priced at the prevailing market implied

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Counterparty Credit Risk, Funding Risk and Central Clearing volatility as it takes a significant weight of remaining value. The derivative dealer would decide to cancel the swap immediately if the implied volatility is low for 1 into 1 receiver swaption (as the remaining value of keeping the swap running is low) and would not cancel if the 1 into 1 receiver swaption implied volatility is high. Since the Bermudan cancellable swap is priced at inception (not at the end of first year as we discussed above), the estimate of the derivative dealer regarding what the 1 into 1 receiver swaption implied volatility will be in 1 year is crucial in determination of the Bermudan cancellable swap price. The higher the 1 into 1 receiver swaption implied volatility will be in 1 year, the more valuable the Bermudan cancellable swap is at inception. The 1 into 1 receiver swaption implied volatility in 1 year (or 1 into 1 implied volatility, 1 year forward) is the market expectation of swaption volatility in the future. This gives a general idea why forward volatility skew has to be priced into Bermudan cancellable swap. A three factor models combined foreign short rate, domestic short rate and FX spot rate has to be used for pricing cross currency swaps and PDRC deals. A SIFI (systematic important financial institution) has millions of portfolios consisting of trades across different product areas and asset classes. All trades should be marked to their corresponding liquid hedging instruments to ensure the consistency of model implied portfolio price.

No arbitrage pricing for credit risk

Pricing and trading counterparty credit risk embedded in derivatives portfolios are similar to pricing and trading of credit risk embedded in bonds. Merton (1974) builds up the structural approach to model default process of a given firm. The default of the specific firm is triggered by the event that the hypothetical asset price 푋푡 falls below a hypothetic liability barrier 퐻푡. 휏 denotes the default time of the firm and is the first time the asset crosses the liability,

휏 = inf {푡 > 0|푋푡 ≤ 퐻푡} (2.4.51)

푉̃푡 is the market price of a defaultable instrument at time 푡. This instrument delivers only one payoff

푉̃푇 at maturity if the firm did not default before maturity and recovery 푅̃휏 at default time 휏. The defaultable instrument price process is a martingale under forward measure with the default free zero coupon bond 푃(푡, 푇) as underlying numeraire,

푉̃푡 푉̃푇1휏>푇 푅̃휏1푡<휏≤푇 (2.4.52) = 피푇 ( + ), 푃(푡, 푇) 푡 푃(푇, 푇) 푃(휏, 푇) where 0 < 푡 < 푇. The recovery 푅̃휏 of the instrument is assumed to be settled at the default time 휏. If we further assume the default process of the firm is independent to the valuation of the derivative in non-default states,

푉̃푡 푅̃휏1푡<휏≤푇 (2.4.53) = 피푇(푉̃ )피푇(1 ) + 피푇 ( ), 푃(푡, 푇) 푡 푇 푡 휏>푇 푡 푃(휏, 푇) and hence,

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푅̃휏1푡<휏≤푇 (2.4.54) 푉̃ = 푃(푡, 푇)피푇(푉̃ )피푇(1 ) + 푃(푡, 푇)피푇 ( ). 푡 푡 푇 푡 휏>푇 푡 푃(휏, 푇) This is an interesting result. Since our current observation time is time 0 and the firm has not defaulted yet (휏 > 0),

푇 푅̃휏1휏≤푇 (2.4.55) 피0 ( ) ̃ 푇 ̃ 푇 푃(휏, 푇) 푇 푉0 = 푃(0, 푇)피0(푉푇)피0(1휏>푇) + 푃(0, 푇) 푇 피0(1휏≤푇). 피0(1휏≤푇) and the price of the default instrument at time 0 could be restated as

푇 푇 (2.4.56) 푉̃0 = 푉0 (1 − ℙ퐷(0, 푇)) + 푅(0, 푇)푉0ℙ퐷(0, 푇), where 푇 푉0 = 푃(0, 푇)피0(푉̃푇), (2.4.57) 푇 푇 ℙ퐷(0, 푇) = 피0(1휏≤푇), 푅̃ 1 피푇 ( 휏 휏≤푇) 0 푃(휏, 푇) 피푇(1 ) 푅(0, 푇) = 0 휏≤푇 . 푉0 푇 Here 푉0 is the default free price of the underlying instrument, ℙ퐷(0, 푇) is the default probability of the firm under forward measure, and 푅(0, 푇) is the effective recovery of the instrument over the investment horizon [0, 푇]. Hence the price discount of this instrument due to default risk of the firm is 푇 푇 푇 푉0 − 푉̃0 = 푉0ℙ퐷(0, 푇) − 푅(0, 푇)푉0ℙ퐷(0, 푇) = (1 − 푅(0, 푇))푉0ℙ퐷(0, 푇). (2.4.58) The above result lays out the general framework for counterparty credit risk pricing. This equation applies for instrument like cash settled forward and option that pays out only one cash flow within the life of this trade. In practice, this general framework has been applied to pricing of corporate (coupon bearing) bonds to calculate an approximation of the credit spread of the entity, (the result is not accurate as this equation serves the single cash instrument), −푆̅퐷 푉̃0 = 푉0푒 , (2.4.59) where 푆̅ is the quoted credit spread over the life of the bond and 퐷 is the effective duration of default free bond with the same tenor and same coupon rate. (Treasury bonds could be used as proxies but the prices have to be marked down by relevant tax benefits and coupons should be adjusted to the same level of the specific corporate bond.) The price of defaultable corporate bond is

푇 (2.4.60) 푉̃0 = 푉0 (1 − (1 − 푅(0, 푇))ℙ퐷(0, 푇)). Therefore, the market implied default probability over the investment horizon is 1 − 푒−푆̅퐷 (2.4.61) ℙ푇 (0, 푇) = . 퐷 1 − 푅(0, 푇)

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The market implied default probability is typically greater the historical default probability. Hull (2009) gives an interesting analysis of excess return through investment into corporate bonds with different credit ratings.

The default probability discussed above is computed under the forward measure. Since the default events discussed here is regarding the same firm, what is the default probability under risk neutral measure?

푁푇 (2.4.62) 푑ℙℚ 푁 ℙℚ(푡, 푇) = 피ℚ(1 ) = 피푇 ( 1 ) = 피푇 ( 푡 1 ), 퐷 푡 푡<휏<푇 푡 푑ℙ푇 푡<휏<푇 푡 푃(푇, 푇) 푡<휏<푇 푃(푡, 푇)

푡 ∫ 푟푠푑푠 where 푁푡 is the money market account and 푁푡 = 푒 0 .

ℚ 푇 푁푇 (2.4.63) ℙ퐷(푡, 푇) = 푃(푡, 푇)피푡 ( 1푡<휏<푇) 푁푡

푇 푁푇 푇 푇 푁푇 = 푃(푡, 푇)피푡 ( ) 피푡 (1푡<휏<푇) + 푃(푡, 푇)퐶표푣 ( , 1푡<휏<푇). 푁푡 푁푡 The money market account does not pay out any intermediate cash flows before maturity 푇 and hence it is a martingale under the forward measure,

푁푡 푇 푁푇 푇 1 푇 푁푇 (2.4.64) = 피푡 ( ) = 피푡 (푁푇) ⇒ = 피푡 ( ). 푃(푡, 푇) 푃(푇, 푇) 푃(푡, 푇) 푁푡 And hence the default probability under risk neutral measure is

ℚ 푇 푇 푁푇 (2.4.65) ℙ퐷(푡, 푇) = ℙ퐷(푡, 푇) + 푃(푡, 푇)퐶표푣 ( , 1푡<휏<푇). 푁푡 Hence, the difference of default probability under risk neutral measure and default probability under forward measure is a covariance term between interest rate and default process. If the interest rate process and default process are mutually independent, ℚ 푇 (2.4.66) ℙ퐷(푡, 푇) = ℙ퐷(푡, 푇). The above analysis could be extended to an instrument that pays out multiple cash flows during the life of the trade. CCDS, credit extinguisher and CVA are examples of pricing the default risk component of a single trade or a portfolio. CCDS is a knock in contract contingent on the default of underlying credit. It will replace the trades pre-specified in the contract with market value (instead of recovery) once the underlying credit defaults. Credit extinguisher is a knock out contract contingent on the default of underlying credit. The holder of credit extinguisher will pay the pre-specified recovery of trades in the contract. It works similar to ISDA set-off provision that allows the surviving party to offset their liabilities by delivering the corporate bonds of the defaulted party. CCDS, shares the same functionality with CVA, provides insurance or hedge to a set of portfolios/trades against the default risk of reference legal entity. Credit extinguisher eliminates the asymmetric treatments of obligations of default such that the surviving party only has to pay the recovery of their liabilities instead of full amount to the default party and hence it helps to cheapen the credit cost or funding

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Counterparty Credit Risk, Funding Risk and Central Clearing cost of the counterparty. However, credit extinguishers potential dilutes the benefits of creditors of a given firm and existence of conflict of make them illegal or unenforceable in some jurisdictions.

How to price CVA/DVA?

How to price a credit (default) contingent instrument like CCDS, credit extinguisher, CVA/DVA etc.?

Suppose the credit contingent instrument pays out a coupon or cash flow 퐶퐷(휏) at the default time of the underlying credit and does not pay anything in non-default states. More specifically, 퐶퐷(휏) is the loss given default (LGD) in CVA/DVA analysis and payoff of the protection leg in the default state in CCDS. Hence the price of credit contingent instrument under risk neutral measure is 푛−1 (2.4.67) ℚ 퐶퐷(휏) 푉푡 = 푁푡피푡 (∑ 1푇푖<휏≤푇푖+1), 푁휏 푖=1 where 0 < 푡 ≤ 푇1 < ⋯ < 푇푛 and 푇푛 is the maturity of this credit contingent instrument. Here 푁푡 is 푡 ∫ 푟푠푑푠 the money market account (푁푡 = 푒 0 ) and serves as the numeraire for the risk neutral measure ℚ. Technically speaking, the credit contingent instrument here is a forward starting contract and it automatically cancels if the underlying credit defaults before the first coupon date 푇1. If we slice the entire streams of contingent cash flows into individual contingent cash flows within corresponding time interval, 푛−1 (2.4.68) 푖 푉푡 = ∑ 푉푡 , 푖=1 and

ℚ 퐶퐷(휏) (2.4.69) 푖 푁푡피푡 ( 1푇푖<휏≤푇푖+1) 0 < 푡 ≤ 푇푖+1 푉푡 = { 푁휏 , 0 푇푖+1 < 푡 or,

푖 ℚ 퐶퐷(휏) (2.4.70) 푉푡 = 1푡≤푇푖+1푁푡피푡 ( 1푇푖<휏≤푇푖+1). 푁휏

We could further define a conditional default time 휏(푇) as the first time asset process 푋푡 crosses the liability process 퐻푡 after time 푇 conditional on the firm has not defaulted before 푇, i.e.

휏(푇) = inf {푡 > 푇|푋푡 ≤ 퐻푡}, (2.4.71) where 푇 ≥ 0. Then

1푇<휏(푇) = 1, (2.4.72)

1푇푖<휏≤푇푖+1 = 1푇푖<휏1푇푖<휏(푇푖)≤푇푖+1 = 1푇<휏1푇푖<휏(푇)≤푇푖+1.

Then the price of individual contingent cash flow (for 0 ≤ 푡 ≤ 푇푖 ≤ 푇푖+1 and hence 1푡≤푇푖+1 = 1) is

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(2.4.73) 퐶퐷(휏) 퐶퐷(휏) 푉푖 = 푁 피ℚ ( 1 ) = 푁 피ℚ (피ℚ (1 1 )). 푡 푡 푡 푇푖<휏≤푇푖+1 푡 푡 푇푖 푇푖<휏 푇푖<휏(푇푖)≤푇푖+1 푁휏 푁휏 Therefore, (2.4.74) 퐶퐷(휏) 푉푖 = 푁 피ℚ (1 피ℚ ( 1 )). 푡 푡 푡 푇푖<휏 푇푖 푇푖<휏(푇푖)≤푇푖+1 푁휏

ℚ Here we define the conditional marginal default probability ℙ퐷(푡, 푇푖, 푇푖+1) as the marginal default probability over [푇푖, 푇푖+1] conditional on the reference credit has not defaulted before 푡, ℚ( ) ℚ (2.4.75) ℙ퐷 푡, 푇푖, 푇푖+1 = 피푡 (1푇푖<휏(푡)≤푇푖+1). Hence, 푖 ℚ ̅ ( ) ℚ( ) (2.4.76) 푉푡 = 푁푡피푡 (1푡<휏1푇푖<휏(푡)퐶퐷 푇푖, 푇푖+1 ℙ퐷 푇푖, 푇푖, 푇푖+1 )

ℚ ̅ ( ) ℚ( ) = 1푡<휏푁푡피푡 (1푇푖<휏(푡)퐶퐷 푇푖, 푇푖+1 ℙ퐷 푇푖, 푇푖, 푇푖+1 ), where 0 ≤ 푡 ≤ 푇푖 ≤ 푇푖+1 and

ℚ 퐶퐷(휏(푇푖)) (2.4.77) 피 ( 1 ( ) ) 푇푖 푁 푇푖<휏 푇푖 ≤푇푖+1 ̅ 휏(푇푖) 퐶퐷(푇푖, 푇푖+1) = ℚ . ℙ퐷(푇푖, 푇푖, 푇푖+1)

Here 퐶퐷̅ (푇푖, 푇푖+1) is the expected payoff contingent on default of reference credit within time interval [푇푖, 푇푖+1] with all available information up to time 푇푖. The dependency between market and credit variables is an integral component in valuation of expected default payoff and contributes to the classic Wrong Way Risk (WWR)/Right Way Risk(RWR) analysis in counterparty risk pricing. An investor is exposed to market and credit correlation if he/she buys a Quanto CDS on an emerging market sovereign USD debt but CDS pays protection 퐶퐷(휏) in domestic currency. The default correlations/dependency between different market participants also have impacts on pricing credit contingent instruments. Now it is the time to discuss the impact of different types of correlations on pricing credit contingent instruments.

The dependency between market and credit variables

Let’s start with the correlation between market and credit variables. Suppose there are 푚 different market variable processes in the market. The market variables include asset process of a given credit, 푗 FX, interest rates, equities etc. 푋푡 is the process of jth market variable, 푗 푗 ℚ ℚ (2.4.78) 푑푋푡 = 휎푡 (훽푗푑푊푀,푡 + 훼푗푑푊푗,푡 ),

ℚ where 푗 = 0,1,2 … 푚 − 1. 푊푗,푡 denotes the 1 dimensional idiosyncratic risk factor to the jth market 푗 ℚ variable 푋푡 and 푊푀,푡 denotes the 푛 dimensional common driving factors which are independent to ℚ 푗 푊푗,푡 . Then the entire spectrum of 푋푡 are mutually independent conditional on all information

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ℚ 푀 generated by 푊푀,푡 (denoted by market filtration ℱ푡 ). The 푛 dimensional common driving factors for market and credit risk components could be further classified as global economy driver and regional and industrial sector growth drivers. Normally, 훼푗 is set to

푇 (2.4.79) 훼푗 = √1 − 훽푗훽푗 , such that

ℚ 푗 푗 2 (2.4.80) 푉푎푟푡 (푑푋푡 ) = (휎푡 ) 푑푡,

푗 푗 ℚ 푇 ℚ 푑푋푡 = 휎푡 (훽푗푑푊푀,푡 + √1 − 훽푗훽푗 푑푊푗,푡 ),

ℚ 푖 푗 푇 퐶표푟푟푡 (푑푋푡, 푑푋푡 ) = 훽푖훽푗 , where 훽푗 denotes 1 × 푛 vector that specifies the loadings/weights of 푛 dimensional common driving factors. Here 훽푖 ≠ 훽푗 . Now suppose the default of firm 푗 is triggered by the event that the 푗 푗 hypothetical asset price 푋푡 falls below a hypothetic liability barrier 퐻푡 . 휏푗 denotes the default time of the firm and is the first time the asset crosses the liability, 푗 푗 (2.4.81) 휏푗 = inf {푡 > 0|푋푡 ≤ 퐻푡 }. It is essential to make the model implied default probabilities consistent with the market implied default probabilities. Brigo and Tarenghi (2005) introduces a constant liability barrier and calibrates the volatility dynamics to fit the market implied default probability, 푗 푗 (2.4.82) 휏푗 = inf {푡 > 0|푋푡 ≤ 퐻 }

Then the marginal default probability over the investment horizon [0, 푇푖] conditional on the reference credit has not default before time 0 is ℚ ( ) ℚ (2.4.83) ℙ푗,퐷 0,0, 푇푖 = 피0 (10<휏≤푇푖). ℚ 푗 The reflection principal of Brownian motion is used to solution to ℙ푗,퐷(0,0, 푇푖). 푋푡 here is an ℚ ℚ arithmetic Brownian motion constructed by two other Brownian motions 푊푀,푡 and 푊푗,푡 . Let’s define another variable 푀푡 as 푗 (2.4.84) 푀푡 = inf 푋푠 0≤푠≤푡 for any given 푡. And the conditional default probability is ℚ ( ) ℚ ℚ 푗 (2.4.85) ℙ푗,퐷 0,0, 푇푖 = 피0 (10<휏≤푇푖) = ℙ (푀푇푖 ≤ 퐻 ).

푗 푗 푗 Given the hypothetical liability barrier 퐻 > 0, and 휏푗 is defined as the first time 푋푡 hits 퐻 as 휏푗 = 푗 푗 푗 푗 inf {푡 > 0|푋푡 = 퐻 } = inf {푡 > 0|푋푡 ≤ 퐻 } (as it is assumed the firm asset is above its liability 푗 푗 barrier, i.e. 푋푡 > 퐻 , in non-default states), 푋푗 − 푋푗 = 푋푗 − 퐻푗 (푠 > 0) (2.4.86) 휏푗+푠 휏푗 휏푗+푠 is a Brownian motion as well. Then

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ℚ 푗 푗 ℚ 푗 푗 푗 ℚ 푗 푗 푗 (2.4.87) ℙ (푋푡 ≤ 퐻 ) = ℙ (푋푡 ≤ 퐻 , 푀푡 ≤ 퐻 ) + ℙ (푋푡 ≤ 퐻 , 푀푡 > 퐻 ). 푗 ℚ 푗 푗 푗 Since 푀푡 ≤ 푋푡 , ℙ (푋푡 ≤ 퐻 , 푀푡 > 퐻 ) = 0. Then ℚ 푗 푗 ℚ 푗 푗 푗 (2.4.88) ℙ (푋푡 ≤ 퐻 ) = ℙ (푋푡 ≤ 퐻 , 푀푡 ≤ 퐻 ) ℚ 푗 푗 푗 ℚ 푗 = ℙ (푋푡 ≤ 퐻 |푀푡 ≤ 퐻 )ℙ (푀푡 ≤ 퐻 ), That is, ℚ 푗 푗 ℚ 푗 푗 ℚ 푗 (2.4.89) ℙ (푋푡 ≤ 퐻 ) = ℙ (푋푡 ≤ 퐻 |휏푗 ≤ 푡)ℙ (푀푡 ≤ 퐻 ). 1 Here ℙℚ(푋푗 ≤ 퐻푗|휏 ≤ 푡) = as 푡 푗 2 ℚ 푗 푗 ℚ 푗 푗 (2.4.90) ℙ (푋푡 ≤ 퐻 |휏푗 ≤ 푡) = ℙ (푋푡 − 퐻 ≤ 0|휏푗 ≤ 푡)

ℚ 푗 푗 = ℙ (푋 − 퐻 ≤ 0|휏푗 ≤ 푡), 휏푗+(푡−휏푗)

푋푗 − 퐻푗 is a Brownian motion as discussed above and hence 휏푗+(푡−휏푗)

ℚ 푗 푗 ℚ 푗 푗 1 (2.4.91) ℙ (푋푡 ≤ 퐻 |휏푗 ≤ 푡) = ℙ (푋 − 퐻 ≤ 0|휏푗 ≤ 푡) = . 휏푗+(푡−휏푗) 2 Therefore, 1 (2.4.92) ℙℚ(푋푗 ≤ 퐻푗) = ℙℚ(푀 ≤ 퐻푗) ⇒ ℙℚ (0,0, 푇 ) = ℙℚ(푀 ≤ 퐻푗) 푡 2 푡 푗,퐷 푖 푇푖 = 2ℙℚ(푋푗 ≤ 퐻푗). 푇푖 Hence, (2.4.93) 퐻푗 − 푋푗 ℙℚ (0,0, 푇 ) = 2ℙℚ(푋푗 ≤ 퐻푗) = 2Φ 0 , 푗,퐷 푖 푇푖 푇 2 √ 푖(휎푗) 푑푡 ( ∫0 푡 ) where Φ(∙) is the cumulative density function of normal distribution. Then the marginal default probability over horizon [푇푖, 푇푖+1] conditional on time 푇푖 shares the same structure, ℙℚ (푇 , 푇 , 푇 ) = 피ℚ(1 ) = 2ℙℚ(푋푗 ≤ 퐻푗|푋푗 ) (2.4.94) 푗,퐷 푖 푖 푖+1 0 푇푖<휏(푇푖)≤푇푖+1 푇푖+1 푇푖

푗 푗 퐻 − 푋푇 = 2Φ 푖 . 푇 +1 2 √∫ 푖 (휎푗) 푑푡 ( 푇푖 푡 )

푖 푗 푗 As discussed above, 푋푡 and 푋푡 (where 푖 ≠ 푗 ) in the entire spectrum of 푋푡 become mutually 푀 independent if they are evaluated conditional on the market filtration ℱ푇푛 which is generated by the ℚ market risk component 푊푀,푡 up to time 푇푛. Here 0 ≤ 푡 ≤ 푇푖 ≤ 푇푛. Then (2.4.95) 푖 ℚ 퐶퐷(휏) ℚ 퐶퐷(휏(푡)) 푉푡 = 푁푡피푡 ( 1푇푖<휏≤푇푖+1) = 푁푡피푡 (1푡<휏 1푇푖<휏(푡)≤푇푖+1), 푁휏 푁휏(푡) And

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(2.4.96) 푖 ℚ 퐶퐷(휏(푡)) 푉푡 = 1푡<휏푁푡피푡 ( 1푇푖<휏(푡)≤푇푖+1) 푁휏(푡)

ℚ ℚ 퐶퐷(휏(푡)) 푀 = 1푡<휏푁푡피푡 (피푡 ( 1푇 <휏(푡)≤푇 |ℱ푇 )). 푁휏(푡) 푖 푖+1 푛

If we define the expected discounted default payoff (discounted back to time 0) as (2.4.97) ℚ 퐶퐷(휏(푡)) 푀 피푡 ( 1푇 <휏(푡)≤푇 |ℱ푇 ) 푁휏(푡) 푖 푖+1 푛 퐶퐷̅ (푇푖, 푇푖+1) = , ℙℚ(푡, 푇 , 푇 , ℱ푀 ) 퐷 푖 푖+1 푇푛 then 푖 ℚ ̅ ( ) ℚ 푀 (2.4.98) 푉푡 = 1푡<휏푁푡피푡 (퐶퐷 푇푖, 푇푖+1 ℙ퐷(푡, 푇푖, 푇푖+1, ℱ푇푛)), where ℚ 푀 ℚ 푀 (2.4.99) ℙ퐷(푡, 푇푖, 푇푖+1, ℱ푇푛) = 피푡 (1푇푖<휏(푡)≤푇푖+1|ℱ푇푛) ℚ 푀 ℚ 푀 = ℙ퐷(푡, 푡, 푇푖+1, ℱ푇푛) − ℙ퐷(푡, 푡, 푇푖, ℱ푇푛). ℚ 푀 푀 Here ℙ퐷(푡, 푇푖, 푇푖+1, ℱ푇푛) is the conditional default probability conditioned on market filtration ℱ푇푛 at time 푡 over investment horizon [푇푖, 푇푖+1]. Here if we further assume the discounted default payoff

퐶퐷(휏(푡)) and conditional default process 1푇푖<휏(푡)≤푇푖+1 are independent conditional on market 푁휏(푡) 푀 filtration ℱ푇푛, (2.4.100) ℚ 퐶퐷(휏(푡)) 푀 피푡 ( 1푇 <휏(푡)≤푇 |ℱ푇 ) 푁휏(푡) 푖 푖+1 푛 퐶퐷̅ (푇푖, 푇푖+1) = ℙℚ(푡, 푇 , 푇 , ℱ푀 ) 퐷 푖 푖+1 푇푛

ℚ 퐶퐷(휏(푡)) 푀 ℚ 푀 피푡 ( |ℱ푇 ) ℙ퐷(푡, 푇푖, 푇푖+1, ℱ푇 ) 푁휏(푡) 푛 푛 = , ℙℚ(푡, 푇 , 푇 , ℱ푀 ) 퐷 푖 푖+1 푇푛 thus, 퐶 (휏(푡)) (2.4.101) ̅ ℚ 퐷 푀 퐶퐷(푇푖, 푇푖+1) = 피푡 ( |ℱ푇 ). 푁휏(푡) 푛 Hence, (2.4.102) 푖 ℚ ℚ 퐶퐷(휏(푡)) 푀 ℚ 푀 푉푡 = 1푡<휏푁푡피푡 (피푡 ( |ℱ푇 ) ℙ퐷(푡, 푇푖, 푇푖+1, ℱ푇 )), 푁휏(푡) 푛 푛 that is, (2.4.103) 푖 ℚ ℚ 퐶퐷(휏(푡)) ℚ 푀 푀 푉푡 = 1푡<휏푁푡피푡 (피푡 ( ℙ퐷(푡, 푇푖, 푇푖+1, ℱ푇 )|ℱ푇 )) 푁휏(푡) 푛 푛

( ) ℚ 퐶퐷(휏 푡 ) ℚ 푀 = 1푡<휏푁푡피푡 ( ℙ퐷(푡, 푇푖, 푇푖+1, ℱ푇푛)). 푁휏(푡)

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Counterparty Credit Risk, Funding Risk and Central Clearing and 푛−1 푛−1 (2.4.104) 푖 ℚ 퐶퐷(휏(푡)) ℚ 푀 푉푡 = ∑ 푉푡 = 1푡<휏푁푡 ∑ 피푡 ( ℙ퐷(푡, 푇푖, 푇푖+1, ℱ푇푛)), 푁휏(푡) 푖=1 푖=1 where ℚ 푀 ℚ 푀 ℚ 푀 (2.4.105) ℙ퐷(푡, 푇푖, 푇푖+1, ℱ푇푛) = ℙ퐷(푡, 푡, 푇푖+1, ℱ푇푛) − ℙ퐷(푡, 푡, 푇푖, ℱ푇푛). In the continuous time limit, the price of this credit contingent instrument is (2.4.106) 푇푛 퐶퐷(푢) 푉 = 1 푁 피ℚ (∫ 푑 ℙℚ(푡, 푡, 푢, ℱ푀 )). 푡 푡<휏 푡 푡 푁 푢 퐷 푇푛 푇1 푢

The above equation is the general pricing framework for credit contingent instruments like CDS protection leg, CVA, CCDS etc. The credit contingent instrument price could be further approximated by introducing convexity adjustment term accounting for the instantaneous covariance

퐶퐷(푢) ℚ 푀 between and 푑푢ℙ퐷(푡, 푡, 푢, ℱ푇 ), and since 푁푢 푛 ℚ ℚ 푀 ℚ ℚ 푀 ℚ (2.4.107) 피푡 (푑푢ℙ퐷(푡, 푡, 푢, ℱ푇푛)) = 피푡 (피푡 (푑푢1푡<휏(푡)<푢|ℱ푇푛)) = 피푡 (푑푢1푡<휏(푡)<푢) ℚ = 푑푢ℙ퐷(푡, 푡, 푢), then,

푇푛 퐶 (푢) (2.4.108) 푉 = 1 푁 (∫ 피ℚ ( 퐷 ) 푑 ℙℚ(푡, 푡, 푢) 푡 푡<휏 푡 푡 푁 푢 퐷 푇1 푢 ℚ 푇푛 푀 퐶퐷(푢) 푑푢ℙ퐷(푡, 푡, 푢, ℱ푇 ) + ∫ 퐶표푣ℚ ( , 푛 ) 푑푢), 푡 푁 푑푢 푇1 푢 that is,

푇푛 ℚ (2.4.109) 퐶퐷(푢) 푑푢ℙ (푡, 푡, 푢) 푉 = 1 푁 ∫ 피ℚ ( ) 퐷 (1 + 휌휎 휎 (푢 − 푡))푑푢, 푡 푡<휏 푡 푡 푁 푑푢 1 2 푇1 푢

ℚ 푀 퐶퐷(푢) 푑푢ℙ퐷(푡,푡,푢,ℱ푇푛) where 휌 is the instantaneous correlation between and . 휎1 denotes the 푁푢 푑푢

퐶퐷(푢) instantaneous percentage volatility of and 휎2 denotes the instantaneous percentage volatility of 푁푢

푑 ℙℚ(푡,푡,푢,ℱ푀 ) 푢 퐷 푇푛 . 푑푢

The above pricing equation is for single name credit contingent instrument. In general, this equation could be extended to multi name credit contingent instrument. The discussion on CVA could be found in section 2.5. The problem becomes much more complex if the multi name credit contingent instrument is highly sensitive to the order of default timing of credits in the default basket. For example, an investor holds a well-diversified portfolio of defaultable high yield bonds. The investor could create a synthetic high yield portfolio with short positions in corresponding CDSs. The overall

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Counterparty Credit Risk, Funding Risk and Central Clearing coupons the investor could receive from these two structures are approximately the same and these two portfolios are not exposed to the order of default timing of underlying credits. Now suppose the investor holds a portfolio of exotic structures like first to default baskets, second to default baskets etc. such that the entire holding of these nth to default baskets is identical to the well diversified portfolio of high yield bonds. The exotic structures here now are highly sensitive to the timing of the default orders in theory and the discrepancy between the model price and the price of underlying defaultable bonds portfolio emerges in reality and exemplifies in some significant market events like the Lehman restructuring process.

The default probability of the jth credit (at or before 푇푗) in the portfolio is denoted by ℚ ℚ ℚ (2.4.110) ℙ푗,퐷(0,0, 푇푗) = 피0 (10<휏푗≤푇푗) = ℙ푗,퐷(푇푗). And the joint default probability of 푀 names governd by Copula function (Li (2000)) is ℚ ℚ (2.4.111) ℙ퐷(푇1, 푇2, … , 푇푀) = ℙ (휏1 < 푇1, 휏2 < 푇2, … , 휏푀 < 푇푀) ℚ ℚ ℚ = 풞 (ℙ1,퐷(푇1), ℙ2,퐷(푇2), … , ℙ푀,퐷(푇푀)), where 풞(∙) is the Copula function. And the 푀 dimensional Gaussian copula is

퐺푎푢푠푠푖푎푛 ℚ ℚ ℚ (2.4.112) 풞푀 (ℙ1,퐷(푇1), ℙ2,퐷(푇2), … , ℙ푀,퐷(푇푀))

−1 ℚ −1 ℚ = Φ푀 (Φ (ℙ1,퐷(푇1)) , … , Φ (ℙ푀,퐷(푇푀)) , 풮), where Φ푀 is 푀 dimensional cumulative density function for normal distribution and 풮 is the correlation matrix for the 푀 individual cumulative density functions. How to structure a transaction such that the correlation between different assets (especially for credit products) is still in debate and un-hedgeability of the correlation skew/dynamics poses significant model risk in the marketplace.

The structural default modelling in the previous sections produce a certain type of implied volatilities of the credit spreads and these volatilities are highly likely to be inconsistent with the implied volatilities of the credit swaptions market. The credit spread volatility does not have an impact in pricing linear instruments like CDS, CDO/CSO but does have a significant contribution to the pricing of CDS Swaptions, CVA (especially with WWR) and CCDS. For CVA on CDS related transactions, the volatilities of credit spreads for counterparty and underlying credit play vital roles in determination of the final price for CVA. CVA of a CDS transaction essentially gives CVA holder the opportunity to enter into a counterparty credit contingent CDS swaption to replace the underlying CDS transaction once if the counterparty defaults. Hence, the credit spread volatility of underlying CDS have a relative greater impact on CDS CVA pricing.

As discussed in pervious sections, the defaultable instrument price process is a martingale under forward measure with the default free zero coupon bond 푃(푡, 푇) as underlying numeraire,

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푉̃푡 푉̃푇1휏>푇 푅̃휏1푡<휏≤푇 (2.4.113) = 피푇 ( + ), 푃(푡, 푇) 푡 푃(푇, 푇) 푃(휏, 푇) where 0 < 푡 < 푇. Therefore, the non-default price of a defaultable instrument at time 푡 conditional the underlying credit has not defaulted yet does not have the following martingale relationship,

푉̃푡 푉̃푇 (2.4.114) ≠ 피푇 ( ), 푃(푡, 푇) 푡 푃(푇, 푇) ̃ ̃ 푉푡 ℚ 푉푇 ≠ 피푡 ( ), 푁푡 푁푇

푡 ∫ 푟푠푑푠 where 푁푡 is the money market account and 푁푡 = 푒 0 . In above equations, the non-default price

푉̃푡 of the defaultable instrument only take the payoff in the survival state at maturity into pricing and fail to price the component contingent on the default of the counterparty. In other words, ̃ ̃ 푇 푉푇 ℚ 푉푇 ̃ 푃(푡, 푇)피푡 ( ) or 푁푡피푡 ( ) is the market price of default risk free asset with payoff 푉푇 at 푃(푇,푇) 푁푇 maturity 푇. Suppose the payoff 푅̃휏 is set at zero at default time 휏 or the transaction automatically knocks out at the default time, the fair Mark to Market (MtM) 푉푡 of the transaction or extinguishing price at time 푡 is

푉푡 = 1푡<휏푉̃푡. (2.4.115)

Hence 푉푡 is the general pricing framework for premium leg or survival leg of the credit contingent instrument as the premium leg is embedded with knock out features. Then if there are no intermediate cash flows between 푡 and 푇, this extinguisher has the following martingale relationship,

1푡<휏푉̃푡 1푇<휏푉̃푇 (2.4.116) = 피푇 ( ), 푃(푡, 푇) 푡 푃(푇, 푇) ̃ ̃ 1푡<휏푉푡 ℚ 1푇<휏푉푇 = 피푡 ( ). 푁푡 푁푇 푁̃ A new pricing measure ℙ could be set up using a survival numeraire asset 푁̃푡 (Schonbucher (2003)).

This credit contingent numeraire asset has a valuation equal to 푁̃푡 in the survival state and zero MtM in the default state. Furthermore, this numeraire asset does not generates any intermediate cash flows between 푡 and 푇. Hence the Randon-Nikodym derivative here is

1푡<휏푁̃푡 (2.4.117) 푁̃ ̃ 푑ℙ푡 10<휏푁̃0 1푡<휏푁푡 ℚ = = , 푁푡 푁 푁̃ 푑ℙ푡 푡 0 푁0 where 10<휏 = 1 and 푁0 = 1. The martingale for extinguisher with survival payoff 푉̃푡 applies for this numeraire asset as well, ̃ ̃ 1푡<휏푁푡 ℚ 1푇<휏푁푇 (2.4.118) = 피푡 ( ). 푁푡 푁푇

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Therefore, 푁̃ (2.4.119) 10<휏푁̃0 1푡<휏푁̃푡 푑ℙ푡 1푡<휏푁̃푡 = 푁̃ = 피ℚ ( ) ⇒ 피ℚ ( ) = 피ℚ ( ) = 1. 푁 0 푡 푁 푡 ℚ 푡 푁 푁̃ 0 푡 푑ℙ푡 푡 0 ℚ ℚ Generally, the default probability ℙ퐷(0,0, 푡) = 피0 (10<휏<푡) > 0 under the risk neutral measure. However, under the new defined survival measure, 푁̃ (2.4.120) 푑ℙ푡 1푡<휏푁̃푡 ℙ푁̃(0,0, 푡) = 피푁̃(1 ) = 피ℚ (1 ) = 피ℚ (1 ) = 0. 퐷 0 0<휏<푡 0 0<휏<푡 ℚ 0 0<휏<푡 푁 푁̃ 푑ℙ푡 푡 0 The disagreement on certain events which have zero probability proves that the survival measure ℙ푁̃ is not equivalent to risk neutral measure ℙℚ. The survival measure ℙ푁̃ is absolutely continuous with respect to risk neutral measure ℙℚ. However, if only survival states are taken into account when

푁̃ 푁̃ pricing 푉̃푇, and 피푡 (1푡<휏) = 피푡 (1 − 1휏≤푡) = 1, 푁̃ (2.4.121) 푁̃ 푉̃푇 푑ℙ푇 피푡 ( |푡 < 휏) 푁̃ 푉̃푇 푁̃ 푉̃ 푑ℙ | 피푁̃ ( ) = 푇 = 피ℚ 푇 푡 푡 < 휏 푡 ̃ 푁̃ 푡 푁̃ ℚ | 푁푇 피푡 (1푡<휏) 푇 푑ℙ푇 ℚ ( 푑ℙ푡 )

푁̃푇 ̃ 1푇<휏 ̃ ℚ 푉푇 푁푡 = 피푡 |푡 < 휏 , 푁̃푇 푁푇 ( 푁푡 ) then,

푉̃푇 1 1 푉̃ 1 푉̃ 푉̃푡 (2.4.122) 피푁̃ = 푁 피ℚ ( 푇<휏 푇 푡 < 휏) = ( 푡<휏 푡 푡 < 휏) = . 푡 ( ) 푡 푡 푁 | ̃ | 푁̃푇 푁̃푡 푇 푁푡 푁̃푡

So what is the martingale measure of credit spread? Suppose 푆̃(푡, 푇푖, 푇푛) denotes the time 푡 forward starting par swap rate of CDS over the investment horizon [푇푖, 푇푛] and 퐴̃(푡, 푇푖, 푇푛) denotes the time 퐴̃ 푡 price of annuity of premium leg of CDS. ℙ is the survival annuity measure with 퐴̃(푡, 푇푖, 푇푛) as the underlying numeraire asset. Here 푛 푛 (2.4.123) ̃( ) ℚ ( ) ℚ ( ) 1푡<휏퐴 푡, 푇푖, 푇푛 = 피푡 (∑ 푃 푡, 푇푖 1푇푖<휏) = 1푡<휏피푡 (∑ 푃 푡, 푇푖 1푇푖<휏(푡)), 푖 푖 that is, 푛 (2.4.124) ̃( ) ( ) ℚ 1푡<휏퐴 푡, 푇푖, 푇푛 = 1푡<휏 ∑ 푃 푡, 푇푖 피푡 (1푇푖<휏(푡)) 푖 푛 ℚ = 1푡<휏 ∑ 푃(푡, 푇푖) (1 − ℙ퐷(푡, 푡, 푇푖)). 푖 The premium leg of the CDS is knocked out conditional on the default event of the underlying reference entity. Therefore premium leg is identical to the extinguisher transaction discussed above and its corresponding martingale under risk neutral measure is

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̃ ̃ ̃ ̃ 1푡<휏푆(푡, 푇푖, 푇푛)퐴(푡, 푇푖, 푇푛) ℚ 1푇<휏푆(푇, 푇푖, 푇푛)퐴(푇, 푇푖, 푇푛) (2.4.125) = 피푡 ( ), 푁푡 푁푇 where 0 ≤ 푡 ≤ 푇 ≤ 푇푖 ≤ 푇푛. If we change from risk neutral measure to survival annuity measure, ̃ ̃ ℚ 1푇<휏푆(푇, 푇푖, 푇푛)퐴(푇, 푇푖, 푇푛) (2.4.126) 피푡 ( ) 푁푇

푁푇 ̃ ̃ 퐴̃ 1푇<휏푆(푇, 푇푖, 푇푛)퐴(푇, 푇푖, 푇푛) 푁푡 = 피푡 , 푁푇 1푇<휏퐴̃(푇, 푇푖, 푇푛) 1 퐴̃(푡, 푇 , 푇 ) ( 푡<휏 푖 푛 ) that is, 1 푆̃(푇, 푇 , 푇 )퐴̃(푇, 푇 , 푇 ) 1 푆̃(푇, 푇 , 푇 ) (2.4.127) ℚ 푇<휏 푖 푛 푖 푛 퐴̃ 푡<휏 푖 푛 ̃ 피푡 ( ) = 피푡 ( 퐴(푡, 푇푖, 푇푛)) 푁푇 푁푡 ̃( ) 1푡<휏퐴 푡, 푇푖, 푇푛 퐴̃ = 피푡 (푆̃(푇, 푇푖, 푇푛)) 푁푡 Therefore, ̃ ̃ ̃ (2.4.128) 1푡<휏푆(푡, 푇푖, 푇푛)퐴(푡, 푇푖, 푇푛) 1푡<휏퐴(푡, 푇푖, 푇푛) 퐴̃ = 피푡 (푆̃(푇, 푇푖, 푇푛)) ⇒ 푆̃(푡, 푇푖, 푇푛) 푁푡 푁푡

퐴̃ = 피푡 (푆̃(푇, 푇푖, 푇푛)). Hence, in the survival state/non-default state at time 푡, i.e. 휏 > 푡, the forward starting CDS spread is a martingale under survival annuity measure,

퐴̃ (2.4.129) 푆̃(푡, 푇푖, 푇푛) = 피푡 (푆̃(푇, 푇푖, 푇푛)), where 0 ≤ 푡 ≤ 푇 ≤ 푇푖 ≤ 푇푛.

Then the Mark to Market of a forward starting CDS fixed receiver (i.e. protection seller of underlying credit) at time 푡 is (2.4.130) 1푇<휏 (퐾 − 푆̃(푇, 푇푖, 푇푛)) 퐴̃(푇, 푇푖, 푇푛) 퐶퐷푆 ℚ 푀푡푀푡 (푇푖, 푇푖+1) = 푁푡피푡 ( ), 푁푇 where 퐾 is the pre-specified fixed premium for the forward starting CDS fixed receiver. The forward starting CDS above resets at time 푇 and provides protection over the period [푇푖, 푇푛]. If we change the pricing measure to survival annuity measure,

퐶퐷푆 퐴̃ (2.4.131) 푀푡푀푡 (푇푖, 푇푖+1) = 1푡<휏퐴̃(푡, 푇푖, 푇푛)피푡 (퐾 − 푆̃(푇, 푇푖, 푇푛))

= 1푡<휏퐴̃(푡, 푇푖, 푇푛) (퐾 − 푆̃(푡, 푇푖, 푇푛)). And the corresponding forward starting CDS receiver swaption is + 퐶퐷푆푠푤푎푝푡푖표푛 ̃ 퐴̃ ̃ (2.4.132) 푀푡푀푡 (푇푖, 푇푖+1) = 1푡<휏퐴(푡, 푇푖, 푇푛)피푡 ((퐾 − 푆(푇, 푇푖, 푇푛)) ).

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Since the forward starting swap rate becomes a martingale under survival annuity measure ℙ퐴̃ (in the survival state), the SDE could be obtained by using martingale representation theorem, 퐴̃ (2.4.133) 푑푆̃(푡, 푇푖, 푇푛) = 휎푆̃푑푊푡 , 퐴̃ where 휎푆̃ is the instantaneous volatility of credit spread (in absolute values) and 푊푡 is the Brownian 퐴̃ motion under survival annuity measure ℙ and correlated to other market factors. 휎푆̃ is calibrated to the CDS swaptions quotes with different assumptions for underlying CDS spread distribution (i.e. normal distribution, log-normal distribution, etc.). The no arbitrage framework in the survival state gives the general interpolation method for deriving forward CDS by using spot contracts

푆̃(0,0, 푇푛)퐴̃(0,0, 푇푛) − 푆̃(0,0, 푇푖)퐴̃(0,0, 푇푖) (2.4.134) 푆̃(0, 푇푖, 푇푛) = , 퐴̃(0, 푇푖, 푇푛) where 0 ≤ 푡 ≤ 푇 ≤ 푇푖 ≤ 푇푛.

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2.5 No Arbitrage Pricing for Unilateral CVA

In essence, counterparty credit risk pricing/CVA pricing is identical to pricing of credit contingent instruments with a given range of Loss Given Default (LGD) on a portfolio of OTC derivatives. The order of default timing is not that crucial in pricing unless either one or both of the two parties are highly likely to default in foreseeable future. The Unilateral CVA is derived here by directly apply the same arguments in pricing single name credit contingent instrument. The nature of CVA is like other tradable assets and has to be properly hedged for trading purposes. CVA measures the expected cost to replace a given trade with a given counterparty and hence the volatility of the Mark to Market (MtM) plays a crucial role in CVA pricing as the MtM calculates the net amount of money to replace that given trade or portfolio in the marketplace. The different products across different asset classes are normally priced with different measures/numeraires and implementation of pricing different product in an integrated environment with relevant netting and margining agreements becomes an integral component of derivative dealer’s internal pricing engine design. The unhedgeable correlation dynamics between counterparty credit and market variables becomes an important pricing component of either one or both parties is/are in financial distress. The generic default contingent instrument pricing equation (2.4.106) discussed in previous section is

푇푛 퐶퐷(푢) 푉 = 1 푁 피ℚ (∫ 푑 ℙℚ(푡, 푡, 푢, ℱ푀 )). 푡 푡<휏 푡 푡 푁 푢 퐷 푇푛 푇1 푢

The credit charge or Unilateral CVA (UCVA) is computed on upfront basis. Here 푡 = 0 and investment horizon is [0, 푇]. Then UCVA/credit charge is 푇 (2.5.1) ℚ 퐶퐷(푡) ℚ 푀 푉0 = 10<휏푁0피0 (∫ 푑푡ℙ퐷(0,0, 푡, ℱ푇 )), 0 푁푡

ℚ ℚ 푀 here if ℙ퐷(푡) = ℙ퐷(0,0, 푡, ℱ푇 ), 푇 (2.5.2) ℚ 퐶퐷(푡) ℚ 푉0 = 피0 (∫ 푑푡ℙ퐷(푡)). 0 푁푡 If we use convexity adjustment to obtain a linear approximation of UCVA/credit charge, 푇 (2.5.3) ℚ 퐶퐷(푡) ℚ 푉0 = 피0 (∫ 푑푡ℙ퐷(푡)) 0 푁푡 푇 ℚ 퐶퐷(푡) ℚ ℚ = ∫ 피0 ( ) 피0 (푑푡ℙ퐷(푡)) 0 푁푡 푇 ℚ ℚ 퐶퐷(푡) 푑푡ℙ퐷(푡) + ∫ 퐶표푣0 ( , ) 푑푡. 0 푁푡 푑푡 Since

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ℚ ℚ ℚ ℚ 푀 ℚ ℚ 푀 (2.5.4) 피0 (푑푡ℙ퐷(푡)) = 피0 (푑푡ℙ퐷(0,0, 푡, ℱ푇 )) = 피0 (피0 (푑푡10<휏(0)<푡|ℱ푇 ))

ℚ ℚ = 푑푡 (피0 (10<휏(0)<푡)) = 푑푡ℙ퐷(0,0, 푡), and if ℚ ℚ ℚ ℚ (2.5.5) ℙ퐷(0, 푡) = ℙ퐷(0,0, 푡) = 피0 (ℙ퐷(푡)), then 푇 ℚ (2.5.6) ℚ 퐶퐷(푡) 푑푡ℙ퐷 (0, 푡) 푉0 = ∫ 피0 ( ) (1 + 휌휎1휎2푡)푑푡, 0 푁푡 푑푡 ℚ 퐶퐷(푡) 푑푡ℙ퐷(푡) where 휌 is the instantaneous correlation between and . 휎1 denotes the instantaneous 푁푡 푑푡 ℚ 퐶퐷(푡) 푑푡ℙ퐷(푡) percentage volatility of and 휎2 denotes the instantaneous percentage volatility of . 푁푡 푑푡

Equation (2.5.6) is useful in pricing CVA with wrong way risk on standalone trade level. For example, if we price a USDRUB FX derivative trade with a Russian counterparty and our counterparty promised to repay USD at the end of this transaction, the correlation between counterparty credit riskiness and exposure profiles could be estimated and back-tested in multiple historical time series. However, when it comes to pricing CVA on portfolio level, equation (2.5.6) has its limitation and more variables have to be introduced to capture the complex dependency structure between different market risk factors and counterparty credit riskiness. Other methods of CVA pricing with wrong way risk are discussed in Brigo, Morini and Pallavicini (2013) and Hull and White (2012).

From the perspective of CVA holder, CVA is the expected amount of net cash flows the derivative dealer needs to replace the ongoing transactions with a defaulted market counterparty by the identical transactions with a surviving market counterparty. Therefore, the net cash flow generated by CVA/credit charge at the default time of counterparty is + 퐶퐷(푡) = 푀푡푀푡 (1 − 푅푡), (2.5.7) where 푀푡푀 refers to the Mark to Market of the ongoing transactions, 푀푡푀+ denotes the exposure + of ongoing transactions (i.e. 푀푡푀 = max(0, 푀푡푀)) and 푅푡 denotes the effective recovery rate of the counterparty at time 푡 . The expression of 퐶퐷(푡) here provides insights of the nature of CVA/credit charge. CVA/credit charge arises because of the asymmetric treatments of the surviving party in the default state of its counterparty. From the perspective of the surviving party, if the ongoing transactions are out of money in the default state of the counterparty, that is 푀푡푀 < 0, the surviving party has to pay the entire MtM to the defaulted counterparty. There is no distinctions in treatments if the ongoing transactions are not in favour of the surviving party. However, if the ongoing transactions are in favour of the surviving party in the default state of its counterparty, that is 푀푡푀 > 0 , the surviving party could only receive a portion of its MtM from the defaulted

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counterparty 푅푡푀푡푀푡. These asymmetric treatments in the default state of counterparty create a material economic loss (1 − 푅푡)푀푡푀푡 for the surviving party in the transactions.

Default State (휏 ≤ 푡) Survival State (푡 < 휏) Net Difference

푀푡푀 < 0 푀푡푀푡 푀푡푀푡 0

푀푡푀 > 0 푅푡 ∙ 푀푡푀푡 푀푡푀푡 (1 − 푅푡)푀푡푀푡

(Table 2.1: Asymmetric Treatments of the Surviving Party in the Default State of Its Counterparty)

+ Therefore, the net cash flow generated by CVA/credit charge is 퐶퐷(푡) = 푀푡푀푡 (1 − 푅푡) as discussed before. Thus, 푇 + (2.5.8) ℚ 푀푡푀푡 (1 − 푅푡) ℚ 푉0 = 피0 (∫ 푑푡ℙ퐷(푡)) 0 푁푡

푇 + ℚ ℚ 퐸푀푡푀푡 (1 − 푅푡) 푑푡ℙ퐷 (0, 푡) = ∫ 피0 ( ) 푑푡. 0 푁푡 푑푡

+ + where 퐸푀푡푀 refers to effective Mark to Market and 퐸푀푡푀푡 = 푀푡푀푡 (1 + 휌휎1휎2푡). Given the existence of correlation between market variables and counterparty credit spreads, the CVA/credit charge pricing engine should be calibrated to the entire volatility and correlation matrix among all variables. Hence, more specifically, CVA/credit charge is 푇 + 퐶푃 (2.5.9) ℚ 푀푡푀푡 (1 − 푅푡 ) ℚ 퐶푉퐴 = 피0 (∫ 푑푡ℙ퐷,퐶푃(푡)) 0 푁푡

푇 + 퐶푃 ℚ ℚ 퐸푀푡푀푡 (1 − 푅푡 ) 푑푡ℙ퐷,퐶푃(0, 푡) = ∫ 피0 ( ) 푑푡, 0 푁푡 푑푡 퐶푃 ℚ where 푅푡 is the effective recovery rate of counterparty at time 푡, ℙ퐷,퐶푃(푡) is the default probability of the counterparty under risk neutral measure ℚ and 푇 is the maturity of the underlying portfolio traded with this counterparty.

A brief discussion on two way CSAs

If both parties signs CSA (credit support annex) at inception and collateral is received or posted during the life of the trade, the Mark to Market for a portfolio of 푁 trades with a netting agreement in place becomes 푁 (2.5.10) + 푖 푀푡푀푡 = max (∑ 푀푡푀푡 − 풞푡 , 0) , 푖=1

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푖 where 푀푡푀푡 refers to the Mark to Market of ith trade at time 푡 and 풞푡 denotes the net amount of collateral received (or posted) at time 푡. Here 풞푡 > 0 the derivative dealer receives a net amount of collateral from the counterparty. From the derivative dealer’s perspective, the CSA in place effectively puts a cap on the exposure of the underlying portfolio with floating 풞푡. The floating strike price could be negative if the net amount of collateral is posted to the counterparty. So the CVA/credit charge could be interpreted as a cap on the underlying portfolio exposure activated by the default of the counterparty. It is effectively a counterparty credit linked knocked in cap with floating strikes 풞푡.

In practice, a derivative dealer charges its counterparty CVA to offset the potential loss due to counterparty credit risk. Actually the derivative dealer sells the counterparty credit linked cap to its counterparty such that the counterparty only has to pay recovery amount of exposure in default state. Therefore, the derivative dealer that shorts CVA implicitly builds up short positions in counterparty credit spread, Vega and Gamma of the cap with floating strikes and the correlation between the counterparty credit spread and underlying portfolio exposure (or cap). The CVA would decline when it comes closer to the maturity of the underlying portfolio and hence the derivative dealer longs Theta via short positions in CVA. The CVA would increase if the correlation is positive (i.e. Wrong Way Risk) and volatility of counterparty credit spread goes up. Hence the derivative dealer shorts the counterparty credit spread volatility if the correlation is positive. And the derivative dealer longs the counterparty credit spread volatility if the correlation is negative (i.e. Right Way Risk) as CVA would decline as volatility of counterparty credit spread goes up. The derivative dealer that sells the CVA to its counterparty therefore shorts the correlation between counterparty credit spreads and market variables.

One of the counterparty credit risk measures is included in CVA pricing equation, + + ℚ 푀푡푀푡 (2.5.11) 퐸퐸0 (푡) = 피0 ( ), 푁푡 + where 0 ≤ 푡 < 푇. 퐸퐸0 (푡) refers to the time 0 expected positive exposure of underlying portfolio at time 푡. The potential future exposure (PFE) is defined as + (2.5.12) ℚ 푀푡푀푡 ℙ퐷 ( < 푃퐹퐸0(푡)) = 훼, 푁푡 where 훼 refers to the confidence level that the present value of exposure at default is smaller than potential future exposure. For example, 훼 = 99% means the present value of exposure at default is smaller than 푃퐹퐸0(푡) in 99% of all possible scenarios. The maximum PFE refers to the maximum value of all 푃퐹퐸0(푡) over a certain time horizon [푠, 푇],

푀푃퐹퐸0(푠, 푇) = sup 푃퐹퐸0(푡), (2.5.13) 푠≤푡≤푇

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Counterparty Credit Risk, Funding Risk and Central Clearing where 0 ≤ 푠 ≤ 푡 ≤ 푇. An alternative risk measure is expected shortfall (ES) + + 푀푡푀푡 푀푡푀푡 (2.5.14) 퐸푆0 = 피 ( | > 푃퐹퐸0(푡)). 푁푡 푁푡 The DVA/credit benefit is the expected net economic gain for the derivative dealer in the default state of itself, 푇 − 퐼 (2.5.15) ℚ 푀푡푀푡 (1 − 푅푡) ℚ 퐷푉퐴 = −피0 (∫ 푑푡ℙ퐷,퐼(푡)) 0 푁푡

푇 − 퐼 ℚ ℚ 퐸푀푡푀푡 (1 − 푅푡) 푑푡ℙ퐷,퐼(0, 푡) = − ∫ 피0 ( ) 푑푡, 0 푁푡 푑푡 퐼 ℚ where 푅푡 is the effective recovery rate of the derivative dealer itself at time 푡, ℙ퐷,퐼(푡) is the default − − probability of the derivative dealer under risk neutral measure ℚ. Here 퐸푀푡푀푡 = 푀푡푀푡 (1 −

휌휎1휎2푡). The asymmetric treatments of Bankruptcy Code gives the defaulted party a material economic gain by paying only the recovery portion of its liabilities. The net negative exposure for the underlying portfolio with CSA (credit support annex) and netting agreements in place is 푁 (2.5.16) − 푖 푀푡푀푡 = min (∑ 푀푡푀푡 − 풞푡 , 0), 푖=1 푖 where 푀푡푀푡 refers to the time 푡 Mark to Market of trade 푖 in the portfolio. As specified before,

풞푡 > 0 indicates the derivative dealer is a net collateral receiver at 푡 and 풞푡 < 0 indicates the derivative dealer is a net collateral poster at 푡. If the overall liability is less than the net collateral posted, the derivative dealer could realise the economic gain via paying less at its default time. Since 푁 푁 (2.5.17) − 푖 푖 −푀푡푀푡 = − min (∑ 푀푡푀푡 − 풞푡 , 0) = max (풞푡 − ∑ 푀푡푀푡 , 0), 푖=1 푖=1

DVA/credit benefit effectively is a floor on a given portfolio with floating strike prices 풞푡 . DVA/credit benefit would reduce the liability of the derivative dealer in its default state at time 푡 to 퐼 푁 푖 the level 푅푡 max(풞푡 − ∑푖=1 푀푡푀푡 , 0). Hence the payoff of DVA/credit benefit in the default state 퐼 푁 푖 of the derivative dealer is (1 − 푅푡) max(풞푡 − ∑푖=1 푀푡푀푡 , 0). Therefore, DVA/credit benefit is knocked in floor with floating strikes 풞푡 triggered by the default event of the derivative dealer itself. The expected negative exposure here is defined as − − ℚ 푀푡푀푡 (2.5.18) 퐸퐸0 (푡) = 피0 ( ). 푁푡 The derivative dealer has to buy DVA from its counterparty (by paying DVA) such that it could pay less in its default state in the future. Therefore, the derivative dealer longs its DVA and hence builds up long positions in its own credit spread and Vega and Gamma of floor on the underlying portfolio. DVA/credit benefit declines as the time approaches the maturity of the portfolio. Hence the derivative dealer shorts Theta. The correlation between the expected negative exposure and the

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Counterparty Credit Risk, Funding Risk and Central Clearing derivative dealer’s own credit spread also has an impact in final price. From the perspective of the derivative dealer, if the credit spread widens, the expected negative exposure goes up due to the positive correlation impact, then the DVA declines as the absolute value of expected negative exposure goes down. This is the Right Way Risk (RWR) scenario from the counterparty’s perspective. The derivative dealer therefore shorts its credit spread volatility if the correlation is positive. If the correlation is negative, the expected negative exposure would decline if credit spread widens. This is the Wrong Way Risk (WWR) scenario from the counterparty’s perspective. Hence the DVA/credit benefit goes up accordingly and therefore derivative dealer longs credit spread volatility if the correlation is negative. Overall, the derivative dealer establishes short positions in the correlation between expected negative exposure and its own credit spread by holding DVA/credit benefits.

A two way CSA is generally required in current OTC market. Thresholds, Minimum Transfer Amount (MTA), Initial Margins (or Independent Amount), Collateral Currency etc. have to be specified in this bilateral collateral agreement. Threshold is defined as limit of uncollateralised portion of exposure both parties could have in transactions. MTA specifies the minimum level of price movements for a margin call. Threshold could be different for two parties in the transactions. Credit + + quality is one of the key determinant of threshold. Here 푉푇ℎ푟푒푠ℎ표푙푑 and 푉푀푇퐴 denote the Threshold and MTA of collateral on the asset side of the transactions, i.e. the portfolio is in the money from the + + derivative dealer’s perspective and the derivative dealer receives collateral with 푉푇ℎ푟푒푠ℎ표푙푑 and 푉푀푇퐴 − − from its counterparty. And 푉푇ℎ푟푒푠ℎ표푙푑 and 푉푀푇퐴 denote the Threshold and MTA of collateral on the liability side of the transactions, i.e. the portfolio is out the money from the derivative dealer’s − − perspective and the derivative dealer pays collateral with 푉푇ℎ푟푒푠ℎ표푙푑 and 푉푀푇퐴 to its counterparty. In practice, the maximum level or cap of loss due to default of the counterparty (from derivative dealer’s + + perspective) is 푉푇ℎ푟푒푠ℎ표푙푑 + 푉푀푇퐴, the maximum level or cap of loss due to default of the derivative − − dealer (from the counterparty’s perspective) is 푉푇ℎ푟푒푠ℎ표푙푑 + 푉푀푇퐴. For example, suppose the initial margin or independent amount is set to 0 and hence the net collateral at time 0 풞0 = 0. If the market experience some movements and the portfolio is in favour of the derivative dealer, i.e. 푀푡푀 > 0, + and 푀푡푀 > 푉푇ℎ푟푒푠ℎ표푙푑 , the derivative dealer will not collect collateral from its counterparty if + + 푀푡푀 − 푉푇ℎ푟푒푠ℎ표푙푑 < 푉푀푇퐴 . Therefore, the effective uncollateralised exposure in a given CSA + + (without initial margin/independent amount in place) is actually 푉푇ℎ푟푒푠ℎ표푙푑 + 푉푀푇퐴. Now suppose + + 푀푡푀 > 푉푇ℎ푟푒푠ℎ표푙푑 + 푉푀푇퐴, the counterparty would receive a margin call from the derivative dealer + + and have to post a collateral amount equal to 푀푡푀 − 푉푇ℎ푟푒푠ℎ표푙푑 > 푉푀푇퐴 to bring the overall + uncollateralised portion back to 푉푇ℎ푟푒푠ℎ표푙푑 level. Generally the exposure could not be zero even if Threshold and MTA are set to 0. This is due to the settlement lag. The derivative dealer would normally wait a few days to receive the collateral after the margin call. The time delay in receiving

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Counterparty Credit Risk, Funding Risk and Central Clearing collateral is settlement lag and creates non zero exposure for ongoing transactions even if zero thresholds and MTAs are specified in CSA.

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2.6 Bilateral CVA without First to Default Features

CVA/credit charge (2.5.9) and DVA/credit benefit (2.5.15) have been discussed in the previous sections. The net credit charge or Bilateral CVA (or Bilateral UCVA as we did not take the first to default into account here) is 퐵퐶푉퐴 = 퐶푉퐴 − 퐷푉퐴. (2.6.1) From the perspective of a bank, the market price of expected loss given default (LGD) if counterparty defaults during the life of the trade is CVA and market price of LGD if the bank defaults is DVA. The bilateral CVA is defined as net positions of CVA and DVA. Therefore the bilateral CVA is the net credit risk components embedded in a derivative transaction. The net CVA or BCVA could be paid as upfront premium equal to the market price of expected loss or a stream of coupons or running spread like a CDS transaction except the payoff at default time is volatile. If the counterparty default before the end of the contract, the counterparty has to pay for the accrual term to cover the gap between default time and last coupon date. The transaction structure is subject to policies of different banks.

The net CVA/ bilateral CVA could be either positive or negative. It depends on the relative credit qualities of two parties and exposures dynamics in this transaction. Under Bankruptcy Code, the default of one party will automatically cancel the trade and hence the surviving party have to liquidate its positions and closeout all trades with this specific counterparty. Therefore, there is a first to default (FTD) optionality embedded in net CVA calculation as discussed in Brigo (2012) and Gregory (2011). This first to default optionality does have a significant impact on evaluation if the credit quality of one of the parties deteriorates to a very low level (Tang and Li (2007)). In reality, the most challenging task for CVA desk is to price CVA on the portfolio level with trades priced by different models, booked under different systems and correlations across different asset classes.

Normally the linear products like forwards and swaps do not have Vegas. Now because the options feature introduced by CVA/DVA, these linear products have exposures to implied volatilities.

Another way to interpret net CVA/bilateral CVA is to treat it as an adjustment to final price due to the relative credit qualities of two parties compared to LIBOR credit quality. If we decompose a derivative transaction into receivables and payables and, and we discount the net receivables using counterparty’s credit curve and discount net payables using the bank’s credit curve, the net difference over default risk free priced at LIBOR flat should be equal to net CVA.

The derivative dealer could choose to charge CVA, BCVA as an upfront charge to its counterparty. Or it can choose to build these CVA, BCVA terms as a funding spread into the funding leg of

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Counterparty Credit Risk, Funding Risk and Central Clearing ongoing transactions with the counterparty. Here comes the question: If the CVA or BCVA funding (퐵)퐶푉퐴 spread is computed in the form of in a standard interest rate swap, the exposure profile would 푃푉01 change of the funding spread is attached to the funding leg and hence CVA or BCVA change accordingly. So what is the equilibrium level of the funding spread such that present value of economic gain (loss) in the interest rate swap is balanced against the present value of economic loss (gain) on its CVA/BCVA?

Suppose the derivative dealer is the fixed receiver in this transaction. The fixed swap rate is set at 푆 at inception. The present value of this fixed receiver swap is denoted by 푃푉0(푆). The CVA desk of the derivative dealer computes the BCVA/net credit charge 퐵퐶푉퐴0(푆) and the trading desk needs to build the net credit charge into the funding spread Δ푆 of the fixed leg (as the counterparty is the fixed payer). Then the present value of the fixed receiver swap becomes 푃푉0(푆 + Δ푆) but the BCVA/net credit charge changes to 퐵퐶푉퐴0(푆 + Δ푆). So what is the fair funding spread Δ푆?

푃푉0(푆) = 푃푉0(푆 + Δ푆) + 퐵퐶푉퐴0(푆 + Δ푆). (2.6.2)

Let’s start with first order Taylor expansion, 휕푃푉 (푆) (2.6.3) 푃푉 (푆 + Δ푆) = 푃푉 (푆) + 0 (푆 + Δ푆 − 푆) + ⋯ 0 0 휕푆

휕푃푉0(푆) = 푃푉 (푆) + (Δ푆) + ⋯, 0 휕푆 휕퐵퐶푉퐴 (푆) (2.6.4) 퐵퐶푉퐴 (푆 + Δ푆) = 퐵퐶푉퐴 (푆) + 0 (푆 + Δ푆 − 푆) + ⋯ 0 0 휕푆

휕퐵퐶푉퐴0(푆) = 퐵퐶푉퐴 (푆) + (Δ푆) + ⋯, 0 휕푆 Technically, as Δ푆 → 0 , first order Taylor Expansion provides good approximations to above

푃푉0(푆 + Δ푆) and 퐵퐶푉퐴0(푆 + Δ푆), therefore,

휕푃푉0(푆) 휕퐵퐶푉퐴0(푆) (2.6.5) 푃푉 (푆) ≈ 푃푉 (푆) + (Δ푆) + 퐵퐶푉퐴 (푆) + (Δ푆), 0 0 휕푆 0 휕푆 and

퐵퐶푉퐴0(푆) (2.6.6) Δ푆 ≈ − . 휕푃푉 (푆) 휕퐵퐶푉퐴 (푆) 0 + 0 휕푆 휕푆 The challenge in implementation above calculation is that the funding spread needs three inputs from 휕푃푉 (푆) two different levels of models. 0 is Delta from micro/trade level models and the other two 휕푆 휕퐵퐶푉퐴 (푆) 퐵퐶푉퐴 (푆) and 0 are price and Delta from portfolio level models. It is not an easy job to 0 휕푆 ensure that the macro/portfolio level models deliver consistent prices and hedging ratios with 푖 micro/trade level models. 푀푡푀0(푡) denotes the time 0 Mark to Market of trade 푖 at time 푡 produced

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ℚ 푀푡푀푡 by the micro/trade level models, 피0 ( ) is the Mark to Market of trade 푖 at time 푡 produced by 푁푡 the macro/portfolio level models, 푖 (2.6.7) 푖 ℚ 푀푡푀푡 푀푡푀0(푡) = 피0 ( ), 푁푡 this procedure ensures the micro/trade level models and macro/portfolio model produce consistent forward curves. The following two equations ensures these two level models produce consistent exposure profiles, 푖+ (2.6.8) 푖 + ℚ 푀푡푀푡 푀푡푀0(푡) = 피0 ( ), 푁푡 and 푖− (2.6.9) 푖 − ℚ 푀푡푀푡 푀푡푀0(푡) = 피0 ( ). 푁푡 And the expected negative exposure could be restated as 푖− 푖 + (2.6.10) 푖 − ℚ 푀푡푀푡 ℚ (0 − 푀푡푀푡) 푀푡푀0(푡) = 피0 ( ) = −피0 ( ), 푁푡 푁푡 Therefore, the expected positive/negative exposure are call/put options on the Mark to Market of underlying transactions with fixed strike price 0 and time to maturity 푡 (in uncollateralised environment) except for the starting time 0. The exposure profiles at inception are time 0 prices of call/put options with time to maturity 0. The forward curve is known at inception and hence the volatility does not have any impact on the starting value of exposure profile. Therefore, the expected positive/negative exposure at inception is contingent present value of the specific trade. If the present value of that trade is positive (like a long position in an option), 푖 + (2.6.11) 푖 + + ℚ 푀푡푀0 ℚ 푖 푀푡푀0(0) = 퐸퐸0 (0) = 피0 ( ) = 피0 (푀푡푀0) > 0, 푁0

푖 − 푖 − − ℚ 푀푡푀0 ℚ 푀푡푀0(0) = 퐸퐸0 (0) = 피0 ( ) = 피0 (0) = 0. 푁0 If the present value of the trade is negative (like a short position in an option), 푖 + (2.6.12) 푖 + + ℚ 푀푡푀0 ℚ 푀푡푀0(0) = 퐸퐸0 (0) = 피0 ( ) = 피0 (0) = 0, 푁0

푖 − 푖 − − ℚ 푀푡푀0 ℚ 푖 푀푡푀0(0) = 퐸퐸0 (0) = 피0 ( ) = 피0 (푀푡푀0) < 0. 푁0 If the present value of the trade is 0 (like trading an interest rate swap at par rate),

푖 + (2.6.13) 푖 + + ℚ 푀푡푀0 ℚ 푀푡푀0(0) = 퐸퐸0 (0) = 피0 ( ) = 피0 (0) = 0, 푁0

푖 − 푖 − − ℚ 푀푡푀0 ℚ 푀푡푀0(0) = 퐸퐸0 (0) = 피0 ( ) = 피0 (0) = 0. 푁0

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Generally volatility are not priced into linear products like interest rate swaps. The exposure calculation on the micro/trade level and CVA/BCVA on the macro/portfolio level takes the volatility into pricing of linear products. Some special cases like trading FX products with emerging markets entity requires to consider dependence between counterparty credit spread and exposure profile. Hence a robust volatility and correlation modelling methodology (which delivers consistent forward curves and exposure profiles with respect to micro/trade level models) is required for CVA/BCVA calculation.

Some standalone CVA calculations have closed from or semi-closed form solutions with certain assumptions. In a Wrong Way Risk free environment, i.e. the correlation between counterparty credit spread and underlying trade dynamics is assumed to be 0, the CVA/credit charge and DVA/credit benefit are 푇 + 퐶푃 (2.6.14) ℚ 푀푡푀푡 (1 − 푅푡 ) ℚ 퐶푉퐴 = 피0 (∫ 푑푡ℙ퐷,퐶푃(푡)) 0 푁푡 푇 퐶푃 + ℚ = (1 − 푅 ) ∫ 퐸퐸0 (푡) 푑푡ℙ퐷,퐶푃(0, 푡), 0 푇 − 퐼 (2.6.15) ℚ 푀푡푀푡 (1 − 푅푡) ℚ 퐷푉퐴 = −피0 (∫ 푑푡ℙ퐷,퐼(푡)) 0 푁푡 푇 퐼 − ℚ = −(1 − 푅 ) ∫ 퐸퐸0 (푡) 푑푡ℙ퐷,퐼(0, 푡), 0 + − + ℚ 푀푡푀푡 − ℚ 푀푡푀푡 퐶푃 ℚ 퐶푃 퐼 ℚ 퐼 where 퐸퐸0 (푡) = 피0 ( ), 퐸퐸0 (푡) = 피0 ( ), 푅 = 피0 (푅푡 ), 푅 = 피0 (푅푡). 푁푡 푁푡

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2.7 Approximations for CVA/DVA

Closed form solutions for CVA/DVA do possibly exist if certain products/trades have closed form + + solutions for 퐸퐸0 (푡) and 퐸퐸0 (푡). For example, CVA and DVA for longing a plain-vanilla option

(cash settled) with maturity 푇, its current price is 푉0, + + ℚ 푀푡푀푡 (2.7.1) 퐸퐸0 (푡) = 피0 ( ) = 푉0, 푁푡 + − ℚ 푀푡푀푡 ℚ 푀푡푀푡 퐸퐸0 (푡) = 피0 ( ) − 피0 ( ) = 푉0 − 푉0 = 0. 푁푡 푁푡 Hence, 푇 (2.7.2) 퐶푃 + ℚ 퐶푃 ℚ 퐶푉퐴 = (1 − 푅 ) ∫ 퐸퐸0 (푡) 푑푡ℙ퐷,퐶푃(0, 푡) = (1 − 푅 )푉0ℙ퐷,퐶푃(0, 푇) 0 푇 ≈ 푆0 ∙ 퐷 ∙ 푉0, 푇 (2.7.3) 퐼 − ℚ 퐷푉퐴 = −(1 − 푅 ) ∫ 퐸퐸0 (푡) 푑푡ℙ퐷,퐼(0, 푡) = 0, 0 where 푆푇 denotes the time 0 observed 푇 years CDS spread for the counterparty and 퐷 is the duration of trade. And it is assumed that the recovery rate of the option transaction is identical to the recovery rate of the counterparty CDS. Furthermore, if the underlying trade is not a plain vanilla option but a derivative with multiple cash flows, the CVA/credit charge and DVA/credit benefit could be approximated by ̅̅̅̅̅+̅ − 퐶푉퐴 = 푆퐶푃 ∙ 퐷 ∙ 퐸퐸 , 퐷푉퐴 = −푆퐼 ∙ 퐷 ∙ 퐸̅̅̅퐸̅̅̅, (2.7.4) where 퐸̅̅̅퐸̅̅+̅ denotes the average expected positive exposure over the entire life of the trade and 퐸̅̅̅퐸̅̅−̅ denotes the average expected negative exposure over the entire life of the trade.

Cesari et al. (2009) provides an approximation for interest rate swaps. It is observed that the exposure profile of an interest rate swap is bell shaped, that is, the exposure starts from 0, increase for a certain period of time, and declines back to 0 at maturity. The gradually declining duration of the swap and increasing volatility of forward curve are identified as main drivers of this bell shaped exposure profile. The duration is assumed to be a fixed proportion 퐴0 of the remaining time to maturity of the swap in Cesari et al. (2009). The underlying interest rate volatility is 휎푁√푡. Then the volatility of the interest rate swap is 푆푤푎푝 (2.7.5) 휎푡 = 퐷 ∙ 휎푁√푡 = 퐴0 ∙ (푇 − 푡) ∙ 휎푁√푡. Hence if we assume the Mark to Market of this at the money or part swap is normally distributed, 푆푤푎푝 (2.7.6) 푀푡푀푡 푆푤푎푝 = 휎푡 푧, 푁푡

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푡 ∫ 푟푠푑푠 where 푧~ℕ(0,1) and 푁푡 is the money market account (i.e. 푁푡 = 푒 0 ). The above equation could be restated as 푆푤푎푝 푆푤푎푝 ℚ (2.7.7) 푑푀푡푀푡 = 푟푡푀푡푀푡 푑푡 + (퐷 ∙ 휎푁 ∙ 푁푡)푑푊푡 , ℚ where 푊푡 is one dimensional Brownian motion under risk neutral measure and 퐷 = 퐴0 ∙ (푇 − 푡). Hence + 푆푤푎푝 ∞ 1 푆푤푎푝 (2.7.8) ℚ 푀푡푀 푆푤푎푝 1 − 푧2 휎 + 푡 2 푡 퐸퐸0 (푡) = 피0 ( ) = 휎푡 푧 ∫ 푒 푑푧 = . 푁푡 0 √2휋 √2휋 Then if we assume the hazard rate curve (default intensity term structure) is flat, 푇 푇 (2.7.9) 퐶푃 + ℚ 1 퐶푉퐴 = (1 − 푅 ) ∫ 퐸퐸0 (푡) 푑푡ℙ퐷,퐶푃(0, 푡) ≈ 푆퐶푃 ∫ 퐴0 ∙ (푇 − 푡) ∙ 휎푁√푡 푑푡, 0 0 √2휋 Therefore, 8 5 (2.7.10) 퐶푉퐴 ≈ 푆 퐴 휎 푇2. 퐶푃 75 0 푁

ℚ 푀푡푀푡 Since the discounted forward Mark to Market 푀푡푀0(푡) = 피0 ( ) is much easier to calculate 푁푡 compared to exposure profiles (as Mark to Market essentially is roll-down analysis and could be generated by micro/trade level models), 퐵퐶푉퐴 = 퐶푉퐴 − 퐷푉퐴 (2.7.11) 푇 퐶푃 + ℚ = (1 − 푅 ) ∫ 퐸퐸0 (푡) 푑푡ℙ퐷,퐶푃(0, 푡) 0 푇 퐼 − ℚ + (1 − 푅 ) ∫ 퐸퐸0 (푡) 푑푡ℙ퐷,퐼(0, 푡), 0 and therefore, 푇 (2.7.12) 퐶푃 ℚ 퐵퐶푉퐴 = (1 − 푅 ) ∫ 푀푡푀0 (푡) 푑푡ℙ퐷,퐶푃(0, 푡) 0 푇 퐶푃 − ℚ − (1 − 푅 ) ∫ 퐸퐸0 (푡) 푑푡ℙ퐷,퐶푃(0, 푡) 0 푇 퐼 − ℚ + (1 − 푅 ) ∫ 퐸퐸0 (푡) 푑푡ℙ퐷,퐼(0, 푡), 0 + − ℚ 푀푡푀푡 ℚ 푀푡푀푡 ℚ 푀푡푀푡 + − as 푀푡푀0(푡) = 피0 ( ) = 피0 ( ) + 피0 ( ) = 퐸퐸0 (푡) + 퐸퐸0 (푡). 푁푡 푁푡 푁푡

푑 ℙℚ (0,푡) 푑 ℙℚ (0,푡) Therefore, if (1 − 푅퐶푃) 푡 퐷,퐶푃 ≈ (1 − 푅퐼) 푡 퐷,퐼 , 푑푡 푑푡 푇 (2.7.13) 퐶푃 ℚ 퐵퐶푉퐴 ≈ (1 − 푅 ) ∫ 푀푡푀0 (푡) 푑푡ℙ퐷,퐶푃(0, 푡). 0

푑 ℙℚ (0,푡) 푑 ℙℚ (0,푡) If (1 − 푅퐶푃) 푡 퐷,퐶푃 < (1 − 푅퐼) 푡 퐷,퐼 , then the derivative dealer is credit risker than its 푑푡 푑푡 counterparty, a conservative BCVA could be calculated as

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푇 (2.7.14) 퐼 ℚ 퐵퐶푉퐴 ≈ (1 − 푅 ) ∫ 푀푡푀0 (푡) 푑푡ℙ퐷,퐼(0, 푡). 0

푑 ℙℚ (0,푡) 푑 ℙℚ (0,푡) If (1 − 푅퐶푃) 푡 퐷,퐶푃 > (1 − 푅퐼) 푡 퐷,퐼 , it is better to perform separate calculations as 푑푡 푑푡 ( 퐶푃) 푇 ( ) ℚ ( ) 퐵퐶푉퐴 ≈ 1 − 푅 ∫0 푀푡푀0 푡 푑푡ℙ퐷,퐶푃 0, 푡 overstates the DVA/credit benefit hence leads to ( 퐼) 푇 ( ) ℚ ( ) lower estimation of actual BCVA and 퐵퐶푉퐴 ≈ 1 − 푅 ∫0 푀푡푀0 푡 푑푡ℙ퐷,퐼 0, 푡 understates the CVA/credit cost and hence leads to lower estimation of actual BCVA. A general rule of thumb here is the actual BCVA should be greater than both numbers calculated above.

The key feature of counterparty credit risk/funding risk pricing is that the calculation is on the macro/portfolio level. It is impossible for micro/trade level models to produce a sensible CVA/DVA/FVA number because the micro/trade level cannot handle the significant portfolio effect. The counterparty risk price for the same trade varies significantly across different counterparties, or more specifically, legal entities, as the different enforceability of netting agreements in different jurisdictions (for example, Russia and EU), different terms in CSAs and other related features. For trades with a given legal entity, it is difficult for the portfolio level model to produce consistent market prices across different assets classes as these individual trades were priced with different trade level models and different systems. Micro/trade level models do take the counterparty risk into final price by discounting the products at a higher rate than risk free rate. For example, the discounting curve for uncollateralised interest rate swaps is LIBOR and LIBOR is greater than the Fed Funds rate or Repo rate. LIBOR credit quality is implicitly assumed for derivatives transactions prior to the great financial crisis. If there is a repo market with abundant liquidity for the underlying assets, the discounting curve or funding curve for that specific asset should be its corresponding repo curve. And derivatives written on that asset would be discounted by LIBOR curve. For example, the options written on Treasury bonds are discounted by LIBOR but the future payoff of Treasuries are discounted by repo rates. Given the nature of repo transaction is collateralised transaction, the repo rate is more sensitive to the credit quality and market volatility of underlying asset rather than the associated counterparty credit risk. For a vast number of assets in the market place, they are priced with a spread attached to the LIBOR rate due to the existence of difference in credit quality, tax polices (like exemptions of certain for Treasury bond holders), liquidity and other factors. For the assets or products whose payoff is highly correlated with the credit risk of underlying like corporate bonds, emerging market debt, etc., additional models have to be used to address the counterparty risk rather than simply quoting a number of funding spreads over LIBOR curve.

Micro/trade level models are generally calibrated to a number of specific parameters to price a certain types of transactions. One model might deliver consistent market prices for a range of products but failed to give good indications of prices of others. Therefore, this type of models should not be used

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Counterparty Credit Risk, Funding Risk and Central Clearing for calculating portfolio level price. As discussed above, the counterparty credit risk pricing is actually a credit linked knock in option pricing problem with floating strikes (linked to its corresponding netting and margining policies) and counterparty portfolios as underlying assets. The calibrations to the volatility skew, correlation skew, forward volatility/correlation dynamics and dependency between counterparty credit risk and market variables etc. are on top of list for macro/portfolio level modellers.

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2.8 General Pricing Formula of Bilateral CVA

The situation that only one of the two parties involved into a transaction is no longer realistic after the credit crunch 2008. The basic accounting rule is that one party’s asset represents another party’s liability. The problem with UCVA is that UCVA breaks this basic rule and creates unbalance between two parties (Albanese, Brigo and Oertel (2012)).

Bilateral credit risk first appeared in Duffie and Huang (1996) and was incorporated into the discount rate (switching discount rate). That is, the payoffs from the swap are discounted by the discount rate of the counterparty that is currently out of money.

The default of the institution itself is also taken into consideration in bilateral credit valuation. A simplified formula used by the industry is the UCVA calculated by the institution itself minus the UCVA calculated by its counterparty. This approach actually ignores the fact the transaction terminates and proceeds into close-out procedures upon the first default. Or in other words, the bilateral credit valuation adjustment heavily depends on which party defaults first.

Here we denote the default time of the institution and counterparty as 휏퐼 and 휏, respectively. Then we have 6 mutually exclusive and exhaustive events:

퐼1 = {휏퐼 < 휏 < 푇}, 퐼2 = {휏퐼 < 푇 ≤ 휏}, 퐼3 = {휏 ≤ 휏퐼 < 푇}, (2.8.1)

퐼4 = {휏 < 푇 ≤ 휏퐼}, 퐼5 = {푇 ≤ 휏퐼 < 휏}, 퐼6 = {푇 ≤ 휏 < 휏퐼}.

푉̃(푡, 푇) is the discounted payoff of an instrument traded with a defautable counterparty at time 푡 and 푉(푡, 푇) is the discounted payoff of an identical instrument traded with a default free counterparty.

By following similar procedures implemented in section 2.2, the general pricing formula for bilateral credit valuation adjustment (BCVA) is + ℚ ̃( ) ℚ ( ) ℚ ( ) (2.8.2) 피푡 (푉 푡, 푇 ) = 피푡 (푉 푡, 푇 ) + 피푡 (퐿퐺퐷퐼1{퐼1∪퐼2}퐷퐹 푡, 휏퐼 (−푀푡푀휏퐼) ) ℚ ( )( )+ − 피푡 (퐿퐺퐷1{퐼3∪퐼4}퐷퐹 푡, 휏 푀푡푀휏 ), where loss given default of the institution 퐿퐺퐷퐼 = 1 − 푅퐼 and loss given default of the counterparty 퐿퐺퐷 = 1 − 푅. Mark to Market (MtM) is defined as ℚ 푀푡푀푡 = 피푡 (푉(푡, 푇)). Here + ( ) ℚ ( ) (2.8.3) 퐷푉퐴 푡, 푇 = 피푡 (퐿퐺퐷퐼1{퐼1∪퐼2}퐷퐹 푡, 휏퐼 (−푀푡푀휏퐼) ), ( ) ℚ ( )( )+ 퐶푉퐴 푡, 푇 = 피푡 (퐿퐺퐷1{퐼3∪퐼4}퐷퐹 푡, 휏 푀푡푀휏 ).

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Detailed proof could be found at Brigo, Morini and Pallavicini (2013). The 6 events defined in (2.8.1) constitute a complete set, then ( ) ( ) ( ) ( ) (2.8.4) 푉 푡, 푇 = 1퐼1∪퐼2푉 푡, 푇 + 1퐼3∪퐼4 푉 푡, 푇 + 1퐼5∪퐼6푉 푡, 푇 The right hand side of (2.8.2) could be rewritten as + ℚ ̃( ) ℚ ( ) ( ) ( ) (2.8.5) 피푡 (푉 푡, 푇 ) = 피푡 (1퐼1∪퐼2푉 푡, 푇 + 1 − 푅퐼 1{퐼1∪퐼2}퐷퐹 푡, 휏퐼 (−푀푡푀휏퐼) ) ℚ ( ) ( ) ( )( )+ + 피푡 (1퐼3∪퐼4푉 푡, 푇 + 푅 − 1 1{퐼3∪퐼4}퐷퐹 푡, 휏 푀푡푀휏 ) ℚ ( ) + 피푡 (1퐼5∪퐼6푉 푡, 푇 ). The first expectation of (2.8.5) could be rewritten as + ( ) ( ) ( ) (2.8.6) 1퐼1∪퐼2푉 푡, 푇 + 1 − 푅퐼 1{퐼1∪퐼2}퐷퐹 푡, 휏퐼 (−푀푡푀휏퐼) + ( ) ( ) = 1퐼1∪퐼2푉 푡, 푇 + 1{퐼1∪퐼2}퐷퐹 푡, 휏퐼 (−푀푡푀휏퐼) + ( ) − 푅퐼1{퐼1∪퐼2}퐷퐹 푡, 휏퐼 (−푀푡푀휏퐼) Since ( ) ( ) ( ) ( ) (2.8.7) 1퐼1∪퐼2푉 푡, 푇 = 1퐼1∪퐼2(푉 푡, 휏퐼 + 퐷퐹 푡, 휏퐼 푉 휏퐼, 푇 ), conditional expectation of (2.8.6) at time 휏퐼 is + + 피ℚ (1 푉(푡, 푇) + 1 퐷퐹(푡, 휏 ) −푀푡푀 − 푅 1 퐷퐹(푡, 휏 ) −푀푡푀 ) (2.8.8) 휏퐼 퐼1∪퐼2 {퐼1∪퐼2} 퐼 ( 휏퐼) 퐼 {퐼1∪퐼2} 퐼 ( 휏퐼)

= 피ℚ 1 (푉(푡, 휏 ) + 퐷퐹(푡, 휏 )푉(휏 , 푇) 휏퐼 ( 퐼1∪퐼2 퐼 퐼 퐼

+ + + 퐷퐹(푡, 휏 ) (−피ℚ (푉(휏 , 푇)) − 푅 퐷퐹(푡, 휏 ) (−피ℚ (푉(휏 , 푇)) ) 퐼 휏퐼 퐼 퐼 퐼 휏퐼 퐼 )

= 1 (푉(푡, 휏 ) + 퐷퐹(푡, 휏 )피ℚ 푉(휏 , 푇) 퐼1∪퐼2 퐼 퐼 휏퐼( 퐼 )

+ + + 퐷퐹(푡, 휏 ) (−피ℚ (푉(휏 , 푇)) − 푅 퐷퐹(푡, 휏 ) −푀푡푀 ) 퐼 휏퐼 퐼 퐼 퐼 ( 휏퐼)

+ = 1 (푉(푡, 휏 ) + 퐷퐹(푡, 휏 ) (피ℚ 푉(휏 , 푇) ) 퐼1∪퐼2 퐼 퐼 휏퐼( 퐼 )

+ ( ) − 푅퐼퐷퐹 푡, 휏퐼 (−푀푡푀휏퐼) )

+ + ( ) ( ) ( ) = 1퐼1∪퐼2 (푉 푡, 휏퐼 + 퐷퐹 푡, 휏퐼 (푀푡푀휏퐼) − 푅퐼퐷퐹 푡, 휏퐼 (−푀푡푀휏퐼) ). If we further condition (2.8.8) on information up to time 푡 and apply law of iterated expectation, we obtain

+ + ℚ ( ) ( ) ( ) (2.8.9) 피푡 (1퐼1∪퐼2 (푉 푡, 휏퐼 + 퐷퐹 푡, 휏퐼 (푀푡푀휏퐼) − 푅퐼퐷퐹 푡, 휏퐼 (−푀푡푀휏퐼) )) which coincides with the second expectation of right hand side of (2.8.2). Similar procedures could be taken for the second term of right hand side of (2.8.5), we obtain

ℚ ( ( ) ( )( )+ ( )( )+) (2.8.10) 피푡 (1퐼3∪퐼4 푉 푡, 휏 + 퐷퐹 푡, 휏 푀푡푀휏 − 푅 ∙ 퐷퐹 푡, 휏 −푀푡푀휏 )

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2.9 CVA/DVA Pricing

The CDS curve of derivative dealer is following:

Term (Years) Spread (bps) 0.5 19 1 30 2 42 3 55 4 68 5 81 7 102 10 122 15 131 20 137 30 143

(Table 2.2: Term Structure of Derivative Dealer’s CDS Curve)

The counterparty is assumed to have the same credit spread term structure of the derivative dealer. Therefore, CVA and DVA could be compared based on the relative dominance of exposure profile as discussed in previous sections.

Here we assume the derivative dealer longs these derivative instruments on standalone basis. The interest rate instruments are EURIBOR swaps and their related swaptions. The floating leg pays 6- month EURIBOR semi-annually on Act/360 basis. The fixed leg pays the fixed rate annually on 30/360 basis. The underlying notional is assumed to be €100 million. The European swaptions can be exercised at the expiry of the swaptions. The difference between cash settlements and physically settlements on exposure profiles are going to be discussed in detail in following sections. The Bermudan swaptions can be exercised on annual basis, that is, they could be exercised every year after the first expiry date of the swaptions. The FX instruments are priced with EURUSD currency pair. For cross currency swaps, the USD leg pays 3-month LIBOR quarterly on Act/360 basis and the EUR leg pays 3-month EURIBOR quarterly on Act/360 basis.

The CVA and DVA reported here are unilateral CVA (2.6.14) and unilateral DVA (2.6.15) as discussed in previous sections. The dependency between the exposure dynamics and default probabilities are not taken into pricing in these examples.

푇 + 퐶푃 푇 ℚ 푀푡푀푡 (1 − 푅푡 ) ℚ 퐶푃 + ℚ 퐶푉퐴 = 피0 (∫ 푑푡ℙ퐷,퐶푃(푡)) = (1 − 푅 ) ∫ 퐸퐸0 (푡) 푑푡ℙ퐷,퐶푃(0, 푡), 0 푁푡 0

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푇 − 퐼 푇 ℚ 푀푡푀푡 (1 − 푅푡) ℚ 퐼 − ℚ 퐷푉퐴 = −피0 (∫ 푑푡ℙ퐷,퐼(푡)) = −(1 − 푅 ) ∫ 퐸퐸0 (푡) 푑푡ℙ퐷,퐼(0, 푡), 0 푁푡 0

+ − + ℚ 푀푡푀푡 − ℚ 푀푡푀푡 퐶푃 ℚ 퐶푃 퐼 ℚ 퐼 where 퐸퐸0 (푡) = 피0 ( ), 퐸퐸0 (푡) = 피0 ( ), 푅 = 피0 (푅푡 ), 푅 = 피0 (푅푡). 푁푡 푁푡

The collateralised derivatives transactions are assumed to be traded under two way credit support annex (CSA). The general terms for two way CSA in this pricing section are

Derivative Dealer Counterparty

Independent Amount 0 0 Initial Margin 0 0 Threshold 0 0 MTA 500,000 500,000 Rounding Amount 10,000 10,000 Rebalancing Interval 1 Day 1 Day CSA Currency EUR EUR

We could start with uncollateralised spot starting payer and receiver interest rate swap. The payer and receiver are priced at par. The par swap rate is a weighted average of forward rates. And the forward rates are market proxies for future LIBOR rates on the floating leg. The technical details could be found in Brigo and Mercurio (2006). The expected Mark to Market (MtM) priced on the forward curve makes fixed paying side in the money during the most of tenor of the swap. This explains why the EPE profile is greater than ENE profile in payer swap exposure (and why the ENE profile is greater than ENE profile in receiver swap exposure).

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(Figure 2.1: Uncollateralised 10 Years Spot Starting IRS Fixed Payer Exposure Profile)

(Figure 2.2: Uncollateralised 10 Years Spot Starting IRS Fixed Receiver Exposure Profile)

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The CVA/DVA of these two trades are shown below.

xVA Price (€) CVA 325,662 DVA -116,818 (Table 2.3: CVA and DVA for Uncollateralised 10 Years Spot Starting IRS Fixed Payer)

xVA Price (€) CVA 116,818 DVA -325,662 (Table 2.4: CVA and DVA for Uncollateralised 10 Years Spot Starting IRS Fixed Receiver)

As these two trades have identical contracts (i.e. par swap) but different directions (i.e. payer and receiver), their CVA and DVA have same size but opposite directions.

Case Study: Interest Rate Derivatives (IRDs) Priced on 5 x 15 Swap Curve

Here we are going to price CVA/DVA on 6 different interest rate products on uncollateralised and collateralised basis. All of these interest rate products are priced on 5x15 swap (i.e. the forward starting interest rate swap that starts in 5 years and matures in 15 years). Let's start with uncollateralised payer and receiver swaps. The exposure profiles of forward starting payer and receiver swaps look similar to those of spot starting counterparts. The fixed paying side dominates the exposure profile (for example, in case of payer swap, the EPE profile is greater than ENE profile).

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(Figure 2.3: Uncollateralised 5x15 Forward Starting IRS Fixed Payer Exposure Profile)

(Figure 2.4: Uncollateralised 5x15 Forward Starting IRS Fixed Receiver Exposure Profile)

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The main difference is the forward end/short end of the curve. The 0x5 tenor of the forward starting swap is effectively a . The forward starting interest rate swap could be understood as a physically settled forward contract on interest rate swap. The front end of the exposure profile is similar to the exposure profile of forward contracts on assets (for example, FX forward which is discussed in next case study). The EPE and ENE are symmetric till year 5 (and the expected Mark to Market profile is almost flat at 0 from year 0 to year 5). The interest rate swap will automatically 'knock in' at the end of year 5. Similar exposure profiles patterns as we observed in spot starting interest rate swaps emerge from year 5 till the expiry of the contract.

The CVA/DVA of these two trades are shown below. The symmetry ratio is the absolute value of the ratio between CVA and DVA.

xVA Price (€) Symmetry Ratio CVA 572,825 1.2 DVA -492,932 1

(Table 2.5: CVA and DVA for Uncollateralised 5x15 Forward Starting IRS Fixed Payer)

xVA Price (€) Symmetry Ratio CVA 492,932 1 DVA -572,825 1.2

(Table 2.6: CVA and DVA for Uncollateralised 5x15 Forward Starting IRS Fixed Receiver)

As these two trades have identical contracts (i.e. forward starting swap priced at par) but different directions (i.e. payer and receiver), their CVA and DVA have same size but opposite directions.

A standard two way CSA with Minimum Transfer Amount 푀푇퐴 = €500퐾 removes a large component of exposure profile. The collateralised exposure profiles are roughly as symmetric as their uncollateralised counterparts. The ratio between CVA and DVA also indicates the existence of symmetry consistency between uncollateralised interest rate swaps and collateralised interest rate swaps.

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(Figure 2.5: Collateralised 5x15 Forward Starting IRS Fixed Payer Exposure Profile)

(Figure 2.6: Collateralised 5x15 Forward Starting IRS Fixed Receiver Exposure Profile)

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xVA Price (€) Symmetry Ratio CVA 54,608 1.2 DVA -43,759 1

(Table 2.7: CVA and DVA for Collateralised 5x15 Forward Starting IRS Fixed Payer)

xVA Price (€) Symmetry Ratio CVA 43,759 1 DVA -54,608 1.2

(Table 2.8: CVA and DVA for Collateralised 5x15 Forward Starting IRS Fixed Receiver)

The effectiveness of exposure reduction by collateral will be addressed at the end of this case study.

Now let's move to discussion of swaptions. Swaption is an option that gives the option buyer/holder a right to enter into a swap at pre-specified in the future. A European swaption could only be exercised at the maturity of the swaption and a Bermudan swaption could be exercised on a range of pre-specified dates before the maturity.

Here a 5 into 10 European swaption is an option that could be exercised at the end of 5 years to enter into a 10-year swap. So effectively this swaption is priced on 5x15 swap curve/forward curve. The mechanics of European swaptions could be found at Brigo and Mercurio (2006). A cash settled swaption pays MtM of underlying swap if MtM is positive and 0 otherwise at maturity. Therefore, the option buyer/holder faces counterparty credit risk as the option seller might not be able to pay off the MtM (if positive) at maturity. The option seller, however, does not have to worry about the counterparty risk of the option buyer as the solvency of the option buyer has no impact of their future payoff in this transaction. Please find exposure profile of uncollateralised cash settled payer and receiver European swaptions below.

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(Figure 2.7: Uncollateralised 5 into 10 Cash Settled European Payer Swaption Exposure Profile)

(Figure 2.8: Uncollateralised 5 into 10 Cash Settled European Receiver Swaption Exposure Profile)

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The CVA/DVA of these two trades are shown below.

xVA Price (€) CVA 311,577 DVA 0

(Table 2.9: CVA and DVA for Uncollateralised 5 into 10 Cash Settled European Payer Swaption)

xVA Price (€) CVA 353,611 DVA 0

(Table 2.10: CVA and DVA for Uncollateralised 5 into 10 Cash Settled European Receiver Swaption)

The physically settled swaption gives the swaption buyer/holder an option to decide to enter into a swap position at maturity of swaption. In our case, the tenor of physically settled swaption’s exposure profile is 15 years which is longer than 5 years of cash settled swaption’s exposure profile. This is logical as the swaption buyer/holder has a positive probability to enter into a 10-year swap at the end of year 5. The ENE is zero till the end of year 5 and falls below zero after year 5 as the potential exercise of the swaption will make the swaption seller face counterparty credit risk. That is, after 5 years, the solvency of physically settled swaption holder will have impact on future payoff of the swaption seller. This is the major difference between cash settled and physically settled swaptions from the perspective of counterparty credit risk.

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(Figure 2.9: Uncollateralised 5 into 10 Physically Settled European Payer Swaption Exposure Profile)

(Figure 2.10: Uncollateralised 5 into 10 Physically Settled European Receiver Swaption Exposure Profile)

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The CVA/DVA of these two trades are shown below.

xVA Price (€) Symmetry Ratio CVA 532,306 18.7 DVA -28,462 1

(Table 2.11: CVA and DVA for Uncollateralised 5 into 10 Physically Settled European Payer Swaption)

xVA Price (€) Symmetry Ratio CVA 504,397 8.3 DVA -60,582 1

(Table 2.12: CVA and DVA for Uncollateralised 5 into 10 Physically Settled European Receiver Swaption)

Bermudan swaptions, compared to European swaptions, provide more opportunities to the swaption buyer/holder. Therefore, they deserve higher valuation from the perspective of swaption holders. Hence, the expected MtM profiles of Bermudan swaptions should be higher than their European counterparts. These could be proved by following graphs. The increase in valuations of Bermudan swaptions in MtM lifts up EPEs and ENEs (compared to European swaptions).

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(Figure 2.11: Uncollateralised 5 into 10 Physically Settled Bermudan Payer Swaption Exposure Profile)

(Figure 2.12: Uncollateralised 5 into 10 Physically Settled Bermudan Receiver Swaption Exposure Profile)

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The CVA/DVA of these two trades are shown below. The absolute value of CVA is greater than that of CVA of European swaption and the absolute value of DVA is less than that of DVA of European swaption as expected.

xVA Price (€) Symmetry Ratio CVA 743,969 75.7 DVA -9,824 1

(Table 2.13: CVA and DVA for Uncollateralised 5 into 10 Physically Settled Bermudan Payer Swaption)

xVA Price (€) Symmetry Ratio CVA 564,974 28.2 DVA -20,069 1

(Table 2.14: CVA and DVA for Uncollateralised 5 into 10 Physically Settled Bermudan Receiver Swaption)

Two way CSA with a tight MTA (here 푀푇퐴 = €500퐾 ) is very effective in reducing the size of exposure profile and bringing symmetry into exposure.

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(Figure 2.13: Collateralised 5 into 10 Physically Settled European Payer Swaption Exposure Profile)

(Figure 2.14: Collateralised 5 into 10 Physically Settled European Receiver Swaption Exposure Profile)

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(Figure 2.15: Collateralised 5 into 10 Physically Settled Bermudan Payer Swaption Exposure Profile)

(Figure 2.16: Collateralised 5 into 10 Physically Settled Bermudan Receiver Swaption Exposure Profile)

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The ratio between CVA and DVA also indicates the symmetry brought by collateralisation.

xVA Price (€) Symmetry Ratio CVA 28,454 1.0 DVA -27,770 1

(Table 2.15: CVA and DVA for Collateralised 5 into 10 Physically Settled European Payer Swaption)

xVA Price (€) Symmetry Ratio CVA 31,872 1 DVA -32,572 1.0

(Table 2.16: CVA and DVA for Collateralised 5 into 10 Physically Settled European Receiver Swaption)

xVA Price (€) Symmetry Ratio CVA 28,297 1 DVA -29,482 1.0

(Table 2.17: CVA and DVA for Collateralised 5 into 10 Physically Settled Bermudan Payer Swaption)

xVA Price (€) Symmetry Ratio CVA 32,011 1.0 DVA -30,538 1

(Table 2.18: CVA and DVA for Collateralised 5 into 10 Physically Settled Bermudan Payer Swaption)

Therefore, collateralisation, as discussed before and will be discussed in more depth and detail in Funding Risk chapter, will move the market risk and counterparty credit risk feature from the transactions. For example, CVA and DVA of uncollateralised Bermudan swaptions are significantly different from CVA and DVA of uncollateralised European swaptions. And this difference is driven by the market risk and its related factors (in our case, early exercise opportunities offered by Bermudan style contracts). However, CVA and DVA of collateralised Bermudan swaptions are very close to CVA and DVA of collateralised European swaptions. Collateralisation does bring homogeneity into counterparty credit risk pricing.

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Collateralised derivative transactions work very similar to repo transactions. The cost of repos is directly linked to the collateral features like haircut etc. instead of the market risk and counterparty risk factors. However, a derivative with standard two way CSAs could be used as collateral or be repoed. We will further explore this issue in Funding Risk chapter and case study in Central clearing chapter.

Case Study: FX Derivatives (FXDs) Priced on 0x15 EURUSD Forward Curve

Here we are going to price CVA/DVA on 6 different FX derivatives contracts. We could start with uncollateralised EURUSD FX forwards. The EUR buyer (i.e. long EUR at maturity) and EUR seller (i.e. short EUR at maturity) are priced at the money at inception. The technical details could be found in Brigo, Morini and Pallavicini (2013) and Clark (2011). The expected Mark to Market (MtM) priced on the forward curve makes a symmetrical exposure profile. This explains why the EPE profile is very close to ENE profile in both transactions.

(Figure 2.17: Uncollateralised 15 Years FX (EURUSD) Forward Long EUR at Maturity Exposure Profile)

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(Figure 2.18: Uncollateralised 15 Years FX (EURUSD) Forward Short EUR at Maturity Exposure Profile)

The CVA/DVA of these two trades are shown below.

xVA Price (€) Symmetry Ratio CVA 1,386,188 1.0 DVA -1,385,380 1

(Table 2.19: CVA and DVA for Uncollateralised 15 Years FX (EURUSD) Forward Long EUR at Maturity)

xVA Price (€) Symmetry Ratio CVA 1,385,380 1 DVA -1,386,188 1.0

(Table 2.20: CVA and DVA for Uncollateralised 15 Years FX (EURUSD) Forward Short EUR at Maturity)

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As these two trades have identical contracts (i.e. EURUSD FX forwards) but different directions (i.e. EUR buyer and seller in the long end), their CVA and DVA have same size but opposite directions.

FX Swaps similar to FX forward except there is a spot transaction in the front end. In our case, if we are going to buy/long EUR at EURUSD forward price at maturity, we sell/short EUR at spot price at inception. Since the front end transaction (i.e. the EUR transaction at inception) is priced at the money, the front end transaction has no impact on expected MtM and exposure profiles. That is, the expected MtM will not be lifted upwards or downwards by the PnL of the front end transaction. Therefore, the FX swaps' exposure profiles are mainly driven by FX forward transaction in the long end. This explains why exposure profiles of FX swaps are very similar to FX forwards discussed earlier.

(Figure 2.19: Uncollateralised 15 Years FX (EURUSD) Swap Long EUR at Maturity Exposure Profile)

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(Figure 2.20: Uncollateralised 15 Years FX (EURUSD) Swap Short EUR at Maturity Exposure Profile)

The CVA/DVA of these two trades are shown below.

xVA Price (€) Symmetry Ratio CVA 1,385,999 1.0 DVA -1,385,136 1

(Table 2.21: CVA and DVA for Uncollateralised 15 Years FX (EURUSD) Swap Long EUR at Maturity)

xVA Price (€) Symmetry Ratio CVA 1,385,136 1 DVA -1,385,999 1.0

(Table 2.22: CVA and DVA for Uncollateralised 15 Years FX (EURUSD) Swap Short EUR at Maturity)

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As these two trades have identical contracts (i.e. EURUSD FX swaps) but different directions (i.e. EUR buyer and seller in the long end), their CVA and DVA have same size but opposite directions.

Collateralisations of FX forwards and FX swaps transaction achieved a significant reduction in size of exposures. This could be observed from following graphs of exposure profiles. The exposure profiles of uncollateralised FX forwards and FX swaps are symmetrical and hence collateralisation doesn't bring more symmetry into these transactions.

(Figure 2.21: Collateralised 15 Years FX (EURUSD) Forward Long EUR at Maturity Exposure Profile)

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(Figure 2.22: Collateralised 15 Years FX (EURUSD) Forward Short EUR at Maturity Exposure Profile)

(Figure 2.23: Collateralised 15 Years FX (EURUSD) Swap Long EUR at Maturity Exposure Profile)

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(Figure 2.24: Collateralised 15 Years FX (EURUSD) Swap Short EUR at Maturity Exposure Profile)

The CVA/DVA of these four trades are shown below.

xVA Price (€) Symmetry Ratio CVA 115,854 1.1 DVA -110,019 1

(Table 2.23: CVA and DVA for Collateralised 15 Years FX (EURUSD) Forward Long EUR at Maturity)

xVA Price (€) Symmetry Ratio CVA 110,019 1 DVA -115,854 1.1

(Table 2.24: CVA and DVA for Collateralised 15 Years FX (EURUSD) Forward Short EUR at Maturity)

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xVA Price (€) Symmetry Ratio CVA 122,025 1.1 DVA -109,370 1

(Table 2.25: CVA and DVA for Collateralised 15 Years FX (EURUSD) Swap Long EUR at Maturity)

xVA Price (€) Symmetry Ratio CVA 109,370 1 DVA -122,025 1.1

(Table 2.26: CVA and DVA for Collateralised 15 Years FX (EURUSD) Swap Short EUR at Maturity)

Cross currency swaps don't have symmetric exposure profiles. Technical details and mechanisms of cross currency swaps could be found in Brigo, Morini and Pallavicini (2013) and Clark (2011). The front end and far/long end notional exchanges are fixed at spot FX rate. Both sides will pay interest rate to each other during the life of the transaction. Cross currency swap could be understood as an exchange of two different interest rate bearing bonds denominated in different currencies. Since long end notional exchange should be priced by FX forward price instead of FX spot price, the exposure profile is not symmetric. As will be discussed in section 3.3 and indicated by equation (3.3.1), (3.3.2) and (3.3.3), by interest rate parity, the currency bearing lower interest rate will appreciate on the FX forward curve. In our case, EUR rates are lower than USD rates. EUR, therefore, appreciates against USD on the forward curve. If we long (short) EUR at maturity in EURUSD cross currency swap, we are going to better (worse) off as we could buy (sell) EUR at spot price (which is significantly lower than the fair price of EURUSD (i.e. EURUSD forward price) at maturity. The expected Mark to Market (MtM), therefore, is going to increase for EUR buyer (at maturity) and decrease for EUR seller (at maturity). And the exposure profiles are driven upwards and downwards accordingly. The exposure profiles of these two trades could be found below.

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(Figure 2.25: Uncollateralised 15 Years EURUSD CCS Long EUR at Maturity Exposure Profile)

(Figure 2.26: Uncollateralised 15 Years EURUSD CCS Short EUR at Maturity Exposure Profile)

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The CVA/DVA of these two trades are shown below.

xVA Price (€) Symmetry Ratio CVA 2,513,569 3.4 DVA -729,266 1

(Table 2.27: CVA and DVA for Uncollateralised 15 Years EURUSD CCS Long EUR at Maturity)

xVA Price (€) Symmetry Ratio CVA 729,266 1 DVA -2,513,569 3.4

(Table 2.28: CVA and DVA for Uncollateralised 15 Years EURUSD CCS Short EUR at Maturity)

Collateralisations of cross currency swaps transaction achieved a significant reduction in size of exposures and symmetrised the exposure profiles. This could be observed from following graphs of exposure profiles.

(Figure 2.27: Collateralised 15 Years EURUSD CCS Long EUR at Maturity Exposure Profile)

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(Figure 2.28: Collateralised 15 Years EURUSD CCS Short EUR at Maturity Exposure Profile)

The CVA/DVA of these two trades are shown below.

xVA Price (€) Symmetry Ratio CVA 119,891 1.1 DVA -109,730 1

(Table 2.29: CVA and DVA for Collateralised 15 Years EURUSD CCS Long EUR at Maturity)

xVA Price (€) Symmetry Ratio CVA 109,730 1 DVA -119,891 1.1

(Table 2.30: CVA and DVA for Collateralised 15 Years EURUSD CCS Short EUR at Maturity)

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Variation margins or two way CSAs are used here to reduce counterparty credit risk exposure. As we could observe from the following tables, the general risk reductions for CVAs are at least 85%. The price of counterparty credit risk is becoming less sensitive to the counterparty credit risk quality but more sensitive to the CSA terms and market risk component of the collateral as we discussed in repo transactions. The inability to repo collateralised derivative positions creates another valuation adjustment called funding valuation adjustment (FVA). FVA will be discussed in detail in next chapter.

Counterparty Credit Risk Analysis of IRDs on 5 x 15 Forward Curve CVA Uncollateralised Collateralised Risk Ratio Fixed Payer 572,825 54,608 9.5% Fixed Receiver 492,932 43,759 8.9% European Payer Swaption 532,306 28,454 5.3% European Receiver Swaption 504,397 31,872 6.3% Bermudan Payer Swaption 743,969 28,297 3.8% Bermudan Receiver Swaption 564,974 32,011 5.7%

DVA Uncollateralised Collateralised Risk Ratio Fixed Payer - 492,932 - 43,759 8.9% Fixed Receiver - 572,825 - 54,608 9.5% European Payer Swaption - 28,462 - 27,770 97.6% European Receiver Swaption - 60,582 - 32,572 53.8% Bermudan Payer Swaption - 9,824 - 29,483 300.1% Bermudan Receiver Swaption - 20,069 - 30,538 152.2%

(Table 2.31: Counterparty Credit Risk Reduction by Variation Margin for IRDs )

There is a significant divergence for DVA risk ratios for European swaptions and Bermudan swaptions from the other risk ratios. This is due to the symmetry brought by collateralisation into these transactions. The swaptions holder of these 4 transactions are deep in the money. Therefore, the swaption holder is going to receive a significant amount collateral from swaption seller. If the market moves against the swaption holder, i.e. the swaptions become less in the money, the ability of swaption holder returning collateral to swaption seller is in doubt. The potential loss of collateralised swaption from swaption seller's perspective therefore is very close to or even much greater than its uncollateralised counterparts.

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Counterparty Credit Risk Analysis of FXDs on 0 x 15 EURUSD Forward Curve CVA Uncollateralised Collateralised Risk Ratio Long EUR Forward 1,386,188 115,854 8.4% Short EUR Forward 1,385,380 110,019 7.9% Long EUR Swap 1,385,999 122,025 8.8% Short EUR Swap 1,385,136 109,370 7.9% Long EUR CCS 2,513,569 119,891 4.8% Short EUR CCS 729,266 109,730 15.0%

DVA Uncollateralised Collateralised Risk Ratio Long EUR Forward - 1,385,380 - 110,019 7.9% Short EUR Forward - 1,386,188 - 115,854 8.4% Long EUR Swap - 1,385,136 - 109,370 7.9% Short EUR Swap - 1,385,999 - 122,025 8.8% Long EUR CCS - 729,266 - 109,730 15.0% Short EUR CCS - 2,513,569 - 119,891 4.8%

(Table 2.32: Counterparty Credit Risk Reduction by Variation Margin for FXDs )

Findings from IRDs and FXDs Case Studies Collateralisation is effective in reducing potential losses due to counterparty credit risk by reducing size of exposures and symmetrising exposure profiles.

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3. Funding Risk and FVA

3.1 Background: Widening LIBOR-OIS Spread

Prior to the Global Financial Crisis (GFC), the cash flows of derivatives and related transactions are discounted by LIBOR curves. CVA/credit charges are taken into account if the cash flows are not ‘credit’ risk free due to the asymmetric treatments of counterparty default stated in Bankruptcy Code. Two assumptions are generally held in the marketplace for the usage of LIBOR curve as the discounting instrument:

1. LIBOR rate is a good proxy for the ‘risk free’ interest rate. The TED spread (spread between 3 month LIBOR rate and 3-month Treasury bills ED (Eurodollar futures)) and LIBOR-OIS spread fluctuate within a certain range of basis points and hence LIBOR is perceived as a reliable indicator of theoretical risk free curve.

2. The derivative dealers are generally funded at LIBOR rates to run their derivative business and there are no further funding costs have to be taken into account to build up their own ‘self- financing’ strategy in pricing and hedging derivatives. The homogeneity of credit quality of large financial institutions/derivative dealers is widely assumed in the marketplace.

The shortcomings of above assumptions are highlighted by the Global Financial Crisis (GFC). The default of Lehman Brothers and other financial institutions reveals the fact lending money to systematically important financial institutions (SIFIs) is credit risky and the credit quality between these institutions is heterogeneous. A completely modernised framework for pricing derivatives with incorporation of credit risk, collateral and funding risk is needed to quantify the real inherent problems in the market.

The difference between collateralised and uncollateralised derivatives transactions has been recognised by the derivative dealers before GFC. The discounting curve for future cash flows generated by collateralised transactions is OIS (overnight index swap) curve. And LIBOR is used to discount the cash flows of uncollateralised transactions. An uncollateralised interest rate swap is priced by projecting the future cash flows of the floating leg by LIBOR curve and discounting both legs by LIBOR curve. A collateralised interest rate swap is priced by projecting floating leg future cash flows by LIBOR curve but discounting both legs by OIS curve. The existence of dual curves or multiple curves (if dual curves of different currencies are taken into account) is not an important issue prior to the GFC as the LIBOR-OIS spread remains consistently around a few basis points. And therefore the swap rates and related pricing and hedging measures (PV01, Delta, Gamma, etc.) of an

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Counterparty Credit Risk, Funding Risk and Central Clearing uncollateralised interest rate swap is not significantly different from its collateralised counterpart. If a derivative dealer intends to hedge an uncollateralised trade by a collateralised trade with a market counterparty, the extra funding cost due to posting collateral the market marker has to bear is negligible and could be absorbed by Bid/Ask spread.

In the aftermath of the Lehman default, the US 3 month LIBOR-OIS spread (i.e. the 3 month LIBOR rate minus 3-month Fed Funds rate) spiked to more than 350 basis points and broke down the old single curve pricing paradigm. It is observed now in the market the swap rate for collateralised interest rate swap is significantly greater than the swap rate of an uncollateralised interest rate swap with identical features (without further adjustments due to counterparty credit risk or funding risk). The swap spread between collateralised and uncollateralised interest rate swap could be attributed to LIBOR-OIS spread. Generally OIS curve is significantly lower than corresponding LIBOR curve in current market paradigm. For example, a derivative dealer is the fixed payer and the LIBOR forward curve is upward sloping here. From the perspective of the derivative dealer, their Mark to Market is initially out of the money before the midterm (denoted as liability side) and becomes significantly in the money after the midterm (denoted as asset side). Then the downward shift of discounting curve by switching from LIBOR discounting to OIS discounting makes the gain of the asset side on the long end greater than the loss of the liability side on the short end. If the swap rate sticks with the uncollateralised scenario, the derivative dealer, as the fixed payer, has a net upfront profit. Therefore, the swap rate of collateralised (OIS discounted) interest rate swap is greater than the swap rate of an uncollateralised (LIBOR discounted) interest rate swap. This example could be proved by a numerical example using the formula in Brigo and Mercurio (2006) that the swap rate is a weighted average of forward LIBOR rates.

Therefore, the extra funding cost in hedging the uncollateralised or partially collateralised derivatives or construction of the theoretical self-financing strategy has to be addressed and quantified in the final price. The underlying currency and type of collateral for a given range of derivatives products specified in a given CSA further complicates the pricing problem. The ongoing revolution of OTC market structure has probably permanently shifted the pricing framework of derivatives and CVA/DVA/FVA components have to be priced into execution prices.

Nowadays the ISDA Credit Support Annex (CSA) requires both sides/parties in derivatives transactions to post assets with high credit equality and high liquidity (like cash and a certain range of sovereign bonds) to mitigate counterparty credit risk. This is a typical two way CSA agreement. Some derivative dealers sign one way CSA of sovereign governments such that the sovereign government does not have to post collateral to the derivative dealer. In return, the derivative dealers could execute the trade at a better price. One way CSA is not within the scope of discussion of this thesis. This

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Counterparty Credit Risk, Funding Risk and Central Clearing thesis mainly discuss the impact and implications of two way CSA. Generally the party that is out of money has to post collateral to its counterparty on daily basis and its counterparty, i.e. the collateral receiver, has to pay overnight index swap (OIS) rate as a compensation to the collateral poster. If the collateral poster has to borrow the high quality assets at their own funding curve, the funding cost for collateral poster is their specific funding rate. Since the collateral receiver pays OIS rate in turn as a compensation, the net funding cost for the collateral poster is their own funding spread, i.e. the difference between the funding rate and OIS rate.

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3.2 FVA Pricing without Counterparty Credit Risk

The rise of Funding Valuation Adjustment (FVA) is due to the change of OTC derivatives market structure after the great financial crisis (GFC). In classic theory, the self-financing strategy for an OTC derivative perfectly replicates the derivative payoff in the survival state. And almost no collateral is required for majority of OTC derivatives transactions in practice. The market participants would have to post collateral in OTC transactions and the negative carry (as funding cost of acquiring collateral with high liquidity is greater than OIS rate which is the return of posting collateral in most cases)on posting collateral becomes an explicit violation of the classic self-financing strategy for replicating derivatives. In other words, the default free price of trading any given OTC derivative should include an extra component on collateral account. In theory, this phenomenon indicates that the no arbitrage price of any derivative becomes entity dependent in the default free world and the entity/counterparty with the significant lower funding rate would have a great advantage in trading business of derivatives.

Bielecki and Rutkowski (2015) used full replication approach on derivation of funding equations. Here we start with a discussion of (collateral) funding in the default free world/survival states. The additional cash flows of collateral account could be treated as dividends in the classic framework as both of them represent the intermediate cash flows within the life of the trade. As discussed in previous sections (equation (2.4.5)), the intermediate cash flows for path dependent option pricing are re-invested into the assets to form up a self-financing strategy,

퐶푡퐷푡 ℕ 퐶푇퐷푇 = 피푡 ( ), 푁푡 푁푇 푡 ∫ 푑푢푑푢 where 0 < 푠 < 푡 < 푇 and 퐷푡 denotes the cumulative dividends up to time 푡, i.e. 퐷푡 = 푒 0 . If we restate the above relationship by the classic capital gain process in Duffie (2003), that is, the dividends are expressed in cash instead of percentage of underlying asset price,

퐶푡 + 퐷푡 ℕ 퐶푇 + 퐷푇 (3.2.1) = 피푡 ( ), 푁푡 푁푇 푡 where 퐷푡 is the separate dividend account and 퐷푡 = ∫0 훿푠 푑푠. If 푁푡 is the money market account and 푡 ∫ 푟푠푑푠 퐶푡+퐷푡 푁푡 = 푒 0 , and 푋푡 = , 푁푡 ℚ (3.2.2) 푋푡 = 피푡 (푋푇),

푋푡 is a martingale under risk neutral measure ℚ. Therefore, by Ito’s lemma,

1 1 1 퐶푡 1 1 (3.2.3) 푑푋푡 = 푑퐶푡 + 푑퐷푡 + (퐶푡 + 퐷푡)푑 ( ) = 푑 ( ) + 푑퐷푡 + 퐷푡푑 ( ). 푁푡 푁푡 푁푡 푁푡 푁푡 푁푡 Since 1 1 1 (3.2.4) 푑퐷푡 = 훿푡퐷푡푑푡, 푑 ( ) = − 2 푑(푁푡) = −푟푡 푑푡, 푁푡 푁푡 푁푡

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Then

퐶푡 퐷푡 (3.2.5) 푑푋푡 = 푑 ( ) + (훿푡 − 푟푡) 푑푡 푁푡 푁푡

푡 푡 푠 − ∫ 푟푠푑푠 − ∫ 푟푠푑푠 ⇒ 푋푡 = 푒 0 퐶푡 + ∫ 푒 0 (훿푠 − 푟푠)퐷푠푑푠. 0

The above equation gives us a generic view on how to price a given consumption asset 퐶푇 with intermediate cash flows 훿푡 generated by dividend account 퐷푡 in absence of default risk (or in survival state) in classic asset pricing theory. The above equation sets up the basis for funding risk analysis in this chapter as the cash flows generated by collateral account are intermediate cash flows in nature. Piterbarg (2010) uses a simple example to derive the Black Scholes PDE in default free world/survival state. The extra cash called by funding account 퐹푡 to be posted into collateral account 퐹 풞푡 is going to be funded by unsecured borrowed by the derivative dealer at rate 푟푡 . The 푂퐼푆 collateral posted in collateral account 풞푡 is reimbursed with OIS rate 푟푡 . And the underlying asset 푅 could be financed in its repo market at rate 푟푡 . The underlying asset process here is

푑푆푡 푅 ℚ (3.2.6) = (푟푡 − 푑푡)푑푡 + 휎푑푊푡 , 푆푡 where 푆푡 is the underlying stock price and 푑푡 is the annual percentage dividend rate. The net position of the option 푉푡 after Delta hedging by the underlying stock is 2 (3.2.7) 휕 1 2 휕 푑푉푡 − Δ푡푑푆푡 = ( + 휎 2) 푉푡푑푡. 휕푡 2 휕푆푡

This net position is going to be financed by the net cash flows generated by the funding account 퐹푡 and collateral account 풞푡 at a rate 푔푡, 푂퐼푆 퐹 푅 푔푡푑푡 = (푟푡 풞푡 + 푟푡 (푉푡 − 풞푡) − 푟푡 Δ푡푆푡 + 푑푡Δ푡푆푡)푑푡. (3.2.8) Therefore, 2 (3.2.9) 휕 1 2 휕 푂퐼푆 퐹 푅 ( + 휎 2) 푉푡 = 푟푡 풞푡 + 푟푡 (푉푡 − 풞푡) − 푟푡 Δ푡푆푡 + 푑푡Δ푡푆푡, 휕푡 2 휕푆푡

휕푉푡 where Δ푡 = . After rearrangements, 휕푆푡 2 (3.2.10) 휕푉푡 푅 휕푉푡 1 2 휕 푉푡 퐹 푂퐼푆 퐹 + (푟푡 − 푑푡)푆푡 + 휎 2 + (푟푡 − 푟푡 )풞푡 − 푟푡 푉푡 = 0, 휕푡 휕푆푡 2 휕푆푡 Or it could be restated by Brigo et al. (2012) structure, 퐹 퐹 푂퐼푆 (휕푡 − 푟푡 + ℒ푡)푉푡 + (푟푡 − 푟푡 )풞푡 = 0, (3.2.11) where ℒ푡 denotes the infinitesimal generator with respect to market risk factors such that ℒ푡푉푡 = 2 푅 휕푉푡 1 2 휕 푉푡 (푟푡 − 푑푡)푆푡 + 휎 2 . The non-linear equation in Brigo et al. (2012) describes the same 휕푆푡 2 휕푆푡 dynamics in a more general setting with counterparty credit risk and asymmetric funding. It will be discussed at the end of this chapter. The solution obtained by applying Feynman-Kac formula is

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푇 푇 퐹 푢 퐹 (3.2.12) ℚ − ∫ 푟푢 푑푢 ℚ − ∫ 푟푣 푑푣 퐹 푂퐼푆 푉푡 = 피푡 (푒 푡 푉푇) + 피푡 (∫ 푒 푡 (푟푢 − 푟푢 )풞푢푑푢), 푡 and therefore, the fair price of the collateralised derivative transaction is equal to its payoff discounted by funding rate and the funding discounted positive funding spread generated by collateral account.

Please note that it is correct informally to apply Feynman Kac formula in presence of nonlinearities as we did above. However, a rigorous analysis with viscosity, classical and other factors is needed to solve PDE/BSDE. Brigo, Francischello and Pallavicini (2016) provides a detailed analysis of this issue.

Here if we use the classic capital gain process (Duffie (2003)), replace the consumption asset price 퐶푡 by the derivative price 푉푡 (i.e. 푉푡 = 퐶푡) and replace the dividend account 퐷푡 by the collateral account

풞푡 (i.e. −풞푡 = 퐷푡), we have the following martingale under risk neutral measure ℚ, 푡 푡 퐹 푢 퐹 (3.2.13) − ∫ 푟푢 푑푢 − ∫ 푟푣 푑푣 푂퐼푆 퐹 푋푡 = 푒 0 푉푡 + ∫ 푒 0 (푟푢 − 푟푢 )(−풞푢)푑푢 0 푡 푡 퐹 푢 퐹 − ∫ 푟푢 푑푢 − ∫ 푟푣 푑푣 퐹 푂퐼푆 = 푒 0 푉푡 + ∫ 푒 0 (푟푢 − 푟푢 )풞푢푑푢. 0 Therefore,

푡 퐹 푡 퐹 푡 퐹 퐹 − ∫ 푟푢 푑푢 − ∫ 푟푢 푑푢 − ∫ 푟푣 푑푣 퐹 푂퐼푆 (3.2.14) 푑푋푡 = −푟푡 푒 0 푉푡푑푡 + 푒 0 푑푉푡 + 푒 0 (푟푡 − 푟푡 )풞푡푑푡.

푡 퐹 ∫ 푟푠 푑푠 Since 푋푡 is a martingale under risk neutral measure ℚ, if we define 푑푀푡 = 푒 0 푑푋푡, 푀푡 is also a martingale under risk neutral measure ℚ, 퐹 퐹 푂퐼푆 푑푀푡 = −푟푡 푉푡푑푡 + 푑푉푡 + (푟푡 − 푟푡 )풞푡푑푡. (3.2.15) Therefore, 퐹 퐹 푂퐼푆 푑푉푡 = 푑푀푡 + 푟푡 푉푡푑푡 − (푟푡 − 푟푡 )풞푡푑푡 (3.2.16) 푂퐼푆 퐹 푂퐼푆 퐹 푂퐼푆 = 푑푀푡 + 푟푡 푉푡푑푡 + (푟푡 − 푟푡 )푉푡푑푡 − (푟푡 − 푟푡 )풞푡푑푡, that is, 푂퐼푆 퐹 푂퐼푆 푑푉푡 = 푑푀푡 + 푟푡 푉푡푑푡 + (푟푡 − 푟푡 )(푉푡 − 풞푡)푑푡. (3.2.17) Then 푂퐼푆 퐹 푂퐼푆 푑푉푡 − 푟푡 푉푡푑푡 = 푑푀푡 + (푟푡 − 푟푡 )(푉푡 − 풞푡)푑푡, (3.2.18)

푡 푂퐼푆 푡 푂퐼푆 − ∫ 푟푠 푑푠 − ∫ 푟푠 푑푠 푂퐼푆 (3.2.19) 푑 (푒 0 푉푡) = 푒 0 (푑푉푡 − 푟푡 푉푡푑푡)

푡 푂퐼푆 푡 푂퐼푆 − ∫ 푟푠 푑푠 − ∫ 푟푠 푑푠 퐹 푂퐼푆 = 푒 0 푑푀푡 + 푒 0 (푟푡 − 푟푡 )(푉푡 − 풞푡)푑푡. If we integrate both sides from 푡 to 푇,

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푇 푂퐼푆 푡 푂퐼푆 − ∫ 푟푠 푑푠 − ∫ 푟푠 푑푠 (3.2.20) 푒 0 푉푇 − 푒 0 푉푡 푇 푢 푂퐼푆 − ∫ 푟푠 푑푠 퐹 푂퐼푆 = ∫ 푒 0 (푟푢 − 푟푢 )(푉푢 − 풞푢)푑푢 푡 푇 푢 푂퐼푆 − ∫ 푟푠 푑푠 + ∫ 푒 0 푑푀푢. 푡

Since 푀푡 is a martingale under risk neutral measure ℚ, ℚ ℚ (3.2.21) 피푡 (푀푇) = 푀푡 ⇒ 피푡 (푑푀푡) = 0. Hence,

푇 푂퐼푆 푡 푂퐼푆 ℚ − ∫ 푟푠 푑푠 − ∫ 푟푠 푑푠 (3.2.22) 피푡 (푒 0 푉푇 − 푒 0 푉푡)

푇 푢 푂퐼푆 ℚ − ∫ 푟푠 푑푠 퐹 푂퐼푆 = 피푡 (∫ 푒 0 (푟푢 − 푟푢 )(푉푢 − 풞푢)푑푢). 푡 After some rearrangements, 푇 푇 푂퐼푆 푢 푂퐼푆 (3.2.23) ℚ − ∫ 푟푠 푑푠 ℚ − ∫ 푟푠 푑푠 퐹 푂퐼푆 푉푡 = 피푡 (푒 푡 푉푇) + 피푡 (∫ 푒 푡 (푟푢 − 푟푢 )(푉푢 − 풞푢)푑푢). 푡 Therefore, the second interpretation of collateralised derivative price is OIS discounted future payoff and its associated OIS discounted positive/negative carry generated by the difference in Mark to Market and collateral account. The above two results does not explicitly depend on theoretical risk free rate but the rates observed in marketplace. The OIS rate could be obtained by bootstrapping OIS swaps across different maturities and funding rate could be extracted from the bonds issued by the derivative dealer.

∗ Here 푉푡 is defined as the price of the derivative with perfect collateralisation and therefore the derivative is not sensitive to counterparty credit risk but sensitive to the market volatility and credit quality of the collateral as we observed in the repo market. The general terms for ‘perfect’ two way CSA are

Derivative Dealer Counterparty

Independent Amount 0 0 Initial Margin 0 0 Threshold 0 0 MTA 0 0 Rounding Amount 0 0 Rebalancing Interval 0 0 CSA Currency EUR EUR

As suggested by Brigo, Capponi and Pallavicini (2014) and Ballotta, Fusai and Marazzina (2015), instantaneous default contagion makes our proposed ‘perfect’ collateralisation ineffective in reducing exposure and the residual gap risk creates a significant CVA. The type of CSA (i.e. continuous

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Counterparty Credit Risk, Funding Risk and Central Clearing collateralisation) shown doesn’t exist in reality. Here we use ‘perfect’ CSA/collateralisation solely for deriving and explaining Piterbarg (2010). We assume counterparty credit risk doesn’t exist for all derivations in section 3.2.

In theory, if the collateral is ‘perfect’ like cash with abundant liquidity and stability, the repo rate should be very close to OIS rate observed in the market. Hence the perfectly collateralised derivative price is

푇 푂퐼푆 ∗ ℚ − ∫ 푟푠 푑푠 (3.2.24) 푉푡 = 피푡 (푒 푡 푉푇). Then FVA, defined as the adjustment term due to partial/imperfect collateralisation, would cost derivative dealer a substantial amount of cash as margin call on their own hedges with full/perfect collateralisation would ask for more collateral than their original position with partial collateralisation. The FVA term implied from Piterbarg (2010) (without counterparty credit risk) is 푇 푢 푂퐼푆 (3.2.25) ∗ ℚ − ∫ 푟푠 푑푠 퐹 푂퐼푆 퐹푉퐴푡 = 푉푡 − 푉푡 = 피푡 (∫ 푒 푡 (푟푢 − 푟푢 )(푉푢 − 풞푢)푑푢). 푡 To be consistent with our earlier work on CVA/credit charge and DVA/credit benefits, FVA at inception is therefore 푇 푇 ℚ 푢 푂퐼푆 ℚ 푀푡푀푡 (3.2.26) − ∫0 푟푠 푑푠 퐹 푂퐼푆 퐹 퐹푉퐴 = 피0 (∫ 푒 (푟푢 − 푟푢 )(푉푢 − 풞푢)푑푢) = 피0 (∫ 푂퐼푆 푠푡 푑푡), 0 0 푁푡 퐹 퐹 퐹 푂퐼푆 where 푠푡 denotes the funding spread and 푠푡 = 푟푢 − 푟푢 . 푀푡푀푡 refers to the Mark to Market (MtM) 푂퐼푆 of given trade combined with its collateral account, i.e. 푀푡푀푡 = 푉푡 − 풞푡. 푁푡 refers to the OIS

푡 푂퐼푆 푂퐼푆 ∫ 푟푠 푑푠 money market account and 푁푡 = 푒 0 . The FVA could be approximated by linear expectation plus a convexity adjustment term, 푇 (3.2.27) ℚ 푀푡푀푡 퐹 퐹푉퐴 = 피0 (∫ 푂퐼푆 푠푡 푑푡) 0 푁푡 푇 푇 ℚ 퐹 ℚ 푀푡푀푡 ℚ 푀푡푀푡 퐹 = ∫ 피0 (푠푡 )피0 ( 푂퐼푆 ) 푑푡 + ∫ 퐶표푣푡 ( 푂퐼푆 , 푠푡 ) 푑푡, 0 푁푡 0 푁푡 Hence, 푇 (3.2.28) ℚ 퐹 ℚ 푀푡푀푡 퐹푉퐴 = ∫ 피0 (푠푡 )피0 ( 푂퐼푆 ) (1 + 휌휎1휎2푡)푑푡, 0 푁푡 where 휌 denotes the correlation between discounted Mark to Market and the funding spread and 휎1 and 휎2 refer to the volatility of funding spread and discounted Mark to Market respectively. If we further define + − (3.2.29) 퐹 ℚ 퐹 + ℚ 푀푡푀푡 − ℚ 푀푡푀푡 푠0 (푡) = 피0 (푠푡 ), 퐸퐸0 (푡) = 피0 ( 푂퐼푆 ) , 퐸퐸0 (푡) = 피0 ( 푂퐼푆 ), 푁푡 푁푡 and assume there is no correlation between the funding spread and Mark to Market (MtM), 푇 푇 + (3.2.30) 퐹 + 퐹 ℚ 푀푡푀푡 퐹퐶퐴 = ∫ 푠0 (푡) ∙ 퐸퐸0 (푡)푑푡 = ∫ 푠0 (푡)피0 ( 푂퐼푆 ) 푑푡, 0 0 푁푡

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푇 푇 − (3.2.31) 퐹 − 퐹 ℚ 푀푡푀푡 퐹퐵퐴 = − ∫ 푠0 (푡) ∙ 퐸퐸0 (푡)푑푡 = − ∫ 푠0 (푡) ∙ 피0 ( 푂퐼푆 ) 푑푡, 0 0 푁푡 where FBA refers to funding benefits adjustment that accounts for positive carry earned during the life of the derivative transaction, and FCA refers to funding cost adjustment accounts for negative carry paid during the life of the derivative transaction. And 퐹푉퐴 = 퐹퐶퐴 − 퐹퐵퐴. (3.2.32) The above result is applied for derivatives transaction collateralised by the same currency. The Piterbarg (2010) framework could be extended to price trades or portfolios collateralised by multiple currencies. For example, the fair price for a derivative denominated in domestic currency but collateralised in foreign currency is 푇 픻 푇 퐹픻 푢 퐹픻 픽 픽 (3.2.33) 픻 ℚ − ∫ 푟푢 푑푢 픻 − ∫ 푟푣 푑푣 퐹푋 퐹 푂퐼푆 픽 푉푡 = 피푡 (푒 푡 푉푇 + ∫ 푒 푡 푆푢 (푟푢 − 푟푢 )풞푢 푑푢), 푡 퐹픻 퐹픽 where 푟푡 and 푟푡 refer to the funding rate of the market marker in domestic and foreign currency, 퐹푋 respectively. 푆푡 refers to the FX spot price of foreign currency denominated in domestic currency, 퐹푋 픽 for example, if USD is foreign currency and EUR is domestic currency, USDEUR rate is 푆푡 . 풞푡 푂퐼푆픽 refers to the collateral account denominated in foreign currency and 푟푡 is the corresponding OIS rate associated with foreign currency. By following the same capital gain process used in single currency case,

픻 픻 퐹픻 픻 퐹푋 퐹픽 푂퐼푆픽 픽 (3.2.34) 푑푉푡 = 푑푀푡 + 푟푡 푉푡 푑푡 − 푆푡 (푟푡 − 푟푡 )풞푡 푑푡,

픻 픻 푂퐼푆픻 픻 퐹픻 푂퐼푆픻 픻 퐹푋 퐹픽 푂퐼푆픽 픽 (3.2.35) 푑푉푡 = 푑푀푡 + 푟푡 푉푡 푑푡 + (푟푡 − 푟푡 )푉푡 푑푡 − 푆푡 (푟푡 − 푟푡 )풞푡 푑푡. Then

픻 푂퐼푆픻 픻 픻 퐹픻 푂퐼푆픻 픻 퐹푋 퐹픽 푂퐼푆픽 픽 (3.2.36) 푑푉푡 − 푟푡 푉푡 푑푡 = 푑푀푡 + (푟푡 − 푟푡 )푉푡 푑푡 − 푆푡 (푟푡 − 푟푡 )풞푡 푑푡.

푡 픻 − ∫ 푟푂퐼푆 푑푠 If we multiply 푒 0 푠 on both sides,

푡 푂퐼푆픻 픻 − ∫ 푟푠 푑푠 픻 푂퐼푆 픻 (3.2.37) 푒 0 (푑푉푡 − 푟푡 푉푡 푑푡)

푡 푂퐼푆픻 − ∫ 푟푠 푑푠 픻 = 푒 0 푑푀푡

푡 푂퐼푆픻 픻 픻 픽 픽 − ∫ 푟푠 푑푠 퐹 푂퐼푆 픻 퐹푋 퐹 푂퐼푆 픽 + 푒 0 ((푟푡 − 푟푡 ) 푉푡 − 푆푡 (푟푡 − 푟푡 ) 풞푡 ) 푑푡.

Therefore,

푡 푂퐼푆픻 − ∫ 푟푠 푑푠 픻 (3.2.38) 푑 (푒 0 푉푡 )

푡 푂퐼푆픻 − ∫ 푟푠 푑푠 픻 = 푒 0 푑푀푡

푡 푂퐼푆픻 픻 픻 픽 픽 − ∫ 푟푠 푑푠 퐹 푂퐼푆 픻 퐹푋 퐹 푂퐼푆 픽 + 푒 0 ((푟푡 − 푟푡 ) 푉푡 − 푆푡 (푟푡 − 푟푡 ) 풞푡 ) 푑푡.

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픻 픻 ℚ픻 픻 Since 푀푡 is a martingale under domestic risk neutral measure ℚ , 피푡 (푑푀푡 ) = 0. If we integrate both sides from 푡 to 푇 and take on expectations on both sides under domestic risk neutral measure ℚ픻,

픻 푇 푂퐼푆픻 푡 푂퐼푆픻 ℚ − ∫ 푟푠 푑푠 픻 − ∫ 푟푠 푑푠 픻 (3.2.39) 피푡 (푒 0 푉푇 − 푒 0 푉푡 )

푇 픻 푢 푂퐼푆픻 픻 픻 ℚ − ∫ 푟푠 푑푠 퐹 푂퐼푆 픻 = 피푡 (∫ 푒 0 ((푟푢 − 푟푢 )푉푢 푡

퐹푋 퐹픽 푂퐼푆픽 픽 − 푆푢 (푟푢 − 푟푢 )풞푢 ) 푑푢).

Hence, after some rearrangements,

픻 푇 푂퐼푆픻 픻 ℚ − ∫ 푟푠 푑푠 픻 (3.2.40) 푉푡 = 피푡 (푒 0 푉푇 )

푇 픻 푢 푂퐼푆픻 픻 픻 ℚ − ∫ 푟푠 푑푠 퐹 푂퐼푆 픻 − 피푡 (∫ 푒 0 ((푟푢 − 푟푢 )푉푢 푡

퐹푋 퐹픽 푂퐼푆픽 픽 − 푆푢 (푟푢 − 푟푢 )풞푢 ) 푑푢).

The perfectly domestic currency collateralised derivative price is

픻 픻 푇 푂퐼푆픻 ∗ ℚ − ∫ 푟푠 푑푠 픻 (3.2.41) 푉푡 = 피푡 (푒 푡 푉푇 ). Then FVA for foreign currency collateralised derivative is 픻 픻 ∗픻 (3.2.42) 퐹푉퐴푡 = 푉푡 − 푉푡 푇 픻 푢 푂퐼푆픻 픻 픻 ℚ − ∫ 푟푠 푑푠 퐹 푂퐼푆 픻 = 피푡 (∫ 푒 0 ((푟푢 − 푟푢 )푉푢 푡

퐹푋 퐹픽 푂퐼푆픽 픽 − 푆푢 (푟푢 − 푟푢 )풞푢 ) 푑푢).

Hence the FVA for foreign currency collateralised derivative at inception becomes 푇 픻 푢 푂퐼푆픻 픻 픻 (3.2.43) 픻 ℚ − ∫ 푟푠 푑푠 퐹 푂퐼푆 픻 퐹푉퐴 = 피0 (∫ 푒 0 ((푟푢 − 푟푢 )푉푢 0

퐹푋 퐹픽 푂퐼푆픽 픽 − 푆푢 (푟푢 − 푟푢 )풞푢 ) 푑푢).

Therefore, 푇 (3.2.44) 픻 ℚ픻 1 퐹픻 픻 퐹푋 퐹픽 픽 퐹푉퐴 = 피 (∫ (푠푡 푉푡 − 푆푡 푠푡 풞푡 ) 푑푡) 0 푂퐼푆픻 0 푁푡 퐹픽 픻 퐹푋 푠푡 픽 푇 푉푡 − 푆푡 퐹픻 풞푡 ℚ픻 푠푡 퐹픻 = 피 ∫ 푠푡 푑푡 , 0 푂퐼푆픻 0 푁푡 ( )

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픻 픻 푡 푂퐼푆픻 픻 푂퐼푆 푂퐼푆 − ∫ 푟푠 푑푠 퐹 where 푁푡 denotes the domestic OIS money market account and 푁푡 = 푒 0 . 푠푡 퐹픽 refers to domestic funding spread and 푠푡 refers to foreign funding spread. If we further define 픽 푠퐹 푀푡푀픻 = 푉픻 − 푆퐹푋 푡 풞픽 푡 푡 푡 퐹픻 푡 , 푠푡 푇 픻 (3.2.45) 픻 ℚ픻 푀푡푀푡 퐹픻 퐹푉퐴 = 피 (∫ 푠푡 푑푡). 0 푂퐼푆픻 0 푁푡 And its linear approximation with convexity adjustment becomes 푇 픻 (3.2.46) 픻 ℚ픻 퐹픻 ℚ픻 푀푡푀푡 퐹푉퐴 = ∫ 피 (푠푡 ) 피 ( ) (1 + 휌휎1휎2푡)푑푡 0 0 푂퐼푆픻 0 푁푡 푇 픻 퐹픻 ℚ픻 푀푡푀푡 = ∫ 푠0 (푡)피 ( ) (1 + 휌휎1휎2푡) 푑푡, 0 푂퐼푆픻 0 푁푡 where 휌 denotes the correlation between domestic funding spread and domestic OIS discounted

Mark to Market and 휎1 and 휎2 refers to volatility of domestic funding spread and domestic OIS

픻 푀푡푀픻 피ℚ ( 푡 ) discounted Mark to Market, respectively. The valuation of 0 푂퐼푆픻 requires further analysis and 푁푡 the dependency of foreign currency funding spread and FX forward rate has to be specified explicitly.

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3.3 Economic Drivers of FVA

Suppose a derivative dealer entered into a derivative transaction with a given counterparty under a two way CSA. The trade now is deeply out of the money from the perspective of the derivative dealer. Hence the derivative dealer is the net collateral poster in this circumstance. If the derivative dealer defaults immediately, the loss to its counterparty is significantly lower than the scenario with no CSA in place. In other words, CVA for a collateralised transaction from the counterparty’s perspective is significantly lower than CVA for an uncollateralised transaction. Or DVA for a collateralised transaction from the derivative dealer’s perspective is significantly low than DVA for an uncollateralised transaction. The reduction in counterparty’s CVA/derivative dealer’s DVA comes at the expense of collateral the derivative dealer has to borrow in the market. Therefore, essentially, the CSA converts the counterparty credit risk into the funding risk for derivative transactions. The impact of CSA and margin requirements (especially initial margins) will be discussed in the Central Clearing Chapter.

The credit quality and reusability of the collateral are key metrics in assessment of the effectiveness in counterparty risk mitigation. The immunity of credit quality of the collateral to the changes in the credit quality of the counterparty would leads to the maximisation of the benefits of collateral in counterparty credit risk mitigation. The positive correlation/negative correlation between these two variables would lead to Wrong/Right Way Risk and caused the CVA to increase/decrease over the investment horizon of the underlying portfolio. Right Way Risk created by the negative dependency of credit quality of collateral on the credit quality of counterparty meets desirable targets from trading/business perspective but reduces the effectiveness of counterparty credit risk mitigation (i.e. the exposure would increase if credit quality of counterparty improves). A classic example of Wrong Way Risk created by collateral is that the counterparty of the derivative dealer is a sovereign entity and eligible collaterals listed in their CSA are the bonds issued by that sovereign entity. If the sovereign runs into the state of financial distress during the life of the trades and the derivative dealer is in the money at that time, the collateral, i.e. the sovereign bonds, would depreciate significantly and cause the exposure to spike/jump to an undesired level. Even if the collateral here is not sovereign bond but their domestic currency, such kind of cash collateral would still expose the derivative dealer to the explicit Wrong Way Risk due to the FX movements. As discussed by Brigo et al. (2013), reusability terms in CSA, i.e. rehypothecation of the collateral, would reduce the effectiveness of collateral in mitigating counterparty risk as the derivative dealer would not be able to receive/call back the full collateral in the default state of their counterparty. However, rehypothecation of the collateral would increase the funding benefits and reduce the funding cost in derivative transactions as the derivative dealer would not have to borrow cash/liquid securities from the capital markets to

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Counterparty Credit Risk, Funding Risk and Central Clearing meet margin calls of a number of its counterparties but instead use the collateral it collects from other counterparties to fulfil its obligations. Rehypothecation of the collateral has a significant impact on funding valuation adjustment (FVA) pricing and would be discussed in detail later this chapter. Some derivative dealers now sets up collateral trading desk or collateral transformation desk to convert the non-cash collateral like sovereign bonds they received from their counterparties into cash collateral via repo and other types of transactions. Rehypothecation of non-cash collateral becomes the prerequisite of collateral transformation activities.

The LIBOR curves were used as the main discounting curves prior to the Global Financial Crisis (GFC). The U.S. Treasury yield curve was perceived as a better proxy for the theoretical ‘risk free’ rate but wasn’t used widely as discounting curve for derivatives. The main reasons were the tax exempts that Treasury bond holders could receive and specialness of liquid Treasury bond holders could use to exploit the market inefficiency. The overall liquidity of LIBOR was better than the Treasury bonds markets globally. The derivative dealers of OTC derivatives could enjoy homogeneity in their funding cost which as closely linked to the level of LIBOR rate as the LIBOR rate is set as the interest rate on an uncollateralised loan by a group of banks in the interbank market on daily basis. The LIBOR market therefore provides pipelines for the derivative dealers to access to short term loans with low rates in uncollateralised environments. Given the nature of LIBOR rate is the benchmark rate at which banks borrows money from capital markets, the implicit credit risk is embedded in the rates and clearly illustrated in the GFC. The spread of EURIBOR 6 month over EURIBOR 3 month jumped above 40 basis points after declaration of bankruptcy of Lehman Brothers in October 2008. The longer tenor EURIBOR was perceived as the instrument carried more credit risk of the banking institutions. The overnight rate, i.e. the rate, carries the least credit risk from this perspective. EONIA, i.e. the OIS rate in Europe, is calculated as a weighted average of uncollateralised interbank overnight lending rates. In a collateralised derivative transaction, the collateral poster would receive OIS rate of the collateral currency from collateral holder as a compensation for posting collateral. The OIS rate for different tenors could be bootstrapped from the OIS Swaps, OIS-LIBOR swaps (which have better liquidity in longer tenors compared to OIS swaps) and LIBOR swaps. The LIBOR-OIS spread was used as a major money market stress indicator by many economists, including former Fed Chairman Alan Greenspan (Brown and Finch (2009)). The banks are less willing to lend in the aftermath of Lehman default during the period the LIBOR-OIS spread went up to above 350 basis points. The OIS rate now, technically, carries a significantly lower credit risk compared to LIBOR rate and represents the fundamental lending/funding rate in a collateralised derivative transaction. Hence, the OIS rate seems to be a natural candidate for discounting the collateralised derivative transactions. Since the OIS is an overnight average of the interbank lending rate, for a collateralised derivative position with long

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Counterparty Credit Risk, Funding Risk and Central Clearing duration like 30 years, the liquidity of OIS swap and LIBOR OIS swap becomes a concern and the derivative dealer could not fund/lend at 30 year OIS rate. CSA agreements of such long tenor transaction would leave some freedoms to other curves like LIBOR curves which further increase the complexity of our analysis as the OIS curve could not give a fair representation of the long term interest rate risk. For example, for a 30 years cross currency swap, the FX forward rate would climb 퐹푋 the forward curve governed by interest rate parity as discussed in section 2.4. Here 푆푡 is the FX spot rate and is the price of one unit of foreign currency denominated in domestic currency, Hence we have

퐹푋 픽 퐹푋 픽 푆푠 푃 (푠, 푇) 푆푡 푃 (푡, 푇) (3.3.1) = 피픻푇 ( ), 푃픻(푠, 푇) 푠 푃픻(푡, 푇) where 푃픻(푡, 푇) is the numeraire for domestic currency forward measure. 푃픻(푡, 푇) and 푃픽(푡, 푇) denote the default free zero coupon bond prices at time 푡 in domestic currency and foreign currency, respectively. Then the interest rate parity is

퐹푋 픻푇 퐹푋 (3.3.2) 퐹 (푠, 푇) = 피푠 (퐹 (푡, 푇)),

푆퐹푋푃픽(푡,푇) where 퐹퐹푋(푡, 푇) = 푡 . If the domestic interest rate products pay out a relatively lower interest 푃픻(푡,푇) rate,

픻 픽 퐹푋 퐹푋 푃 (푡, 푇) < 푃 (푡, 푇) ⇒ 퐹 (푡, 푇) > 푆푡 , (3.3.3) that is, the domestic price of one unit of foreign currency is going to depreciate in the future. In other words, the domestic currency (currency with lower interest rate) is expected to appreciate against the foreign currency (currency with higher interest rate). Therefore, if the derivative dealer enters into a cross currency swap with a counterparty and the derivative dealer pays the higher interest rate and its currency (and receives the lower interest rate and its currency), the Mark to Market from the derivative dealer’s perspective is going to be significantly positive over the majority of 30 years and the significantly greater positive exposure as a result of interest rate parity would create a persistent long term funding cost for the derivative dealer. An example of this cross currency swap would be discussed in detail later this chapter.

There are a number of assets qualified to be posted as collateral in a typical CSA agreements. Generally, cash, sovereign bonds and corporate bonds, and some specific Asset Backed Securities (ABSs), equities etc. are all eligible collaterals. The quantitative terms like threshold, minimum transfer amount etc. are driving factors of the shapes of exposure profiles. The impact of other important factors like netting agreements on CVA/DVA and rehypothecation on FVA will be discussed in detail later this chapter. One question asked by many people regarding the choice of collateral is that if there is any quantifiable advantage to choose a specific currency over other currencies?

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As discussed before, the return on posted cash collateral is the OIS rate of that collateral currency. In other words, the return on posted collateral is determined by the specific choice of the collateral currency. It is optimal for the derivative dealer to choose the highest yielding currency (or the currency with highest interest rate) listed in the CSA as the underlying collateral to maximise its returns. And highest yielding currency is going to depreciate against other currencies in the future under no arbitrage framework (which contradicts with classic carry trade experiment in economics). Hence the highest yielding currency becomes the natural choice of ‘cheapest to deliver’ selection at inception of the transactions. Therefore, if the derivative dealer has to choose one from a set of currencies as the underlying collateral for their portfolio with a given counterparty and the portfolio + − has a symmetric exposure profile (i.e. 퐸퐸0 (푡) = −퐸퐸0 (푡), 푡 ∈ [0, 푇]), ceteris paribus, the derivative dealer should choose the currency with highest yields/interest rates. Its counterparty would follow the same logic to choose the currency with the highest yields as well. If the derivative dealer decides to choose non-cash collateral, normally a haircut would be attached to the collateral such that the market price of the non-cash collateral posted is greater than its cash equivalent. In order to maximise the funding benefits (or minimise the funding cost), rehypothecation has to be put in place and the non-cash collateral would be transformed into cash via Repo or Synthetic Repo transactions. Another haircut is attached in the Repo arm of this collateral trading/transformation process, therefore, if the derivative dealer receives the non-cash collateral from its counterparty and decides to repo this non- cash collateral into cash equivalent in the corresponding currency (to post to other counterparties), the derivative dealer should make adjustments to repo rate they are going to pay, i.e. the repo rate

1−퐻푅푒푝표 should be multiplied by the factor (퐻 denotes the percentage of haircuts), and 1−퐻푛표푛−푐푎푠ℎ 푐표푙푙푎푡푒푟푎푙 compare with the OIS rate they are going to receive from other counterparties by posting the cash collateral. And the collateral switch option, i.e. the right to substitute the collateral within the life of the underlying portfolio, has a direct impact on optimisation of collateral management. For example, if the derivative dealer is a net collateral poster with a given counterparty and observes the OIS rate of a given currency widens with respect to the rest of currencies, it is optimal for the derivative dealer to switch/substitute the collateral of on-going transactions to the currency with highest yield (i.e. OIS rate). It might be difficult to implement in practice as the collateral transactions inherently are ‘zero sum’ games and the counterparty is aware of the fact that collateral with highest yield has a higher probability to depreciate against other currencies in non-arbitrage framework and therefore would not consent the change of collateral in certain market scenarios. If the collateral switch option is not allowed in the CSA, the derivative dealer could choose the highest yield currency as the optimal collateral.

Evaluation of collateral and its related options are path dependent problems. It has to be priced at netting sets level (for collateral switch option pricing and rehypothecation within the netting set etc.)

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Counterparty Credit Risk, Funding Risk and Central Clearing and funding sets level (for rehypothecation across the netting sets). The exact amount of collateral cost/benefit depends on the future evolution of the exposure profile which is driven by the market variables dynamics and the development of business (i.e. trades and other sorts of financial transactions) with given counterparties. Collateral trading/transformation desks have been set up within different derivative dealers to facilitate market and regulatory structure changes. The overall significant collateral benefit is not easy to exploit due to the bilateral nature of CSAs and it heavily depends on the avaibility of financial assets on the derivative dealer’s book.

The global financial crisis (GFC) has called for solutions for counterparty credit risk. A certain range of OTC derivatives now are required/mandated to be centrally cleared via CCPs by the Dodd-Frank Wall Street Reform and European Market Infrastructure Regulation (EMIR). The margin requirements set by central counterparties (CCPs) follow the general structure of two way CSAs set by ISDA but with improved collateral requirements and initial margins. The detailed discussion would be found in the Central Clearing chapter. One of the important sides of CCPs is the central clearing mandate potentially creates the a channel to spread systematic risk within the global financial system as the CCPs retains the right to increase the initial margin requirements in market turmoil which would lead to knock on effects in the form of margin calls to different market participants. As the transactions with CCPs are well collateralised, OIS curve is the standard discounting tool for OTC products pricing. The embedded multi-currency collateral problems and collateral switch options are removed from the collateral agreements in CCPs and hence significantly reduce the evaluation difficulties in assessing the pricing of different OTC portfolios. The detailed collateral currency choices principles and cross product netting would be discussed in Central Clearing chapter. Piterbarg (2010) presents a generic pricing framework for OTC derivatives with a collateral account and constructs a self-financing strategy via the combination of Delta, collateral and funding accounts under no-arbitrage condition. This framework sets the OIS curve as theoretical appropriate way to discount derivatives. Continuously settled cash collateral under a symmetric two way CSA with zero threshold, MTA (minimum transfer amount) and rounding amount is assumed in Piterbarg (2010). And the collateral poster is assumed to receive the OIS rate of the collateral currency on the net amount of collateral from the collateral receiver as a compensation for posting collateral. Although the continuous collateral settlements with key collateral parameters set at zero have a reasonable divergence from the market practice in bilateral OTC market, this setting could be served as a sound pricing basis for counterparty credit risk and funding risk analysis for derivative dealers, especially those have been GCM (general clearing member) of CCPs.

The OIS yield curve has to be built on LIBOR-OIS swaps with OIS swaps and LIBOR swaps simultaneously. This is due to the LIBOR-OIS swaps are perceived to have a better liquidity compared to the standard OIS swaps. Given the fact that the derivative dealers normally could not be

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Counterparty Credit Risk, Funding Risk and Central Clearing funded at the OIS yield curve, some further adjustments have to be taken into account for funding risk in derivatives pricing, even in absence of counterparty credit risk. This adjustment is called funding valuation adjustment (FVA).

In the post GFC (global financial crisis) era, the OTC derivatives market, especially the inter-dealer market, has experienced a significant increase in the use of collateral in transactions. However, the other side of market participants like corporates and sovereigns do not have the capacity to meet daily margin calls and therefore a large portion of the OTC derivatives market remained uncollateralised or partially collateralised. As discussed in pervious sections, collateral has a significant impact on CVA/DVA pricing as it could be used to reduce the exposure. It will be shown in this section that the collateral has a significant impact in FVA as well. The asymmetric collateral requirements/treatments in these two sides of OTC derivatives market give rise to the funding risk pricing and analysis. From the derivative dealer’s perspective, FVA, i.e. funding valuation adjustment, is used to quantify the associated funding costs and benefits in trading OTC derivatives to these two sides of market with asymmetric collateral requirements.

In theory, the derivative dealer should run a flat OTC derivatives book (i.e. an OTC derivatives book with perfect hedges in place) and profit from the bid/ask spread in buying/selling derivatives and its hedging positions. The derivative dealers has to borrow/fund its operations at rate higher than the theoretical risk ‘free’ rate. The market structure and regulatory changes cause the derivative dealer to adjust its business model and manage its collateral positions with different counterparties. If the derivative dealer could buy/sell derivatives with a fully collateralised counterparty and hedge these transactions with back-to-back trades with another fully collateralised counterparty, the cash flows of underlying transactions and their associated collateral accounts would match perfectly with the cash flows of the hedging trades and their associated collateral accounts, and the derivative dealer here does not have to borrow/lend any extra money to facilitate these classic self-financing business models. That is, FVA, as the price of these extra cash flows, is set at 0. If the derivative dealer buys/sells a certain type of derivatives with an uncollateralised or partially collateralised counterparty and hedges these derivatives with back-to-back transactions with a fully collateralised counterparty, the cash flows of the underlying transactions could match perfectly with the cash flows of the hedging trades, but the cash flows of the collateral account of the underlying transactions would not match with cash flows of the collateral account of the hedging trades. If the derivative dealer is − expected to have a negative Mark to Market, or the expected negative exposure (i.e. 퐸퐸0 (푡)) + dominates the expected positive exposure (i.e. 퐸퐸0 (푡)), that is,

− + −퐸퐸0 (푡) > 퐸퐸0 (푡), (3.3.4) the derivative dealer would receive positive cash flows from the collateral account of the hedging trades and post a portion of the cash flows to the collateral account of the underlying transactions

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Counterparty Credit Risk, Funding Risk and Central Clearing with partially collateralised counterparty. (The derivative dealer would not post anything to the collateral account if the counterparty is uncollateralised.) These positive cash flows are termed as funding benefit and its corresponding price is FBA (funding benefits adjustment). If the derivative + dealer is expected to have a positive Mark to Market, or the expected positive exposure (i.e. 퐸퐸0 (푡)) − dominates the expected negative exposure (i.e. 퐸퐸0 (푡)), that is,

+ − 퐸퐸0 (푡) > −퐸퐸0 (푡), (3.3.5) the derivative dealer would post positive cash flows to the collateral account of the hedging trades and receive a portion of the cash flows from the collateral account of the underlying transactions with partially collateralised counterparty. (The derivative dealer would not receive anything from the collateral account if the counterparty is uncollateralised.) The derivative dealer therefore experience a cash outflow or negative cash flow during the trading process. And the derivative dealer has to borrow money to pay out these cash flows. These negative cash flows are termed as funding cost and its corresponding price is FCA (funding cost adjustment). The FVA, funding valuation adjustment, is a combination of FCA and FBA. The existence of mismatch of cash flows of collateral accounts breaks down the classic self-financing strategy as the derivative dealer now has to borrow/lend extra money generated by market risk neutral positions. Piterbarg (2010) adds the collateral account to market neutral positions and makes market risk neutral strategy ‘self-financing’ again. Therefore, a proper quantification of funding components due to the existence of mismatch of collateral requirements is essential in determination of hedging strategies for the uncollateralised or partially collateralised portfolios. As we will discuss later this chapter, the FCA (funding cost adjustment) is + driven by the exposure (i.e. 퐸퐸0 (푡)) and FBA (funding benefit adjustment) is driven by negative − exposure (i.e. 퐸퐸0 (푡)). The similarities and difference in pricing FCA/FBA and CVA/DVA would be addressed in the end of this chapter.

In the classic market structure, the swap spread (i.e. the 푛 year swap rate minus the 푛 year Treasury yield) could be approximated by LIBOR-GC spread. GC, i.e. general collateral rate (Repo rate), reflects the generic funding costs/benefits the derivative dealers have to take into account in pricing interest rate derivatives in absence of collateral account. Suppose the derivative dealer pays fixed in an interest rate swap transaction and hedges by buying a Treasury with identical time to maturity. The purchase of Treasury bond could be financed by Repo transactions. Therefore, the effective running cost of the derivative dealer in hedging positions is the GC (general collateral) repo rate. Then the process of running interest rate swap trading business becomes an integrated business chain. The derivative dealer earns yield to maturity of the specific Treasury bond and pays out the Treasury yield plus swap spread to the counterparty. The counterparty pays LIBOR rate to the derivative dealer in exchange for the Treasury yield plus the swap spread (i.e. the fixed rate). The derivative dealer has to use the LIBOR rate receivables to pay GC repo rates on its funding repo positions. Therefore, in a competitive market, the swap spread the derivative dealer losses on the fixed leg could be

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Counterparty Credit Risk, Funding Risk and Central Clearing compensated by the LIBOR-GC spread the market marker earns on the floating leg. That is, the LIBOR-GC spread is a sensible approximation of the swap spread in the classic market structure without a collateral account. The current market structure and regulatory environment requires improved usage of collateral to reduce counterparty credit risk. The additional funding cost/benefits with the usage of collateral account in the underlying transaction would have a direct impact on the swap spread and the well collateralised hedging position (i.e. repo transaction of Treasury bond) makes no contributions to the funding components in this case. For an uncollateralised fixed payer interest rate swap, the exposure dominates the negative exposure and the net funding component would result in a net funding cost term. The derivative dealer would build the funding cost into swap spread by paying less on the fixed leg. In other words, the fixed payer interest rate swap produces net funding costs and the receiver swap, on the contrary, produces net funding benefits. The details of this analysis could be found later this chapter.

Gregory (2012) suggests a simple approach to price FVA by discounting the cash flows of underlying transactions by the derivative dealer’s funding curve instead of the OIS curve. The result price differential with respect to the OIS discounted derivative price is FVA. The drawbacks of ‘discounting curve’ approach, as discussed by Gregory (2012), are implicit symmetric funding costs and funding benefits assumption and disallowance of common features of CSA like threshold, MTA (minimum transfer amount), etc.

Hunt and Lemonnier (2014) extends the discount curve method by including the collateral account into a simple one period model. Suppose the derivative dealer sells one option with contingent payoff at maturity to the uncollateralised or partially collateralised counterparty and simultaneously enters into a back-to-back hedging position with full collateralisation. Hence, at inception 푡 = 0 , the ∗ ∗ derivative dealer posts collateral 풞푡 to the counterparty and receives collateral 푉푡 ( 푉푡 =

푇 푂퐼푆 ℚ − ∫ 푟푠 푑푠 ∗ 피푡 (푒 푡 푉푇)) from the fully collateralised hedging position. The collateral difference 푉푡 − 풞푡

푂퐼푆 is posted to the treasury. At maturity 푡 = 1, the derivative dealer receives 풞푡(1 + 푟푡 ) from the ∗ 푂퐼푆 partially collateralised or uncollateralised counterparty and posts 푉푡 (1 + 푟푡 ) to the fully ∗ 퐹 collateralised hedging counterparty. And the treasury would deliver (푉푡 − 풞푡)(1 + 푟푡 ) to the derivative dealer. The derivative dealer is assumed to borrow or lend extra cash from/to the treasury. The difference in borrowing and lending rates are going to be discussed later this chapter. The appropriate discounting curve therefore is the derivative dealer’s funding curve as the derivative dealer could only resort to its treasury to borrow money in case of shortage of cash collateral. It is plausible that the derivative dealer could place the excess cash through reverse repo transactions and GC (general collateral) curve seems to be a valid candidate for discounting curve. Given the fact it is impossible to repo the receivables of derivatives contracts in the hence GC curve is

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Counterparty Credit Risk, Funding Risk and Central Clearing not a good candidate curve to discount the shortage of cash collaterals. Therefore, the present value of net cash flows of this one period transaction is

1 ∗ 퐹 ∗ 푂퐼푆 푂퐼푆 (3.3.6) 퐹 ((푉푡 − 풞푡)(1 + 푟푡 ) − 푉푡 (1 + 푟푡 ) + 풞푡(1 + 푟푡 )) 1 + 푟푡

1 퐹 푂퐼푆 ∗ = 퐹 ((푟푡 − 푟푡 )(푉푡 − 풞푡)), 1 + 푟푡 that is,

1 퐹 푂퐼푆 ∗ (3.3.7) 퐹푉퐴 = 퐹 ((푟푡 − 푟푡 )(푉푡 − 풞푡)). 1 + 푟푡 In the continuous time limit,

푇 푢 퐹 (3.3.8) ℚ − ∫ 푟푠 푑푠 퐹 푂퐼푆 ∗ 퐹푉퐴푡 = 피푡 (∫ 푒 푡 (푟푢 − 푟푢 )(푉푢 − 풞푢)푑푢). 푡 Piterbarg (2010) suggests that

푇 푢 푂퐼푆 (3.3.9) ℚ − ∫ 푟푠 푑푠 퐹 푂퐼푆 퐹푉퐴푡 = 피푡 (∫ 푒 푡 (푟푢 − 푟푢 )(푉푢 − 풞푢)푑푢), 푡 and exhibits difference in terms of discounting rate and Mark to Market valuation. These two expression actually equal to each other after some rearrangement. Here FVA is interpreted as the price differential between a partially collateralised derivative transaction and fully collateralised identical derivative transaction. As defined by Piterbarg (2010), the partially collateralised derivative price is

푇 푇 퐹 푢 퐹 (3.3.10) ℚ − ∫ 푟푢 푑푢 ℚ − ∫ 푟푣 푑푣 퐹 푂퐼푆 푉푡 = 피푡 (푒 푡 푉푇) + 피푡 (∫ 푒 푡 (푟푢 − 푟푢 )풞푢푑푢), 푡 and the fully collateralised derivative price is

푇 ℚ 푇 푂퐼푆 ℚ 푢 푂퐼푆 (3.3.11) ∗ − ∫푡 푟푠 푑푠 − ∫푡 푟푠 푑푠 퐹 푂퐼푆 푉푡 = 피푡 (푒 푉푇) + 피푡 (∫ 푒 (푟푢 − 푟푢 )(푉푢 − 풞푢)푑푢) 푡

ℚ 푇 푂퐼푆 − ∫푡 푟푠 푑푠 = 피푡 (푒 푉푇), as 푉푢 = 풞푢 for 푢 ∈ [푡, 푇] in fully collateralised or perfect collateralised transaction. Piterbarg (2010), as discussed in previous sections, has proved that

푇 푢 푂퐼푆 (3.3.12) ∗ ℚ − ∫ 푟푠 푑푠 퐹 푂퐼푆 퐹푉퐴푡 = 푉푡 − 푉푡 = 피푡 (∫ 푒 푡 (푟푢 − 푟푢 )(푉푢 − 풞푢)푑푢). 푡 ∗ Here let’s use 푉푡 and 푉푡 above to calculate FVA.

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ℚ 푇 푂퐼푆 ℚ 푇 퐹 ∗ − ∫푡 푟푠 푑푠 − ∫푡 푟푢 푑푢 (3.3.13) 퐹푉퐴푡 = 푉푡 − 푉푡 = 피푡 (푒 푉푇) − 피푡 (푒 푉푇)

푇 ℚ 푢 퐹 − ∫푡 푟푣 푑푣 퐹 푂퐼푆 − 피푡 (∫ 푒 (푟푢 − 푟푢 )풞푢푑푢). 푡 Therefore, the price differential could be reduced to

푇 푇 푂퐼푆 푇 퐹 푢 퐹 (3.3.14) ℚ − ∫ 푟푠 푑푠 − ∫ 푟푢 푑푢 ℚ − ∫ 푟푣 푑푣 퐹 푂퐼푆 피푡 (푒 푡 푉푇 − 푒 푡 푉푇) − 피푡 (∫ 푒 푡 (푟푢 − 푟푢 )풞푢푑푢), 푡 that is,

푇 퐹 푂퐼푆 푇 푂퐼푆 (3.3.15) ℚ − ∫ (푟푠 −푟푠 )푑푠 − ∫ 푟푠 푑푠 피푡 ((1 − 푒 푡 ) 푒 푡 푉푇)

푇 푢 퐹 ℚ − ∫ 푟푣 푑푣 퐹 푂퐼푆 − 피푡 (∫ 푒 푡 (푟푢 − 푟푢 )풞푢푑푢). 푡 Since

푇 푢 퐹 푂퐼푆 푇 퐹 푂퐼푆 (3.3.16) 퐹 푂퐼푆 − ∫ (푟푠 −푟푠 )푑푠 − ∫ (푟푠 −푟푠 )푑푠 ∫ (푟푢 − 푟푢 ) 푒 푡 푑푢 = 1 − 푒 푡 , 푡 then the price differential becomes

푇 푢 퐹 푂퐼푆 푇 푂퐼푆 (3.3.17) ℚ 퐹 푂퐼푆 − ∫ (푟푠 −푟푠 )푑푠 − ∫ 푟푠 푑푠 피푡 (∫ (푟푢 − 푟푢 ) 푒 푡 푒 푡 푉푇푑푢) 푡 푇 푢 퐹 ℚ − ∫ 푟푣 푑푣 퐹 푂퐼푆 − 피푡 (∫ 푒 푡 (푟푢 − 푟푢 )풞푢푑푢). 푡 And it is equal to

푇 푢 퐹 푇 푂퐼푆 (3.3.18) ℚ 퐹 푂퐼푆 − ∫ 푟푠 푑푠 − ∫ 푟푠 푑푠 피푡 (∫ (푟푢 − 푟푢 ) 푒 푡 푒 푢 푉푇푑푢) 푡 푇 푢 퐹 ℚ − ∫ 푟푣 푑푣 퐹 푂퐼푆 − 피푡 (∫ 푒 푡 (푟푢 − 푟푢 )풞푢푑푢). 푡 By law of iterated expectations,

푇 ℚ 푢 퐹 푇 푂퐼푆 (3.3.19) 퐹 푂퐼푆 − ∫푡 푟푠 푑푠 − ∫푢 푟푠 푑푠 피푡 (∫ (푟푢 − 푟푢 ) 푒 푒 푉푇푑푢) 푡 푇 ℚ 푢 퐹 ℚ 푇 푂퐼푆 퐹 푂퐼푆 − ∫푡 푟푠 푑푠 − ∫푢 푟푠 푑푠 = 피푡 (∫ (푟푢 − 푟푢 ) 푒 피푢 (푒 푉푇) 푑푢), 푡 hence,

푇 푢 퐹 푇 푂퐼푆 (3.3.20) ℚ 퐹 푂퐼푆 − ∫ 푟푠 푑푠 − ∫ 푟푠 푑푠 피푡 (∫ (푟푢 − 푟푢 ) 푒 푡 푒 푢 푉푇푑푢) 푡 푇 푢 퐹 ℚ 퐹 푂퐼푆 − ∫ 푟푠 푑푠 ∗ = 피푡 (∫ (푟푢 − 푟푢 ) 푒 푡 푉푢 푑푢), 푡

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푇 푂퐼푆 ∗ ℚ − ∫ 푟푠 푑푠 given 푉푢 = 피푢 (푒 푢 푉푇). Therefore,

푇 ℚ 푢 퐹 (3.3.21) ∗ 퐹 푂퐼푆 − ∫푡 푟푠 푑푠 ∗ 퐹푉퐴푡 = 푉푡 − 푉푡 = 피푡 (∫ (푟푢 − 푟푢 ) 푒 푉푢 푑푢) 푡 푇 ℚ 푢 퐹 − ∫푡 푟푣 푑푣 퐹 푂퐼푆 − 피푡 (∫ 푒 (푟푢 − 푟푢 )풞푢푑푢). 푡 Hence,

푇 푢 퐹 (3.3.22) ∗ ℚ − ∫ 푟푠 푑푠 퐹 푂퐼푆 ∗ 퐹푉퐴푡 = 푉푡 − 푉푡 = 피푡 (∫ 푒 푡 (푟푢 − 푟푢 ) (푉푢 − 풞푢)푑푢). 푡 The market is eventually taken over by the derivative dealer has the lowest FVA and hence the best credit quality and lowest funding cost as projected by Hull and White (2014).

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3.4 FVA Pricing with Counterparty Credit Risk

If we further expand the framework of funding analysis into a realistic setting, that is, the funding would terminate or ‘knock out’ at the default time of the derivative dealer, the funding pricing problem could be reduced to defaultable instruments (credit extinguisher) pricing as discussed in 퐼 I section (2.4). 휏 denotes the default time of the derivative dealer and is the first time its 푋푡 falls below 퐼 the liability 퐻푡 , 퐼 퐼 퐼 휏 = inf {푡 > 0|푋푡 ≤ 퐻푡 } (3.4.1) 퐼 푉̃푡 is the market price of a defaultable or knock out instrument triggered by default of the derivative dealer at time 푡. The defaultable instrument price process is a martingale under forward measure as before, 퐼 퐼 퐼 푉̃푡 푉̃푇1 퐼 푅̃휏1 퐼 (3.4.2) = 피푇 ( 휏 >푇 + 푡<휏 ≤푇), 푃(푡, 푇) 푡 푃(푇, 푇) 푃(휏, 푇) where 0 < 푡 < 푇. The recovery 푅̃휏 of the funding component is assumed to be 0 at the default time 휏퐼. If the default process of the derivative dealer is independent to the valuation of the derivative in non-default states, ̃ 퐼 (3.4.3) 푉푡 푇 퐼 푇 퐼 푇 퐼 푇 = 피 (푉̃ )피 (1 퐼 ) ⇒ 푉̃ = 푃(푡, 푇)피 (푉̃ )피 (1 퐼 ). 푃(푡, 푇) 푡 푇 푡 휏 >푇 푡 푡 푇 푡 휏 >푇 Hence the price of the credit extinguisher at time 0 could be restated as

퐼 퐼 푇 (3.4.4) 푉̃0 = 푉0 (1 − ℙ퐷,퐼(0, 푇)), Where 퐼 푇 퐼 푉0 = 푃(0, 푇)피0(푉̃푇), (3.4.5) 푇 푇 ℙ퐷,퐼(0, 푇) = 피0(1휏퐼≤푇). 푇 Here 푉0 is the default free price of the underlying instrument, ℙ퐷(0, 푇) is the default probability of the firm under forward measure. And the default probability under risk neutral measure could be obtained by the same procedures in CVA/DVA chapter,

ℚ 푇 푇 푁푇 (3.4.6) ℙ퐷,퐼(푡, 푇) = ℙ퐷,퐼(푡, 푇) + 푃(푡, 푇)퐶표푣 ( , 1푡<휏퐼<푇). 푁푡 ℚ 푇 And hence ℙ퐷,퐼(푡, 푇) = ℙ퐷<퐼(푡, 푇) if the interest rate process and default process are independent.

Here the credit extinguisher pays out a coupon or cash flow 퐶푆(푡) in the survival (non-default) state of the underlying credit at time 푡 and does not pay anything at or after 푡 in default states (i.e. 휏 < 푡).

More specifically, 퐶푆(푡) is the funding cost/benefit in FVA analysis and payoff of the premium leg in the survival states of credit extinguisher. Hence the price of credit extinguisher under risk neutral measure is

푛 퐼 (3.4.7) ℚ 퐶푆(휏 ) 푉 = 푁 피 (∑ 1 퐼 ), 푡 푡 푡 푇푖<휏 푁휏퐼 푖=1

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where 0 < 푡 ≤ 푇1 < ⋯ < 푇푛 and 푇푛 is the maturity of this credit extinguisher. Here 푁푡 is the money 푡 ∫ 푟푠푑푠 market account 푁푡 = 푒 0 . The credit extinguisher here is a forward starting contract and it automatically cancels if the derivative dealer defaults before the first coupon date 푇1. If we slice the entire streams of contingent cash flows into individual contingent cash flows within corresponding time interval, 푛 (3.4.8) 푖 푉푡 = ∑ 푉푡 , 푖=1 and 퐼 ℚ 퐶푆(휏 ) (3.4.9) 푁 피 ( 1 퐼) 0 < 푡 ≤ 푇 푖 푡 푡 푇푖<휏 푖 푉푡 = { 푁휏퐼 , 0 푇푖 < 푡 Or, 퐼 (3.4.10) 푖 ℚ 퐶푆(휏 ) 푉 = 1 푁 피 ( 1 퐼). 푡 푡≤푇푖 푡 푡 푇푖<휏 푁휏퐼 We could further define a conditional default time 휏퐼(푇) as 퐼 퐼 퐼 휏 (푇) = inf {푡 > 푇|푋푡 ≤ 퐻푡 }, (3.4.11) where 푇 ≥ 0. Then

1푇<휏퐼(푇) = 1, (3.4.12)

1 퐼 = 1 퐼1 퐼 = 1 퐼 1 퐼 . 푇푖<휏 푇푖<휏 푇푖<휏 (푇푖) 푇<휏 푇푖<휏 (푇)

Then the price of individual cash flow (for 0 ≤ 푡 ≤ 푇푖) of the credit extinguisher is

퐼 퐼 (3.4.13) 푖 ℚ 퐶푆(휏 ) ℚ ℚ 퐶푆(휏 (푇푖)) 푉푡 = 푁푡피푡 ( 1푇 <휏퐼) = 푁푡피푡 (피푇 (1푇 <휏퐼 1푇 <휏퐼(푇 ))). 푁 퐼 푖 푖 푖 푁 퐼 푖 푖 휏 휏 (푇푖)

Therefore,

퐼 (3.4.14) 푖 ℚ ℚ 퐶푆(휏 (푇푖)) 푉푡 = 푁푡피푡 (1푇 <휏퐼 피푇 ( 1푇 <휏퐼(푇 ))) 푖 푖 푁 퐼 푖 푖 휏 (푇푖)

퐼 ℚ ℚ 퐶푆(휏 (푇푖)) = 푁푡피푡 (1푇 <휏퐼 피푇 ( )). 푖 푖 푁 퐼 휏 (푇푖)

Hence,

푖 ℚ ℚ (3.4.15) 푉 = 푁 피 (1 퐼 1 퐼 퐶̅ (푇 )) = 1 퐼푁 피 (1 퐼 퐶̅ (푇 )), 푡 푡 푡 푡<휏 푇푖<휏 (푡) 푆 푖 푡<휏 푡 푡 푇푖<휏 (푡) 푆 푖

퐶 (휏퐼(푇 )) ̅ ℚ 푆 푖 ̅ where 0 ≤ 푡 ≤ 푇푖 and 퐶푆(푇푖) = 피푇 ( ). Here 퐶푆(푇푖) is the discounted expected payoff in 푖 푁 퐼 휏 (푇푖) the survival state of the derivative dealer at time 푇푖 with all available information up to time 푇푖. If we

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푖 further assume the existence of dependency between market and credit variables in 푉푡 = 퐼 ℚ 퐶푆(휏 ) 푁 피 ( 1 퐼), 푡 푡 푇푖<휏 푁휏퐼 퐼 (3.4.16) 푖 ℚ 퐶푆(휏 (푡)) 푉 = 1 퐼 푁 피 ( 1 퐼 ) 푡 푡<휏 푡 푡 푇푖<휏 (푡) 푁휏퐼(푡)

퐼 ℚ ℚ 퐶푆(휏 (푡)) 푀 = 1푡<휏퐼푁푡피푡 (피푡 ( 1푇 <휏퐼(푡)|ℱ푇 )) 푁휏퐼(푡) 푖 푛

ℚ ℚ as 1 퐼 = 1 퐼 1 퐼 and 피 (1 퐼) = 1 퐼피 (1 퐼 ). Here 0 ≤ 푡 ≤ 푇 ≤ 푇 . 푇푖<휏 푡<휏 푇푖<휏 (푡) 푡 푇푖<휏 푡<휏 푡 푇푖<휏 (푡) 푖 푛 Therefore,

푖 ℚ ℚ 푀 (3.4.17) 퐼 ̅ ( ) 푉푡 = 1푡<휏 푁푡피푡 (퐶푆 푇푖 ℙ푆,퐼(푡, 푡, 푇푖, ℱ푇푛)), where 퐼 (3.4.18) ℚ 퐶푆(휏 (푡)) 푀 피푡 ( 1푇 <휏퐼(푡)|ℱ푇 ) 푁휏퐼(푡) 푖 푛 퐶푆̅ (푇푖) = , ℙℚ (푡, 푡, 푇 , ℱ푀 ) 푆,퐼 푖 푇푛 ℚ 푀 ℚ 푀 ℙ푆,퐼(푡, 푡, 푇푖, ℱ푇푛) = 1 − ℙ퐷,퐼(푡, 푡, 푇푖, ℱ푇푛). ℚ 푀 And ℙ푆,퐼(푡, 푡, 푇푖, ℱ푇푛) is the survival probability of the market marker over [푡, 푇푖] conditional on the 푀 market filtration ℱ푇푛 . Here we define the conditional marginal default probability ℚ 푀 [ ] ℙ퐷,퐼(푡, 푇푖, 푇푖+1, ℱ푇푛) as the marginal default probability over 푇푖, 푇푖+1 conditional on the derivative dealer has not defaulted before 푡,

ℚ 푀 ℚ 푀 ℙ (푡, 푇 , 푇 , ℱ ) = 피 (1 퐼 |ℱ ). (3.4.19) 퐷,퐼 푖 푖+1 푇푛 푡 푇푖<휏 (푡)≤푇푖+1 푇푛

Then the conditional marginal default probability over [푡, 푇푖] becomes ℚ 푀 ℚ 푀 ℙ (푡, 푡, 푇 , ℱ ) = 피 (1 퐼 |ℱ ) (3.4.20) 퐷,퐼 푖 푇푛 푡 푡<휏 (푡)≤푇푖 푇푛 ℚ 푀 = 피 (1 퐼 + 1 퐼 + ⋯ + 1 퐼 |ℱ ), 푡 푡<휏 (푡)≤푇1 푇1<휏 (푡)≤푇2 푇푖−1<휏 (푡)≤푇푖 푇푛 that is,

ℚ 푀 ℚ 푀 ℚ 푀 ℙ (푡, 푡, 푇 , ℱ ) = 피 (1 퐼 |ℱ ) + ⋯ + 피 (1 퐼 |ℱ ) (3.4.21) 퐷,퐼 푖 푇푛 푡 푡<휏 (푡)≤푇1 푇푛 푡 푇푖−1<휏 (푡)≤푇푖 푇푛 푖 ℚ 푀 = ∑ ℙ퐷,퐼(푡, 푇푗−1, 푇푗|ℱ푇푛 ), 푗=1 ℚ 푀 ℚ 푀 where 푡 = 푇0. And hence the survival probability is ℙ푆,퐼(푡, 푡, 푇푖, ℱ푇푛) = 1 − ℙ퐷,퐼(푡, 푡, 푇푖, ℱ푇푛). 퐶 (휏(푡)) Here if the discounted survival payoff 푆 of the premium leg of credit extinguisher and 푁휏(푡) conditional survival process 1 퐼 (or conditional default process 1 ) are assumed to be 푇푖<휏 (푡) 푡<휏(푡)≤푇푖 푀 independent conditional on market filtration ℱ푇푛,

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퐼 (3.4.22) ℚ 퐶푆(휏 (푡)) 푀 피푡 ( 1푇 <휏퐼(푡)|ℱ푇 ) 푁휏퐼(푡) 푖 푛 퐶푆̅ (푇푖) = ℙℚ (푡, 푡, 푇 , ℱ푀 ) 푆,퐼 푖 푇푛 퐼 ℚ 퐶푆(휏 (푡)) 푀 ℚ 푀 피푡 ( |ℱ푇 ) ℙ푆,퐼(푡, 푡, 푇푖, ℱ푇 ) 푁휏퐼(푡) 푛 푛 = , ℙℚ (푡, 푡, 푇 , ℱ푀 ) 푆,퐼 푖 푇푛 thus, 퐶 (휏퐼(푡)) (3.4.23) ̅ ℚ 푆 푀 퐶푆(푇푖) = 피푡 ( |ℱ푇 ). 푁휏퐼(푡) 푛 Hence, 퐼 (3.4.24) 푖 ℚ ℚ 퐶푆(휏 (푡)) 푀 ℚ 푀 푉푡 = 1푡<휏퐼 푁푡피푡 (피푡 ( |ℱ푇 ) ℙ푆,퐼(푡, 푡, 푇푖, ℱ푇 )), 푁휏퐼(푡) 푛 푛 that is, 퐼 (3.4.25) 푖 ℚ ℚ 퐶푆(휏 (푡)) ℚ 푀 푀 퐼 푉푡 = 1푡<휏 푁푡피푡 (피푡 ( ℙ푆,퐼(푡, 푡, 푇푖, ℱ푇푛)|ℱ푇푛)) 푁휏퐼(푡)

퐼( ) ℚ 퐶푆(휏 푡 ) ℚ 푀 퐼 = 1푡<휏 푁푡피푡 ( ℙ푆,퐼(푡, 푡, 푇푖, ℱ푇푛)), 푁휏퐼(푡)

And

푛 푛 퐼 (3.4.26) 푖 ℚ 퐶푆(휏 (푡)) ℚ 푀 퐼 푉푡 = ∑ 푉푡 = 1푡<휏 푁푡 ∑ 피푡 ( ℙ푆,퐼(푡, 푡, 푇푖, ℱ푇푛)). 푁휏퐼(푡) 푖=1 푖=1 In the continuous time limit, the price of this credit contingent instrument is

푇푛 (3.4.27) ℚ 퐶푆(푢) ℚ 푀 푉 = 1 퐼푁 피 (∫ ℙ (푡, 푡, 푢, ℱ )푑푢), 푡 푡<휏 푡 푡 푁 푆,퐼 푇푛 푇1 푢 where ℚ 푀 ℚ 푀 (3.4.28) ℙ푆,퐼(푡, 푡, 푢, ℱ푇푛) = 1 − ℙ퐷,퐼(푡, 푡, 푢, ℱ푇푛). The above equation lays out the general pricing framework for credit extinguishers like CDS premium leg, FVA, etc. Convexity adjustment term accounting for the instantaneous covariance

퐶푆(푢) ℚ 푀 between and ℙ푆,퐼(푡, 푡, 푢, ℱ푇 ) could be introduced to make a linear approximation of its price, 푁푢 푛 and since

ℚ ℚ 푀 ℚ ℚ 푀 ℚ ℚ ( ) (3.4.29) 피푡 (ℙ푆,퐼(푡, 푡, 푢, ℱ푇푛)) = 피푡 (피푡 (1푢<휏(푡)|ℱ푇푛)) = 피푡 (1푢<휏(푡)) = ℙ푆,퐼 푡, 푡, 푢 , then,

푇푛 ℚ 퐶푆(푢) ℚ (3.4.30) 푉 = 1 퐼 푁 (∫ 피 ( ) ℙ (푡, 푡, 푢) 푑푢 푡 푡<휏 푡 푡 푁 푆,퐼 푇1 푢

푇푛 퐶푆(푢) + ∫ 퐶표푣ℚ ( , ℙℚ (푡, 푡, 푢, ℱ푀 )) 푑푢), 푡 푁 푆,퐼 푇푛 푇1 푢

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푇푛 퐶푆(푢) (3.4.31) 푉 = 1 푁 ∫ 피ℚ ( ) ℙℚ (푡, 푡, 푢)(1 + 휌휎 휎 (푢 − 푡)) 푑푢, 푡 푡<휏 푡 푡 푁 푆,퐼 1 2 푇1 푢

퐶푆(푢) ℚ where 휌 is the instantaneous correlation between and ℙ푆,퐼(푡, 푡, 푢). 휎1 denotes the instantaneous 푁푢

퐶푆(푢) ℚ percentage volatility of and 휎2 denotes the instantaneous percentage volatility of ℙ푆,퐼(푡, 푡, 푢). 푁푢 The funding valuation adjustment (FVA) is computed on upfront basis. Here 푡 = 0 and investment horizon is [0, 푇]. Then FVA is 푇 (3.4.32) ℚ 퐶푆(푡) ℚ 푀 푉0 = 10<휏퐼푁0피0 (∫ ℙ푆,퐼(0,0, 푡, ℱ푇 )푑푡), 0 푁푡 ℚ ℚ 푀 here if ℙ푆,퐼(푡) = ℙ푆,퐼(0,0, 푡, ℱ푇 ), 푇 ℚ 퐶푆(푡) ℚ (3.4.33) 푉0 = 피0 (∫ ℙ푆,퐼(푡)푑푡). 0 푁푡 If we use convexity adjustment to obtain a linear approximation of UCVA/credit charge, 푇 ℚ 퐶푆(푡) ℚ (3.4.34) 푉0 = 피0 (∫ ℙ푆,퐼(푡)푑푡) 0 푁푡 푇 푇 ℚ 퐶푆(푡) ℚ ℚ ℚ 퐶푆(푡) ℚ = ∫ 피0 ( ) 피0 (ℙ푆,퐼(푡)) 푑푡 + ∫ 퐶표푣0 ( , ℙ푆,퐼(푡)) 푑푡. 0 푁푡 0 푁푡 Since

ℚ ℚ ℚ ℚ 푀 ℚ ℚ 푀 ℚ (3.4.35) 피0 (ℙ푆,퐼(푡)) = 피0 (ℙ푆,퐼(0,0, 푡, ℱ푇 )) = 피0 (피0 (1푡<휏(0)|ℱ푇 )) = 피0 (1푡<휏(0))

ℚ = ℙ푆,퐼(0,0, 푡), and if ℚ ℚ ℚ ℚ (3.4.36) ℙ푆,퐼(0, 푡) = ℙ푆,퐼(0,0, 푡) = 피0 (ℙ푆,퐼(푡)), then 푇 ℚ 퐶푆(푡) ℚ (3.4.37) 푉0 = ∫ 피0 ( ) ℙ푆,퐼(0, 푡)(1 + 휌휎1휎2푡)푑푡, 0 푁푡

퐶푆(푡) ℚ where 휌 is the instantaneous correlation between and ℙ푆,퐼(0, 푡). 휎1 denotes the instantaneous 푁푡

퐶푆(푡) ℚ percentage volatility of and 휎2 denotes the instantaneous percentage volatility of ℙ푆,퐼(0, 푡). 푁푡 FVA is the expected amount of net cash flows the derivative dealer needs to post/receive on the perfect collateralised hedging counterparty with respect to the ongoing transactions the derivative dealer has with uncollateralised or partially collateralised counterparty. Therefore, the net cash flow generated by FVA with the uncollateralised or partially collateralised counterparty is 퐹 퐶푆(푡) = 푀푡푀푡푠푡 , (3.4.38) 퐹 where 푀푡푀푡 refers to the Mark to Market of the ongoing transactions at time 푡, and 푠푡 denotes the 퐹 퐹 푂퐼푆 funding spread of the derivative dealer 푠푡 = 푟푢 − 푟푢 . Therefore,

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푇 (3.4.39) ℚ 푀푡푀푡 퐹 ℚ 푉0 = ∫ 피0 ( ∙ 푠푡 ) ℙ푆,퐼(0, 푡)(1 + 휌휎1휎2푡)푑푡. 0 푁푡

푀푡푀푡 퐹 Here if we define the correlation between and 푠푡 as 휌푀푡푀,푆퐹 and the volatility of funding 푁푡

퐹 푀푡푀푡 spread 푠푡 and volatility of discounted Mark to Market as 휎푆퐹 and 휎푀푡푀, respectively. Then the 푁푡 퐶 (푡) volatility of the 푆 satisfies 푁푡

2 2 (3.4.40) 휎1 = √휎푆퐹 + 휎푀푡푀 + 2휌푀푡푀,푆퐹휎푆퐹 휎푀푡푀. Hence, 푇 (3.4.41) 퐹 ℚ 푀푡푀푡 ℚ 푉0 = ∫ 푠0 (푡)피0 ( ) ℙ푆,퐼(0, 푡)(1 + 휌휎1휎2푡)(1 + 휌푀푡푀,푆퐹휎푆퐹휎푀푡푀푡)푑푡, 0 푁푡 퐹 ℚ 퐹 where 푠0 (푡) = 피0 (푠푡 ). If we further assume 휌 = 휌푀푡푀,푆퐹 = 0, 푇 (3.4.42) 퐹 ℚ 푀푡푀푡 ℚ 퐹푉퐴 = ∫ 푠0 (푡)피0 ( ) ℙ푆,퐼(0, 푡)푑푡. 0 푁푡 And + − (3.4.43) + ℚ 푀푡푀푡 − ℚ 푀푡푀푡 퐸퐸0 (푡) = 피0 ( ) , 퐸퐸0 (푡) = 피0 ( ) , 푁푡 푁푡

+ − ℚ 푀푡푀푡 퐸퐸0 (푡) + 퐸퐸0 (푡) = 피0 ( ), 푁푡 Then 푇 + (3.4.44) 퐹 ℚ 푀푡푀푡 ℚ 퐹퐶퐴 = ∫ 푠0 (푡)피0 ( ) ℙ푆,퐼(0, 푡)푑푡, 0 푁푡 푇 − 퐹 ℚ 푀푡푀푡 ℚ 퐹퐵퐴 = − ∫ 푠0 (푡)피0 ( ) ℙ푆,퐼(0, 푡)푑푡, 0 푁푡 퐹푉퐴 = 퐹퐶퐴 − 퐹퐵퐴. It is necessary to include the survival state of the uncollateralised or partially collateralised counterparty into the pricing of FVA as the underlying transactions would close out at the default time of the counterparty and hence FVA term would disappear accordingly. This would be addressed in the following sections.

Gregory (2012) develops an intuitive approach to structure FVA problem. The post GFC (global financial crisis) OTC derivatives market regime brings the payoffs to both parties in the default states in bilateral OTC transactions (i.e. CVA and DVA) into pricing framework. The asymmetric treatments of collateral in today’s OTC market structure suggests the potential negative carry (FCA) and positive carry (FBA) in the survival states (or non-default states) should be built into the pricing regime as well to complete the classic self-financing strategy under no arbitrage framework. In bilateral CVA structure, CVA/credit charge calculates the price of loss given default (LGD) of the

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Counterparty Credit Risk, Funding Risk and Central Clearing counterparty in the survival state of the derivative dealer with its probability density

ℚ ℚ ℚ ℚ ℙ퐷,퐶푃(0, 푡)ℙ푆,퐼(0, 푡) (i.e. ℙ퐷,퐶푃(0, 푡) (1 − ℙ퐷,퐼(0, 푡))), and DVA/credit benefit calculates the price of loss given default (LGD) of the derivative dealer itself in the survival state of the its counterparty

ℚ ℚ ℚ ℚ with its probability density ℙ퐷,퐼(0, 푡)ℙ푆,퐶푃(0, 푡) (i.e. ℙ퐷,퐼(0, 푡) (1 − ℙ퐷,퐶푃(0, 푡))). (Gregory (2012) suggests that the survival probability terms could be removed to account for potential impact of substitution closeout, we keep them here for completeness.) FVA here calculates the price of carry of collateral accounts in market marking derivatives (i.e. trading and hedging) with probability

ℚ ℚ ℚ ℚ ℙ푆,퐶푃(0, 푡)ℙ푆,퐼(0, 푡) (i.e. (1 − ℙ퐷,퐼(0, 푡)) (1 − ℙ퐷,퐶푃(0, 푡))) completes the missing component of the probability space. Here we assume the double defaults at the same time in bilateral markets is ℚ ℚ impossible, i.e. ℙ퐷,퐶푃(0, 푡)ℙ퐷,퐼(0, 푡) = 0 . Gregory (2012) suggests that FVA is equal to the following linear term with independence between market and credit variables in place, 푇 (3.4.45) 퐹,+ + ℚ ℚ 퐹푉퐴0 = ∫ 푠0 퐸퐸0 (푡)ℙ푆,퐶푃(0, 푡)ℙ푆,퐼(0, 푡)푑푡 0 푇 퐹,− − ℚ ℚ + ∫ 푠0 퐸퐸0 (푡)ℙ푆,퐶푃(0, 푡)ℙ푆,퐼(0, 푡)푑푡, 0 퐹,+ 퐹,− where 푠0 and 푠0 denotes the funding spread for borrowing and lending, respectively. Brigo et al. (2012) suggests that mathematically it is not correct to have a linear representation of FVA with asymmetric funding spreads. Therefore, the linear FVA term derived by Burgard and Kjaer (2012) is used in the following sections of FVA, 푇 (3.4.46) 퐹 ℚ 푀푡푀푡 ℚ ℚ 퐹푉퐴0 = ∫ 푠0 (푡)피0 ( 푂퐼푆 ) ℙ푆,퐶푃(0, 푡)ℙ푆,퐼(0, 푡)푑푡, 0 푁푡 and 푇 + (3.4.47) 퐹 ℚ 푀푡푀푡 ℚ ℚ 퐹퐶퐴0 = ∫ 푠0 (푡)피0 ( 푂퐼푆 ) ℙ푆,퐶푃(0, 푡)ℙ푆,퐼(0, 푡)푑푡, 0 푁푡 푇 − 퐹 ℚ 푀푡푀푡 ℚ ℚ 퐹퐵퐴0 = − ∫ 푠0 (푡)피0 ( 푂퐼푆 ) ℙ푆,퐶푃(0, 푡)ℙ푆,퐼(0, 푡)푑푡, 0 푁푡

퐹푉퐴0 = 퐹퐶퐴0 − 퐹퐵퐴0.

Further approximations could be made with additional assumptions and restrictions on the funding spread and survival probabilities. These approximations will not be discussed here. Interested readers could find the relevant discussions in Gregory (2012).

Derivatives are not term funded in practice. In other words, the derivative dealer would not finance a 푛 years interest rate swap by issuing 푛 years bond. Therefore, although the funding components discussed so far are short term funds used to fill the gaps created by asymmetric collateral requirements in OTC derivatives trading, these short terms funds do carry more risks like liquidity

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Counterparty Credit Risk, Funding Risk and Central Clearing risk and credit risk etc. compared to the collateral return rate, i.e. OIS rate. The asymmetric funding rate problem is well addressed in Gregory (2012) and Brigo et al. (2012). The derivative dealer is not able to borrow the short term fund at the same rate it is able to lend in the marketplace. More specifically, given the collateral funding is short term in nature and derivatives repo market is not available, the derivative dealer has to contact its treasury department to finance its margin calls. This indicates the derivative dealer would borrow at its own funding curve in face of shortage of collateral cash. If the derivative dealer has excess collateral cash, and the treasury normally does not use the short term funds to retire long term debt, the derivative dealer has to enter into reverse repo market to lend short term cash. The return the derivative dealer earns on the reverse repo is GC (general collateral) rate and the collateralised GC rate normally is expected to be lower than uncollateralised OIS rate. Kaminska (2010) reported GC was trading cheaper to OIS, i.e. GC rate was higher than the OIS rate during that period of time, the upside of investing short term excess collateral is almost 퐺퐶 푂퐼푆 nothing (푟푡 − 푟푡 → 0). Therefore, the theoretical funding benefits term FBA does not materially exist in practice. This forms up the basis for FVA/FDA accounting methodology proposed by Albanese and Andersen (2014). The detailed accounting treatments of CVA/DVA/FVA would be discussed at the end of this chapter.

Let’s work through an example to explain the intricacies of FVA. Suppose party A enters into a fixed payer interest rate swap with party B. If the current yield curve is monotonic upward sloping, the forward curve derived from the yield curve is upward sloping as well and the floating leg LIBOR rate is expected to ‘climb up’ the forward curve under risk neutral measure. This leads to positive Mark to Market (and negative carry) in the first half of the term and negative Mark to Market (and positive carry) in the second half of the term. Party A here is assumed to have symmetric funding spreads. By taking discounting factor into account, overall, the exposure dominates the negative exposure (i.e. + − 퐸퐸̅̅̅̅0 > −퐸퐸̅̅̅̅0 ) and hence party A faces a net funding cost for this transaction under risk neutral measure. As discussed in previous sections, under real world measure, the floating LIBOR rate is expected to fluctuate around the current spot LIBOR rate level within a given range of volatilities. This would result in a significant negative Mark to Market from party A’s (fixed payer’s) perspective − under real world measure. Therefore, the negative exposure dominates the exposure (i.e. −퐸퐸̅̅̅̅0 > + 퐸퐸̅̅̅̅0 ) and party A faces a net funding benefit for this transaction under real world measure. Since funding cost are un-hedgeable variables in current pricing framework, it is not necessary to price or analyse FVA under risk neutral measure.

Current market practice is the derivative dealers tend to transfer the funding components to its counterparties by building FVA into final prices. Given the example above, the derivative dealers would charge the counterparties net funding cost for paying fixed in a plain-vanilla interest rate swap. The derivative dealers with better credit qualities/ratings or more collateral assets would be able to

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Counterparty Credit Risk, Funding Risk and Central Clearing offer more competitive prices in this scenario. An uncollateralised or partially collateralised counterparty could potentially benefits from paying fixed in an interest rate swap with a derivative dealer with lower credit qualities or higher funding spreads. This is because the derivative dealer is receiving fixed in this transaction and fixed receiver swap would bring net funding benefits to the derivative dealer. The net funding benefits would be greater if the derivative dealer is funding at a higher rate than its competitors. The portfolio netting effect is not taken into account in above discussions. In terms of assessment of future funding cost, the derivative dealer could charge its counterparty lower funding cost if there is enough evidence that its funding spread is going to drop substantially within the life of transaction. Gregory (2012) suggests that the funding cost, unlike CVA, does not have to be Mark to Market. The hedging of funding cost is impossible in current stage of market development as it requires the derivative dealer to be able to trade their own funding spread on a forward basis which calls for a liquid forward starting corporate bond/bond options and its related swaptions market.

Before moving on to discuss the mathematically linkages between CVA/DVA/FVA, we have a brief discussion of the economics of these three terms here. Suppose party A traded a portfolio of OTC derivatives with party B and party A is in the money. In other words, party A has net receivables from party B and effectively party A is lending short term extensible loan to party B. Since there does not exist a repo market for OTC derivatives receivables, party A cannot get financed from the market and has to charge party B for this short term lending. Hence the pricing of funding cost, from this perspective, follows the same pricing structure of CVA except using party A’s funding curve instead of party B’s CDS curve. Party A’s funding curve is extracted from senior unsecured bonds issued by party A with additional funding instruments party A uses in running its business. Economically speaking, the net funding cost party A charging party B is a served as a compensation for senior bond holders and additional funding instruments providers of party A for using their capital to run short term extensible loan to party B. Party A, as a derivative dealer for certain types of derivatives, normally price portion of its expected funding cost into bid-ask spread of the final price. With the reform of market regime, the actual carrying cost of holding derivatives positions due to asymmetries in collateral requirements in trading business and portfolio effects may be significantly more than the bid-ask spread for general trading purpose. Party A may experience negative carry cost over time and a funding reserve therefore could be set up equal to the expected net funding cost at inception to offset the potential funding costs and amortise this funding reserve by actual funding costs over time. Now suppose party A is out of money and therefore has net payables to party B. Party A effectively is borrowing a short term extensible loan from party B and party B has to reallocate its available financial resources to fund this short term loan. These financial resources party B have could be used to fund party B’s receivables on other contracts, meet margin calls or retire pari passu bonds. Party A therefore has a net funding benefit in this circumstance. The pricing of funding benefits follows the

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Counterparty Credit Risk, Funding Risk and Central Clearing same pricing structure of DVA except using party A’s funding spread instead of its CDS spread. The party A’s funding spread here is the spread of party A’s pari passu bonds. Therefore, the DVA and funding benefits shares a large component of value and common economic origin except on CDS bond basis spread. And it would result in double counting problems for pricing funding benefits into final execution price if DVA is already included in the fair price of OTC derivatives portfolios. Funding benefits, therefore, is a positive carry earned over time in the survival state of party A to offset the loss of value due to time decay (or Theta) of DVA in the default state of party A. This result holds if rehypothecation is allowed within netting sets but disallowed across netting sets. The detailed discussion would be found later this chapter. From an accounting perspective, the fair price of a bilaterally traded OTC derivatives portfolio should take the default risk of both parties into final price without further adjustments of funding conditions of both parties as suggested by Modigliani and Miller (1958) and Hull and White (2013). From a business perspective, trading OTC derivatives do take a handful amount of funding costs and this should be built into the final price even in a competitive market.

Netting and collateral have a significant impact in the final pricing and risk management of counterparty credit risk and funding risk. Netting of OTC derivatives in a given portfolio with a counterparty (or more specifically, legal entity) could substantially reduce counterparty risk and hence CVA and DVA. Netting is not always available for trading OTC derivatives. For example, the netting agreements do not have legal enforceability in some regions like Russia and some Middle East countries, and trading with counterparties domiciled within these regions could not have netting effect in pricing CVA/DVA/FVA. Normally a large banking corporation would have different legal entities around the global to optimise its business operations. The OTC derivatives of these different legal entities are generally not netted against each other. Collateralisation could reduce CVA and DVA significantly as it reduces the size of overall exposure. Its impact on funding costs/benefits is not direct and depends on if the collateral posted/received could be rehypothecated or reused. Suppose the market marker receives collateral from its counterparty and this collateral could be rehypothecated. This means the market marker could post these received collateral to other counterparties and obtain the funding of collateral at OIS rate instead of its much higher unsecured funding rate. Hence, rehypothecation could significantly reduce the funding cost of the derivative dealer.

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3.5 CVA, DVA and FVA at Trade Level

Pricing and modelling funding costs and benefits generally use the same modelling framework of CVA/DVA (under certain assumptions). FVA pricing and analysis could help the derivative dealer price its cost of running certain types of derivatives business and align the interests of individual trading desks with overall interest of the derivative dealer. For example, the trading desk could enter into an uncollateralised long term derivative position at day 1 and the whole firm has to finance the long term funding cost of running this position. Given the similarities in pricing CVA/DVA and FVA, it becomes a natural choice for the derivative dealer to use its CVA desk to price and analyse funding costs and funding benefits. The funding function (within the CVA desk or a separate central funding desk) should aim to optimise the collateral usage (especially rehypothecation across different counterparties), reduce the funding cost and improve balance sheet management. Mutual break clauses, re-couponing and trade compressions could be introduced to optimise the portfolio holdings. Netting and collateralisation, especially non-cash collateral, could be used to enhance the counterparty credit quality. The non-cash collaterals include stocks and bonds issued by the counterparty and further analysis on the Wrong Way Risk of collateral should be built into the integrated collateral transformation/trading function. The derivatives portfolios could be re- engineered and repackaged into structured notes like credit linked notes (CLNs) to reduce consumption of balance sheet.

It seems natural to use FBA to replace DVA under certain assumptions. On the trade/micro level, double counting issues make the co-existence of FBA and DVA problematic. The bilateral CVA (from the derivative dealer’s perspective) is 푇 + 퐶푃 (3.5.1) ℚ 푀푡푀푡 (1 − 푅푡 ) ℚ ℚ 퐶푉퐴 = 피0 (∫ 푂퐼푆 ℙ푆,퐼(푡) 푑푡ℙ퐷,퐶푃(푡)), 0 푁푡

푇 − 퐼 ℚ 푀푡푀푡 (1 − 푅푡) ℚ ℚ 퐷푉퐴 = −피0 (∫ 푂퐼푆 ℙ푆,퐶푃(푡) 푑푡ℙ퐷,퐼(푡)), 0 푁푡 and FVA (from the derivative dealer’s perspective) is 푇 + (3.5.2) ℚ 퐹 푀푡푀푡 ℚ ℚ 퐹퐶퐴 = 피0 (∫ 푠푡 푂퐼푆 ℙ푆,퐶푃(푡)ℙ푆,퐼(푡)푑푡), 0 푁푡 푇 − ℚ 퐹 푀푡푀푡 ℚ ℚ 퐹퐵퐴 = −피0 (∫ 푠푡 푂퐼푆 ℙ푆,퐶푃(푡)ℙ푆,퐼(푡)푑푡), 0 푁푡 ℚ ℚ 푀 ℚ ℚ 푀 where ℙ푆,퐼(푡) = ℙ푆,퐼(0,0, 푡, ℱ푇 ) and ℙ퐷,퐼(푡) = ℙ퐷,퐼(0,0, 푡, ℱ푇 ) are the survival probability and 푀 ℚ default probability of the derivative dealer conditional on the market filtration ℱ푇 , and ℙ푆,퐶푃(푡) = ℚ 푀 ℚ ℚ 푀 ℙ푆,퐶푃(0,0, 푡, ℱ푇 ) and ℙ퐷,퐶푃(푡) = ℙ퐷,퐼(0,0, 푡, ℱ푇 ) are survival probability and default probability 푀 of the counterparty conditional on the market filtration ℱ푇 . The conditional marginal default

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ℚ 푀 probability ℙ퐷,퐼(푡, 푇, 푇 + Δ푇, ℱ푇 ) of the derivative dealer (conditional on the derivative dealer has not defaulted before 푡) is defined as

ℚ 푀 ℚ 푀 ℚ 퐼 퐼 ℚ 퐼 ℙ (푡, 푇 , 푇 + Δ푇 , ℱ ) = 피 (1 퐼 |ℱ ) = ℙ (휏 (푡) ≤ 푇 + Δ푇 |휏 (푡) > 푇 ) = 피 (휆 Δ푇 ). 퐷,퐼 푖 푖 푖 푇 푡 푇푖<휏 (푡)≤푇푖+Δ푇푖 푇 푖 푖 푖 푡 푇푖 푖

Here 0 < 푡 < 푇푖 < 푇. Therefore the conditional marginal survival probability of the derivative dealer is

ℚ 푀 ℚ 푀 ℙ (푡, 푇 , 푇 + Δ푇 , ℱ ) = 피 (1 퐼 퐼 |ℱ ), (3.5.3) 푆,퐼 푖 푖 푖 푇 푡 푇푖+Δ푇푖<휏 (푡)|푇푖<휏 (푡) 푇 that is, ℚ 푀 ℚ 퐼 퐼 푀 (3.5.4) ℙ푆,퐼(푡, 푇푖, 푇푖 + Δ푇푖, ℱ푇 ) = ℙ (휏 (푡) > 푇푖 + Δ푇푖|휏 (푡) > 푇푖, ℱ푇 ) ℚ 퐼 = 피푡 (1 − 휆푇푖Δ푇푖).

If we slice the time horizon [푡, 푇푖] into equal size time windows with band width Δ푇, then the survival probability of the derivative dealer over [푡, 푇푖] is ℚ 푀 ℚ 퐼 퐼 퐼 푀 (3.5.5) ℙ푆,퐼(푡, 푡, 푇푖, ℱ푇 ) = ℙ (휏 (푡) > 푇푖|휏 (푡) > 푇푖−1, … , 휏 (푡) > 푡, ℱ푇 ), that is, ℚ 푀 ℚ 퐼 퐼 푀 (3.5.6) ℙ푆,퐼(푡, 푡, 푇푖, ℱ푇 ) = ℙ (휏 (푡) > 푇푖|휏 (푡) > 푇푖−1, ℱ푇 ) ℚ 퐼 퐼 푀 ∙ ℙ (휏 (푡) > 푇푖−1|휏 (푡) > 푇푖−2, ℱ푇 ) ∙ … ℚ 퐼 퐼 푀 ∙ ℙ (휏 (푡) > 푡|휏 (푡) > 푡, ℱ푇 ). Therefore,

ℚ ( 푀) ℚ 퐼 퐼 ( 퐼 ) (3.5.7) ℙ푆,퐼 푡, 푡, 푇푖, ℱ푇 = 피푡 ((1 − 휆푇푖−1Δ푇) ∙ (1 − 휆푇푖−2Δ푇) ∙ … ∙ 1 − 휆푡Δ푇 ). Hence in the continuous time limit Δ푇 → 0,

ℚ ℚ −휆퐼 Δ푇 −휆퐼 Δ푇 퐼 ℚ 푖 퐼 푀 푇 푇 −휆푡Δ푇 − ∑ 휆푗Δ푇 (3.5.8) lim ℙ (푡, 푡, 푇푖, ℱ푇 ) = 피 (푒 푖−1 푒 푖−2 … 푒 ) = 피 (푒 ) Δ푇→0 푆,퐼 푡 푡

ℚ 푇푖 퐼 − ∫푡 휆푠푑푠 = 피푡 (푒 ). 푀 Therefore, the survival probability conditional on market filtration ℱ푇 over [0, 푡] for the derivative dealer is

ℚ ℚ ℚ 푡 퐼 푀 − ∫0 휆푠푑푠 (3.5.9) ℙ푆,퐼(푡) = ℙ푆,퐼(0,0, 푡, ℱ푇 ) = 피푡 (푒 ). Then the its default probability over [0, 푡] is

ℚ ℚ ℚ ℚ 푡 퐼 푀 푀 − ∫0 휆푠푑푠 (3.5.10) ℙ퐷,퐼(푡) = ℙ퐷,퐼(0,0, 푡, ℱ푇 ) = 1 − ℙ푆,퐼(0,0, 푡, ℱ푇 ) = 1 − 피푡 (푒 ), then the marginal default probability or default probability density is

ℚ ℚ 푡 퐼 ℚ 푡 퐼 ℚ (3.5.11) − ∫0 휆푠푑푠 퐼 − ∫0 휆푠푑푠 퐼 푑푡ℙ퐷,퐼(푡) = 푑푡 (1 − 피푡 (푒 )) = 휆푡피푡 (푒 ) 푑푡 = 휆푡ℙ푆,퐼(푡)푑푡.

퐹 The funding spread of derivative dealer 푠푡 is extracted from the senior unsecured pari passu bond issued by the derivative dealer and in theory the recovery of the derivatives contract is equal to the recovery of senior unsecured bond. If CDS bond basis spread is 0, the funding spread here is equal

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Counterparty Credit Risk, Funding Risk and Central Clearing to the CDS spread trading in the market. Brigo and Mercurio (2006) and O’Kane (2008) derived the classic credit triangle relationship with a flat hazard rate curve, 푠퐹 = (1 − 푅퐼)휆퐼. (3.5.12) Then the DVA term is

푇 − 퐼 ℚ 푀푡푀푡 (1 − 푅푡) ℚ ℚ (3.5.13) 퐷푉퐴 = −피0 (∫ 푂퐼푆 ℙ푆,퐶푃(푡) 푑푡ℙ퐷,퐼(푡)) 0 푁푡 푇 −( 퐼) ℚ 푀푡푀푡 1 − 푅푡 ℚ 퐼 ℚ = −피0 (∫ 푂퐼푆 ℙ푆,퐶푃(푡) 휆푡ℙ푆,퐼(푡)푑푡). 0 푁푡 And it could be rewritten as 푇 − (3.5.14) ℚ 퐹 푀푡푀푡 ℚ ℚ 퐷푉퐴 = −피0 (∫ 푠푡 푂퐼푆 ℙ푆,퐶푃(푡) ℙ푆,퐼(푡)푑푡) = 퐹퐵퐴. 0 푁푡 This result gives rise to two pricing frameworks for counterparty credit risk and funding risk. If the derivative dealer ignores the DVA term as it is illogical for the derivative dealer (as a whole) to benefit from its own default, the entire pricing adjustment becomes CVA plus FVA (i.e. 퐶푉퐴 + 퐹퐶퐴 − 퐹퐵퐴 ). If the derivative dealer is able to monetise its DVA term (via charging UCVA to its counterparty or selling put options on itself and CDSs of its peers), the entire pricing adjustment becomes BCVA plus FCA (i.e. 퐶푉퐴 − 퐷푉퐴 + 퐹퐶퐴). Current accounting standards requires CVA and DVA have to be marked to market but FVA could be computed under accrual accounting principle. Therefore, BCVA plus FCA framework is implicitly supported by current accounting standard and leaves some room to the derivative dealer to retain benefits of DVA. Funding costs and benefits, as discussed above, are carry terms in running trading business and hence qualified as contributing components to PnL. BCVA plus FCA framework actually uses the funding benefits terms to offset CVA instead of FCA. Essentially this method uses expected positive carry to offset potential insurance liabilities without touching negative carry and leaves the expected negative carry on accrual accounting terms. This would potentially cause conflict of interests between CVA desk and central funding desk. If the derivative dealer enters into a derivative transaction with a counterparty under SCSA and hedges the underlying trade with back to back transaction under identical SCSA terms, the funding costs and benefits would perfectly offset with each other and hence the central funding desk has zero net funding costs. CVA desk of the derivative dealer in this case could not recognise DVA contribution. These results generally hold at counterparty level (or netting set level) but fail to capture the potential benefits of rehypothecation on the funding sets level (or cross netting sets level) in the funding costs and benefits analysis. In practice, the perfect cancellation of funding costs and benefits discussed above is hard to achieve due to the existence of mismatch of collateral terms like thresholds, MTAs (minimum transfer amounts), and rounding etc. between trading counterparty and hedging counterparty and time gap in between receiving and posting collateral (from the derivative dealer’s perspective). Since CVA and FCA are driven by expected positive exposures, the funding cost component and CVA is created by the exposure

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Counterparty Credit Risk, Funding Risk and Central Clearing roughly up to the threshold (as the first margin call would be made if the exposure cross the sum of threshold and MTA mark). The economics of funding analysis indicates the funding of derivatives trading is collateral funding, therefore, the collateral terms improvement with certain counterparties (for example, from uncollateralised counterparties to partially collateralised counterparties) would effectively reduce the funding cost of the derivative dealer. The collateral transformation/trading desk could work out further plans and innovative solutions to convert/repo the securities collateral held by counterparties into reusable cash collateral to facilitate the ongoing reform of the derivatives trading process. An interesting example on funding analysis would be the net funding costs and benefits of trading a derivative with uncollateralised counterparty and hedging with a partially collateralised counterparty. Suppose the underlying trade is a receiver swap. In terms of mathematics, the receiver swap on the uncollateralised side ends up with a net funding benefit. The derivative dealer trades a payer swap on the hedging side and this leaves with a net funding cost. The net funding benefits would be significantly smaller compared to the funding benefits as the exposure on the hedging side are cap with threshold of CSA. Hence these transactions would leave the derivative dealer a net funding benefit. In terms of economics, this net funding benefits could be perceived as the potential benefits of receiving collateral from the hedging side if the market sells off (and hence the rates go up) and Mark to Market rises above the threshold but not posting any collateral on the original trade net of the cost of posting on the hedge but not receiving collateral if the reverse scenario happens.

Collateral could significantly reduce size of exposure profile and have direct impacts on the funding profile and regulatory capital profile. For uncollateralised transactions, CVA regulatory capital charges are expensive and market risk hedges could not release capital and CDS index hedges could give a capital relief up to 50% in most cases (unless the derivative dealer has sufficient evidence the existence of significant high correlation between the counterparty credit quality and CDS index). For collateralised transactions, CVA regulatory capital charges would be less compared to uncollateralised trades. The precise quantification of funding cost/benefit of collateral (i.e. the funding rate of the derivative dealer minus the OIS rate of the collateral currency) with full usage of rehypothecation remains to be a challenge. Collateral switch optionality and cheapest to deliver option embedded in CSAs would give rise to different optimisation strategies to improve the overall collateral operations mechanism and material contributions to PnL. A significant portion of OTC derivatives now moves into central counterparties (CCPs). Centrally cleared OTC derivatives could achieve a significant capital relief at the expense of initial margin requirements. Given the initial margin has to be posted in segregated accounts could not be rehypothecated or reused, the cost of initial margin would be a significant contributor to overall funding cost of the derivative dealer. The general clearing members now provide margin lending service to non-clearing members. The margin lending market would easily go frozen in stressed market conditions and might create a new source of systematic risk in the

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Counterparty Credit Risk, Funding Risk and Central Clearing global financial system. The regulators intend to push the OTC derivatives market from uncollateralised phase to collateralised (two way CSA) phase and over-collateralised (CCP) phase. Since CCPs require both parties to post initial margins, central clearing provides a methodology to minimise counterparty credit risk (through reductions in CVA and DVA) at the expense of significant higher funding cost. CCP, therefore, is a mechanism to convert counterparty credit risk into funding risk. The credit risk or too big to fail problem of CCP will be discussed in the central clearing chapter. Initial margin, or independent amount, could be viewed as a negative threshold in exposure calculation with no netting effect as the initial margin or independent amount are posted into segregated accounts. The initial margin is like one way CSA, that is, the general clearing members have to post to CCP but CCP does not post initial margin to general clearing members. With strict variation margin and initial margin in place, the entire exposure get shifted up such that there is tiny expected negative exposure left. DVA or FBA would decline to close to 0. CVA will be technically made to 0 due to the assumption of excellent credit quality of CCP and general clearing members (GCMs) do not charge CCP for CVA in practice. FCA now becomes the dominant component of entire valuation adjustment term. The optimisation of trading price adjustments (CVA/DVA/FVA), regulatory capital charges for CVA, liquidity and leverage ratios by varying initial margins and other parameters is and remains to be a huge challenge in the next few years due to the complexity of counterparty risk, funding risk and their interactions with regulatory capitals. The market is undergoing tremendous changes in many different aspects. SCSA, i.e. standard CSA, is produced by ISDA as an update on CSA to remove the optionality embedded in CSAs and create a homogenous collateral valuation framework. For example, the cheapest to deliver option depends on future exposure profile, LIBOR OIS spread, and other features linked to the collateral like haircuts etc. Details of SCSA and comparison to CCP margin requirements will be presented in central clearing chapter.

In the post GFC (global financial crisis) era, the push to collateralisation reduces counterparty credit risk but requires the market participants to raise enough liquidity to fulfil their collateral calls. The FVA is the price of cost and benefits of trading with not fully collateralised counterparty and hedging with fully collateralised counterparty. Here we could define a new measurement called Funding of Collateral (FOC) and use it to compare the associated costs and benefits for the collateral requirements evolvements. The self-financing strategy in classic Black Scholes framework without a collateral account gives us the price of an option with a contingent payoff 푉푇 at maturity,

푇 퐹 퐹 ℚ − ∫ 푟푠 푑푠 (3.5.15) 푉푡 = 피푡 (푒 푡 푉푇).

퐹 푉푡 is no arbitrage price of an OTC derivative by following self-financing strategy as the derivative dealer has to fund their hedging instrument (with the general held assumption the receivables of OTC 퐹 derivatives could not be repoed). And therefore, 푉푡 is the fair price of an uncollateralised derivative.

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Piterbarg (2010) presents the no arbitrage framework of pricing an OTC derivative with a collateral account in place. The fair price of the OTC derivative (in absence of counterparty risk) is 푇 푇 퐹 푢 퐹 (3.5.16) ℚ − ∫ 푟푢 푑푢 ℚ − ∫ 푟푣 푑푣 퐹 푂퐼푆 푉푡 = 피푡 (푒 푡 푉푇) + 피푡 (∫ 푒 푡 (푟푢 − 푟푢 )풞푢푑푢). 푡

Then the funding of collateral (FOC) is therefore 푇 푢 퐹 (3.5.17) 퐹 ℚ − ∫ 푟푣 푑푣 퐹 푂퐼푆 퐹푂퐶 = 푉푡 − 푉푡 = 피푡 (∫ 푒 푡 (푟푢 − 푟푢 )풞푢푑푢). 푡

Since collateral posting/receiving is a zero sum game, an increase of the magnitude of absolute value of collateral account 풞푢 indicates either one of the two parties in a given derivative transaction would have to raise more collateral. This essentially increase the funding liquidity risk of the whole financial system. Two way CSA requires both parties to set up collateral accounts with pre-agreed thresholds, MTAs, etc. SCSA removes the optionality like collateral switch option and sets the threshold at 0. Central clearing margining requirements further improve the collateral requirement by initial margins. The counterparty credit risk would be gradually removed from the system (or concentrated into one ‘default remote’ hub) and replaced by funding liquidity risk as the market participants have to raise more and more collaterals to meet tougher margin requirements.

The incompleteness of Repo Market makes it costly for derivative dealers to run trading business in current regulatory regime. The asymmetry of collateral requirements in derivatives trading makes the derivative dealers systematically short of collateral and creates a significant negative carry for them to procure collateral. The funding valuation adjustment (FVA) discussed above is the price of costs and benefits of posting/receiving collateral in trading process (i.e. trading and hedging a derivatives portfolio). And DVA is the price discount the derivative dealer grants to its counterparty as a compensation for not delivering the full Mark to Market (or the full amount of shortfall of collateral) at the default time of the derivative dealer before the expiry of the underlying transaction. Albanese and Iabichino (2013) further develops the linear representation form of FVA (with symmetric funding spreads assumptions discussed in Brigo et al. (2012)) with specifications of variation margins and initial margins accounts. Details would be found later this chapter. DVA, has been identified overlapped with FBA, raises a controversial but interesting question. Suppose the credit spread widens significantly and the derivative dealer is heading towards its default, DVA here would surge and create positive PnL in this scenario. Economically speaking, should the DVA PnL contribute to the wealth of shareholders of the derivative dealer? In theory, it should, as suggested by Modigliani and Miller (1958), the shareholders could realise this profit (the increase in DVA) by buying back the outstanding debt at a cheap price. Effectively, this is the strategy to monetise FBA. The difference between FBA and DVA will be discussed later. However, this strategy is problematic and hard to implement in practice as the firm is short of cash at that time. And if the derivative dealer does

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Counterparty Credit Risk, Funding Risk and Central Clearing default, DVA is realised as a writing off of the liabilities to the derivatives counterparties and hence comes as a benefit for the senior unsecured bond holders. The shareholders’ wealth would be completely wiped out in this circumstance.

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3.6 CVA, DVA and FVA at Portfolio Level

Hull and White (2013) suggests that FVA would break the Modigliani and Miller (1958) general equilibrium of valuation of a given derivative dealer as the overall valuation of a derivative dealer or a project should not be affected by its funding strategies. Albanese and Iabichino (2013) points out funding strategies of collateral account in OTC derivatives trading would transfer wealth across the capital structure of the derivative dealer from its shareholders to bondholders but the overall valuation of the derivative dealer should remain immune to funding strategies. As discussed above, shareholders, in theory, should capture the arbitrage opportunity and use equity capital to buy back the derivative dealer’s debt. Given it is problematic to implement this arbitrage strategy, the derivative dealer could choose to transfer these funding costs to its counterparties as a compensation for its shareholders or complete the Repo market by creating synthetic repo trading market structure for OTC derivatives receivables. Margin lending suggested by Albanese and Iabichino (2013) is perceived as one possible approach to create synthetic repo market but the opacity of the derivatives trading book held by a derivative dealer and potential moral hazard or adverse selection problems would be detrimental to the development of this market. Margin lending, as suggested by Gregory (2012), is vulnerable to market volatility and could be frozen in market distressed situations. In theory, the OTC derivatives insured by the CVA desk of the derivative dealer is hedged against counterparty credit risk and could be pledged as the underlying asset of a repo transactions with a haircut accounting for its market volatility. The repo rate is not sensitive to the counterparty credit risk but the quality of the underlying asset. The derivative dealer in principle would have to pay DVA to the counterparty at inception of an OTC transaction for paying less if the counterparty is in the money in the default state of the derivative dealer and receive FVA from the counterparty at inception of that transaction for extra costs and benefits in hedging with a fully collateralised counterparty. Then the derivative dealer would pay (and receive) excess spreads to the collateral lender (and collateral borrower) on accrual basis during the life of the trade. Generally the collateral lender has better bargaining power than the counterparty from the perspective of the derivative dealer in practice and FVA therefore caused more problems for the derivative dealers than CVA/DVA. Since the nature of funding benefits is short term excess cash, it is impractical for the derivative dealer to assume it could earn funding spread on the net received collateral via reverse repo transactions. Albanese and Iabichino (2013) extends the FVA framework to account for specificities of variation margin and initial margins with rehypothecation in the following way, here the funding benefits term FBA is assumed to be zero and no counterparty risk is priced in this structure, + (3.6.1) 푡 푂퐼푆 푉푀 퐼푀 ℚ − ∫ 푟푣 푑푣 퐹 푖 퐹 푖 퐹푉퐴 = 피0 (푒 0 (푠푡 (∑ 푉푀푡) + 푠푡 (∑ 퐼푀푡)) 푑푡), 푖 푖

푖 푖 where 푉푀푡 and 퐼푀푡 denote the time 푡 net variation margin and initial margin for ith netting set (or legal entity). The initial margin works like a cash buffer saved in a segregated account to cover the

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Counterparty Credit Risk, Funding Risk and Central Clearing potential shortfall of collateral in variation margin in the default state of the counterparty. The detailed discussions of mechanics of initial margins is included in the central clearing chapter. Here

퐹푉푀 퐹퐼푀 푠푡 and 푠푡 denote the funding spread of the derivative dealer of variation margin and initial 퐹푉푀 퐹퐼푀 margin, respectively. Generally 푠푡 is expected to be greater than 푠푡 due to stricter collateral

푖 + assets requirements. (∑푖 푉푀푡) calculates the net variation margin shortfall (or net collateral required to be posted to different counterparties to mitigate the loss given default of the derivative dealer) for the derivative dealer over its entire trading book as rehypothecation is available for variation margins. 푖 ∑푖 퐼푀푡 calculates the net collateral required for the segregated initial margins. The FVA could be rewritten as + 푇 (3.6.2) 푡 푂퐼푆 푉푀 퐼푀 퐼푀 ℚ − ∫ 푟푣 푑푣 퐹 푖 퐹 푖 퐹푉퐴 = 피0 (∫ 푒 0 (푠푡 (∑ 푀푡푀푡) + 푠푡 (∑ 풞푡 )) 푑푡). 0 푖 푖 Albanese, Andersen and Iabichino (2014) further expands the FVA framework with counterparty credit risk and a separate margin valuation adjustment (accounting for the price of initial margins). FVA then becomes the price of costs and benefits of variation margins held by the derivative dealer, + 휏퐼 (3.6.3) ℚ 푡 푂퐼푆 푉푀 − ∫0 푟푣 푑푣 퐹 푖 퐹푉퐴 = 피0 (∫ 푒 푠푡 (∑ 푀푡푀푡 1휏푖>푡) 푑푡), 0 푖 and 휏퐼 (3.6.4) ℚ 푡 푂퐼푆 퐼푀 퐼푀 − ∫0 푟푣 푑푣 퐹 푖 푀푉퐴 = 피0 (∫ 푒 푠푡 (∑ 풞푡 1휏푖>푡) 푑푡), 0 푖 where 휏퐼 denotes the default time of the derivative dealer and 휏푖 denotes the default of ith counterparty or legal entity. And then FVA could be reduced to + 푇 푖 ℚ (3.6.5) (∑푖 푀푡푀푡 ℙ (푡)) ℚ 퐹푉푀 푆,퐶푃(푖) ℚ 퐹푉퐴 = 피0 (∫ 푠푡 푂퐼푆 ℙ푆,퐼(푡)푑푡), 0 푁푡

ℚ ℚ where ℙ푆,퐶푃(푖)(푡) and ℙ푆,퐼(푡) denote the survival probability of the ith counterparty/legal entity and the derivative dealer, respectively. FBA term vanishes due to the inability of the derivative dealer retiring long term debt by using short term collateral cash.

The unit of funding analysis expands from the netting sets in CVA/DVA to a larger ‘funding’ set. The funding set could be defined as a collection of OTC derivatives trades whose variation margins received (from the derivative dealer’s perspective) could be rehypothecated to other trades held by the derivative dealer. For example, if a derivative dealers signs CSA with 10 different counterparties and these 10 counterparties agree their variation margins could be rehypothecated, the derivative dealer is eligible to use the collateral received from one counterparty to post to the other counterparties and only has to fund the collateral shortfall in case of net shortage of collateral (i.e. the

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Counterparty Credit Risk, Funding Risk and Central Clearing collateral received from some of these 10 counterparties is not sufficient to post to other counterparties). Therefore, the funding cost is significant reduced due to the existence of rehypothecation clause embedded in CSA and the derivative dealer has to borrow collateral at its own 푖 funding curve if and only if ∑푖 푀푡푀푡 1휏푖>푡 > 0. (The funding cost associated with initial margins is going to be addressed in central clearing chapter.) If the derivative dealer becomes a net receiver of variation margins, these excess collateral is more likely to be invested in short term funding instruments like reverse repos and funding benefits from holding these excess collateral is deemed to 퐺퐶 푂퐼푆 be zero as the GC OIS spread is negligible (i.e. 푟푡 − 푟푡 → 0).

Here if we use the credit triangle relationship, 퐹푉푀 퐼 퐼 (3.6.6) 푠푡 = (1 − 푅푡)휆푡, Then + 푇 푖 ℚ (3.6.7) (∑푖 푀푡푀푡 ℙ (푡)) ℚ 퐼 퐼 푆,퐶푃(푖) ℚ 퐹푉퐴 = 피0 (∫ (1 − 푅푡)휆푡 푂퐼푆 ℙ푆,퐼(푡)푑푡). 0 푁푡

That is, + (3.6.8) 푇 푖 ℚ (∑푖 푀푡푀푡 ℙ (푡)) ℚ ℚ 퐼 퐼 푆,퐶푃(푖) ℚ 퐹푉퐴 = 피0 피푡 (∫ (1 − 푅푡)휆푡 푂퐼푆 ℙ푆,퐼(푡)푑푡) . 0 푁푡 ( )

ℚ ℚ 푡 퐼 ℚ 푡 퐼 ℚ − ∫0 휆푠푑푠 퐼 − ∫0 휆푠푑푠 퐼 Since 푑푡ℙ퐷,퐼(푡) = 푑푡 (1 − 피푡 (푒 )) = 휆푡피푡 (푒 ) 푑푡 = 휆푡ℙ푆,퐼(푡)푑푡,

+ 푇 푖 ℚ (3.6.9) (∑푖 푀푡푀푡 ℙ (푡)) ℚ 퐼 푆,퐶푃(푖) ℚ 퐼 ℚ 퐹푉퐴 = 피0 (∫ (1 − 푅푡) 푂퐼푆 피푡 (휆푡ℙ푆,퐼(푡)푑푡)), 0 푁푡

Hence, + 푇 푖 ℚ (3.6.10) (∑푖 푀푡푀푡 ℙ (푡)) ℚ 퐼 푆,퐶푃(푖) ℚ 퐹푉퐴 = 피0 (∫ (1 − 푅푡) 푂퐼푆 푑푡ℙ퐷,퐼(푡)), 0 푁푡

퐼 where 푅푡 is the deterministic recovery rate of the senior unsecured bond issued by the derivative dealer. If we compare the above FVA expression to classic bilateral CVA pricing equations (on netting sets level), 푇 + 퐶푃 (3.6.11) ℚ 푀푡푀푡 (1 − 푅푡 ) ℚ ℚ 퐶푉퐴 = 피0 (∫ 푂퐼푆 ℙ푆,퐼(푡) 푑푡ℙ퐷,퐶푃(푡)), 0 푁푡

푇 − 퐼 ℚ 푀푡푀푡 (1 − 푅푡) ℚ ℚ 퐷푉퐴 = −피0 (∫ 푂퐼푆 ℙ푆,퐶푃(푡) 푑푡ℙ퐷,퐼(푡)), 0 푁푡

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FVA is driven by the default risk of the derivative dealer (like DVA) and the positive exposure profile (like CVA). It is a bit confusing as FVA shares the credit risk component with DVA and market risk component with CVA and CVA/DVA could be interpreted as the knock out option/protection sold to the counterparty/derivative dealer such that the counterparty/derivative dealer could pay the amount up to the recovery in the its default state. What is the economic interpretation of FVA here then?

FVA here is a knock out option triggered by the default event of the derivative dealer itself. The payoff of this knock out option at the default time of the derivative dealer is + (3.6.12) 퐼 푖 (1 − 푅푡) (∑ 푀푡푀푡 1휏푖>푡) 푖 where 푡 = 휏퐼. Therefore, FVA, or the price of the funding cost here, is equal to the loss given default of the underlying portfolio when it is in the money at the default time of the derivative dealer. This is very confusing as the counterparty has to pay for the full amount of exposure the derivative dealer has at the default time of the derivative dealer and the derivative dealer as a firm does not suffer from any loss in OTC derivatives transactions if the underlying portfolio is in the money as regulated by the Bankruptcy Code. So what is the ‘correct’ economic interpretation of this FVA expression

퐹푉푀 퐼 퐼 derived by credit triangle relationship (i.e. 푠푡 = (1 − 푅푡)휆푡)?

Let’s think in this way. The asymmetric default treatments imposed by Bankruptcy Code makes the defaulted legal entity could have an ‘advantage’ by paying only the recovery amount on their liabilities

퐼 푖 + but receiving full amount on their assets in derivatives transactions. (1 − 푅푡)(∑푖 푀푡푀푡 1휏푖>푡) given 푡 = 휏퐼 represents the net exposure (or assets) the derivative dealer has at its default time 휏퐼. If the derivative dealer enters into back-to-back transactions to hedge the existing portfolio, the hedging portfolio would have exactly the opposite exposure profile, i.e. the negative exposure profile of the hedging portfolio is equal to + (3.6.13) 퐼 푖 −(1 − 푅푡) (∑ 푀푡푀푡 1휏푖>푡) . 푖 Therefore, + ℚ 푇 푖 (3.6.14) (∑푖 푀푡푀푡 ℙ (푡)) ℚ 퐼 푆,퐶푃(푖) ℚ 피0 (∫ (1 − 푅푡) 푂퐼푆 푑푡ℙ퐷,퐼(푡)) 0 푁푡

+ ℚ 푇 푖 (∑푖 푀푡푀푡 ℙ (푡)) ℚ 퐼 푆,퐶푃(푖) ℚ = −피0 (− ∫ (1 − 푅푡) 푂퐼푆 푑푡ℙ퐷,퐼(푡)) 0 푁푡

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Counterparty Credit Risk, Funding Risk and Central Clearing is the price of the knock out option the derivative dealer bought from its ‘hedging’ counterparties. In Hull and White (2014), FVA is recognised as the DVA for the derivative dealer’s hedging instruments variation margins (Hull and White (2014) named 퐷푉퐴2) and the DVA for the underlying instruments

(Hull and White (2014) named 퐷푉퐴1) combined with the DVA for the hedging instruments are the net benefits the derivative dealer has in their default state. The bondholders of the derivative dealer are the recipients of these two benefits as the reduced liabilities in underlying and hedging instruments would increase the expected recovery rate of the senior unsecured bonds issued by the derivative dealer. Although Hull and White (2014) did not present an explicit expression for DVA for hedging instruments variation margins and FVA, the economic interpretation of these two terms is correct and intuitive. By following the same structure above, + 푇 푖 ℚ (3.6.15) (∑푖 푀푡푀푡 ℙ (푡)) ℚ 퐼 푆,퐶푃(푖) ℚ 퐹푉퐴 = 퐷푉퐴2 = −피0 (− ∫ (1 − 푅푡) 푂퐼푆 푑푡ℙ퐷,퐼(푡)), 0 푁푡 with

푇 푖− 퐼 (3.6.16) 푀푡푀푡 (1 − 푅푡) 퐷푉퐴 = − ∑ 피ℚ (∫ ℙℚ (푡) 푑 ℙℚ (푡)). 1 0 푁푂퐼푆 푆,퐶푃(푖) 푡 퐷,퐼 푖 0 푡 Hence, the FVA is a cost item and DVA for hedging instruments variation margins is a benefit item (from the perspective of the derivative dealer). Therefore, the fair price for an OTC derivatives portfolio is ∗ 푉0 = 푉0 − 퐶푉퐴 + 퐷푉퐴1 − 퐹푉퐴 + 퐷푉퐴2, (3.6.17) where (3.6.18) 푉푡 푉∗ = 피ℚ (∑ ), 0 0 푁푂퐼푆 푡 푡

+ 퐶푃(푖) 푇 푖 푀푡푀푡 (1 − 푅푡 ) 퐶푉퐴 = ∑ 피ℚ (∫ ℙℚ (푡) 푑 ℙℚ (푡)), 0 푁푂퐼푆 푆,퐼 푡 퐷,퐶푃(푖) 푖 0 푡

푇 푖− 퐼 푀푡푀푡 (1 − 푅푡) 퐷푉퐴 = − ∑ 피ℚ (∫ ℙℚ (푡) 푑 ℙℚ (푡)), 1 0 푁푂퐼푆 푆,퐶푃(푖) 푡 퐷,퐼 푖 0 푡 + 푇 푖 ℚ (∑푖 푀푡푀푡 ℙ (푡)) ℚ 퐹푉푀 푆,퐶푃(푖) ℚ 퐹푉퐴 = 피0 (∫ 푠푡 푂퐼푆 ℙ푆,퐼(푡)푑푡), 0 푁푡

+ 푇 푖 ℚ (∑푖 푀푡푀푡 ℙ (푡)) ℚ 퐼 푆,퐶푃(푖) ℚ 퐷푉퐴2 = −피0 (− ∫ (1 − 푅푡) 푂퐼푆 푑푡ℙ퐷,퐼(푡)). 0 푁푡

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푡 푂퐼푆 푂퐼푆 푂퐼푆 ∫ 푟푠 푑푠 Here 푁푡 denotes the OIS money market account and 푁푡 = 푒 0 . FVA therefore is the funding cost of variation margins the derivative dealer has to borrow on an unsecured basis at a substantial spread over the OIS rate due to the non-existence of derivatives receivables repo market. DVA on hedging instruments is the benefit the senior creditors of the derivative dealer could receive in the default state of the derivative dealer. Since 퐹푉퐴 = 퐷푉퐴2, ∗ 푉0 = 푉0 − 퐶푉퐴 + 퐷푉퐴1. (3.6.19) The current market practice is the derivative dealer charges FVA to its counterparty although the theory proposed by Hull and White (2014) suggests the of a derivatives portfolio is free of funding components. Albanese and Andersen (2014) suggests FVA charged by derivative dealers could be perceived as a compensation for their shareholders as FVA represents a wealth transfer from the shareholders to the bondholders as FVA or DVA for hedging instruments would provide benefits only to the bondholders. CVA is the wealth transferred from the derivative dealer to its counterparty by allowing its counterparty to pay the recovery amount of its liabilities (or the derivative dealer’s assets) in the default state of the counterparty. DVA is the wealth transferred from the counterparty to the derivative dealer by allowing the derivative dealer to pay the recovery amount of the derivative dealer’s liabilities in the default state of the derivative dealer. More specifically, DVA are the benefits to the bondholders of the derivative dealer as the shareholders will lose their entire equity in the default state of the derivative dealer. FVA is the wealth transferred internally from the shareholders to the senior unsecured bondholders as the shareholders cover the operating cost of hedging portfolio (as FVA is a PnL component) and senior bondholders could receive the benefits of FVA by paying recovery amount on the net liabilities on the hedging instruments. Therefore, DVA and FVA are benefits to the bondholders of the derivative dealer. The detailed accounting treatments is discussed in the following accounting part of this chapter.

Brigo et al. (2012) provides a robust modelling framework for analysing OTC derivatives portfolios and expands Piterbarg (2010) structure by allowing existence of counterparty credit risk. The final pricing PDE allows for asymmetric funding spreads and OIS bid-ask spreads,

(∂푡 + 푓̃푡 − 휆푡 + ℒ푡)푉̅푡(퐶; 퐹) + (푓̃푡 − 푐푡̃ )퐶푡 + 휆푡휃푡(퐶, 휀) = 0 (3.6.20) with boundary condition

푉̃푇(퐶; 퐹) = 0. (3.6.21) The only drawback of this pricing framework is that it is very difficult to obtain a closed form solution for FVA. The funding components interacts with the CVA/DVA and market dynamics and it could be numerically solved by an American Monte Carlo algorithm. It could be treated as a master equation for FVA and the linear form of FVA representations (under different assumptions) should satisfy this non-linear PDE.

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Detailed FVA pricing examples are provided together with MVA (Margin Valuation Adjustment) at the end of Central Clearing chapter.

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3.7 xVA Accounting

Albanese, Andersen and Iabichino (2014) and Albanese and Andersen (2014) laid out a general framework for FVA analysis for accounting, risk management and collateral trading. Typically, the derivative dealers would assign trades with positive MtM (net receivables) to an asset account (A) and trades with negative MtM (net payables) to a liability account (L). Then the portfolio fair value (PFV) without taking default risk of derivative dealer and its counterparty into account is 푃퐹푉 = 퐴 − 퐿. (3.7.1)

Then the CVA term, as the wealth transferred from shareholders of the derivative dealer to the counterparties of the derivative dealer, should be deducted from the asset side. The entire CVA of a given derivative dealer against all counterparties is booked under a contra asset account (CA). Similarly, the DVA term, as the wealth transfer from the counterparty’s shareholders to the derivative dealer, should be added to the liability side. Given the fact that DVA is a beneficiary term to the derivative dealer, the entire DVA of all counterparties holding against the derivative dealer is booked under a contra liability account (CL). And the portfolio fair value could be updated as 푃퐹푉 = (퐴 − 퐶퐴) − (퐿 − 퐶퐿) = 퐴 − 퐿 − 퐶퐴 + 퐶퐿. (3.7.2) We could reconcile the equity of an OTC derivatives portfolio with PFV and a retained earnings (RE) account 퐸푞푢푖푡푦 = 푅퐸 + 푃퐹푉 = 푅퐸 + 퐴 − 퐿 − 퐶퐴 + 퐶퐿. (3.7.3) DVA is a terms accounts for wealth transfer from shareholders of counterparty to the derivative dealer and the derivative dealer could only exercise (realise) its cash payoff of DVA at its own default time (via paying recovery amount of net payables). Hence DVA could neither be used to absorb general losses of the derivative dealer’s operating activities prior to the derivative dealer’s default, nor increase the wealth of the shareholders (whose positions in the derivative dealer would be entirely wiped out by the default of the derivative dealer). Basel Committee and Banking Supervision intends to remove the DVA impact from valuation of derivatives such that the banks could not have a net increase in its common equity tier 1 capital by declines in their own credit quality. In other words, the present value of the cash flows benefit the bank after its default should be excluded from regulatory capital. As discussed in previous sections, DVA for underlying portfolio and DVA for hedging instruments only benefit bondholders of the derivative dealer. Therefore, the amendments the regulators made to financial regulations implicitly coincides with the interests of the shareholders. The same principle of exclusion of benefits from widening credit spreads should be applied for the bilateral CVA and FVA as well. Details would be discussed later this section. From regulatory capital perspective, the common equity tier 1 capital (CET1) of the derivative dealer is defined as 퐶퐸푇1 = 퐸푞푢푖푡푦 − 퐶퐿 = 푅퐸 + 퐴 − 퐿 − 퐶퐴. (3.7.4)

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Funding transfer policy (FTP) determines the cost or benefits the derivative dealer has to pass to its counterparties to break even when initiating new transactions. CVA/DVA and FVA are key components in determination of FTP in current market structure. Suppose the derivative dealer initiates a new transaction with a counterparty. FTP measures the marginal increments of CVA/DVA/FVA the marker maker has to charge (at a minimum level) to break even by taking on additional counterparty credit risk and funding risk of the new trade. The change to the portfolio fair value at inception of the new trade on Mark to Market basis is Δ푃퐹푉 = Δ(퐴 − 퐶퐴) − Δ(퐿 − 퐶퐿) = Δ퐴 − Δ퐿 − Δ퐶퐴 + Δ퐶퐿, (3.7.5) The derivative dealer has discretion over its funding transfer policy. For example, if the derivative dealer decides to take DVA related components in contra liability account into determination of incremental charges, the FTP is 퐹푇푃 = Δ퐶퐴 − Δ퐶퐿, (3.7.6) Then the change in the retained earnings account becomes a fair measurement of the default free valuation of the new transaction (on accrual accounting basis), Δ푅퐸 = 퐹푇푃 − (Δ퐴 − Δ퐿), (3.7.7) The change in equity and change in common tier 1 equity are Δ퐸푞푢푖푡푦 = Δ푅퐸 + Δ푃퐹푉 = 0, (3.7.8) Δ퐶퐸푇1 = Δ퐸푞푢푖푡푦 − Δ퐶퐿 = −Δ퐶퐿. That is, the derivative dealer has to mark down its regulatory capital by change in contra liability account. And the change in common equity tier 1 capital is the wealth transferred from the shareholders to the bondholders. If the derivative dealer decides to ignore the DVA related components in funding transfer policy, 퐹푇푃 = Δ퐶퐴, (3.7.9) Then the change in the retained earnings account is Δ푅퐸 = 퐹푇푃 − (Δ퐴 − Δ퐿), (3.7.10) The change in equity and change in common tier 1 equity are Δ퐸푞푢푖푡푦 = Δ푅퐸 + Δ푃퐹푉 = Δ퐶퐿, (3.7.11) Δ퐶퐸푇1 = Δ퐸푞푢푖푡푦 − Δ퐶퐿 = 0. The shareholders in this case are not affected by DVA related components as the derivative dealer decide to monetise the DVA components by charging its counterparties Δ퐶퐴. The common practice regarding DVA term for a derivative dealer is to charge DVA to its counterparty. Since DVA term could not benefit the shareholders, DVA term charged by the derivative dealer would turn into trading gains and hence ultimately increase CET1 and Equity. The problem of this practice is that the price offered by the derivative dealer may not be competitive to keep hold of the business flows. In the following section, we are going to give examples of current funding transfer policies.

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3.8 FCA/FBA Accounting (푭푻푷 = 횫푪푨 − 횫푪푳)

The funding valuation adjustment (FVA) emerges from the fact derivative dealers have to post margins on their fully collateralised hedging positions against uncollateralised or partially collateralised positions with their counterparties. The derivative dealers have to go to the market to borrow money at their own funding curve and receive interest on their margin account approximated by OIS curve. The symmetric funding valuation adjustment (SFVA) on the netting sets level derived in section 3.4 is 푇 푖 (3.8.1) 푖 ℚ 퐹 푀푡푀푡 ℚ ℚ 푆퐹푉퐴 = 피0 (∫ 푠푡 푂퐼푆 ℙ푆,퐶푃(푖)(푡)ℙ푆,퐼(푡)푑푡). 0 푁푡 퐹 ℚ ℚ Here 푠푡 denotes the derivative dealer’s funding spread and ℙ푆,퐶푃(푖)(푡) and ℙ푆,퐼(푡) denote the survival probability of the counterparty and the market, respectively. And the SFVA is decomposed of asset and liabilities as well. The liability side is FBA and it overlaps with DVA, 푇 푖− (3.8.2) 푖 ℚ 퐹 푀푡푀푡 ℚ ℚ 퐹퐵퐴 = −피0 (∫ 푠푡 푂퐼푆 ℙ푆,퐶푃(푖)(푡)ℙ푆,퐼(푡)푑푡), 0 푁푡

푇 푖− 퐼 (3.8.3) 푖 ℚ 푀푡푀푡 (1 − 푅푡) ℚ ℚ 퐷푉퐴 = −피0 (∫ 푂퐼푆 ℙ푆,퐶푃(푖)(푡) 푑푡ℙ퐷,퐼(푡)), 0 푁푡

퐹 퐼 퐼 ℚ 퐼 ℚ 푖 푖 where 푠푡 = (1 − 푅푡)휆푡 and 푑푡ℙ퐷,퐼(푡) = 휆푡ℙ푆,퐼(푡)푑푡. Hence 퐷푉퐴 = 퐹퐵퐴 . 푇 푖+ (3.8.4) 푖 푖 푖 ℚ 퐹 푀푡푀푡 ℚ ℚ 퐹퐶퐴 = 푆퐹푉퐴 + 퐹퐵퐴 = 피0 (∫ 푠푡 푂퐼푆 ℙ푆,퐶푃(푖)(푡)ℙ푆,퐼(푡)푑푡). 0 푁푡 The FCA and FBA for the derivative dealer under FCA/FBA Accounting is 푇 푖+ (3.8.5) 푀푡푀푡 퐹퐶퐴 = ∑ 퐹퐶퐴푖 = ∑ 피ℚ (∫ 푠퐹 ℙℚ (푡)ℙℚ (푡)푑푡), 0 푡 푁푂퐼푆 푆,퐶푃(푖) 푆,퐼 푖 푖 0 푡 푇 푖− 푀푡푀푡 퐹퐵퐴 = ∑ 퐹퐵퐴푖 = − ∑ 피ℚ (∫ 푠퐹 ℙℚ (푡)ℙℚ (푡)푑푡). 0 푡 푁푂퐼푆 푆,퐶푃(푖) 푆,퐼 푖 푖 0 푡 As discussed in previous sections, symmetric FVA and FCA/FBA Accounting take rehypothecation within a netting set (i.e. the collateral received on a given trade could be posted to other trades within the same netting set or counterparty) into pricing as CVA and DVA. The rehypothecation among netting sets however are not priced in SFVA framework and hence SFVA overprices FCA and FBA due to classic Jensen’s inequality.

Basel Committee on Banking Supervision requires the items with benefits after the bank’s default should be deducted from the regulatory capital (BSBC (2013)). Therefore, the terms with DVA features should be placed in contra liability (CL) account. The common way employed by derivative dealers to monetise DVA is to pass it to its counterparty by charging unilateral CVA (not bilateral CVA). The unilateral CVA here is

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+ 퐶푃(푖) (3.8.6) 푇 푖 푀푡푀푡 (1 − 푅푡 ) 푈퐶푉퐴 = ∑ 피ℚ (∫ 푑 ℙℚ (푡)). 0 푁푂퐼푆 푡 퐷,퐶푃(푖) 푖 0 푡

In symmetric funding pricing framework, the overall pricing adjustments is 퐶푉퐴 + 퐹퐶퐴 − 퐹퐵퐴, (3.8.7) with 퐹퐵퐴 = 퐷푉퐴. If the derivative dealer decides to charge UCVA to its counterparty, the pricing adjustment becomes 푈퐶푉퐴 + 퐹퐶퐴 − (푈퐶푉퐴 − 퐶푉퐴) − 퐹퐵퐴. (3.8.8) Here

+ 퐶푃(푖) (3.8.9) 푇 푖 푀푡푀푡 (1 − 푅푡 ) 퐶푉퐴 = ∑ 피ℚ (∫ ℙℚ (푡) 푑 ℙℚ (푡)). 0 푁푂퐼푆 푆,퐼 푡 퐷,퐶푃(푖) 푖 0 푡

Therefore,

푇 푖+ 퐶푃(푖) (3.8.10) 푀푡푀푡 (1 − 푅 ) 푈퐶푉퐴 − 퐶푉퐴 = ∑ 피ℚ (∫ 푡 (1 − ℙℚ (푡)) 푑 ℙℚ (푡)). 0 푁푂퐼푆 푆,퐼 푡 퐷,퐶푃(푖) 푖 0 푡 That is,

+ ( ) (3.8.11) 푇 푖 퐶푃 푖 푀푡푀푡 (1 − 푅푡 ) 푈퐶푉퐴 − 퐶푉퐴 = ∑ 피ℚ (∫ ℙℚ (푡)푑 ℙℚ (푡)). 0 푁푂퐼푆 퐷,퐼 푡 퐷,퐶푃(푖) 푖 0 푡

푈퐶푉퐴 − 퐶푉퐴 represents the component of CVA increases as the default probability of the derivative dealer goes up. Under FCA/FBA Accounting or symmetric FVA Accounting, 퐶퐴 = 푈퐶푉퐴 + 퐹퐶퐴, (3.8.12) and 퐶퐿 = (푈퐶푉퐴 − 퐶푉퐴) + 퐹퐵퐴. (3.8.13) Therefore, the portfolio fair value becomes 푃퐹푉 = 퐴 − 퐿 − 퐶퐴 + 퐶퐿 = 퐴 − 퐿 − 푈퐶푉퐴 − 퐹퐶퐴 + (푈퐶푉퐴 − 퐶푉퐴) + 퐹퐵퐴, (3.8.14) that is, 푃퐹푉 = 퐴 − 퐿 − (퐶푉퐴 + 푆퐹푉퐴). (3.8.15) IFRS 13, Fair Value Measurement, requires all derivatives transactions should be priced at their exit prices in financial statements. The exit price here is marked to the market. CVA is priced by market variables like credit curves. FVA is the price of funding costs and benefits of a given derivative dealer and hence it is entity specific item. The FCA/FBA Accounting or symmetric FVA Accounting therefore breaks the exit price principle and caused explicit pricing disputes on OTC derivative portfolios.

Then the equity of the derivative dealer is

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퐸푞푢푖푡푦 = 푅퐸 + 푃퐹푉 = 푅퐸 + 퐴 − 퐿 − (퐶푉퐴 + 푆퐹푉퐴). (3.8.16) And the common equity tier 1 capital is 퐶퐸푇1 = 퐸푞푢푖푡푦 − 퐶퐿 = 푅퐸 + 퐴 − 퐿 − (퐶푉퐴 + 푆퐹푉퐴) − (푈퐶푉퐴 − 퐶푉퐴) − 퐹퐵퐴. (3.8.17) That is, 퐶퐸푇1 = 퐸푞푢푖푡푦 − 퐶퐿 = 푅퐸 + 퐴 − 퐿 − (푈퐶푉퐴 + 퐹퐶퐴). (3.8.18) If the derivative dealer takes a new OTC derivative transaction into its portfolio, the change to the portfolio fair value at inception of the new transaction is Δ푃퐹푉 = Δ퐴 − Δ퐿 − Δ퐶퐴 + Δ퐶퐿 = Δ퐴 − Δ퐿 − (Δ퐶푉퐴 + Δ푆퐹푉퐴), (3.8.19) The funding transfer policy (FTP) used by the derivative dealer is to charge changes in UCVA and SFVA to its counterparty, i.e. 퐹푇푃 = Δ푈퐶푉퐴 + Δ푆퐹푉퐴, (3.8.20) Then the change in the retained earnings account is Δ푅퐸 = 퐹푇푃 − (Δ퐴 − Δ퐿), (3.8.21) The change in equity and change in common equity tier 1 capital are Δ퐸푞푢푖푡푦 = Δ푅퐸 + Δ푃퐹푉 = 퐹푇푃 − Δ퐶퐴 + Δ퐶퐿 = Δ푈퐶푉퐴 − Δ퐶푉퐴, (3.8.22) Δ퐶퐸푇1 = Δ퐸푞푢푖푡푦 − Δ퐶퐿 = −Δ퐹퐵퐴. Generally the first to default impact in CVA pricing is not significant unless the credit spread of either the counterparty or the derivative dealer per se is trading at significantly high level. Therefore, in most cases, under FCA/FBA Accounting, Δ퐸푞푢푖푡푦 → 0, (3.8.23) Δ퐶퐸푇1 = Δ퐸푞푢푖푡푦 − Δ퐶퐿 = −Δ퐹퐵퐴. The regulatory capital therefore, has to be marked down by incremental FBA or DVA of the new transaction.

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3.9 FVA/FDA Accounting (푭푻푷 = 횫푪푨)

The FBA term discussed above in theory should be used to buy back or retire some portion of derivative dealer’s long term debt. The derivative dealer’s bonds price rallies and funding spread narrows. This is not pragmatic as volatility of MtM may switch the exposure direction in a short period of time and therefore the funding benefit is actually a term with short term nature. The excess collateral from variation margin account usually is invested into repo market to make sure the derivative dealer could access to cash in a short notice. Then the lending rate in funding benefits term 퐹 퐹 푟푒푝표 푂퐼푆 is not 푠푡 but 푠푡 = 푟푡 − 푟푡 . It could be further assumed at 0 from prudent perspective. Albanese, Andersen and Iabichino (2014) points out the derivative dealer normally funds at funding set level (i.e. legal entity) as long as the cash received and derivatives funding could be re- hypothecated. Then the initial symmetric FVA could be updated as asymmetric FVA in the following way, + 푇 푖 ℚ (3.9.1) (∑푖 푀푡푀푡 ℙ (푡)) ℚ 퐹푉푀 푆,퐶푃(푖) ℚ 퐹푉퐴 = 피0 (∫ 푠푡 푂퐼푆 ℙ푆,퐼(푡)푑푡). 0 푁푡

This FVA term recognises the derivative dealer’s funding activities are conducted at aggregate accounts level. Given the fact that cash received and derivatives funding could be rehypothecated, it will be a funding cost to a derivative dealer if and only if the aggregate accounts calls for margin and the derivative dealer becomes a net poster of collateral in this scenario. As discussed in previous sections, FVA, or DVA for hedging instruments, represents the price of knock out option the derivative dealer bought from its fully collateralised hedging counterparties and it would benefit bondholders at default time with higher recovery rate, + 푇 푖 ℚ (3.9.2) (∑푖 푀푡푀푡 ℙ (푡)) ℚ 퐼 푆,퐶푃(푖) ℚ 퐷푉퐴2 = −피0 (− ∫ (1 − 푅푡) 푂퐼푆 푑푡ℙ퐷,퐼(푡)). 0 푁푡

So FVA is an internal wealth transfer from the shareholders of the derivative dealer to the bondholders of the derivative dealer. This FVA term is recorded in contra asset account as a cost to shareholders and the double entry bookkeeping system requires this internal transfer of wealth to the bondholders should be recognised as a benefit term on the liability side. Albanese and Andersen (2014) introduces the Funding Debt Adjustment (FDA) booked under the contra liability account,

퐹퐷퐴 = 퐷푉퐴2. (3.9.3) And hence 퐹푉퐴 = 퐹퐷퐴. (3.9.4) Under FVA/FDA Accounting or symmetric FVA Accounting, 퐶퐴 = 푈퐶푉퐴 + 퐹푉퐴, (3.9.5) And

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퐶퐿 = (푈퐶푉퐴 − 퐶푉퐴) + 퐷푉퐴 + 퐹퐷퐴, (3.9.6) where

+ ( ) (3.9.7) 푇 푖 퐶푃 푖 푀푡푀푡 (1 − 푅푡 ) 푈퐶푉퐴 − 퐶푉퐴 = ∑ 피ℚ (∫ ℙℚ (푡)푑 ℙℚ (푡)), 0 푁푂퐼푆 퐷,퐼 푡 퐷,퐶푃(푖) 푖 0 푡

푇 푖− 퐼 푀푡푀푡 (1 − 푅푡) 퐷푉퐴 = 퐷푉퐴 = − ∑ 피ℚ (∫ ℙℚ (푡) 푑 ℙℚ (푡)). 1 0 푁푂퐼푆 푆,퐶푃(푖) 푡 퐷,퐼 푖 0 푡 Although we have new entries like FDA and FVA in contra liability and contra asset accounts, they have no impact on the portfolio fair value as FDA and FVA cancel in the portfolio fair value equation, 푃퐹푉 = 퐴 − 퐿 − 퐶퐴 + 퐶퐿 (3.9.8) = 퐴 − 퐿 − (푈퐶푉퐴 + 퐹푉퐴) + (푈퐶푉퐴 − 퐶푉퐴) + 퐷푉퐴 + 퐹퐷퐴, that is, 푃퐹푉 = 퐴 − 퐿 − (퐶푉퐴 − 퐷푉퐴). (3.9.9) The portfolio fair value in FVA/FDA Accounting framework preserves asset liability symmetry and Modigliani Miller equilibrium where fair price to the derivative dealer is free from the funding impacts. The drawback here is deviations of entry prices by FVA/FDA Accounting from fair derivatives price with a funding component.

Then the equity of the derivative dealer is 퐸푞푢푖푡푦 = 푅퐸 + 푃퐹푉 = 푅퐸 + 퐴 − 퐿 − (퐶푉퐴 − 퐷푉퐴). (3.9.10) And the common equity tier 1 capital is 퐶퐸푇1 = 퐸푞푢푖푡푦 − 퐶퐿 (3.9.11) = 푅퐸 + 퐴 − 퐿 − (퐶푉퐴 − 퐷푉퐴) − (푈퐶푉퐴 − 퐶푉퐴) − 퐷푉퐴 − 퐹퐷퐴. That is, 퐶퐸푇1 = 퐸푞푢푖푡푦 − 퐶퐿 = 푅퐸 + 퐴 − 퐿 − (푈퐶푉퐴 + 퐹퐷퐴). (3.9.12) Or 퐶퐸푇1 = 퐸푞푢푖푡푦 − 퐶퐿 = 푅퐸 + 퐴 − 퐿 − (푈퐶푉퐴 + 퐹푉퐴). (3.9.13) If the derivative dealer takes a new OTC derivative transaction into its portfolio, the change to the portfolio fair value at inception of the new transaction is Δ푃퐹푉 = Δ퐴 − Δ퐿 − Δ퐶퐴 + Δ퐶퐿 = Δ퐴 − Δ퐿 − (Δ퐶푉퐴 + Δ퐹푉퐴), (3.9.14) The funding transfer policy (FTP) used by the derivative dealer is to charge changes in UCVA and FVA to its counterparty, i.e. 퐹푇푃 = Δ푈퐶푉퐴 + Δ퐹푉퐴. (3.9.15)

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That is, 퐹푇푃 = Δ퐶퐴. (3.9.16)

Since the change in the retained earnings account is Δ푅퐸 = 퐹푇푃 − (Δ퐴 − Δ퐿), (3.9.17) the change in equity and change in common equity tier 1 capital are Δ퐸푞푢푖푡푦 = Δ푅퐸 + Δ푃퐹푉 = 퐹푇푃 − Δ퐶퐴 + Δ퐶퐿 = Δ퐶퐿, (3.9.18) Δ퐶퐸푇1 = Δ퐸푞푢푖푡푦 − Δ퐶퐿 = 0. Therefore, the common equity tier 1 capital becomes insensitive to new transactions in FVA/FDA Accounting framework.

Findings from xVAs Accounting Analysis If the derivative dealer initiates new trades in his/her portfolio, then the two different accounting frameworks will have different impacts on the dealer’s equity and regulatory capital. FCA/FBA Accounting FVA/FDA Accounting 훥퐸푞푢푖푡푦 0 Δ(푈퐶푉퐴 − 퐶푉퐴) + Δ퐷푉퐴 + Δ퐹퐷퐴 훥퐶퐸푇1 −Δ퐹퐵퐴 or −Δ퐷푉퐴 0

If we further assume first to default is removed from our calculation and rehypothecation across different netting sets is disallowed in our portfolios, the results become more explicit. FCA/FBA Accounting FVA/FDA Accounting 훥퐸푞푢푖푡푦 0 Δ퐷푉퐴 + Δ퐹퐶퐴 훥퐶퐸푇1 −Δ퐷푉퐴 0

That is, if the derivative dealer recognises DVA or funding benefits in pricing and passes on these benefits to counterparties, his/her CET1 (Common Tier 1 Equity) is going to be marked down by changes in DVA or funding benefits. If the derivative dealer decides not to recognise DVA or funding benefits in pricing and does not pass on these benefits to counterparties, his/her CET1 is immune from any changes brought by taking on new transactions.

Therefore, it is logical for derivative dealers to monetise their DVAs (for example, charge UCVAs to counterparties) or hedge their DVAs (for example, charge BCVA to counterparties to keep their prices competitive and hedge DVAs by other means).

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Can we find evidence of this practice in reality? We are going to explore answers to this question in next section.

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3.10 DVA/FBA Hedging

Mathematical foundations of empirical studies on xVA

As discussed in section 3.8 and section 3.9, under FCA/FBA Accounting (where rehypothecation of variation margins is not allowed across different netting sets/counterparties), the entire pricing adjustment (in absence of initial margins) is 퐶푉퐴 − 퐷푉퐴 + 퐹퐶퐴, (3.10.1) where

+ ( ) (3.10.2) 푇 푖 퐶푃 푖 푀푡푀푡 (1 − 푅푡 ) 퐶푉퐴 = ∑ 피ℚ (∫ ℙℚ (푡) 푑 ℙℚ (푡)), 0 푁푂퐼푆 푆,퐼 푡 퐷,퐶푃(푖) 푖 0 푡

푇 푖− 퐼 (3.10.3) 푀푡푀푡 (1 − 푅푡) 퐷푉퐴 = − ∑ 피ℚ (∫ ℙℚ (푡) 푑 ℙℚ (푡)), 0 푁푂퐼푆 푆,퐶푃(푖) 푡 퐷,퐼 푖 0 푡 and 푇 푖+ (3.10.4) 푀푡푀푡 퐹퐶퐴 = ∑ 피ℚ (∫ 푠퐹 ℙℚ (푡)ℙℚ (푡)푑푡). 0 푡 푁푂퐼푆 푆,퐶푃(푖) 푆,퐼 푖 0 푡 Under FVA/DVA Accounting (where rehypothecation of variation margins is allowed across different netting sets/counterparties), the entire pricing adjustment (in absence of initial margins) is 퐶푉퐴 − 퐷푉퐴 + 퐹푉퐴, (3.10.5) where

+ ( ) (3.10.6) 푇 푖 퐶푃 푖 푀푡푀푡 (1 − 푅푡 ) 퐶푉퐴 = ∑ 피ℚ (∫ ℙℚ (푡) 푑 ℙℚ (푡)), 0 푁푂퐼푆 푆,퐼 푡 퐷,퐶푃(푖) 푖 0 푡

푇 푖− 퐼 (3.10.7) 푀푡푀푡 (1 − 푅푡) 퐷푉퐴 = − ∑ 피ℚ (∫ ℙℚ (푡) 푑 ℙℚ (푡)), 0 푁푂퐼푆 푆,퐶푃(푖) 푡 퐷,퐼 푖 0 푡 and + 푇 푖 ℚ (3.10.8) (∑푖 푀푡푀푡 ℙ (푡)) ℚ 퐹 푆,퐶푃(푖) ℚ 퐹푉퐴 = 피0 (∫ 푠푡 푂퐼푆 ℙ푆,퐼(푡)푑푡). 0 푁푡

The FCA term in FCA/FBA Accounting and FVA term in FVA/FDA Accounting are essentially DVA terms of the hedging instruments (under the same CSA terms of the underlying derivative transactions) as discussed in Hull and White (2014). Albanese, Andersen and Iabichino (2014) extends these two accounting frameworks and formalises the funding (FTP) policy for CVA/DVA/FVA. If the derivative dealers intend to avoid marking down their Common Equity Tier 1 Capital (CET1) by their DVAs, they have to monetise DVAs by charging their counterparties

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Suppose the derivative dealer trades an OTC derivative with a given counterparty and hedges by a back- to-back transaction with a CCP. The common feature of the two accounting frameworks above is that the derivative dealer should monetise its DVA and funding cost components (in hedging instrument) with its counterparty from the perspective of stabilising CET1. The nature of funding costs, as discussed by Hull and White (2014), is the DVA of the hedging positions (under the identical CSA terms with its counterparty). The monetisation or hedging of DVA, therefore, becomes a crucial part in running OTC derivatives trading business. (Analysis of FVA could follow the same structure we present in this chapter.)

Data Description

The data of DVAs of different financial institutions are provided by Bloomberg. Banks generally report their DVA positions on quarterly basis and divide their DVAs into DVA for fixed income (field name ARDR_DVA_FIXED_INCOME) and DVA for equity (field name ARDR_DVA_EQUITY).

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Goldman Sachs

We have 13 observations for DVA for Equity and DVA for Fixed Income. These observations are crude data and no filtering techniques are applied.

(Figure 3.1: GS DVAs Sources: data – Bloomberg)

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Morgan Stanley

We have 25 observations for DVA for Equity and DVA for Fixed Income. These observations are crude data and no filtering techniques are applied. DVA for Fixed Income is far more volatile than DVA for Equity from 2011 third quarter to 2012 third quarter. Morgan Stanley did not give any specific reasons for these movements in these 5 quarters financial statements. A reasonable guess is Morgan Stanley pushed for heavier rebalancing for their fixed income portfolios.

(Figure 3.2: MS DVAs Sources: data – Bloomberg)

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J.P. Morgan

We have 7 observations for DVA for Equity and DVA for Fixed Income. These observations are crude data and no filtering techniques are applied.

(Figure 3.3: JPM DVAs Sources: data – Bloomberg)

Explanatory variables of DVAs

The generic DVA formula is

푇 푖− 퐼 푀푡푀푡 (1 − 푅푡) 퐷푉퐴 = − ∑ 피ℚ (∫ ℙℚ (푡) 푑 ℙℚ (푡)). 0 푁푂퐼푆 푆,퐶푃(푖) 푡 퐷,퐼 푖 0 푡 The default dependency between counterparties and the institution/derivative dealer itself and correlation between negative exposure and default probabilities of the institution do make significant contributions to the final DVA price under certain scenarios. If we further assume these three factors (i.e. the survival probabilities of counterparties, default probabilities of the institution and the negative exposure profile) are mutually independent, 푇 푖− 퐼 푀푡푀푡 (1 − 푅푡) (3.10.9) 퐷푉퐴 = − ∑ ∫ 피ℚ ( ) ℙℚ (0, 푡) 푑 ℙℚ (0, 푡). 0 푁푂퐼푆 푆,퐶푃(푖) 푡 퐷,퐼 푖 0 푡

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Principal component analysis (PCA) assumes a set of uncorrelated linear factors drives the market and therefore the real market structure could be revealed through PCA. The DVAs for fixed income and DVAs for Equity tends to move in the same direction (i.e. move up and down together) as depicted in the three figures above. PCA, therefore, could be used to identify and extract the common driving factors of DVAs for fixed income and equity of each individual derivative dealer. It is difficult to achieve a significant reduction in dimensions by doing PCA on a number of market markers’ DVAs due to the limited availability of market data at current stage of market development. In this chapter, we would focus on identification and interpretation of principal components of DVAs. The practical hedging strategies for CVA/DVA/FVA books based on PCA eigenvectors could be further studied in future research.

푡 The general PCA structure is that a number 푛 of highly correlated market data 푦푖 (where 푖 = 1, … , 푛) like DVAs could be explained by a number 푘 of uncorrelated linear factors (i.e. principal components) 푡 푘 푡 (3.10.10) 푦1 푥푖1 휀1 푡 ( ⋮ ) = ∑ 훼푖 ∙ ( ⋮ ) + ( ⋮ ), 푡 푡 푦푛 푖=1 푥푖푛 휀푛

푥푖1 푡 where 훼푖 is the i-th principal component at time 푡, ( ⋮ ) is i-th component factor loading and 푥푖푛 푡 휀1 ( ⋮ ) is the residual of the market data unexplained by the 푛 principal components. Here if we apply 푡 휀푛 PCA to two dimensional DVA time series for Goldman Sachs, Morgan Stanley and J.P. Morgan, we could have the following graphs.

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(Figure 3.4: GS DVAs PCA Variance Analysis Sources: data – Bloomberg)

(Figure 3.5: MS DVAs PCA Variance Analysis Sources: data – Bloomberg)

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(Figure 3.6: JPM DVAs PCA Variance Analysis Sources: data – Bloomberg)

The variance explained by principal components are

Bank Goldman Sachs Morgan Stanley J.P. Morgan 1st Principal Component 87.93% 98.60% 96.56% 2nd Principal Component 12.07% 1.40% 3.44%

(Table 3.1: Explanatory Power of Principal Components of GS, MS and JPM)

The first component of two dimensional DVAs of these three banks explains more than 85% of the total variation of the entire data. The linkage between the first component and a fundamental variable or economic indicator enables the derivative dealers and risk managers to better understand and manage their own risk profile.

DVA could be expressed as 푇 푖− 퐼 푀푡푀푡 (1 − 푅푡) (3.10.11) 퐷푉퐴 = − ∑ ∫ 피ℚ ( ) ℙℚ (0, 푡) 푑 ℙℚ (0, 푡) 0 푁푂퐼푆 푆,퐶푃(푖) 푡 퐷,퐼 푖 0 푡 with certain assumptions discussed earlier this section. If the derivative dealers do take on OTC derivative positions with certain counterparties under certain CSAs and place corresponding hedging

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Counterparty Credit Risk, Funding Risk and Central Clearing positions with hedging counterparties, the resulted exposure profiles of the derivative dealers are expected to be relatively flat and a significant portion of DVA variations could be attributed to the ℚ variations in the default probabilities (i.e. ℙ퐷,퐼(0, 푡)). Therefore, it is economic intuitively to use the CDS spreads as the external explanatory variable for the first principal component of two dimensional DVAs.

(Figure 3.7: GS DVA Analysis Sources: data – Bloomberg)

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(Figure 3.8: MS DVA Analysis Sources: data – Bloomberg)

(Figure 3.9: JPM DVA Analysis Sources: data – Bloomberg)

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It seems that there exists a strong correlation between the CDS spreads dynamics and the first principal component of the DVAs. Let’s start with a single factor linear regression by using CDS running spreads as the explanatory variable.

(Figure 3.10: Prediction Test of CDS as Explanatory variable for GS Sources: data – Bloomberg)

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(Figure 3.11: Prediction Test of CDS as Explanatory variable for MS Sources: data – Bloomberg)

(Figure 3.12: Prediction Test of CDS as Explanatory variable for JPM Sources: data – Bloomberg)

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The explanatory power of plain CDS running spread is not significant from this perspective. The 푅2s of 퐶퐷푆푡 for these three banks are

Bank Goldman Sachs Morgan Stanley J.P. Morgan 퐶퐷푆푡 5.73% 4.66% 22.60%

(Table 3.2: Summary of 푅2of CDS as Explanatory Variable for GS, MS and JPM)

If 퐶퐷푆푡−1 is selected as the explanatory variable,

(Figure 3.13: Prediction Test of 1 Lag in CDS as Explanatory variable for GS Sources: data – Bloomberg)

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(Figure 3.14: Prediction Test of 1 Lag in CDS as Explanatory variable for MS Sources: data – Bloomberg)

(Figure 3.15: Prediction Test of 1 Lag in CDS as Explanatory variable for JPM Sources: data – Bloomberg)

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The explanatory power of one lag of plain CDS running spread is significantly greater than that of plain CDS running spread but the slope of the regressions reverts in this circumstance. This poses a significant problem to the interpretation of DVAs. The details could be found later this section. The 2 푅 s of 퐶퐷푆푡−1 for these three banks are

Bank Goldman Sachs Morgan Stanley J.P. Morgan 퐶퐷푆푡−1 35.29% 19.68% 60.30%

(Table 3.3: Summary of R^2 of 1 Lag in CDS as Explanatory Variable for GS, MS and JPM)

Here the first order changes in CDS spreads is defined as Δ퐶퐷푆푡 and Δ퐶퐷푆푡 = 퐶퐷푆푡 − 퐶퐷푆푡−1.

If Δ퐶퐷푆푡 is selected as the explanatory variable,

(Figure 3.16: Prediction Test of 1st Order Changes in CDS as Explanatory variable for GS Sources: data – Bloomberg)

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(Figure 3.17: Prediction Test of 1st Order Changes in CDS as Explanatory variable for MS Sources: data – Bloomberg)

(Figure 3.18: Prediction Test of 1st Order Changes in CDS as Explanatory variable for JPM Sources: data – Bloomberg)

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The explanatory power of first order changes in CDS running spreads is significantly greater than the other two candidates but the slope of the regressions remains negative in this circumstance. The 푅2s of Δ퐶퐷푆푡 for these three banks are

Bank Goldman Sachs Morgan Stanley J.P. Morgan 훥퐶퐷푆푡 73.49% 83.83% 98.34%

(Table 3.4: Summary of R^2 of 1st Order Changes in CDS as Explanatory Variable for GS, MS and JPM)

The summary table of 푅2s for these three different explanatory variables is

Bank Goldman Sachs Morgan Stanley J.P. Morgan 퐶퐷푆푡 5.73% 4.66% 22.60% 퐶퐷푆푡−1 35.29% 19.68% 60.30% 훥퐶퐷푆푡 73.49% 83.83% 98.34%

(Table 3.5: Summary of R^2 of three different explanatory variables as Explanatory Variable for GS, MS and JPM)

Do banks really hedge their DVAs?

Here if we expand explanatory power analysis to the DVA numbers (not just the first principal component of two dimensional DVAs),

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(Figure 3.19: Prediction Test of 1st Order Changes in CDS as Explanatory variable for GS DVAs Sources: data – Bloomberg)

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(Figure 3.20: Prediction Test of 1st Order Changes in CDS as Explanatory variable for MS DVAs Sources: data – Bloomberg)

(Figure 3.21: Prediction Test of 1st Order Changes in CDS as Explanatory variable for JPM DVAs Sources: data – Bloomberg)

2 The 푅 s of Δ퐶퐷푆푡 for these three banks DVAs are

Bank Goldman Sachs Morgan Stanley J.P. Morgan 훥퐶퐷푆푡 68.14% 83.49% 98.37%

(Table 3.6: Summary of R^2 of 1st Order Changes in CDS as Explanatory Variable for DVAs GS, MS and JPM)

Δ퐶퐷푆푡 yields highest explanatory power and exhibits the best fit to the DVA data series. Generally,

DVA is expected to exhibit positive correlation with Δ퐶퐷푆푡, that is, DVA is expected to increase if CDS spread widens (i.e. the credit quality worsens). Our regression analysis rebuts our expectation and presents a negative slope, i.e. the DVA is expected to decrease if CDS spread widens. Why?

The DVA terms in financial statements is normally DVA net of hedges. That is,

퐷푉퐴푁푒푡 표푓 퐻푒푑푔푒푠 = 퐴푐푡푢푎푙 퐷푉퐴 − 퐷푉퐴 퐻푒푑푔푒푠 (표푟 퐷푉퐴 푀표푛푒푡푖푠푎푡푖표푛푠). (3.10.12)

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If the banks use the funding transfer policy (FTP) as discussed in Albanese and Andersen (2014), they would charge their counterparties unilateral CVA (UCVA) as a way to monetise DVA or (perfectly) hedge DVA. Therefore the DVA net of hedges would tend to exhibit a significant high 푅2 as the DVA hedging positions are fairly priced by the bank’s own CDS curve. If the bank employs other DVA hedging strategies like selling protections on its peers as discussed in Brigo (2015), the DVA net of hedges would tend to exhibit a relatively lower 푅2. As DVA hedging positions are positively correlated with the bank’s own CDS curve, the DVA net of hedges should have a negative slope against the first order changes in the bank’s CDS spread.

As predicted by linear regressions discussed above, if the CDS curves of the three banks shift upwards by 1 basis point, the PnL would increase by

Bank Goldman Sachs Morgan Stanley J.P. Morgan 훥퐶퐷푆푡 = +1 푏푝 $1.9843 MM $10.525MM $10.1451 MM

(Table 3.7: DVA Book Sensitivity to CDS changes for GS, MS and JPM)

(Note: The PCA analysis for CVAs is included in appendix.)

How do investors profit from DVA hedging activities?

In theory, the increase in DVA is interpreted as a benefit term to the bank itself as the senior bondholders of the bank would benefit by paying less on the bank’s liabilities on OTC derivatives positions in the default state of the bank. Given the bondholders could only realise their DVA gains at the default time of the bank, the DVA hedging positions or monetised DVA components are booked as net profits for the shareholders. Therefore, monetised DVA components could be perceived as the implicit default protection the shareholders sold to the senior bondholders. And the entire realised DVA gains at the default time of the bank would be allocated to the senior bondholders. Hence, as long as the banks are not in their default states, the increase in DVA would lead to an increase in PnL for the banks’ shareholders and the equity prices of the banks would therefore correspondingly go up. Since the monetised DVA components or DVA hedging positions are well explained by Δ퐶퐷푆푡, we could use Δ퐶퐷푆푡 as a proxy for monetised DVA components and build up a long short equity trading strategies by utilising Δ퐶퐷푆푡 as trading signals.

The banks’ equity prices and CDS spreads are provided by Bloomberg. The banks have to report their DVA positions on quarterly basis and the long short equity strategies are built on quarterly data series. The 20 banks used to construct the portfolio are

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Goldman Sachs Deutsche Bank Barclays Capital BNP Paribas Morgan Stanley UBS Standard Charted Lloyds Banking Group J.P. Morgan & Chase HSBC Nomura Crédit Agricole Citi Group RBS Group Societe Generale Santander BofA Merrill Lynch Commerzbank Credit Suisse Wells Fargo

(Table 3.8: DVA Hedging Equity Portfolio Holdings)

The investors should expect the equity price of the banks increase if they observe the CDS spreads widen. Hence the rationale investors would long the banks with greatest Δ퐶퐷푆푡 and short banks with lowest Δ퐶퐷푆푡. The investment horizon is from 2008/12/31 to 2015/03/31. The annualised risk return profiles are summarised as

Risk/Return Profile Long/Short 6 Banks Long/Short 10 Banks Long/Short 20 Banks Return 22.7% 23.0% 14.1% Volatility 31.9% 23.3% 17.7% Sharpe Ratio 0.71 0.99 0.80

(Table 3.9: Risk/Return Profile of DVA Hedging Equity Portfolio)

Investors could profit from constructing DVA hedging portfolios. And this strategy indeed provides an alternative way for banks to monetise their DVAs by trading equities of their industry peers.

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4. Central Clearing

4.1 Introduction to Central Clearing

The global financial crisis which caused by failures of large financial institutions highlighted the importance of risk management in over the counter (OTC) derivative and regulations in financial markets. Although the financial crisis was not caused by OTC derivatives, the OTC derivatives had been deemed as the channels to propagate systematic risk and amplify various problems.

The counterparty credit risk created by derivative transactions due to risk of insolvency of one party creates knock on impact and systematic risk in the financial system due to the market structure of OTC derivatives market. The over-reliance of the whole financial system on a few key nodes was fragile in the financial crisis and the distress of some of the key financial institutions exacerbated the instability of the system. The view that the financial shocks we experienced in the crisis were transmitted and amplified by counterparty credit risk, opacity in OTC market and interconnectedness of large financial institutions was formed in this context.

Regulatory changes aiming at moving the risk of global banks from bilateral OTC market were embarked by policymakers in the aftermath of financial crisis. Greater capital requirements for OTC derivatives were introduced to strengthen bank’s operating capability in distressed market environment. A certain number of standardised products were moved from bilateral OTC market to mandatory central clearinghouse. The main task of a central counterparty (CCP) is to interpose itself between counterparties by acting as a buyer to sellers and a seller to buyers of certain products. The aim of this chapter here is to give an introduction to this new trading paradigm for OTC products.

Brigo and Pallavicini (2014) provided a complete analysis of nonlinear xVA pricing under central clearing regime and made a detailed analysis of comparisons of derivatives pricing under central clearing and derivatives pricing with standard CSAs.

Duffie and Zhu (2011) pointed out that CCPs are efficient in reducing exposures under the condition that an unrealistically high number of participants were involved in the CCPs. The exposures here are IID and homogenous Gaussian exposures. Rama et al. (2012) consider more realistic exposure distribution and suggested that netting benefits under CCPs are much higher than the one suggested by Duffie and Zhu (2011). The main discussion of CCPs at the current stage focus on the netting benefits on bilaterally clearing and central clearing. Gregory (2012) suggested the auction regime of CCPs should be taken into account in the CCP analysis. CCPs could auction the positions of a defaulted member among the surviving members. While under a bilateral setting, the trades would be

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Counterparty Credit Risk, Funding Risk and Central Clearing terminated immediately. Any critics over the CCP regime should at least provide a better choice for unwinding or replacing positions than auction if a major financial intermediary defaults. Acemoglu et al. (2013) provides a framework for studying the magnitude and number of negative shocks on financial system. From a bank’s perspective, it provides a potential way to understand the default amount contribution in a CCP regime.

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4.2 A Brief Review of Clearing Mechanisms of Exchanges

A financial product generally takes a long time to move to be traded in an exchange. It has to meet certain requirements for trading volume, fungibility and liquidity. The product per se has to be standardised to improve its liquidity and exchanges therefore becomes a central trading facility for trading and hedging a given set of financial risks. The trading facility and reporting service provided by exchanges provides a channel for price discovery and greater transparency of market prices.

The key counterparty risk mitigation methodology used by exchanges is ‘clearing’. Clearing is the full term service provided by exchanges on transactions between different counterparties throughout the lifecycle of trades. Margining and netting have been developed to reduce counterparty risk that a counterparty is not able to settle underlying trades.

Three different forms of clearing have been developed in the market place.

1. Direct Clearing Direct clearing is like netting in OTC markets. It uses standardisation of derivatives terms which makes the underlying trades fungible and therefore two counterparties would apply payment of difference instead of delivery of multiple streams of cash flows to reduce the exposure. Exchanges in direct clearing do not interpose themselves between counterparties but serve as monitor and mediator in case of any trades disputes. Further developments of standardisation of products enhance the liquidity and reduce difficulties of closing out positions.

2. Clearing Rings Exchanges would invite multiple market participants to join a trade compression ring to offset their positions. This kind of multilateral netting could effectively reduce the overall exposure in the whole system and all the participants in the ring have to reach agreements on settlement prices for hundreds of different contracts which would be facilitated by exchanges. However, this overall reduction in terms of collective exposures is not welcomed by all market participants as some market participants would prefer having a large exposures to a certain set of counterparties rather than other counterparties.

The aforementioned two clearing structures did not set up a mechanism to stop the spreading of domino effect caused by failure of a counterparty.

3. Complete Clearing

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The complete clearing comes to the stage and a central counterparty is placed between different market participants and becomes a central counterparty to all transactions in the marketplace. In extreme market distressed scenarios, contracts managed by defaulted counterparties would be taken over by CCP and CCP would halt the spread of failures of certain market participants leading to creation of cascading effect which results in further defaults of seemingly unrelated institutions. The key feature of CCP is homogeneity in counterparty risk mitigation principles. All clearing members have to adhere to strict margin requirements which requires daily variation margin to offset daily PnL and segregated initial margin accounts to cover the potential shortfall in closing out positions in case of default of a clearing member. Additional buffers like default funds are also introduced to cover the potential losses. The details would be discussed in the following sections.

The homogeneity in margining requirements caused resistance from clearing members with stronger credit quality as the cost of clearing would be almost the same for clearing member with lower credit quality and the cost of clearing (and cost of maintaining a better credit quality) outweighs the potential benefits to some clearing members in some markets.

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4.3 Clearing for OTC Derivatives

The standardisation of exchange trade products have been developed for many years and it has improved price transparency and market accessibility for general market participants. OTC derivatives, on the contrary, have been transacted on bilateral basis and each party in the transaction has to bear its counterparty risk and apply its own methodology to manage counterparty risk. The key driver of popularity of OTC markets is flexibility of OTC products. Market markets could use their financial engineering techniques to create tailored or customised risk management contracts to support their clients. The OTC products therefore has a clear advantage over exchange trade products from risk management perspective as they will be tailored to the specificities of clients’ demand and hence the clients will not be exposed to the classic ‘basis risk’. This advantage, of course, comes with a price. Given the fact the OTC product is customised to the client’s need, it becomes difficult for the client to offload this trade from their book. The client may receive an uncompetitive price for unwinding the transaction with original counterparty, and have to be granted permission from the original counterparty for assigning or novating the trade to a different counterparty.

The significant growth of OTC market was fuelled by deregulations in financial industry and the profitability of these transactions had attracted many market participants. However, OTC derivatives market was still transacted on bilateral basis and much less transparent than exchange market. The OTC derivatives trades generally have long maturities and normally the exchange traded products are short maturity products. Hence CCP, as the party interposing itself between different clearing members, changes the practice of clearing and settlement in bilateral market and propose default mutualisation among clearing members as its method to manage counterparty credit risk.

The cascading impact of Lehman’s default raised a great concern regarding systematic risk of OTC markets. The main methodologies proposed by regulators and academia could be categorised into two main families: 1. Reduce the default probability of systematically important financial institutions (SIFIs); 2. Streamline the default process and reduce the shockwaves impact in the financial system.

Bilateral clearing

In bilateral OTC derivatives market, each side of a given transaction (or a portfolio) should hold sufficient capital on their book as a buffer against potential losses due to the default of the counterparty. Market participants therefore take on counterparty credit risk by holding a fair amount of capital and absorb the losses in the default states of their counterparties. The ‘survivor’ pays for the losses in bilateral OTC market. The regulatory capital is generally computed by converting the

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Counterparty Credit Risk, Funding Risk and Central Clearing entire exposure profile into one year exposure with additional attached parameters linked to the credit quality of the counterparties. Given its time horizon and VaR (value at risk) approach used on capital calculation, the regulatory capital is not sensitive to the short term market volatility and hence the capital is not pro-cyclical. CVA is introduced to price the exposure the market participant bears in OTC transactions but the losses of counterparty credit risk are mainly absorbed by the capital in general due to un-hedgable issues in CVA hedging activities. If one of the key market participant in bilateral OTC market defaults, the default process of the market participant is un-coordinated (as we have seen in Lehman default) and the counterparties would have to liquidate or close out their positions by themselves. Two way CSAs are now widely used in bilateral OTC derivatives trading in current market regime. Derivative dealers have been developing their in-house platforms to transform uncollateralised positions into partially/fully collateralised positions with certain thresholds and MTAs (minimum transfer amounts) for reductions in regulatory capital.

Central clearing

Central clearing improve the margining requirements to a different level by imposing initial margins on OTC derivatives transactions. The counterparty default loss that is absorbed by capital of survivors in bilateral OTC market is absorbed by initial margins posted by the defaulting parties and defaulted parties in central clearing regime. The ‘defaulter’ pays for the losses in central clearing system. The initial margins are calculated on a short time horizon and not sensitive to changes in counterparty credit quality but sensitive to the changes in the market volatilities. CCPs therefore are likely to increase their initial margins in turbulent market scenarios which in turn transforms into heavier funding liquidity cost of market participants and may further amplify chaos in the marketplace. The procyclicality exhibited by initial margin mechanisms could be mitigated by introducing stressed market data into initial margin calculation and expanding the time horizon of the time series of market data. If the initial margins posted by the defaulted clearing member is not sufficient to absorb the entire losses in the system, the remaining losses would be mutualised via default funds among the surviving clearing members. Rather than closing out the defaulter’s portfolios in an uncoordinated manner in bilateral markets, the defaulter’s portfolios will be centralised auctioned in central clearing system. It is hard to say the initial margins requirements imposed by central counterparties are ‘sound and safe’ market practices as the procyclicality of initial margins mechanisms might lead further failures in the financial system. However, it is clear that the underlying risk sensitivities and many other aspects of OTC derivatives have been changed by central clearing. For examples, if one of the key market participants defaults, the losses are absorbed by survivors’ capital in uncollateralised OTC transactions and absorbed by survivors’ capital and variation margin posted by the defaulter in partially/fully collateralised OTC transactions. In central clearing system, the losses are shared jointly with initial margins and variation margins posted by the

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Counterparty Credit Risk, Funding Risk and Central Clearing defaulter, default funds of CCPs and capital held by survivors to cover counterparty credit risk. It is not obvious if central clearing is a better risk management mechanism compared to bilateral clearing with two way CSA but is clear that the underlying risk dynamics and interactions between different variables have been changed and even more complicated (especially in terms of initial margin calculation). Generally the derivative dealer have two exposure profiles in bilateral clearing mechanism for every single trading business, i.e. exposure of underlying trade and exposure of hedging instrument. Typically the hedging transaction is traded with two way CSA in current market structure. In central clearing mechanism, especially for client clearing business, the market marker would have two exposure profiles for the underlying transaction. The derivative dealer has the transaction with the end user and places the same trade with a CCP for central clearing. If the derivative dealer decides to hedge this underling transaction, it has to enter into a hedging position with another clearing member or even another CCP. Therefore, the typical two exposure profiles in bilateral clearing marking making business now evolves into three exposure profiles in central clearing trading business. The initial margin requirements and default fund contributions further proliferates the costs of running trading business in central clearing mechanism. The co-existence of bilateral clearing market and centrally clearing market bifurcates the trading and hedging structure and creates netting inefficiencies in the OTC derivatives market. This issue would be discussed in detail later this chapter.

Market structure of bilateral clearing and central clearing

ISDA (2014) reports 56% of current OTC interest rate derivatives are centrally cleared and it could be further increased to 80% if all eligible interest rate transactions are cleared in the foreseeable future. The remaining portion of the interest rate derivatives that are deemed non-clearable are cross currency swaps, interest rate swaptions and non-clearable currencies denominated interest rate derivatives. CCPs, like SwapClear etc., are affiliated to exchanges in the history. Their main responsibilities are clearing standardised exchange traded products. The initial margins for exchange traded products are normally calculated on a one day time horizon with a single underlying risk factor like equity index. The long dated complex OTC derivatives without standardisation and liquidity are relative difficult to be centrally cleared and the quantification or calculation of initial margin requirements becomes a far more complex job compared to the initial margins calculations for exchange traded products. Given the broad ranges of OTC derivatives, only a small fraction (i.e. relatively standardised OTC products) of all types of derivatives is eligible to be cleared. The interesting fact is this small fraction of all types of derivatives makes up the majority of the OTC derivatives market. It seems to reach an agreement in the marketplace that many OTC derivatives would not be central cleared as these customised OTC derivatives without abundant liquidity are more suitable to be risk managed on a bilateral basis (Duffie et al. (2010)). Due to the inability of

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Counterparty Credit Risk, Funding Risk and Central Clearing central clearing exotic structures, it is easy to understand the motivation to introduce enhanced margin requirements (with initial margins in particular) into bilateral OTC market. Some exotic products are deemed to be too risky and complicated to price and margin and CCPs therefore do not clear these exotic products as these exotics might damage the security of a clearing house. In other words, the derivatives that are ‘dangerous’ and might drag the market into the next financial crisis are too ‘exotic’ for CCPs to clear. There is no evidence that the exotic products are going to be cleared in CCPs in short notice at the time of writing. However, if the regulations do push these exotic products to be cleared via CCPs, the initial margins and other related costs would be enough to ‘kill’ the market to that type of products as CCPs should make sufficient conservative assumptions regarding the valuation and margining of the risks embedded in exotics. This might lead us back to some discussions of the origin of the global financial crisis (GFC). If some of the risks embedded in those exotic products are proved to be too complex to value and CCPs decide to not to clear these products as they might harm or pose significant downside risk to the overall stability of CCPs operations, is it ‘safe’ to leave these products in the bilateral clearing markets since we ‘believe’ the derivative dealers have better risk managements skills to manage these exotic products?

Standardisation of OTC derivatives is one of the key steps in moving OTC products into CCPs. The products per se could be standardised to enhance their liquidity but the complexity of risks are still embedded in these products. Standardisation therefore might create an illusion that enhanced liquidity of these products are proofs of reduced complexity in the structure. Index based products are classic examples of standardisations. The derivative dealers, those with better knowledge on risks in these standardised products, would be incentivised to select the riskiest ones to be centrally cleared. This is an example of adverse selection problems in central clearing.

What products or asset classes should be centrally cleared?

It naturally comes to one question: should plain vanilla OTC products be cleared through CCPs since these products are simple and liquid? Or should these products stay in the bilateral cleared markets since they do not pose excessive risks into the marketplace? For example, FX products are exempt from central clearing as the majority of FX products are short maturity products and these products with short maturity normally exhibit relatively small exposure profile. However, this argument does not hold in distressed market scenarios as FX rates are heavily linked to sovereign credit risk and jump in volatility would produce a significant increase in exposure even for short dated FX products.

The current market trend on CCP products coverage developments are on the opposite side of the initial motivation or fundamental view of mandatory clearing in the aftermath of global financial crisis (GFC). IMF (2010) claimed that the bailout of AIG could have been prevented by central clearing

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Counterparty Credit Risk, Funding Risk and Central Clearing system as the uncollateralised exposure would not be possible to be accumulated to systematically risky level in central clearing system. AIG would probably still face funding liquidity risk of margin calls but this problem would be found out much earlier if there is mandatory clearing in place for AIGFP’s credit derivatives positions. The regulatory move to central clearing has been perceived as a result of bailout of AIG. However, if we review the AIGFP (AIG Financial Products) credit derivatives positions, these positions are highly complex and customised to the protection buyers’ need and therefore not qualified to be cleared in current CCP regime. In that booming period of credit derivatives, AIGFP did not have to post any collateral in their variation margins due to their high credit ratings. Interest rate swaps and other standardised OTC products which are relatively simple and take up a majority of outstanding OTC contracts in the marketplace are now being centrally cleared. The remaining part, i.e. the customised OTC contracts are relatively complex and hard to price, remains in bilateral clearing market and poses a potential trigger to the future financial crisis. The risk associated with non-clearable OTC transactions could be mitigated by enhanced CSAs with initial margins in bilateral clearing markets at the expenses of increasing funding liquidity cost for market participants. CCPs might expand their product coverage by further standardising certain types of OTC derivatives if the potential profits from clearing these products is expected to dominate the costs of developing the clearing and margining facilities for these products.

Duffie and Zhu (2010) indicates that a large number of CCPs would reduce the multilateral netting benefits and the loss of netting benefits might outweighs the benefits gained by clearing through CCPs. For example, if we have two CCPs clear the same product in one jurisdiction, the clearing members (and end users of their clearing service) would face higher initial margin requirements and associated costs and increase the clearing members close out costs as original offsetting positions (underlying trading position and its corresponding hedging instrument) could not be netted against each other and have to be replaced by other transactions (via auctions etc.). Market fragmentation is the most important factor in co-existence of multiple CCPs on the same products. Generally the OTC products traded within a given region is ought to be cleared in the CCPs domiciled in that specific region. The difference of Bankruptcy Codes makes it even harder to clear the same products across different jurisdictions. The problems would be further complicated along with increased closing out costs in fragmented markets due to the lack of coordination of different CCPs close out processes in case of default of a major market participant.

There might be some problems if the CCPs could handle clearing of multiple asset classes or types of OTC derivatives. For example, suppose a clearing member just went into bankruptcy and the CCP handling its OTC derivatives portfolio now has to close out its existing positions with this defaulted member. The products or transactions with better liquidity like interest rate swaps (IRS) in that portfolio would be closed out more quickly and the CCPs and associated trading counterparties could

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Counterparty Credit Risk, Funding Risk and Central Clearing therefore access to the initial margins and default fund contributions in a shorter notice compared to illiquid positions like credit default swaps (CDS). These asymmetric treatments of transactions with different liquidity conditions gives an unfair advantage for the liquid positions holders. Product segregations could be introduced to reduce this problem at the expense of higher operating costs and reduced netting efficiency. The problem could also be alleviated by clearing only one asset class or different types of products with similar liquidity for a given CCP. Current market structure development is at this stage and OTC CCPs normally provide clearing service for a single asset class.

Market competition between CCPs also have impacts on the number of CCPs in the marketplace. A CCP generally starts with clearing a single asset class within a geographical region and grow with expansion in geographical bases and products/services coverage. The economics of scale of running central clearing business will be obtained until a sufficient portion of certain products is cleared through that CCP such that the clearing members of that CCP could realise multilateral netting (similar trade compression in bilateral clearing markets) benefits. Competition between CCPs would incentivise to CCPs to lower its clearing cost especially in the initial margin requirements and this type of competition would make CCPs loosen their risk management practice to attract more business flows. Policymakers, regulators and market participants would have to trade off on certain terms to reach an agreement on functions of CCPs and the CCPs across different jurisdictions should work out plans to further enhance multilateral netting (across asset classes if possible) and develop common close out procedures to minimise the negative impacts on the market by divergences on geopolitical issues.

From derivative dealer’s perspective, it is critical to choose a CCP to clear a certain type of products in a jurisdiction. Only the CCP recognised by regulators of a given region would have impacts on regulatory capital the derivative dealer should hold to maintain operations in that region. The membership requirements like initial margins, default fund contributions and products coverage offered by the CCP would have significant impacts on running trading business as funding liquidity risk and netting benefits should be taken into consideration. The specificities of margin requirements like qualified securities to be posted as variation margin and initial margin, initial margin calculation methods and potential cross margining policies with other CCPs would determine FVA and MVA price as discussed in FVA chapter. If a derivative dealer finds itself has built a substantial exposure or size of portfolio with a given CCP, the derivative dealer may decide to clear some portion of its portfolio with another CCP to diversify its potential losses at expense of reduction in netting benefits. In stressed testing scenarios, segregation of margin accounts and portability of the portfolio (i.e. the right to transfer its portfolio holdings with the defaulted clearing member to surviving members) are criterions used in assessing the safety of CCP operations.

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4.4 Advantages and Disadvantages of Central Clearing

Central clearing has certain advantages compared to classic bilateral clearing regime. Multilateral netting regarding certain products are automatically achieved without a third party like TriOpitma intervention to achieve trade compression effects. The clearing members would have extra flexibility to initiate new transactions and hedge the transactions within clearing members of CCP without incurring further costs on regulatory capital and margins as the exposure of these two trades with different parties have been perfectly netted by CCP. The strict initial margin and variation margin policies set by CCPs in theory reduce the burden of the clearing members assessing the credit qualities of certain counterparties as over-collateralised transactions (like certain repo trades with haircuts) per se are not sensitive to the counterparty risk but sensitive to the market volatility of collateral assets. If the loss due to default of one clearing member exceeds its initial margin, the defaulted member’s default fund contribution will be on the frontline of defence of CCP. If the loss exceeds the entire commitments (i.e. initial margin and default fund contribution) of the defaulted member, the remaining loss is distributed and mutualised by the surviving clearing members. Then the counterparty default loss is shared across surviving members of the CCP without forming up into a dramatic disappearance of a key node of the entire financial system. CCPs increase the pricing transparency of OTC derivatives to reduce the pricing disputes regarding Mark to Market of certain trades and their associated variation margins. The fungibility introduced by standardisation process of OTC products makes it easier for market participants to trade, hedge and close out their positions. The cooperation between CCPs and local jurisdictions would reduce the legal risk in running OTC derivatives business and consolidate available resources to improve operating efficiency. Now it is possible to trade anonymously on CCPs. The enhanced market entry, fungibility of products, and other benefits actually increase the liquidity of centrally cleared OTC derivatives. Daily margining improves the price transparency and mitigates the counterparty credit risk. The activities of clearing members are actively monitored by CCPs to prevent the exposure profile of certain market participants builds up to an excessive level for the financial system. The monitoring regime used by CCPs could improve the transparency of market participants’ activities, rectify the severe asymmetric information problems due to short of information regarding exposure profiles of certain market participants presented in bilateral clearing markets and disperse market panic in distressed scenarios. If one of the clearing members does default, the end users of the defaulted clearing member’s clearing service are able to port/transfer their portfolio to surviving clearing members of CCP. Their existing portfolios with the defaulted clearing member therefore do not have to be closed out and continue to operate with surviving clearing members. This portability/transferability function of CCPs would be served as a stabiliser of the financial system in the default states of clearing members.

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The following up coordinated default management by CCPs like central auctions would reduce the price distortions due the default of market participants in crisis period. Multilateral netting would significantly reduce the outstanding positions and the number of positions have to be closed out in the default state of counterparties. CCPs therefore could resolve the problem of ‘uncoordinated’ actions by different market participants in OTC markets and is expected to reduce the level of chaos in turbulent market conditions.

These advantages of CCPs discussed above may be questionable under certain circumstances. The clearing members are incentivised by loss mutualisation scheme to clear more ‘dangerous’ trades with CCPs due to inherent asymmetric information problems in OTC markets. This moral hazard and adverse selection problems could be mitigated by increase in margin requirements (especially initial margins) and default fund contributions at the expenses of increasing funding liquidity costs of market participants. The multilateral netting feature of central clearing might be perceived as a redundant function of CCPs as trade compressions in OTC markets could achieve the same netting effects in normal market conditions. However, the multilateral netting in the default state of a clearing member under central clearing regime would be proved to be invaluable as the trade compression is vulnerable in that circumstance and it becomes difficult to ask the third party to step in to compress the existing portfolios with defaulted market participants in OTC market.

Apart from the benefits and advantages of CCPs, central clearing have some disadvantages over bilateral clearing. The cost of margin requirements, default fund contributions and associated capital requirements would lead to a substantial overall cost. The segregation of initial margin accounts and portability/transferability would increase the overall cost even further and make the market participants expensive to trade derivatives with CCPs. The multilateral netting benefits by central clearing may not dominate the loss of netting benefits across different asset classes (as FX products are exempt from central clearing) or across multiple CCPs (as cross margining and cooperation between different CCPs is not prevalent at current stage of developments of central clearing). The initial margins imposed by CCPs in derivatives transactions are sensitive to market volatility and a relatively small increase in initial margin requirements as a response to moderate increase in market volatilities is reasonable in normal market conditions. When it comes to turbulent market conditions, i.e. the market comes to its stress periods, the initial margins would be expected to experience an upward jump to stabilise CCPs and increase its safety. The additional funding liquidity risk created by the procyclicality of initial margins regime would be amplified during this period and probably result in contagion effects like classic ‘liquidity traps’ experienced by different market participants in the marketplace. The expansion of historical observation window including the 2008-2009 period would reduce the impact of procyclicality problem on initial margins calculation at the expense of maintaining the initial margins at a high level in normal market conditions. The large derivative

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Counterparty Credit Risk, Funding Risk and Central Clearing dealers have their own settlements, risk management and margining systems and the functions of CCPs generally duplicates these existing functions or systems existed in bilateral clearing markets. The clearing members entry requirements are set a level restrained market competition due to the associated cost (like default fund contributions) of clearing.

In bilateral OTC derivatives market, the price of any derivative is sensitive to counterparty credit risk (CVA component), CSA terms (or margin requirements) and funding curves (FVA component). CCPs, like repo markets, remove these price components specific to entities and homogenise the prices for market participants regardless of their trading counterparties. Repo markets, as discussed before, implement collateralisation strategies to remove the counterparty dependent components out of the price and the repo rate is sensitive to the qualities of underlying collateral like market volatilities instead of credit quality of counterparty. The overcollateralization scheme implemented by CCPs via initial margin requirements removes the counterparty dependent pricing components by the same strategy and enhance the fungibility of centrally cleared products (as the market counterparties become interchangeable) but introduce CCP dependent pricing components like margin valuation adjustment (MVA) accounting for the funding cost of initial margins. As discussed in previous sections, if a SIFI (systematically important financial institutions) defaults in bilateral cleared market, the size and magnitude of overall losses are difficult to estimate due to lack of knowledge of holdings of the defaulted institution and contagion effect would circulate throughout the financial system via exposed positions of counterparties of defaulted institution. The classic ‘too connected to fail’ problem is expected to be circumvented by loss mutualisation regime of CCPs. The CCPs would have solid information of clearing members’ exposure profile and the damage of uncertainty of losses created by asymmetric information problem could be mitigated by dispersion of residual losses (after absorption by initial margin and default fund contribution posted by the defaulted clearing member) among the surviving members such that the shockwave created by the default would not concentrate on a single or a few counterparties but mutualised by the entire financial system. The systematic risk is expected to be reduced by CCPs from this perspective. Homogenisation of statuses of all clearing members give the chance to derivative dealers or institutions with relative worse credit qualities an advantage in trading with CCPs as they would not face the significant counterparty credit risk charges (CVA) in bilateral OTC markets as long as these institutions are able to maintain their collateral operations adhered to the CCPs margining requirements. Significant CVA charges in bilateral OTC markets would push the institutions with worse credit qualities to improve their credit qualities but such kind of incentives are removed by central cleating mechanism. The clearing members would therefore be incentivised to reduce their monitoring costs on individual member’s credit qualities. And the institutions with worse credit qualities would be able to build up greater positions in central clearing regime than in bilateral clearing markets. The governance and internal risk control of CCPs therefore are allocated with greater responsibilities on maintaining the stability of CCPs. It seems like

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Counterparty Credit Risk, Funding Risk and Central Clearing the homogeneity in statuses of clearing members do create some problems and CCPs should reasonably allow heterogeneity exists among the statues of clearing members. This sounds like a reasonable suggestion but may faces significant challenges in practical implementation as what we have observed in problems created by heterogeneity in membership statuses and treatments in European Union.

Asymmetric information problem discussed above would become more severe if more complex OTC derivatives like exotics without abundant liquidity could be cleared through CCPs. The clearing members, i.e. derivative dealers and leading banking institutions, have a clear information advantage of these products over CCP risk managers. The potential adverse selection problem could be acute as clearing members are incentivised to place the transactions with highest pricing divergence in CCPs to ‘arbitrage’ profits. CCPs could impose higher initial margins to account for potential jumps in market volatilities and mispricing to mitigate the asymmetric information problem. And it turn this would make central clearing too expensive for certain market participants. Loss mutualisation give the clearing members a channel to shift risk from their own balance sheet to the balance sheet of CCP. The moral hazard and adverse selections problems brought by standardisation of OTC derivatives or fungibility of centrally cleared derivatives makes it worthwhile for CCPs to segregate certain portfolios to mitigate the potential problems embedded in clearing certain products (Pirrong (2011)).

Mandatory central clearing for certain OTC products creates some side effects. These unintended consequences have significant impacts on competition between different CCPs and services and products offered by CCPs in the future developments. Futurisation of OTC derivatives becomes one of the ‘innovative’ solutions provided by CCPs as a measure to reduce clearing costs. Futures contracts written on assets generally could enhance the liquidity of trading underlying risks of those assets. The initial margin for future contracts is calculated on one day horizon and trading futures on OTC derivatives therefore could reduce the funding costs of initial margins (as the initial margin for OTC derivatives is generally calculated on five day horizon). The drawback of trading futures on OTC derivatives is that the cash flows generated by the may not match perfectly with the underlying OTC derivatives. The residual risk due to the mismatch between futures contracts and underlying OTC derivatives might magnify under certain market circumstances which will be discussed later. The celebrated CME deliverable swap futures is an example of futurisation of OTC derivatives. The time to maturity of this futures contract is typically set at three months and it promises to deliver OTC swap cleared by CME at maturity. The margin period of risk for futures is two days which is shorter compared to (at least) five days margin period risk for underlying OTC derivatives (Gregory (2014)). Futurisation of OTC products has been a controversial issue as the motivation for CCPs to provide futures clearing service is to reduce funding costs for end users and provide more competitive prices without a material reduction in risk of the transaction.

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Convexity adjustments created by central clearing

An interesting aspect of futurisation of OTC derivatives is the convexity adjustments embedded in swaps futures pricing. Dolmetsch and Leising (2011) reports the Jefferies sued Nasdaq clearing unit International Derivatives Clearing Group (IDCG) for discrepancy created by underlying interest rate swaps and its futurisation traded with IDCG. That is, the futurisation of interest rate swaps did not provide economic equivalents of underlying interest rate swaps. A large component of the price difference is due to the convexity adjustment. Let’s first have a brief review of convexity adjustments in trading uncollateralised interest rate swap and hedging by Eurodollar futures. An interest rate swap fixed receiver positions is a positively convex position as long position in fixed coupon bond. That means, the fixed receiver would benefit more if market rallies than the loss if the market sells off (with the same magnitude movement in yield to maturity). However, the Eurodollar futures is a linear instrument of underlying 3 month rates. In other words, a long position in Eurodollar futures is equivalent to an interest rate swap fixed receiver position without any convexity. A basis point move in underlying rate is equivalent to $25 movement in the valuation of Eurodollar futures (as 0.01% ∙ 3 ∙ 1,000,000 = 25) regardless of the directions of rates movements. If the market sells off (i.e. the 12 rates go up), the settlements prices of Eurodollar futures decline and the trader profits $25 if the rate moves up by 1 basis point. If the market rallies (i.e. the rates go down), the settlements prices of Eurodollar futures increase and the trader loses $25 if the rate moves down by 1 basis point. Hence, the trader receives cash in a high interest rate environment (and the cash could be reinvested at a high (short term) interest rate via reverse repo market) and pays cash in a low interest rate environment (and the cash could be borrowed at a low (short term) interest rate via repo market or the unsecured funding rate of the trader as discussed in FVA chapter ).Therefore, the net positions constructed by receiving fixed in interest rate swap and shorting Eurodollar futures therefore have a positive correlation with the underlying rates. Therefore, if the fixed rate is computed as a weighted average of the forward rates implied by Eurodollar futures, it is always better to receive fixed rate in an interest rate swap and hedge by shorting Eurodollar futures. This is one of the classic examples discussed in Hull (2000). The fixed receiver here therefore is expected to receive a lower rate than the swap rates implied by Eurodollar futures prices and the spread the fixed receiver accepts is expected to equal to the convexity benefits he/she could receive in fixed receiver swap. Therefore the trade shorts forward rate agreements (FRAs) and shorts Eurodollar futures to arbitrage for convexity benefits. For a centrally cleared interest rate swap, a convexity adjustment should be added to fix remove the systematic bias introduced by daily margining procedures. Suppose a clearing member enters into an interest rate swap fixed payer position. If the market sells off (i.e. the underlying rates go up), the clearing member profits from his fixed payer position and receives variation margins from CCPs in cash to account for the changes in Mark to Market of this position. The cash received is then invested

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Counterparty Credit Risk, Funding Risk and Central Clearing at a higher short term interest rate via reverse repo market. Now if the market rallies (i.e. the underlying rates go down), and the swap rates drop to its previous level, the clearing member therefore has to return the identical amount of cash back to variation margin. The clearing member could borrow the cash from repo market at a lower rate. The net profits here would be a positive number as the clearing member receives the variation margin and invests the cash in a high rates environment and pays the variation margin and borrows the cash in a low rates environment. And the overall profits of a cleared interest rate swap fixed payer position (including a variation margin account) is positively correlated with underlying interest rates. A convexity adjustment therefore should be introduced to properly account for this positive correlation term. Hence, a cleared interest rate fixed payer swap has a greater value over its uncleared interest rate fixed payer swap. The clearing member therefore should pay a higher fixed rate to the fixed receiver as a compensation for the convexity adjustment. The spread of cleared interest rate swap rate over uncollateralised interest rate swap rate in a theoretical environment without counterparty credit risk and funding risk therefore has two components. The first component is spread accounts for change of discounting curve from LIBOR curve to OIS curve as discussed in CVA/DVA/FVA chapters. The second component is the convexity adjustment discussed here accounting for the positive correlation between the net value of fixed payer swap and underlying interest rates.

In order to remove the second component spread due to positive correlation between net value of fixed payer swap and underlying interest rates, some CCPs like LCH introduces the price alignment interest (PAI) rate on the variation margin. If the clearing member receives variation margin from CCP, the clearing member has to pay PAI rate on the net variation margin he/she received to the CCP. If the clearing member pays variation margin to CCP, the CCP has to pay PAI rate on the net variation margin he/she received to the clearing member. The PAI rate is normally set at OIS rate of major currencies. This discussion briefly answers why the funding spread is calculated as the funding rate over OIS rate in the linear representation of FVA in Piterbarg (2010).

It is clear that the margining requirements do play an important role in valuation of interest rate swaps. The suit Jefferies brought on IDCG regarding mismatch in terms of valuation of interest rate swaps and futurisation products is partially due to the convexity adjustment created by different margins requirements in OTC markets and CCPs. In the aftermath of global financial crisis (GFC), margining has evolved to be one the key pricing component of derivatives. The standardisation procedures of OTC derivatives implemented by CCPs have made derivatives more liquid and fungible at the expenses of changes in underlying economic characteristics of original OTC derivatives.

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OTC derivatives are initially designed or customised to perfectly replicate the cash flows and other related features the end user intends to hedge. Futurisation of OTC derivatives produce financial products with similar economic characteristics of the OTC deals at a lower price but cannot have perfect matches of cash flows and margining requirements etc. of the OTC derivatives. The end users of derivatives, therefore, have to balance the benefits of reduction in costs and the costs of mismatches of cash flows in trading futurisation of original deals. Additional liquidity and fungibility of derivatives clearly come at a price and end users of derivatives therefore have to make the decision to take it or not in current regulatory regime.

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4.5 Regulatory Arbitrages Created by Central Clearing

Multiple CCPs domiciled in different geographical regions give rise to regulatory arbitrage opportunities as difference exists in global regulatory regimes like bankruptcy laws in different jurisdictions and exemptions for mandatory clearing and margin requirements. Futurisation of OTC derivatives discussed above could be treated as an attempt of regulatory arbitrage as futurisation per se does not change the economic drivers of the deals but reduce its associated margining costs. One of classic regulatory arbitrage strategies in the early days of mandatory clearing is to structure the OTC derivative into a non-standard derivative such that the deal is not clearable. This comes down to the definition of ‘standardised’ OTC derivatives. Regulators therefore introduce margin requirements for bilateral OTC derivatives market to eliminate these arbitrage opportunities. Under US Dodd Frank regulation, non-financial end users those enter into derivatives transactions for hedging purposes are exempt from clearing requirements. And the initial margins for hedging positions are allowed to be rehypothecated once such that the derivative dealer could enter into back to back transaction with CCPs to offload the end user’s transaction and use initial margin collected from the end users to post to CCPs. However, the definition of ‘hedging transaction’ is ambiguous and subjective. In theory, regulators are assumed to have a perfect knowledge of the activities of market participants and the features like time to maturity and contract type of hedging instruments could be qualified and quantified to some extent. It turns out to be some kind of mission impossible as it leaves enough room for market participants to turn their hedging positions into speculative positions. Globalisation of economy in the past three decades has made tremendous contributions to the growth of financial markets. Difference in regulatory treatments on mandatory clearing and margining requirements incentivise the market participants to choose the ‘optimal’ CCP and market competition among these CCPs would push some of CCPs to loosen the margin requirements to keep hold of business flows. The effective way to mitigate this effect is globalisation of legislations on mandatory clearing and margining requirements. FX transactions are exempt from mandatory clearing requirements under Dodd-Frank and FX forwards and swaps with physical settlements are exempt from initial margin requirements under BCBS-IOSCO. Cross currency swaps are not clearable at current stage of market development but subject to bilateral margin requirements. Given the fact that cross currency swap has a significant FX component, the FX component of cross currency swaps is exempt from initial margins requirements under the Basel Committee and International Organisation of Securities Commissions (BCBS-IOSCO). The initial margin requirements for cross currency swaps therefore are equivalent to the initial margins for the interest rate swap component. EMIR allows non-financial entities not to clear their OTC derivatives positions if the Mark to Market of their positions are below a pre-specified threshold. Here if derivative dealer trades clearable OTC derivatives with several different legal entities and these legal entities belong to the same parent company, the derivative dealer has to consolidate these different

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Counterparty Credit Risk, Funding Risk and Central Clearing legal entities into a group, assign a threshold to the group and allocate the individual thresholds of initial margins for these different legal entities. Some financial entities and systematically important non-financial institutions are exempt from initial margins requirements if their positions are below a threshold under BCBS-ISOCO.

Regulators have been aware of the existence of regulatory arbitrage opportunities and measures have been taken to remove or limit these opportunities as discussed above. The mandatory clearing and margin requirements intend to create a safe environment via moving the claimed ‘dangerous’ uncollateralised OTC derivatives into CCPs with strict margin requirements. The implicit market entry requirements of CCPs clearing service restrains the access for general market participants and enables leading derivative dealers profits from providing client clearing service.

Netting efficiencies are reduced in theory if the portfolios with the same counterparty have to be bifurcated into cleared portfolios and bilateral portfolios. The existence of multiple CCPs as studied by Duffie and Zhu (2010) would reduce the overall netting efficiencies even further. Market participants, therefore, have to find a solution to optimise their netting strategy and improve the netting efficiencies in the bilateral clearing markets to make up for the loss in this fragmented clearing market structure (i.e. co-existence of central clearing and bilateral clearing markets). Some third parties like TriOptima in the marketplace now provides trade compression services to market participants to achieve multilateral netting and reduce overall exposure in bilateral and central clearing markets in order to reduce the net demand for margin and capital. These netting optimisation strategies could be implemented via trades unwinds, novations etc. to achieve a global reduction in counterparty credit risk, funding risk and regulatory capital. Given the existence of CCPs providing clearing service for the same asset classes and products, it has been suggested that interoperability of CCPs (for example, cross margining) could significantly reduce the exposure of market participants and retain the netting benefits lost due to multiple CCPs. The reduction of exposures of market participants by interoperability comes at the expense of increase of exposures between different CCPs. This leads to another interesting question how the CCPs manage counterparty credit risk of their competitors. Cox et al (2013) suggests that the netting benefits would be enhanced if the reduction of exposure of market participants dominates the increased exposures between CCPs. However, interoperability of CCPs (or CCP linkage) alleviates the diversification effect of trading with different CCPs of market participants and is likely to pose greater systematic risks of entire financial markets and legal disputes of CCPs domiciled in different jurisdictions over the interpretations of the mandatory clearing and margin requirements and bankruptcy laws. Several key assumptions of derivatives pricing theory like credit linked instruments pricing by senior unsecured pari passu bonds (Gelpern (2015)) and legal procedures like Collective Actions Clauses (CACs) (Gulati (2015)) have proved to be problematic in practical implementation in recent sovereign debt

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Counterparty Credit Risk, Funding Risk and Central Clearing restructuring process. CCPs, therefore, have to cooperate with not only market participants to improve netting efficiencies in the financial systems but also policy makers and legislators across different jurisdictions to work out an implementable and effective interoperability plan for CCPs.

The implicit leverage embedded in derivatives instruments have been accused one of the main reasons of global financial crisis and regulators therefore impose margin requirements on derivatives market participants to mitigate leverage effect and its associated credit risk component. In current market structure, margin requirements do make derivatives trading safer than before but this reduced leverage phenomenon does not apply for the entire balance sheet of market participants. In other words, it is possible that the leverage of derivatives positions declines but the leverage of rest items goes up. The funding liquidity risk of collateral asset is an explicit channel to transform the implicit leverage in derivatives positions into leverage of the rest of balance sheet via borrowing eligible collateral assets. The ‘re-leveraging’ effect applied to the rest of balance sheet created by mandatory clearing and margining requirements do make market participants changing their capital structures. Therefore, the reduction in leverage in derivatives positions does not indicate the reduction of leverage in the entire financial system. The market participants would probably increase leverage in other aspects of balance sheet as a response to reduced leverage in derivatives exposures. It becomes difficult to assess the impact of leverage reduction of derivatives positions on the financial markets and it will give rise to the new instruments aiming to retain the leverage in the future.

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4.6 xVA and Central Clearing

CVA have been built into pricing of OTC derivatives instruments for a few years to account for the potential loss with a given counterparty in trading bilateral OTC derivatives. The additional pricing components like FVA (and MVA) are added into the pricing equation of derivatives to quantify the impact of mandatory margining requirements. Different costs would be applied to client clearing services provided by leading derivative dealers as different CCPs have different clearing and margining requirements. The end users of un-clearable OTC derivatives might be affected by the additional pricing components of clearing. For example, if a counterparty trades an un-clearable OTC derivative (i.e. the derivative is not subject to mandatory clearing and margining requirements, for example, an interest rate swaption or an interest rate swap with a systematically important non- financial institution whose positions are below the specified clearing threshold (under EMIR) and simultaneously below specified initial margin threshold (under BCBS-IOSCO)) with a clearing member and the clearing member executes a hedging position with CCP for this derivative, the end user would be charged with additional initial margin cost (MVA or initial margin funding cost component of FVA) of that hedging instrument in current market regime. The initial margin funding cost component might be even higher for this transaction than the clearable derivative as the hedging instrument executed by the clearing member may not fully offset with existing portfolio the clearing member has with CCP like clearable trade. (The initial margin funding cost might be even greater if the exposure exceeds a specified threshold and a margin multiplier would be applied for initial margin calculation for large exposures.) The pricing of initial margin funding cost is more complex than CVA, DVA and FVA structure as initial margin per se does depend on subjective opinions of each individual CCP and detailed calculation parameters (like haircuts), methodologies could be changed at discretion of CCPs over time. The initial margin is generally calculated on the base framework of Value at Risk (VaR) and projected over the entire life of the portfolio (or individual transactions). The clearing members (or derivative dealers) therefore have to decide the initial margin funding cost calculation methodology as the long dated derivative transaction is highly likely to be unwound before the maturity of contract but initial margin is calculated throughout the life of the transaction. It seems like clearing members have to develop their own strategies to identify the optimal unwinding time of long dated transactions and build this unwinding time into initial margin calculation. Another interesting aspect of initial margin related cost calculation is the frontloading problems in mandatory clearing and margining procedures. Only the OTC derivatives transactions done after effective date of mandatory clearing and margining requirements have to be adhered to the clearing and margining requirements. The market participants have the discretion on whether to clear existing OTC derivatives portfolios (or transactions) via CCPs. The CCPs would talk to the regulators of a different jurisdictions on products variety of clearing service and regulators would decide if a certain type of derivatives are authorised to be cleared or not. It is not clear if an executed un-clearable long dated

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OTC derivatives would be authorised to be cleared or not during the time period between the discussion and implementation. If the final decision of clearing applies to the trades with time to maturity over a specified threshold, the long dated OTC derivatives should be frontloaded with relevant initial margin funding costs. The pricing and trading business of certain types of OTC derivatives therefore would be transformed by the expansion of product coverage of clearing mandate in the future. The end users of clearable OTC derivatives now have to use the client clearing service provided by clearing members in current clearing regime. The non-financial firms may find it difficult to post margins on daily basis and clearing members would facilitate their trading activities by providing short term funding of margins. This kind of client clearing margin lending activities could be used as a way to hedge funding benefits adjustment (FBA) component. The end users may not have qualified collateral assets (for variation margins and initial margins) and the collateral trading/transformation desks of clearing members could transform the unqualified assets like stocks or bonds issued by the end users into qualified collateral assets at a service fee. Implicitly, the default risk of the end users of derivatives is transferred to the clearing members via mandatory clearing requirements. The market entry for clearing and default risk transferring make those ‘too big to fail’ or ‘too connected to fail’ intuitions even bigger and better connected in today’s regulatory regime.

Bailouts of large financial institutions and moral hazard problems of these bailouts have been heavily criticised by the general public. Bankruptcy of Lehman reflects different aspects of inherent problems of OTC derivatives markets and calls market participants to look for solutions to make the financial system safer in the aftermaths of this historical default event. Central clearing is definitely not a perfect solution to the problems of OTC derivatives market revealed in global financial crisis but at least it provides a valid alternative to create a safer financial market. Multilateral netting offered by CCPs could reduce overall exposures across different market participants. Margining practices implemented by CCPs do significantly reduce the leverage embedded in bilateral OTC derivatives transactions. CCPs standardise certain types of OTC derivatives such that their standardised versions could gain additional liquidity and fungibility and bring greater transparency to the pricing area of OTC derivatives. In case of default of clearing member, CCPs implements loss mutualisation scheme to absorb the loss beyond the defaulted member’s initial margin and default fund contribution to stabilise the market activities. The internal default management process like auctions of defaulted portfolios and portability reduces the overall cost (as some portfolios could be transferred to surviving clearing members and hence do not have to be closed out) created by default of clearing members and reduce the potential shockwave or domino effect across the financial system.

It has been suggested some of the benefits of central clearing could be realised in bilateral OTC markets without interventions of CCPs and some functions of CCPs are actually redundant as the bilateral OTC markets could perform well under new regulatory regime. For example, trade

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Counterparty Credit Risk, Funding Risk and Central Clearing compressions service provided by third parties like TriOptima in the market could net their clients’ portfolios on multilateral basis. Mandatory margining requirements in bilateral OTC markets could also significantly reduce the leverage of OTC derivatives trading by CSAs. These views reveal the CCPs do replicate some functions of the bilateral OTC markets but fail to acknowledge that the CCPs could deliver a better and more efficient performance on these functions compared to bilateral clearing markets. Although it is still early to reach a conclusion which derivatives market structure is better, as the eventual costs and benefits of central clearing depend on some many different aspects like clearing service coverage and number of competitive CCPs, CCPs do make fundamental changes to the trading business of derivatives and such kind of market structure changes is likely to be not reversible.

Loss mutualisation scheme implemented by CCPs does not have its duplicates in bilateral markets. The well-known credit contagion effect studied by many finance literatures might be prevented or reduced at least by this scheme as loss mutualisation scheme ‘mutualises’ the shockwave impact created by the default event with different clearing members and hence avoids harsh hits on several entities at the same time without any protection other than regulatory capital (and CVA reserves) on their books. The side effect of loss mutualisation scheme is moral hazard problem as the clearing member now has the chance to share his/her problems with the rest of clearing members. It is very difficult to assess the price of benefits and costs of loss mutualisation scheme. The claimed ‘diversification’ created by competitive CCPs may turn out to be a flawed point as the high standard of market entries of CCPs membership makes the multiple CCPs across the globe do share a large components of list of clearing members. It is more likely in a crisis that several market participants default or come close to default in bilateral OTC markets and the losses due to inability of these market participants to post margins are passed to clearing members by client clearing service these members provide. The CCPs would, therefore, mutualise these losses among surviving members and increase initial margin requirements (or haircuts) to enhance its stability. The feedback loop created by these channels would be studied later this chapter. The default management practice of CCPs, from this perspective, is controversial as it does provide a clear mandate to defend the crisis of financial markets but might not be effective in reality. CCPs, like CME and LCH.Clearnet, do provide a good example of defence against Lehman default in global financial crisis (GFC). Their success, heavily relies on the fact the initial margins are sufficient to offset the losses of the related positions. However, this mechanism might prove to be problematic if the CCPs expand their product coverage and compete by lowering initial margin requirements. The obvious benefit of central clearing is the centralised auction scheme of managing defaulted portfolios. The uncoordinated default management practice in bilateral OTC markets could not provide a more effective and efficient solution at this stage.

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One of the key features of mandatory clearing and margining requirements is the sharp increase in margin requirements. The increase in margins, as discussed in CVA/DVA chapter, do have a significant impact on reduction of exposure profile and hence associated counterparty credit risk in derivatives trading business. The funding cost of margins, especially variation margins as discussed in FVA chapter, is priced by the controversial FVA term.

The market marking of derivatives excessively relies on short term funding like repo market prior to global financial crisis (GFC) and this reliance has been proved to be vulnerable to shockwaves in the long dated derivatives market like subprime mortgage. The regulatory response to GFC, after bailouts of ‘too big to fail’ institutions by governments, aims to remove counterparty credit risk from the bilateral OTC derivatives market by imposing mandatory clearing and margining requirements. The significant funding costs created by mandatory margining requirements, especially initial margin components, make the market participants having to use their long term funding instruments to finance short term variation margin calls and long term initial margin with short term ‘return’ (i.e. the compensation of posting initial margin account is still the short term OIS rate). The funding liquidity risks associated with margining requirements would be discussed in detail later this chapter.

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4.7 Risk Conversion by Margin Requirements

Let’s have a brief review of conversion process from counterparty credit risk to funding liquidity risk. The classic Black Scholes framework lays out the general structure to price an uncollateralised OTC derivative. The funding component of Black Scholes is to facilitate the dynamic delta hedging strategy by construction of a self-financing portfolio. The counterparty credit risk is high but the funding liquidity risk is low in that scenario. If both parties decide to sign two way CSA to upgrade their OTC derivatives trading business, the daily variation margin would significantly reduce counterparty credit risk. The counterparty credit risk could be reduced further by initial margins in bilateral OTC markets and CCPs and even further by default fund of CCPs. The reduction of counterparty credit risk is accompanied by increase in margining requirements and the increased margining requirements would poses greater funding liquidity risk entire financial system during turbulent periods as market participants have to fund these margin calls at significant high rate (for example, the funding rate for the defaulted market participant is technically infinite). Due the distinct nature of variation margin and initial margin, the funding liquidity risks created by these two margins are different.

As discussed in FVA chapter, the FVA (the funding cost and benefits for variation margins) is effectively the wealth transfer from shareholders to the senior unsecured bondholders of a derivative dealer. The funding benefits component is the DVA of the underlying transaction and funding costs component the DVA of the hedging instrument (under the same CSA terms with the underlying transaction) if the default risk is assumed to be the only driver of funding spread. Hull and White (2012) points out the FVA should not affect the valuation of the derivative dealer by the classic capital structure theorem proposed by Modigliani and Miller (1958) that the firm’s valuation is not affected by its capital structure. But FVA does have an impact on building up the no arbitrage price of a self-financing portfolio. The FVA or variation margins management could be proved to be costly as the eligible collateral for variation margins have to be cash (for CCPs) and cash or liquid securities with strong credit quality like US Treasury bonds or German Bunds etc. A great number of market participants may not have liquid securities in hand and find it very expensive to post variation margins on daily basis as the variation margin calls could not be funded by receivables of their operations especially for non-financial end users of derivatives. Therefore, it seems to be plausible that US regulators grant non-financial end users of derivatives exemption from clearing their derivatives portfolio if derivatives positions are for hedging purposes. (The exemption of non- financial end users derivatives position is different in European Union as discussed in previous sections.) Another aspect of costs associated with variation margins is that the potential delays in receiving (collateral from one party) and posting (collateral to another party) in operations. This delay would be significant in turbulent periods of the financial markets and poses significant funding costs for general market participants. CCPs have a significant advantage of margin management over the

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Counterparty Credit Risk, Funding Risk and Central Clearing rest of market participants and may be incentivise to amplify this delay or even interrupt the flows of variation margins for concerns of their own survivability in a distressed market condition. (The CME and Hong Kong Futures Exchange variation margins managements on Black Monday 1987 (or 1987 Crash on 19 October 1987) are examples of delays of CCPs on posting variation margins.)

The introduction of initial margins into OTC derivatives market by mandatory clearing and margining requirements effectively remove the counterparty credit risk from OTC derivatives in theory as the underlying derivatives positions are overcollateralised. One of key advantages of trading derivatives prior to global financial crisis is that generally it costs little to enter into a given position. Now the initial margin requirement simply breaks down this condition and asks both parties to post additional amount of collateral in segregated accounts in order to remove the counterparty credit risk. Initial margin, therefore, could be perceived as an insurance for the potential losses generated by the variation margin. If the counterparty is out of money and not able to post enough variation margin required before the expiry of the underlying transaction, initial margin could be liquidated to fill in the gap between Mark to Market of underlying transaction and net collateral received in the variation account. That is one of the reasons initial margin is calculated by stressed scenarios by Value at Risk (VaR) or expected shortfall (ES) of market movements instead of counterparty credit quality. Initial margin has been perceived to be expensive as it cannot be rehypothecated and the compensation for posting initial margin (normally applies for a long dated portfolio) is tied to the short term interest rate like OIS curve. Initial margins calculation is more complex than existing methods for derivatives pricing (generally priced under risk neutral measure) as initial margin pricing is priced under real world measure (or a combination of real world and risk neutral world) and subjective to individual CCP’s assessment of future exposures at specified confidence interval.

Bernanke (1990) points out the margining operations of clearing houses relies on the extension of credit and ex-post insurance provided by the Fed plays the key role in protecting clearing and settlements systems. Margining, originally designed to remove counterparty credit risk, leads to other sorts of risk like funding liquidity risk and these risks finally may be converted back to credit risk. The insolvency of AIG in 2008 was due to the significantly large margin call (the celebrated ‘death spiral’ or liquidity trap) triggered by downgrading AIG’s credit rating. The funding liquidity risk taken by AIG was eventually converted back to credit risk and cost tax payers $85 billion to pay off AIG’s margin liabilities and bailout AIG. This would naturally lead to the question if it is necessary to have margining practice in place as eventually the funding risk would be converted back to credit risk (or the firm would default if the funding spread jumps into infinity). The credit rating linked margin trigger now is removed from standard CSAs or margin agreements due to its obvious drawback in protection against exposure and ‘cliff edge’ effect on the distressed counterparty. Gulf of Mexico oil spill in 2010 put BP in similar scenarios on their derivatives positions. BP’s credit rating was

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Counterparty Credit Risk, Funding Risk and Central Clearing downgraded by the rating agencies and credit rating linked margin calls were triggered. BP took the loan from the banks (as OTC derivatives counterparties of BP) to pay off its liabilities on margin calls triggered by credit downgrading. The banks viewed that BP was experience a short term operation risk and would not default due to oil spill. The extension of credit in the form of loans essentially converted the short term funding liquidity risks faced by these banks (on derivatives positions) back to long term credit risk embedded in loans. Initial margins requirements could be justified in these cases as initial margins could provide an extra layer of defence against the counterparty credit risk. It would not be possible under US Dodd-Frank if BP could provide sufficient evidence that the derivatives positions of were for hedging purposes.

It has been discussed in previous sections and CVA/DVA chapter that margin reduces counterparty credit risk and initial margin could reduce the entire exposure profile down to zero (if the initial margin is set large enough). Regulators therefore impose high initial margin requirements in central clearing and bilateral clearing OTC market to eradicate counterparty credit risk that has been identified as one of the key causes of global financial crisis (GFC). As indicted by examples in previous sections, the side effects of margining practice could be significant and make the original margining requirements ineffective in some distressed market scenarios. Generally if the market participant deposit collateral cash into margin accounts to meet margin calls, the market participant would receive OIS rate of that collateral currency as a compensation for posting collateral. Variation margin is generally calculated on daily basis (i.e. the holding period of a given amount of collateral cash in variation margin is one day) and then it is reasonable to set compensation for daily margining process or the return on daily deposit collateral cash as OIS rate of that specific collateral currency. However, the holding period of collateral cash of initial margin is generally much longer and could be assumed to be equal to the time to maturity of the underlying OTC derivative transaction (without any break clauses in place). It seems it does not make sense to receive only OIS rate as compensation for depositing collateral assets in initial margin accounts over a long dated OTC transaction. Given the initial margin collateral asset could not be rehypothecated (except for client clearing service) and the initial margin has to be segregated from variation margin, the initial margins could not generate higher returns than OIS rates as the receiver of initial margins could not be used in other means. The clearing mandate requires the initial margins should be able to be liquidated at fairly stable value to offset shortfall in variation margins in turbulent periods of financial markets and therefore the collateral assets may not be invested in certain assets to generate higher returns. Although OIS rate, i.e. the short term interest rate, is not appropriate to compensate a long term deposit, it is probably the only practical rate for this ‘special’ long term deposits due to the existing restrictions placed on initial margin account. Initial margins are expensive because the market participant posting initial margins has to fund the collateral assets at its own long term funding curve and receive short term rates like OIS rate in return for depositing initial margin assets.

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The negative carry experienced by banks and financial institution in running trading business could be priced by FVA (or FVA for variation margins and MVA for initial margins) as discussed in FVA chapter. The derivative dealers have to fund their positions at their funding curve (typically significantly higher than OIS rate) but receive OIS rate as return for a long term period. If the collateral assets posted in initial margin accounts are not cash but other types of securities like sovereign bonds, a conservative haircut would be attached to the valuation to account for the loss of liquidation in stressed market conditions. The optimisation of margins, used to be a middle or back office function of derivative dealers, now becomes a front office function as margins emerge as significant pricing components of OTC derivatives. The embedded options in collateral agreements like cheapest to deliver option, collateral switch option, rehypothecation and different haircuts have to be priced into the execution price of derivatives as these features materially change the self- financing strategies for replicating the derivative dynamics. For example, cash collateral denominated in different currencies would earn OIS rates of these specific currencies and non-cash collateral would be reimbursed with different repo rates. The impact of these features have been discussed in FVA chapter.

The actual demand for collateral created by mandatory clearing and margining requirements is hard to estimate due to netting bifurcation created by separation of bilateral clearing market and central clearing market and dispersion of netting benefits due to the existence of multiple CCPs. The main criticism of clearing mandates clearing and margining mandates is the significant increase in costs of trading derivatives. Heller and Vause (2012) produced a rough estimate of sensitivities of margin costs with respect to market clearing structure. The total margin costs are expected to be reduced by at least 25% for interest rate swaps and CDSs and 50% for single name CDSs and CDS indices if these products combination could be cleared via a single CCP. The level of market fragmentation is one of the contributing factors to the total costs of margining. The variation margins provide symmetric funding costs and benefits in theory under certain assumptions (like symmetric margin requirements on trading and hedging positions and symmetric funding curves for borrowing and lending etc.) and therefore the net funding liquidity risk created by variation margins is immaterial and should be considered as second order contributing factor to the net demand of collateral (BCBS- ISOCO (2013) and CGFS (2013)). The operational delays in margining process in reality requires the market participants to hold extra collateral in place to meet potential margin calls. The reduced usage of collateral rehypothecation combined with declined short term securities lending business significantly reduced the velocity of margin and increased the liquidity costs of collateral in the aftermath of global financial crisis (GFC). The Modigliani Miller theorem implies the initial margins costs would be insignificant for market participants as institutions just have to borrow more to meet margin requirements as suggested by Hull and White (2014). The extra fund borrowed in the form of

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Counterparty Credit Risk, Funding Risk and Central Clearing senior unsecured bonds to meet margin requirements is the funding cost for the market participants. The extra margin the counterparty received aims to reduce the loss given default of that market participant. Therefore, the market price of the reduction in loss given default of the market participant should be equal to the funding cost of that market participant as discussed in FVA chapter. The margin lending business model is built on this simple result. The market participants like pension funds or asset management firms with extra qualified margin could post margins on behalf of market participants those are short of eligible collateral in exchange for the margin lending fees (i.e. the funding cost adjustment) for providing margin lending service. Essentially the margin lenders in the marketplace are rolling over their own short term funding curve to invest in their long term derivatives portfolio positions. The collateral rich market participants funded at a low rate could lend collateral out to reach a high return. This kind of market practice is vulnerable to market volatilities and market lending fees could rocket into sky in stressed market conditions. The margin costs could be significantly reduced if the scope eligible collateral assets of initial margins requirements could be expanded to include illiquid assets and securities. The illiquid assets and securities could be used to reduce exposure profile and hence the risk weighted assets usage allocated to counterparty credit risk. Conservative haircuts should be assigned to these illiquid assets to account for liquidity risk (or risk of loss in value due to liquidation) in stressed market scenarios. The obvious problem associated with this type of initial margin is that these assets may not be liquidated in turbulent periods of financial markets and hence the initial margin could not act as the buffer for in variation margins and fully absorb the shortfall in collateral. These illiquid assets could reduce the overall risk exposure by overstating its ability in absorbing potential credit risk in stressed conditions. Therefore, the eligible collateral assets for initial margins should have small market volatility with abundant liquidity such that the loss of liquidation in crises would be insignificant and the shortfalls in variation margins could be fully protected by initial margins. These highly liquid collateral assets are expensive and the funding costs of acquiring these liquid collateral assets are significantly higher compared illiquid assets for general market participants.

The variation margin calls could be costly associated with significant funding liquidity risk especially in crises or market crashes like Black Monday Crash 1987 as studied by Bernanke (1990) and Brady (1988). Heller and Vause (2012) suggests that the variation margins could increase to a substantial level within a few weeks during turbulent periods of financial markets. The market volatility could spike to a substantial level and cumulated daily variation margin calls creates a significant short term shortfall of liquid collateral assets. The velocity of variation margin is time varying and tends to be slow in crises as market participants would like to delay their margin payments due to concerns of credit and liquidity conditions in the marketplace. This reduced velocity of margins is normally accompanied with increased funding costs of market participants as market participants would not like to lend capital to others during stressed market conditions. The large variation margin volatilities

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Counterparty Credit Risk, Funding Risk and Central Clearing created by huge swings in underlying asset prices would make market participants to liquidate other positions to meet variation margin calls. This classic feedback effect creates fire sales in other segments of markets. The variation margin, initially intends to remove counterparty credit risk and hence reduce the probability of emergence of crisis, converts the counterparty credit risk into funding liquidity risk and would lead to potential systematic risk in stressed market conditions. If clearing member does default under this circumstance, CCP has to back up its defaulted clearing member’s position and the resulted short term funding risk for CCP magnify to a significant amount as the CCP has to pay out the losses on the defaulted clearing member’s portfolio. The CCPs, therefore, should stockpile a significant amount of liquid collateral assets to defend themselves against potential large swings of asset prices in the market.

The collateral asset of variation margin of central clearing is required to be cash and the liquidity shock due to a jump in asset prices in one segment of financial market would be amplified via variation margin mechanism of CCPs and transferred to other segments of the market. The Repo market is normally frozen in turbulent periods and the overall funding cost of market participants would increase substantially due to inability of fundamental funding instruments of derivatives in stressed market conditions.

The mandatory clearing and margining requirements make every market participant (except certain groups exempt from these requirements under certain conditions) exposed Mark to Market volatility of derivatives. As indicated in AIG bailout example discussed in previous sections, the volatility created by variation margin effectively increases the default probability of market participants in crises. As indicated by Bernanke (1990) and BP case, the extension of credit plays the vital role in funding liquidity risk management in these scenarios. The funding liquidity risk created by margin calls faced by distressed market participants is effectively converted back to counterparty credit risk. Either this counterparty credit risk is absorbed by extension of credit like BP case (the derivatives counterparties and banks granted BP loans to offset BP’s liabilities in margins), or the distressed market participant is pushed out of business and forced to declare bankruptcy like Lehman Brothers in 2008. The term ‘liquidity trap’ precisely describes the phenomenon of conversion from liquidity risk into credit risk. If the distressed market participants are not lucky (for example, Lehman Brothers) enough to receive any form of extension of credit, the forced variation margin mechanism of CCPs for better managing risks would lead to more ‘virtual’ defaults as discussed in Kenyon and Green (2013).

Further to the potential price distortion impacts created by variation margins, the procyclicality problem of initial margin would amplify the price distortion and is expected to enhance the feedback effect discussed in above section. Generally the initial margin calculation is based on an internal model set by CCPs tracking market volatilities with penalising factors linked to stress testing

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Counterparty Credit Risk, Funding Risk and Central Clearing parameters like Value at Risk (VaR) and Expected Shortfall (ES). If the market volatility spikes in one segment (i.e. a certain derivative product or asset class) of financial market, besides corresponding variation margin calls from CCPs (to make up for the net changes in Mark to Market of underlying positions), an initial margin call might be sent out as a result of a significant increase in VaR or ES and the extra initial margin is expected to make up for the increase in expected magnified losses in the variation margins (during turbulent periods). The feedback effects force market participants to liquidate their positions in other segments of the market to meet variation and initial margin calls and therefore the market volatilities in other segments of markets would increase as well. The spill over effect of market volatilities across different segments of market is one of key drawbacks of initial margin mechanism. The initial margin calculation methodology, therefore, should be set to produce stable results even under stressed markets. This could be solved by extending the historical estimation window to include global financial crisis (GFC). Then the problem becomes if the initial margins calculated by these dataset would be too high for current market conditions. The expected funding liquidity cost per unit of initial margin is lower than that of variation margin as non-cash collateral is allowed for initial margins. The objective of initial margin is protect CCPs and market participants against shortfalls in variations margins in derivatives trading business. The key criteria for assessment of eligible collateral assets for initial margins therefore should be the non-cash securities should be able to be liquidated without any significant loss in value in stressed market conditions. This criteria effectively narrows down the range of potential candidates of collateral assets for initial margins. However, competition between different CCPs would lead to relaxed requirements for initial margins. Therefore, CCPs have to balance the business flows generated by relaxing initial margin requirements and the potential risk of failure in distress conditions created by inability of liquidation or huge losses in value in liquidation of initial margins. The segregation of initial margins provides another layer of defence of CCPs against default risk of clearing members and exacerbates the scarcity problem of eligible collateral assets in the marketplace. This indeed poses a significant challenge to derivatives trading business and calls for innovative solutions from market participants.

The margining process effectively converts counterparty credit risk into funding liquidity risk and this risk conversion process creates several interesting questions on optimisation of portfolio allocation between centrally cleared portfolios and bilateral cleared portfolios. Suppose a market participant is exempt from mandatory clearing and margining requirements. The market participant finds it difficult to adhere to margining requirements due to scarcity of available liquid collateral assets and potential procyclicality of margin requirements. A third party in the market provides margin lending service such that the market participant could pay periodic fees in exchange for the service that the third party manages all the related margining process on behalf of the market participant. The margin lending fee, however, is expected to increase in stressed market conditions or in scenarios where the credit quality of the market participant declines. Thereby, the market participant is exposed to its own

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Counterparty Credit Risk, Funding Risk and Central Clearing credit risk volatility. The interesting aspect of such phenomenon is that market participant does not face margin process volatility via margin lending service (as the market participant completely offloads its collateral posting obligations to the margin lender) but exposes to the volatility of its own credit risk. The market participant could mitigate the difficulty of managing its own credit risk volatility by signing a two way CSA with a clearing member and posting non-eligible collateral assets to the clearing member. The collateral trading/transformation desk of that clearing member would synthetically repo the non-eligible collateral assets into eligible collateral assets for clearing and charge a periodic fee insensitive to the credit quality of the market participant for providing collateral transformation/upgrading service. The market participant, in this case, partially exposes to the variation margining volatilities. And the partial counterparty credit risk and funding liquidity risk of the market participant is now taken over by the clearing member. If the clearing member provides credit like loans to the market participant to fund the margin posting activities of the market participant to the clearing member, the clearing member would face less regulatory capital charge but convert the funding liquidity risk of the market participant into counterparty credit risk. This is a regulatory arbitrage strategy and the overall credit risk taken by the clearing member remains unchanged as the counterparty credit risk of derivatives trading business is taken over by the counterparty credit risk of the treasury department or commercial banking and related financing functions arm. Hence the counterparty credit risk reduction of clearing member via margining process leads to counterparty credit risk increase to its own bondholders.

The risk transformation process could be further exemplified by changes in CVA and FVA. Here we assume symmetric FCA/FBA Accounting (discussed in FVA chapter) and rehypothecation across different netting sets is disallowed. CVA, the market price of the counterparty credit risk, gradually declines along with the collateral requirements upgrading process from uncollateralised bilateral OTC derivatives trading, bilateral OTC derivatives trading with variation margins, bilateral OTC derivatives trading with variation margins and initial margins, to the central cleared derivatives trading. The funding components FVA could split into two terms, the FVA for variation margins and FVA for initial margins or MVA. The FVA for variation margins gradually declines along with the collateral upgrading process. The FVA for initial margins works the opposite direction and it increases along with collateral upgrading process. Given the initial margin account has to be segregated and the magnitude of initial magnitude is expected to be significantly greater than variation margin (such that it could offset the variation margin shortfall in the default state of the counterparty), the FVA for initial margins dominates the FVA for variation margins. The capital charge peaks on uncollateralised OTC derivatives positions and gradually reduces on the process with variation margins and initial margins included in OTC transactions. The centrally cleared OTC transactions would increase the capital charges above the bilateral margined (initial margins included) level due to the increased exposure on default fund contribution. From pricing perspective, the bilateral clearing with variation

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Counterparty Credit Risk, Funding Risk and Central Clearing margins would deliver the most competitive price among these 4 different margining practices. As discussed in previous sections, margining process merely converts counterparty credit risk into different sorts of financial risks like funding liquidity risks etc. It is not clear such kind of risk conversion would benefit the entire financial market and general economy. The non-financial firms have built up their expertise in managing counterparty credit risk with banking institutions and it takes time to develop their expertise in managing funding liquidity risk under current regulatory regime.

The funding liquidity risk of mandatory clearing and margining requirements could be traced back to variation margins, initial margins and default fund contributions. The market participants are expected to receive multiple variation margin calls for large amounts of collateral asset in turbulent periods of financial markets. The substantial amount of net collateral assets demand combined with increased funding difficulty would pose significant funding liquidity risks for market participants. At the same time, the initial margins are sensitive to market volatilities, especially to large price movements in the market, and the initial margins requirements are expected to increase in stressed market scenarios. The initial margins calculations are subject to CCPs internal models and CCPs have discretion over changing stressed testing parameters. The potential conflict of interests (between CCPs, clearing members and non-clearing members) in increasing initial margins requirements in stressed markets create a significant challenge to the overall market stability and potentially lead to detrimental impacts on the social welfares. Default fund contributions per se are subject to Wrong Way Risk and are likely to increase in stressed markets conditions as the CCPs are likely to increase default funds when the default probabilities of their clearing members significantly increase in turbulent periods. The funding liquidity risk is more likely to be experienced by market participants when funding is severely difficult.

The generic idea of mandatory clearing and margining requirements is that central clearing plus initial margins would create a safer OTC derivatives market and reduce the probability of financial crises. As discussed in previous sections, the counterparty credit risk is not eradicated by this practice and is merely transformed into different sorts of financial risks. The procyclicality of margins and potential feedback effects created by margining requirements would exasperate the funding liquidity risks for market participants and funding liquidity risks are highly likely to be converted into counterparty credit risk in extreme scenarios and lead to systematic shocks across the entire financial market.

Variation margins have been used in derivatives trading to adjust for the changes in the market prices of derivatives positions on frequent regular basis. If one of the clearing members of a CCP defaults, the CCP is exposed to the market risk for the period between receiving the last variation margin from the defaulted clearing member and closing out that clearing member’s portfolio. This period is

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Counterparty Credit Risk, Funding Risk and Central Clearing defined as margin period of risk (MPR). The initial margin is designed to be served as a buffer to the adverse market movements during the margin period of risk within a certain range of confidence interval (the minimum confidence interval set by European Securities and Markets Authority (ESMA) for centrally cleared OTC derivatives is 99.5% (ESMA (2012)). One of the drawback of the design of initial margin is that the initial margin is measured based on probability not the size of the loss. The default fund is therefore introduced to absorb or mutualise (at least some portion of) the loss beyond initial margins.

The assessment of MPR is complex and the determination of MPR is subjective to CCPs and their products coverage. The market volatility over the turbulent period during which at least one of clearing members defaults is going to be significantly higher than the normal markets. The other factors like liquidation costs and bid ask spread should be taken into account in determination of MPR. All the factors affecting the close out process are generally not modelled explicitly by CCPs (Gregory (2014)) but integrated into the choice of MPR. Variation margin reduces current exposure and initial margin covers the future exposure under extreme scenarios. There had been disputes over variation margins calculations in OTC derivatives market as discussed in previous sections. The more complex Value at Risk (VaR) based initial margins calculations are likely to cause more disputes as initial margin per se is a complex subjective estimation of future exposure dynamics under market stressed scenarios. The CCPs normally have their in house initial margin calculation methodologies and they could enforce their calculations as regulations implicitly give them greater bargaining powers. ISDA (2013) introduced the concept of the standard initial margin model (SIMM) into bilateral OTC derivatives market. The entire portfolio is driven by a number of pre-specified factors and the initial margins could be calculated via the simulations of the suite of factors. This implantation of SIMM is subjective to the approval of all jurisdictions.

BCBS-IOSCO (2013) introduced standardised initial margin schedule for initial margin calculations.

Asset class Initial margin requirement (% of notional exposure) Credit: 0–2 year duration 2 Credit: 2–5 year duration 5 Credit 5+ year duration 10 Commodity 15 Equity 15 Foreign exchange 6 Interest rate: 0–2 year duration 1 Interest rate: 2–5 year duration 2 Interest rate: 5+ year duration 4 Other 15

(Table 4.1: BCBS-ISOCO (2013) Standardised Initial Margin Schedule Source: BCBS-ISOCO (2013))

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The netting effects between different trades within the same portfolio is calculated by the net gross ratio (NGR), and the net standardised initial margin (NSIM) defined in BCBS-ISOCO (2013) is 푁푆퐼푀 = 0.4 ∙ 퐺푟표푠푠 퐼푛푖푡푖푎푙 푀푎푟푔푖푛 + 0.6 ∙ 푁퐺푅 ∙ 퐺푟표푠푠 퐼푛푖푡푖푎푙 푀푎푟푔푖푛. (4.7.1) We follow standardised initial margin schedule in this chapter. Here the net gross ratio is defined as the net replacement cost (i.e. net default risk free price) over the gross replacement cost (i.e. gross default risk free price). As discussed in Gregory (2014), for example, if we have two interest rate 10−4 6 derivatives with replacement costs 10 and -4, respectively. Then 푁퐺푅 = = = 60%. 10 10

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4.8 CVA, DVA, FVA and MVA Pricing

In this section, we follow section 3.8 FCA/FBA Accounting (where rehypothecation of variation margins is not allowed across different netting sets/counterparties). The entire pricing adjustment is 퐶푉퐴 − 퐷푉퐴 + 퐹퐶퐴 + 푀푉퐴, (4.8.1) The corresponding funding transfer policy (FTP) is 퐹푇푃 = 퐶푉퐴 + 퐹퐶퐴 + 푀푉퐴. (4.8.2) The term structure of derivative dealer’s funding spread is following:

Term (Years) Funding Spread (bps) 0.5 11 1 17 2 24 3 31 4 38 5 45 7 57 10 68 15 73 20 77 30 80

(Table 4.2: Term Structure of Derivative Dealer’s Funding Spread)

Case Study: Interest Rate Derivatives (IRDs) Priced on 5 x 15 Swap Curve and Cross Currency Swaps Priced on 0 x 15 FX Forward Curve

This part serves as a continuation of our xVA pricing analysis of section 2.9 with extension into FVA and MVA pricing. The product features have been discussed in detail in section 2.9. Our analysis here will build up a complete picture of collateralisation development and its impact on derivative pricing. Our analysis in section 2.9 shows standard two way CSAs could effectively reduce size of exposure and symmetrise exposure profiles. We are going to continue on this direction.

The collateralisation developments are divided into 4 stages: 1. Uncollateralisation: Both sides do not post any collateral at all; 2. Collateralisation: Both sides sign standard two way CSAs term in section and post collateral accordingly; 3. Overcollateralisation with Symmetric Variation Margin (VM) and Asymmetric Initial Margin (IM): The underlying transaction is moved to CCP;

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4. Overcollateralisation with Symmetric Variation Margin (VM) and Symmetric Initial Margin (IM): Both sides have to post initial margins to segregated accounts.

By standardised initial margin schedule for initial margin of BCBS-IOSCO (2013), the initial margins for 8 different transactions are equal to €4 million (as the underlying nationals are €100 million and the tenor of these transactions are all longer than 4 years). Since the FX component of cross currency swaps are exempt from initial margin requirements, the cross currency swaps in this section will need €4 million for initial margins as this amount is the requirement for their interest rate components.

The ‘Total’ calculates the entire xVA as in (4.8.1). It is assigns the price of the risks other than market risk that the underlying transaction incurs. The ‘FTP’ calculates the price that the derivative dealer would transfer to his/her counterparty such that his/her CET1 will not change.

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(Figure 4.1: Exposure Profile Evolvements of 5 x 15 Forward Starting IRS Fixed Payer (Notional = EUR 100 MM))

5x15 Froward Starting Fixed Payer xVA Uncollateralised Collateralised Central Clearing Bilateral Clearing with IMs CVA 572,825 54,608 - - DVA - 492,932 - 43,759 - - FCA 392,240 24,782 - - FBA - 333,413 - 22,234 - - MVA - - 620,577 620,577 Total 472,133 35,631 620,577 620,577 FTP 965,065 79,390 620,577 620,577

(Table 4.3: xVA for 5x15 Froward Starting Fixed Payer under Different Margin Requirements)

Findings for 5x15 Froward Starting Fixed Payer under Different Margin Requirements

Total xVA and FTP reaches their minimum at second stage of collateralisation, i.e. standard two way CSA or symmetric VM. From the perspective of derivative dealer, central clearing and bilateral clearing with IMs assigns the highest prices for the risk but charges his/her counterparty less than uncollateralised scenario.

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(Figure 4.2: Exposure Profile Evolvements of 5 x 15 Forward Starting IRS Fixed Receiver (Notional = EUR 100 MM))

5x15 Froward Starting Fixed Receiver xVA Uncollateralised Collateralised Central Clearing Bilateral Clearing with IMs CVA 492,932 43,759 - - DVA - 572,825 - 54,608 - - FCA 333,413 22,234 - - FBA - 392,240 - 24,782 - - MVA - - 620,577 620,577 Total 253,520 11,385 620,577 620,577 FTP 826,345 65,993 620,577 620,577

(Table 4.4: xVA for 5x15 Froward Starting Fixed Receiver under Different Margin Requirements)

Findings for 5x15 Froward Starting Fixed Receiver under Different Margin Requirements

Total xVA and FTP reaches their minimum at second stage of collateralisation, i.e. standard two way CSA or symmetric VM. From the perspective of derivative dealer, central clearing and bilateral clearing with IMs assigns the highest prices for the risk but charges his/her counterparty less than uncollateralised scenario.

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(Figure 4.3: Exposure Profile Evolvements of 5 into 10 European Payer Swaption (Notional = EUR 100 MM))

5 into 10 European Payer Swaption xVA Uncollateralised Collateralised Central Clearing Bilateral Clearing with IMs CVA 532,306 28,454 - - DVA - 28,462 - 27,770 - - FCA 356,177 11,515 - - FBA - 22,321 - 11,258 - - MVA - - 620,577 620,577 Total 860,021 12,199 620,577 620,577 FTP 888,483 39,969 620,577 620,577

(Table 4.5: xVA for 5 into 10 European Payer Swaption under Different Margin Requirements)

Findings for 5 into 10 European Payer Swaption under Different Margin Requirements

Total xVA and FTP reaches their minimum at second stage of collateralisation, i.e. standard two way CSA or symmetric VM. From the perspective of derivative dealer, central clearing and bilateral clearing with IMs assigns less price for the risk and less charges to his/her counterparty than uncollateralised scenario.

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(Figure 4.4: Exposure Profile Evolvements of 5 into 10 European Receiver Swaption (Notional = EUR 100 MM))

5 into 10 European Receiver Swaption xVA Uncollateralised Collateralised Central Clearing Bilateral Clearing with IMs CVA 504,397 31,872 - - DVA - 60,582 - 32,572 - - FCA 331,343 12,590 - - FBA - 45,791 - 14,053 - - MVA - - 620,577 620,577 Total 775,158 11,890 620,577 620,577 FTP 835,740 44,462 620,577 620,577

(Table 4.6: xVA for 5 into 10 European Receiver Swaption under Different Margin Requirements)

Findings for 5 into 10 European Receiver Swaption under Different Margin Requirements

Total xVA and FTP reaches their minimum at second stage of collateralisation, i.e. standard two way CSA or symmetric VM. From the perspective of derivative dealer, central clearing and bilateral clearing with IMs assigns less price for the risk and less charges to his/her counterparty than uncollateralised scenario.

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(Figure 4.5: Exposure Profile Evolvements of 5 into 10 Bermudan Payer Swaption (Notional = EUR 100 MM))

5 into 10 Bermudan Payer Swaption xVA Uncollateralised Collateralised Central Clearing Bilateral Clearing with IMs CVA 743,969 28,297 - - DVA - 9,824 - 29,483 - - FCA 497,596 14,443 - - FBA - 9,035 - 16,875 - - MVA - - 620,577 620,577 Total 1,231,741 13,257 620,577 620,577 FTP 1,241,565 42,740 620,577 620,577

(Table 4.7: xVA for 5 into 10 Bermudan Payer Swaption under Different Margin Requirements)

Findings for 5 into 10 Bermudan Payer Swaption under Different Margin Requirements

Total xVA and FTP reaches their minimum at second stage of collateralisation, i.e. standard two way CSA or symmetric VM. From the perspective of derivative dealer, central clearing and bilateral clearing with IMs assigns less price for the risk and less charges to his/her counterparty than uncollateralised scenario.

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(Figure 4.6: Exposure Profile Evolvements of 5 into 10 Bermudan Receiver Swaption (Notional = EUR 100 MM))

5 into 10 Bermudan Receiver Swaption xVA Uncollateralised Collateralised Central Clearing Bilateral Clearing with IMs CVA 564,974 32,011 - - DVA - 20,069 - 30,538 - - FCA 376,889 15,883 - - FBA - 14,599 - 14,986 - - MVA - - 620,577 620,577 Total 921,794 17,356 620,577 620,577 FTP 941,863 47,894 620,577 620,577

(Table 4.8: xVA for 5 into 10 Bermudan Receiver Swaption under Different Margin Requirements)

Findings for 5 into 10 Bermudan Receiver Swaption under Different Margin Requirements

Total xVA and FTP reaches their minimum at second stage of collateralisation, i.e. standard two way CSA or symmetric VM. From the perspective of derivative dealer, central clearing and bilateral clearing with IMs assigns less price for the risk and less charges to his/her counterparty than uncollateralised scenario.

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(Figure 4.7: Exposure Profile Evolvements of 15 Years EURUSD Cross Currency Swap Long EUR at Maturity)

15 Years EURUSD Cross Currency Swap Long EUR at Maturity xVA Uncollateralised Collateralised Central Clearing Bilateral Clearing with IMs CVA 2,513,569 119,891 - - DVA - 729,266 - 109,730 - - FCA 1,917,141 50,393 - - FBA 529,847 - 47,981 - - MVA - - 620,577 620,577 Total 3,701,444 60,554 620,577 620,577 FTP 4,430,710 170,284 620,577 620,577

(Table 4.9: xVA for 15 Years EURUSD Cross Currency Swap Long EUR at Maturity under Different Margin Requirements)

Findings for 15 Years EURUSD Cross Currency Swap Long EUR at Maturity under Different Margin Requirements

Total xVA and FTP reaches their minimum at second stage of collateralisation, i.e. standard two way CSA or symmetric VM. From the perspective of derivative dealer, central clearing and bilateral clearing with IMs assigns less price for the risk and less charges to his/her counterparty than uncollateralised scenario.

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(Figure 4.8: Exposure Profile Evolvements of 15 Years EURUSD Cross Currency Swap Short EUR at Maturity)

15 Years EURUSD Cross Currency Swap Short EUR at Maturity xVA Uncollateralised Collateralised Central Clearing Bilateral Clearing with IMs CVA 729,266 109,730 - - DVA - 2,513,569 - 119,891 - - FCA 529,847 47,981 - - FBA - 1,917,141 - 50,393 - - MVA - - 620,577 620,577 Total - 1,254,456 37,820 620,577 620,577 FTP 1,259,113 157,711 620,577 620,577

(Table 4.10: xVA for 15 Years EURUSD Cross Currency Swap Short EUR at Maturity under Different Margin Requirements)

Findings for 15 Years EURUSD Cross Currency Swap Short EUR at Maturity under Different Margin Requirements

Total xVA and FTP reaches their minimum at second stage of collateralisation, i.e. standard two way CSA or symmetric VM. From the perspective of derivative dealer, central clearing and bilateral

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Counterparty Credit Risk, Funding Risk and Central Clearing clearing with IMs assigns less price for the risk and less charges to his/her counterparty than uncollateralised scenario.

Findings from IRDs and FXDs Case Study These 4 different stages of collateralisation have direct impact on pricing.

Standard two way CSA or symmetric variation margins produces lowest xVA prices and FTP charges across 8 different transactions. The impact of market risk factors of underlying transactions is significantly reduced by standard two way CSAs.

Initial margins significantly increase the xVA prices and FTP charges. And the impact of market risk factors of underlying transactions is further diminished. The final total xVA price and FTP charge for central clearing and bilateral clearing with initial margins of these 8 transactions are all equal to €620,577. Homogeneity in xVA pricing and FTP charge finally achieves in these two stages.

It turns out the counterparty credit risk could be turned to 0 with a conservative initial margin requirement. (Here we assume CCP is a default remote entity.) And no matter what underlying transactions are, the pricing becomes homogenous. Further analysis could be conducted on portfolio basis.

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Summary

The negative impact of global financial crisis (GFC) on the financial markets and the general economy has led to serious and lengthy discussions of the roots and causes of the failure of financial system. The counterparty credit risk and opacity of OTC derivatives markets have been identified as key drivers or catalysts of the global financial crisis and the light regulations on these two areas of OTC markets were heavily blamed by the general public. The systematic mitigation of counterparty credit risk, therefore, comes to the centre of the stage post global financial crisis regulation reform. The banks are required to hold a greater amount of capital against counterparty credit risk like CVA capital charge in Basel III and therefore are prevented from taking excessive exposure in pre global financial crisis era by imposing substantial regulatory capital on these types of OTC transactions. CCPs, like CME and LCH.Clearnet, performed well during the financial crisis period and were compared with chaos and dysfunctions of bilateral OTC markets in Lehman default. The centralised auction process for defaulted clearing members’ OTC derivatives portfolios and strict initial margins did deliver an effective performance in managing default of Lehman Brothers. The reduction in counterparty credit risk combined with increase in transparency in derivatives pricing by central clearing mechanism are recognised by policy makers and regulators and clearing mandates for standardised OTC derivatives are introduced and expected to be fully implemented in a progressive timescale. The significant challenges in pricing and trading long dated OTC derivatives faced by CCPs would create different opportunities to further develop the financial theory. The initial margin, i.e. the extra amount of collateral posted at inception of transactions to cover potential additional costs like shortfalls in variation margins in the default state of clearing members, significantly increases the costs of trading standardised OTC derivatives and presents an obvious regulatory arbitrage opportunity for market participants to trade non-standard OTC derivatives in traditional bilateral OTC markets. Mandatory bilateral margin requirements are therefore introduced into remove this regulatory arbitrage opportunity and market participants are required to post initial margins in bilateral OTC markets. Given the prevalence of variation margins in CSAs in bilateral OTC derivatives market, the most significant change brought by regulations in OTC derivatives trading is initial margins.

The mandatory clearing of standardised OTC derivatives, bilateral margin requirements for non- standardised OTC derivatives and CVA regulatory capital are three main methods implemented by regulatory to remove counterparty credit risk from the OTC derivatives market. The costs of three mandates have to be assessed and balanced with the benefits to the general economy. The initial margins (in CCPs and bilateral OTC markets) are required to be liquid high quality securities and have to be segregated from variation margins. The reduction in rehypothecation associated with increased scarcity of liquid high quality securities in the marketplace push up the costs of trading

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OTC derivatives. The loss of netting benefits due to existence of multiple CCPs and bifurcation between CCPs and bilateral clearing markets makes it necessary for market participants to implement optimisation strategies like trade compressions and cross margining (or interoperability)between different CCPs to reduce costs. Banks are incentivised to optimise their holding of bilateral cleared portfolios and centrally cleared portfolios and provide collateral/margin transformation service (or create synthetic repo markets) for end users of derivatives to upgrade their non-eligible collateral assets into eligible collateral assets. Several new sorts of risks created by these mandates and client clearing services might create potential shocks in financial market in the future.

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5. Conclusion

The initial aim of financial engineering is to develop techniques help investors better manage their risks. While in fact this series of financial crisis is partial due to the global exposure to US mortgages market created by financial engineering. It was at the banks’ benefit to not get too concerned with counterparty risk and this had led them to accumulate significant uncollateralised exposures to monoline insurers. The downgrading of these monolines’ credit ratings triggered margin calls and banks had to face billions of loss due to counterparty risk. The wrong way risk could be directly observed in this financial crisis. US government had to further expand their liability side on their balance sheet to absorb these losses and this level of debt posed a potential systematic risk to the entire financial market. The failure of Lehman wiped out around 8000 different counterparties and led to years of litigations. Troubled Asset Relief Program (TARP) assisted Citi Group in 2008 defended the systematic shockwaves caused by the failures of the giant financial institutions. CVA, used to be a technical term, became a buzzword associated with financial markets. Basel III, Dodd-Frank Wall Street Reform and Consumer Protection Act, and European Market Infrastructure Regulation (EMIR), aiming at improving the stability of financial markets, came into act and shaped the new banking industry. It seemed that active hedging of CVA to obtain capital relief was encouraged by the regulations. However, the use of CDS, which perceived as major ways of counterparty credit risk transfers, introduced wrong way risk into the market. Central Counterparties (CCPs) replaced the prime brokers like Lehman, AIG as the new hubs of financial market and brought a greater level of transparency into the system but did not eradicate the counterparty risk. Substantial funding cost, named as Funding Value Adjustment (FVA), combined with counterparty risk made the banks tighten up their collateral policies and then transferred to the end-users of derivatives in the form of knock-on effect. Whether CCPs would make the financial markets safer remains to be a questions and could be answered till another financial crisis hits the markets. CCPs have obtained a privileged position in OTC derivatives market by mandatory clearing and margining requirements and they have discretion over the margining rules and default management process. In stressed markets, CCPs are incentivised to take actions to increase its survival probability and may have negative impacts on the rest of market participants. The counterparty credit risk isn’t gone. It is just converted into other forms of financial risks like funding liquidity risk etc. by clearing and margining mandates. The probability of next financial crisis induced by counterparty credit risk and OTC derivatives market is expected to be decreased via recent regulation reforms. However, at the same time, it is becoming more likely that the next financial crisis would be catalysed by the different forms of risk converted from counterparty credit risk in the financial system.

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6. Appendix

Empirical Studies on CVA

Data Description

The data of CVAs of different financial institutions and credit indices are provided by Bloomberg. The two firms reported CVAs on relatively regular basis are J.P. Morgan and Citi. Bank of America are added into the group but the time series are further curtailed due to mismatches in time series.

(Figure A.1: JPM & C CVAs Sources: data – Bloomberg)

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(Figure A.2: JPM, C & BAC CVAs Sources: data – Bloomberg)

Explanatory variables of CVAs

The generic CVA formula is

+ ( ) 푇 푖 퐶푃 푖 푀푡푀푡 (1 − 푅푡 ) 퐶푉퐴 = ∑ 피ℚ (∫ ℙℚ (푡) 푑 ℙℚ (푡)). 0 푁푂퐼푆 푆,퐼 푡 퐷,퐶푃(푖) 푖 0 푡

Here if we assume default probabilities of counterparties, survival probabilities of the institution and exposure profile are mutually independent,

+ ( ) 푇 푖 퐶푃 푖 푀푡푀푡 (1 − 푅푡 ) 퐶푉퐴 = − ∑ ∫ 피ℚ ( ) ℙℚ (0, 푡) 푑 ℙℚ (0, 푡). 0 푁푂퐼푆 푆,퐼 푡 퐷,퐶푃(푖) 푖 0 푡 Here if we apply PCA to two dimensional CVA time series for JP Morgan and Citi and three dimensional CVA time series for JP Morgan, Citi and Bank of America, we have the following graphs.

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(Figure A.3: JPM & C CVAs PCA Variance Analysis Sources: data – Bloomberg)

(Figure A.4: JPM, C & BAC PCA Variance Analysis Sources: data – Bloomberg)

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Two Credit Indices are used in this analysis:

1. iTraxx SenFin: Senior Financials 2. iTraxx Xover: Crossovers

And these credit indices are used as proxies of the counterparties of the derivative dealers. Here if we run a simple linear regression of 1st principal component with respect to these two credit indices,

(Figure A.5: Prediction Test of iTraxx SenFin as Explanatory variable for JPM & C Sources: data – Bloomberg)

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(Figure A.6: Prediction Test of iTraxx Xover as Explanatory variable for JPM & C Sources: data – Bloomberg)

(Figure A.7: Prediction Test of iTraxx SenFin as Explanatory variable for JPM, C & BAC Sources: data – Bloomberg)

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(Figure A.8: Prediction Test of iTraxx Xover as Explanatory variable for JPM, C & BAC Sources: data – Bloomberg)

The explanatory power of plain credit indices (CDX) running spread is not significant from this 2 perspective. The 푅 s of 퐶퐷푋푡 for these banks are

Bank J.P. Morgan and Citi J.P. Morgan, Citi and Bank of America 푖푇푟푎푥푥푆푒푛퐹푖푛푡 21.81% 3.65% 푖푇푟푎푥푥푋표푣푒푟푡 20.14% 2.79%

(Table A.1: Summary of 푅2of CDX as Explanatory Variable for J.P. Morgan and Citi and J.P. Morgan, Citi and Bank of America)

If 퐶퐷푋푡−1 is selected as the explanatory variable,

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(Figure A.9: Prediction Test of One Lag in iTraxx SenFin as Explanatory variable for JPM & C Sources: data – Bloomberg)

(Figure A.10: Prediction Test of One Lag in iTraxx Xover as Explanatory variable for JPM & C Sources: data – Bloomberg)

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(Figure A.11: Prediction Test of One Lag in iTraxx SenFin as Explanatory variable for JPM, C & BAC Sources: data – Bloomberg)

(Figure A.12: Prediction Test of One Lag in iTraxx Xover as Explanatory variable for JPM, C & BAC Sources: data – Bloomberg)

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2 The 푅 s of 퐶퐷푋푡−1 for these banks are

Bank J.P. Morgan and Citi J.P. Morgan, Citi and Bank of America 푖푇푟푎푥푥푆푒푛퐹푖푛푡−1 0.25% 32.48% 푖푇푟푎푥푥푋표푣푒푟푡−1 23.90% 26.79%

(Table A.2: Summary of 푅2of One Lag in CDX as Explanatory Variable for J.P. Morgan and Citi and J.P. Morgan, Citi and Bank of America)

Here the first order changes in CDX spreads is defined as Δ퐶퐷푋푡 and Δ퐶퐷푋푡 = 퐶퐷푋푡 − 퐶퐷푋푡−1.

If Δ퐶퐷푋푡 is selected as the explanatory variable,

(Figure A.13: Prediction Test of 1st Order Changes in iTraxx SenFin as Explanatory variable for JPM & C Sources: data – Bloomberg)

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(Figure A.14: Prediction Test of 1st Order Changes in iTraxx Xover as Explanatory variable for JPM & C Sources: data – Bloomberg)

(Figure A.15: Prediction Test of 1st Order Changes in iTraxx SenFin as Explanatory variable for JPM, C & BAC Sources: data – Bloomberg)

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(Figure A.16: Prediction Test of 1st Order Changes in iTraxx Xover as Explanatory variable for JPM, C & BAC Sources: data – Bloomberg)

2 The 푅 s of Δ퐶퐷푋푡 for these banks are

Bank J.P. Morgan and Citi J.P. Morgan, Citi and Bank of America 훥푖푇푟푎푥푥푆푒푛퐹푖푛푡 30.03% 46.37% 훥푖푇푟푎푥푥푋표푣푒푟푡 64.17% 38.51%

(Table A.3: Summary of 푅2of 1st Order Changes in CDX as Explanatory Variable for J.P. Morgan and Citi and J.P. Morgan, Citi and Bank of America)

The summary table of 푅2s for iTraxx SenFin based explanatory variables is

Bank J.P. Morgan and Citi J.P. Morgan, Citi and Bank of America 푖푇푟푎푥푥푆푒푛퐹푖푛푡 21.81% 3.65% 푖푇푟푎푥푥푆푒푛퐹푖푛푡−1 0.25% 32.48% 훥푖푇푟푎푥푥푆푒푛퐹푖푛푡 30.03% 46.37%

(Table A.4: Summary of 푅2of iTraxx SenFin Based Explanatory Variables for J.P. Morgan and Citi and J.P. Morgan, Citi and Bank of America)

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The summary table of 푅2s for iTraxx Xover based explanatory variables is

Bank J.P. Morgan and Citi J.P. Morgan, Citi and Bank of America 푖푇푟푎푥푥푋표푣푒푟푡 20.14% 2.79% 푖푇푟푎푥푥푋표푣푒푟푡−1 23.90% 26.79% 훥푖푇푟푎푥푥푋표푣푒푟푡 64.17% 38.51%

(Table A.5: Summary of 푅2of iTraxx SenFin Based Explanatory Variables for J.P. Morgan and Citi and J.P. Morgan, Citi and Bank of America)

It is clear that first order changes in both credit indices give higher explanatory power of 1st principal component of these banks. Here if we expand explanatory power analysis to the plain CVA numbers,

(Figure A.17: Prediction Test of 1st Order Changes in iTraxx SenFin as Explanatory variable for JPM CVAs Sources: data – Bloomberg)

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(Figure A.18: Prediction Test of 1st Order Changes in iTraxx SenFin as Explanatory variable for C CVAs Sources: data – Bloomberg)

(Figure A.19: Prediction Test of 1st Order Changes in iTraxx SenFin as Explanatory variable for BAC CVAs Sources: data – Bloomberg)

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(Figure A.20: Prediction Test of 1st Order Changes in iTraxx Xover as Explanatory variable for JPM CVAs Sources: data – Bloomberg)

(Figure A.21: Prediction Test of 1st Order Changes in iTraxx Xover as Explanatory variable for C CVAs Sources: data – Bloomberg)

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(Figure A.22: Prediction Test of 1st Order Changes in iTraxx Xover as Explanatory variable for BAC CVAs Sources: data – Bloomberg)

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