On the Proxy Modelling of Risk-Neutral Default Probabilities

DEGREE PROJECT IN MATHEMATICS, SECOND CYCLE, 30 CREDITS STOCKHOLM, SWEDEN 2020

EDVIN LUNDSTRÖM

KTH ROYAL INSTITUTE OF TECHNOLOGY SCHOOL OF ENGINEERING SCIENCES

On the Proxy Modelling of Risk- Neutral Default Probabilities

EDVIN LUNDSTRÖM

Degree Projects in Financial Mathematics (30 ECTS credits) Master's Programme in Industrial Engineering and Management KTH Royal Institute of Technology year 2020 Supervisor at Svenska Handelsbanken AB: Niklas Karlsson Supervisor at KTH: Camilla Landén Examiner at KTH: Camilla Landén

TRITA-SCI-GRU 2020:071 MAT-E 2020:034

Royal Institute of Technology School of Engineering Sciences KTH SCI SE-100 44 Stockholm, Sweden URL: www.kth.se/sci

Abstract

Since the default of Lehman Brothers in 2008, it has become increasingly important to measure, manage and price the default risk in financial derivatives. Default risk in financial derivatives is referred to as counterparty credit risk (CCR). The price of CCR is captured in Credit Valuation Adjustment (CVA). This adjustment should in principle always enter the valuation of a derivative traded over-the-counter (OTC). To calculate CVA, one needs to know the probability of default of the counterparty. Since CVA is a price, what one needs is the risk-neutral probability of default. The typical way of obtaining risk-neutral default probabilities is to build credit curves calibrated using Credit Default Swaps (CDS). However, for a majority of a bank’s counterparties there are no CDSs liquidly traded. This constitutes a major challenge. How does one model the risk-neutral default probability in the absence of observable CDS spreads? A number of methods for constructing proxy credit curves have been proposed previously. A particularly popular choice is the so-called Nomura (or cross-section) model. In studying this model, we find some weaknesses, which in some instances lead to degenerate proxy credit curves. In this thesis we propose an altered model, where the modelling quantity is changed from the CDS spread to the hazard rate. This ensures that the obtained proxy curves are valid by construction. We find that in practice, the Nomura model in many cases gives degenerate proxy credit curves. We find no such issues for the altered model. In some cases, weseethat the differences between the models are minor. The conclusion is that the altered model is a better choice since it is theoretically sound and robust.

Proxymodellering av riskneutrala fallissemangssannolikheter

Sammanfattning

Sedan Lehman Brothers konkurs 2008 har det blivit allt viktigare att mäta, hantera och prissätta kreditrisken i finansiella derivat. Kreditrisk i finansiella derivat benämns ofta motpartsrisk (CCR). Priset på motpartsrisk fångas i kreditvärderingsjustering (CVA). Denna justering bör i princip alltid ingå i värderingen av ett derivat som handlas över disk (eng. over-the-counter, OTC). För att beräkna CVA behöver man veta sannolikheten för fallissemang (konkurs) hos motparten. Eftersom CVA är ett pris, behöver man den riskneutrala sannolikheten för fallissemang. Det typiska tillvägagångsättet för att erhålla riskneutrala sannolikheter är att bygga kreditkurvor kalibrerade med hjälp av kreditswappar (CDS:er). För en majoritet av en banks motparter finns emellertid ingen likvid handel i CDS:er. Detta utgör en stor utmaning. Hur ska man modellera riskneutrala fallissemangssannolikheter vid avsaknad av observerbara CDS-spreadar? Ett antal metoder för att konstruera proxykreditkurvor har föreslagits tidigare. Ett särskilt populärt val är den så kallade Nomura- (eller cross-section) modellen. När vi studerar denna modell hittar vi ett par svagheter, som i vissa fall leder till degenererade proxykreditkurvor. I den här uppsatsen föreslår vi en förändrad modell, där den modeller- ade kvantiteten byts från CDS-spreaden till riskfrekvensen (eng. hazard rate). Därmed säkerställs att de erhållna proxykurvorna är giltiga, per konstruktion. Vi finner att Nomura-modellen i praktiken i många fall ger degenererade proxykreditkurvor. Vi finner inga sådana problem för den förändrade modellen. I andra fallservi att skillnaderna mellan modellerna är små. Slutsatsen är att den förändrade modellen är ett bättre val eftersom den är teoretiskt sund och robust.

Acknowledgements

This thesis was written during the spring of 2020 at the Royal Institute of Technology (KTH) and Svenska Handelsbanken. I would like to thank my supervisors Camilla Landén at KTH and Niklas Karlsson at Handelsbanken for their support and guidance. I would also like to thank Fredrik Bohlin at Handelsbanken for introducing the research topic.

Stockholm, May 2020 Edvin Lundström

Contents

List of Figures IV

List of Tables V

1 Introduction 1 1.1 Background ...... 1 1.2 Problem statement ...... 2 1.3 Purpose and research question ...... 3 1.4 Outline ...... 3

2 Counterparty credit risk 4 2.1 Financial risk ...... 4 2.1.1 Profit and loss ...... 5 2.2 Counterparty credit risk ...... 5 2.2.1 Exposure ...... 5 2.2.2 Loss given default ...... 7 2.2.3 Probability of default ...... 7 2.3 The typical situation ...... 7 2.4 Risk mitigating measures ...... 8 2.4.1 Netting ...... 8 2.4.2 Collateral ...... 9 2.5 CVA and DVA ...... 9 2.5.1 Regulatory CVA ...... 10 2.6 Default and spread risk ...... 11

3 Introduction to credit modelling 12 3.1 Credit models ...... 12 3.1.1 Firm value models ...... 12 3.1.2 Reduced form models ...... 13 3.2 P versus Q ...... 14 3.2.1 Real-world default probabilities ...... 14 3.2.2 Risk-neutral default probabilities ...... 15 3.3 The credit risk premium ...... 15 3.4 Shortage of liquidity problem ...... 16 3.5 Proposed proxy methods ...... 17 3.5.1 Single-name ...... 18 3.5.2 Index ...... 18 3.5.3 Intersection ...... 19 3.5.4 Cross-section (Nomura) ...... 19

I 3.5.5 Third party ...... 20 3.5.6 Others ...... 20

4 Interest rate modelling 22 4.1 Bank account and discount factor ...... 22 4.2 Zero bonds and zero rates ...... 23 4.3 Forward rates ...... 24

5 Reduced form credit modelling 25 5.1 Default and survival probabilities ...... 25 5.2 The Poisson process ...... 26 5.2.1 Constant intensity ...... 27 5.2.2 Time-varying and deterministic intensity ...... 28 5.2.3 Time-varying and stochastic intensity ...... 29 5.2.4 Summary ...... 29 5.3 Building blocks ...... 30 5.3.1 Introducing the building blocks ...... 30 5.3.2 Pricing ...... 31 5.3.3 Pricing under simplifying assumptions ...... 31 5.4 Interest rate versus credit modelling ...... 32

6 Credit Default Swaps 34 6.1 Basic structure ...... 34 6.1.1 Big and small bang ...... 35 6.2 Valuation ...... 35 6.2.1 The premium leg ...... 36 6.2.2 The protection leg ...... 37 6.2.3 Full value and breakeven spread ...... 37

7 Model 38 7.1 Constructing the survival curve ...... 38 7.1.1 Desirable properties ...... 38 7.1.2 Interpolation ...... 39 7.1.3 Bootstrap ...... 40 7.2 Proxy models ...... 41

8 Data 46 8.1 Credit Default Swap quotes ...... 46 8.2 Interest rates ...... 48

9 Results and discussion 49 9.1 Initial results ...... 49 9.2 Additive models ...... 49 9.3 The Nomura model collapses ...... 51 9.4 Arbitrage in the curve ...... 52 9.5 Large differences ...... 53 9.6 Rating monotonicity ...... 53 9.7 Volatility ...... 55

II 10 Conclusion 57 10.1 Further research ...... 57

References 59

Appendix A Alternative characterisations 61 A.1 Hazard rate ...... 61 A.2 Implied quantities ...... 62 A.2.1 Relation to forward rates ...... 63

III List of Figures

2.1 The impact of the sign of the value at default ...... 6 2.2 Probability of default curve ...... 8 2.3 The typical bank setup ...... 8

3.1 The difference between real-world and risk-neutral default probabilities. . 16

5.1 Risky ZCB paying one unit of currency if no default, zero otherwise . . . 30 5.2 Instrument paying one unit of currency if default before T , zero otherwise 31

6.1 Basic structure of a credit default swap ...... 34 6.2 CDS cash flows ...... 35

9.1 Survival curve and hazard rate for the proxy curve for BBB-rated European industrial companies ...... 50 9.2 Nomura method failing ...... 51 9.3 Nomura method resulting in negative hazard rates ...... 53 9.4 Large differences between the Nomura and the alternative method . 54 9.5 5Y survival probabilities for European Financial corporates ...... 55 9.6 15-year survival probabilities over time for the two proxy models...... 56

IV List of Tables

5.1 Analogies between interest rate and credit modelling ...... 33

7.1 Sample dummy variable matrix ...... 43

8.1 Summary of CDS data ...... 48

9.1 Survival probabilities for European industrial companies rated BBB according to the two models...... 50 9.2 Statistics for proxy credit curves built with different versions of the models 50

V Glossary

CCR Counterparty Credit Risk CDS Credit Default Swap CVA Credit Valuation Adjustment DVA Debit Valuation Adjustment LGD Loss Given Default MTM Mark-To-Market OTC Over-The-Counter PD Probability of Default ZCB Zero Coupon Bond

VI Notation

General

1{· } Indicator function, with condition inside brackets R The set of real numbers

Interest rate modelling B(t) The value of the bank account at time t ≥ 0 rt Stochastic short rate prevailing at t D(t, T ) The stochastic discount factor between t and T Z (t, T ) Value at t of a zero-coupon bond maturing at T with unit notional R(t, T ) Continuously compounded zero rate at t for maturity T F (t; T1,T2) The simply compounded forward rate at t, for the period [T1,T2] f(t, T ) The instantaneous forward rate prevailing at t for maturity T > t δ (t, T ) Accrual factor for the period t to T , according to some day count convention

Credit modelling τ Time of default F (t, T ) Probability of default in the interval between t and T Q (t, T ) Probability of survival over the interval between t and T Zˆ (t, T ) Value at t of a defaultable zero-coupon bond maturing at T with unit notional and zero recovery λ Constant hazard rate (intensity) λ(t) Deterministic and time-varying hazard rate Γ(s, t) Deterministic hazard function (cumulative intensity) for the interval [s, t] λt Stochastic hazard rate Λ(s, t) Stochastic hazard function for the interval [s, t] h(t) Hazard rate (instantaneous forward default rate) extracted from market quotes

Probability P A probability measure P The real-world (physical) probability measure Q The risk-neutral probability measure E [·] Expectation EQ [·] Expectation under the measure Q

VII

Chapter 1

Introduction

1.1 Background

The default of Lehman Brothers in 2008 showed that no entity can be assumed to be free from the risk of default. During the financial crisis that followed, the market realised that the default risk in financial derivatives must be measured, managed and priced. Lenders of money have always been aware that borrowers may default on their obligations. This has traditionally been the area of credit risk. Banks have studied and analysed the risk of default of the counterparties in mortgages, credit cards, etc. for a long time. However, before 2008 the default risk in financial derivatives was not really recognised, in particular for counterparties considered ”default remote”, such as banks (e.g. Lehman) and insurance companies (e.g. AIG). The default risk in derivatives is often called counterparty credit risk (CCR) or just counterparty risk. Counterparty risk is the risk that a counterparty owing us some payments defaults, which means that we can only expect to receive a fraction of the value of the payments.1 In principle, this risk should always enter the valuation of a derivative traded over-the-counter (OTC). The price of counterparty risk is captured in what is known as Credit Valuation Adjustment (CVA). CVA has been a key issue for banks in recent years due to accounting and capital requirements in addition to the volatility of credit spreads. There is also a converse adjustment: Debit Value Adjustment (DVA), accounting for one’s own risk of default. CVA and DVA are two of the valuation adjustments collectively referred to as the xVAs. These are different adjustments applied to the ”raw” value of a financial derivative, aiming to account for the credit, funding and capital costs that institutions may face in trading derivatives. In general, one can think of counterparty credit risk in terms of three components:

• Exposure: what a defaulting counterparty potentially owes us at default

• Loss given default: the fraction of the exposure that will be lost at default

• Default probability: the probability that a counterparty will be in default at a given point in time

As an example, a basic formula for CVA is given by (Gregory, 2015):

m X CVA = −LGD [EE(ti) × PD(ti−1, ti)] , (1.1) i=1

1I.e. one has a claim on the amount owed against the bankrupt estate

1 where LGD denotes loss given default, EE is the expected exposure and PD probability of default. t0 denotes today, tm is the maturity of the longest trade in the portfolio and m is the number of discretisation steps. This thesis is focused on the modelling of the default probabilities. We see that CVA is proportional to PD and is therefore very sensitive to the default probability. In Equation (1.1), PD(ti−1, ti) denotes the (marginal) default probability for the interval ti−1 to ti. Therefore, it is clear that we need to model the term structure of default probabilities (cf. interest rate curves). This means, in essence, what we need is the probability of default curve, or the cumulative default distribution function, F (t) = P [τ ≤ t], where τ denotes the random time of default. To calculate the value of some derivative using an interest rate curve, the curve must first be calibrated. Likewise, the probability of default curve must also be calibrated before we can use e.g. Equation (1.1). There are two fundamentally different methodologies in use:

• Observe historical credit events for entities2 of the same credit quality. Then employ some statistical procedure to arrive at the estimates. The estimated probabilities are called real-world probabilities or the probability of default under the physical probability measure P. • Use market quotations on credit instruments and extract the market expectations. Probabilities estimated this way are thus market implied and referred to as risk- neutral probabilities, or the probability of default under the risk-neutral probability measure Q. Note that these two methodologies are independent of the actual model used for the default time. Instead, they merely represent two ways of calibration. The two methodologies can be compared to estimating the volatility for a stock by either looking at the historical time series (realised volatility) or calibrating the volatility from market quotes on options (implied volatility). In traditional credit risk it has been standard to use real-world probabilities. However, when calculating pricing components (such as CVA) risk-neutral probabilities are typically used, since derivatives pricing is performed in a risk-neutral setting. Moreover, there has been regulatory pressure towards the universal use of risk-neutral probabilities. Recent versions of accounting standards (FAS 157 and IFRS 13) and Basel III all advocate the use of observable market data (i.e. risk-neutral probabilities) when determining the fair values of derivatives. Thus, there is clearly a need to be able to produce risk-neutral probabilities.

1.2 Problem statement

This thesis is written in cooperation with Handelsbanken. As most banks, Handelsbanken trade financial derivatives with numerous counterparties. For Handelsbanken these are primarily medium to large sized corporations based in Sweden. To calculate CVA, etc., default probabilities are needed for all of the bank’s counterparties. In particular, as stated above, one needs a risk-neutral probability of default curve. However, it should be added that Handelsbanken is using real-world probabilities for credit risk. 2Assuming that one has a view of the credit quality of the entity in question

2 In the literature, the standard approach for obtaining risk-neutral default probabilities is to bootstrap the credit curve using market quotes on single-name credit default swaps (CDSs). However, there are seldom CDSs traded with sufficient liquidity for the vast majority of counterparties, especially in the Nordic region. Hence, we need to rely on a proxy method. The issue of estimating default probabilities for counterparties without liquid CDS quotations is sometimes referred to as the shortage of liquidity problem. The Basel Com- mittee acknowledges that this is indeed a challenge and states in the consolidated Basel framework (BCBS, 2019):

Whenever the credit default swap (CDS) spread of the counterparty is available, this must be used. Whenever such a CDS spread is not available, the bank must use a proxy spread that is appropriate based on the rating, industry and region of the counterparty.

Similar formulations are present in European legislation, see e.g. Commission Delegated Regulation (EU) No 526/2014 (2014). Thus, one can conclude that any proxy method should take into account region, industry and rating.

1.3 Purpose and research question

The purpose of the thesis is to explore and analyse viable proxy modelling approaches for risk-neutral default probabilities in the absence of direct observables. The main question that the thesis aims to answer is: Which modelling framework is most suitable to model risk-neutral default probabilities for entities without liquidly traded credit default swaps? Additionally, we will investigate: How can we improve upon previously suggested modelling approaches?

1.4 Outline

The thesis is structured as follows: In Chapter 2 we will review some fundamental concepts in counterparty credit risk. In Chapter 3 we introduce credit modelling. Chapter 4 provides an overview of basic interest rate theory. Chapter 5 goes through reduced form credit models in some detail. Chapter 6 takes a brief look at the characteristics and valuation of Credit Default Swaps. In Chapter 7 we describe the proxy models studied and Chapter 8 introduces the data set. We present and discuss the results in Chapter 9. The thesis is concluded in Chapter 10.

3 Chapter 2

Counterparty credit risk

In this chapter we will introduce counterparty credit risk. First we establish counterparty credit risk in relation to other types of financial risk. Then we explore the three main components of counterparty credit risk. We also take a brief look at the typical situation a bank finds itself in. We will look at some standard counterparty credit risk mitigating actions. Then, CVA and DVA is introduced. The chapter ends with a discussion on the differences between default and spread risk.

2.1 Financial risk

The term risk denotes the possibility that some (unknown) future event might result in negative consequences. The future is uncertain, and thus risk is strongly coupled with the probability that an uncertain event will materialise. Financial risk generally relates to the possibility for some entity to lose money. Losses can be incurred for a variety of reasons. Traditionally, the different sources of losses have been divided into different risk types. Some of these are explained below.

Market risk: Before the financial crisis, the risk type attracting the most attention was market risk. Market risk refers to the effect on a portfolio resulting from changes in market prices. For example, an FX forward will gain or lose value depending on fluctuations in the spot exchange rate.

Credit risk: The default risk in cash products such as mortgages and loans is classified as credit risk or lending risk. For these products, the exposure (i.e. the value at risk) is often easy to compute and known in advance.

Counterparty credit risk: CCR deals with the default risk in financial derivatives. A characteristic feature is that the future value of financial derivatives is often uncertain.

For a traditional credit exposure, such as a mortgage, the amount owed in the event of default is easily estimated. For example, the amount owed at default is the principal amount on the loan. However, for financial derivatives, the situation is a bit more complicated. Consider a ten-year interest rate swap entered today with zero present value. We cannot tell at the trade date whether the swap will have a positive or negative present value in five years, since this will depend on the fluctuations of interest rates. Thus,it is apparent that one may view counterparty credit risk as an intersection of market risk and credit risk.

4 Furthermore, lending risk is always unilateral, whereas counterparty risk is typically bilateral. That is, a buyer of a bond is exposed to considerable credit risk. On the contrary, the issuer of the bond does not face a loss if the bond buyer defaults. This can be compared with the case of an interest rate swap, for which we do not know whether the future value will be positive or negative from our perspective.

2.1.1 Profit and loss Financial risk refers to the possibility of losing money. We can, however, distinguish between two different concepts of ”loss”:

Cash losses: With the term cash losses, we mean losses that are actually realised. Specif- ically, losses arising from actual defaults are cash losses.

Paper losses: The term paper losses is used to reflect balance sheet losses due to fluctuations in market prices, that will not necessarily materialise.

An example of paper losses could be mark-to-market losses due to price fluctuations for bonds in a hold-to-maturity portfolio. These losses never materialise, since the bonds are held to maturity and not sold in the secondary market. Similarly, one may incur balance sheet losses from CVA. CVA can be defined as the market price of counterparty credit risk. Thus, the value of the CVA will be sensitivity to fluctuations in the market price of credit risk (i.e. credit spreads). The Basel Committee reports that two-thirds of the credit losses during the financial crisis were CVA losses (i.e. paper losses), and only the remaining third stemming from actual defaults, i.e. cash losses (BCBS, 2011; Kenyon and Stamm, 2012).

2.2 Counterparty credit risk

Counterparty credit risk can be divided into three main components: Exposure, Loss given default and Probability of default:

• Exposure: an estimate of what a defaulting counterparty might owe us in the event of default

• Loss given default: an estimate of the percentage of the exposure that will be lost in case of counterparty default

• Probability of default: an estimate of the probability that a counterparty will default at a certain point in time.

It is important to note that all three components are time-dependent.

2.2.1 Exposure Credit exposure (hereafter just exposure) is the key component in counterparty credit risk since it is the value that may be at risk in default scenarios. Furthermore, exposure of some sort is included in all of the xVAs. The exposure is in turn driven by market risk. Thus, we repeat: Counterparty risk is a combination of market and credit risk. A defining feature of counterparty risk is an asymmetry of potential losses thatisnot present in market risk. This asymmetry stems from what happens at the default of the counterparty. Following default, one needs to determine the net value of the outstanding

5 transactions, after considering netting and collateral agreements.1 Now, the question is whether the net value is positive or negative. If the net value is negative, the surviving party is still obliged to pay the amount to the defaulting counterparty. That is, one cannot walk away from the amount owed. Thus, the position is largely unchanged from a valuation perspective. If the net value is positive, the defaulting counterparty will be unable to fulfil its obligations. The surviving party will thus have a claim on the positive value at the time of default and can only expect to recovery a fraction of the claim. The recovery value is unknown and is typically not included in the definition of exposure.

Positive value Claim on owed amount

Counterparty default

Negative value Need to pay owed amount

Figure 2.1: The impact of the sign of the value at default

To summarise: the surviving party loses if the value is positive but does not gain if the value is negative. This is illustrated in Figure 2.1. Formally: Definition 2.1 (Exposure). Let V (t) denote the present value of a contingent claim at time t. The exposure at time t is defined as: E(t) = max {V (t), 0} = V (t)+ This asymmetrical pay-off can be likened to a short option position (Hull, 2012). From classical options pricing theory, we can conclude two things: • Pricing of options is relatively complex. Therefore, quantifying exposure even for simple instruments may turn out to be complex. • A key determinant of an option price is volatility. Thus, exposure will be affected by the volatility of the underlying contracts and collateral. However, Gregory (2015) argues that treating xVA quantification as a giant exotic option pricing problem is potentially misleading and tends to draw focus from other important aspects. One reason is that it is difficult to explicitly write down the payoff of the option. This, in turn, due to the ambiguities relating to the ”value” or ”close-out amount” at default. One may also be interested in what a counterparty will owe us on average. Definition 2.2 (Expected Exposure). Let V (t) be the value of a contingent claim at time t. Let V (t)+ denote the exposure, and let Ψ be the probability density function of V at time t, then (Ruiz, 2015): Z ∞ EE(t) = V (t)+Ψ(t)dV (t) −∞ We see that only positive values of the claim contribute to the expected exposure. We can interpret the EE as ”how much we can be owed on average” which is not the same thing as ”how much we can be owed on average, given that we are owed something”. This is because the negative values of the claim contribute in terms of their probability. 1Netting and collateral are introduced in Section 2.4

6 2.2.2 Loss given default The loss given default (LGD) refers to the percentage amount that is lost in the event of counterparty default. Equivalently, one also talks about the recovery rate, which is simply one minus the LGD. The LGD depends on the seniority of the claim. Normally an over-the-counter derivative claim is of the same seniority (i.e. pari passu) as senior unsecured debt. Which in turn is what is referenced by typical credit default swaps. Gregory (2015) reports that LGD has varied significantly historically, depending on sector, economic conditions and the seniority of the claim. However, the quantification of LGD is not of primary importance for CVA computations due to a cancellation effect, meaning that the estimation of the probability of default is more important. One should also keep in mind that the CDS market uses the recovery rate as part of the quoting mechanism (Kenyon and Stamm, 2012).

2.2.3 Probability of default To assess the counterparty risk, we must assess the credit quality of the counterparty over the lifespan of the relevant transactions. Naturally, these time horizons may be very long, spanning several decades. There are two important aspects to consider, given a certain time horizon:

• What is the probability of the counterparty defaulting?

• What is the probability of the counterparty’s credit quality deteriorating? (E.g. credit rating downgrade, or widening of credit spreads)

Note that we will use the term ”default” to refer to any ”credit event” affecting the counterparty. For a more complete treatment of credit events, see O’Kane (2008) or Brigo, Morini, and Pallavicini (2013). Suppose that the default probability of a counterparty between the current time and a year from now has been determined (that is, the annual default probability). It is crucial to consider what the future annual probability of default is. E.g. the probability between year five and six from now. In other words, we need the full term structure ofdefault probabilities. The term structure of the probability of default is also known as a probability of default (PD) curve. The PD curve is the cumulative distribution function for the default time. A conceptual PD curve is shown in Figure 2.2. F (t) is the probability in the interval [0, t]. PD(t1, t2) denotes the (unconditional) probability of default from t1 to t2, i.e. PD(t1, t2) = F (t2) − F (t1). Another name for PD curve is credit curve. A rigorous mathematical definition of probability of default is given in Section 5.1.

2.3 The typical situation

A bank typically finds itself in the situation depicted in Figure 2.3. Banks trade withend- users of derivatives, such as corporations, hedge funds, pension funds and other financial institutions. Moreover, banks commonly aim to run flat books. That is, they try to maintain trading portfolios hedged from a market risk perspective. Thus, any transactions made with end-users of derivatives will be hedged with other market participants (with hedging transactions done either on a one-to-one basis or on portfolio level) (Gregory, 2015).

7 F (t)

PD(t1, t2)

t1 t2

Figure 2.2: Probability of default curve

In the case when the bank executes an offsetting transaction with another market participant, the market risk will be completely hedged. However, the bank still has counterparty risk exposure to both counterparties. Furthermore, if either of the counterparties would default the bank is now left with the open market risk from the remaining trade with the surviving counterparty. Figure 2.3 shows another typical feature, namely that the bank faces asymmetry in the collateral. End users of derivatives (such as medium sized corporations) will most often not post collateral. Collateral will be explained in Section 2.4.2.

2.4 Risk mitigating measures

This section will briefly introduce some common counterparty risk mitigating actions.

2.4.1 Netting A central and important counterparty credit risk mitigant is netting. Netting means that a collection of trades will be grouped into a netting set, for which only the net amount will be owed in the event of default. That is, positive and negative values will offset each other, reducing the total exposure. Netting sets represent separate entities from a credit exposure perspective. Netting sets are defined by legal agreements between counterparties. These often follow recommendations from the International Swaps and Derivatives Association (ISDA) (Ruiz, 2015). The agreements are called master agreements or ISDA agreements. It is common that multiple netting sets exist between two counterparties. For example, one might have different netting sets for different asset classes.

Uncollateralised Collateralised Client Bank Hedge

Figure 2.3: The typical bank setup

8 The benefit of netting is easy to see: Let Vi be the value of contract i and let R be the recovery rate. Without netting, the loss in case of default is:

N X (1 − R) max(Vi, 0) i=1

However, with netting the loss is:

N ! X (1 − R) max Vi, 0 i=1

Since what a defaulting party owes each of its counterparties is dependent on the usage of netting, CCR measures should be computed per legally enforceable netting set.

2.4.2 Collateral To further reduce counterparty risk, counterparties add a Credit Support Annex (CSA) to their Master Agreements. The CSA concerns collateral, which refers to an asset offsetting default risk in a legally enforceable way. The CSA stipulates that a party will have to post collateral to the counterparty when the netting set is in the money. That is, against any negative exposure a party has, it will send collateral to the counterparty. The collateral is usually in the form of cash or securities. In practice: If one party in a trade defaults, the surviving party will keep the collateral, to offset the loss incurred. Clearly, collateral can dramatically reduce the counterparty credit risk. Banks typically set trading limits on counterparty risk. Collateralisation increases the capacity to trade, since the trading limit utilisation for each trade is reduced. The typical collateral agreement is bilateral, meaning that both parties are required to post collateral when the value of their transactions is negative. There are also unilateral agreements where only one of the parties post collateral. Furthermore, many parties do not enter into collateral agreements. This is typical for the end users of derivatives. They might not have the resources to set up a collateral management operation, or do not have access to the cash or securities to be posted. Thus, collateral agreements are more common in the interbank market.

2.5 CVA and DVA

CVA is in essence today’s price of the default risk of the counterparty. Historically, there existed a view that CVA should be seen as an actuarial ”risk reserve” rather than a ”hedging price”. CVA as a hedging price is the dominating view as of now. CVA can be defined by the relationship (Gregory, 2015):

Risky value = Risk-free value + CVA. (2.1)

A complexity not explicitly stated in Equation (2.1) is that CVA is not additive across trades, due to risk mitigants such as collateral and netting. CVA should be computed for all trades covered by the same arrangements, e.g. in the same netting set. As stated in the introduction, the standard formula for CVA is (Gregory, 2015):

m X CVA = −LGD [EE(ti) × PD(ti−1, ti)] , (2.2) i=1

9 where t0 is today, tm is the longest maturity in the netting set and m is the number of discretisation steps. LGD was introduced in Section 2.2.2, EE is the expected exposure as in Definition 2.2 and PD(ti−1, ti) is the marginal probability of default for the time interval [ti−1, ti] We see in the CVA formula that CVA is proportional to the marginal probability of default, thus the CVA is highly sensitive to the PD estimate. In the pre-crisis world, banks considered themselves risk-free. This opinion was also shared by the clients of the banks. Hence, banks could charge clients for the clients’ counterparty credit risk through CVA but did not take into account the banks’ own risk of default. This was consistent with the accounting concept ”going concern”, which means that financial statements should be based on the assumption that the reporting entity will remain in existence for eternity. The above definition of CVA reflects this type of unilateral CVA. Lehman’s default showed that even the large banks cannot be regarded as default free. Thus, the counterparty of the bank might want to take into account the default risk of the bank (i.e. calculating the CVA from their side). Since the calculated value of a derivative differs depending on the calculating entity, this leads to a violation ofthelaw of one price. In turn meaning that counterparties will not agree on the fair price of a derivative. This led to Debt Valuation Adjustment (DVA), namely today’s price of one’s own default risk. Along the lines of Equation (2.2), DVA is defined as:

m X DVA = −LGD [NEE(ti) × PD(ti−1, ti)] , i=1 where NEE is the negative expected exposure. With DVA also considered, symmetry in pricing is restored. Bilateral CVA is then defined as:

BCVA = CVA + DVA

2.5.1 Regulatory CVA For banks with internal model method approval, the formula to be used for CVA when calculating the capital requirements for CVA risk is (Article 383 in Regulation (EU) No 575/2013 of the European Parliament and of the Council (2013)):

CVA =LGDMKT m X si−1ti−1 siti · max 0, exp − − exp − LGD LGD i=1 MKT MKT EE D + EE D · i−1 i−1 i i , 2 where again t0 is today, tm is the longest maturity in the netting set, m is the number of discretisation steps and si is the credit spread at time ti. The CDS spread of the counterparty is to be used if it is available. Otherwise, the institution should use a ”proxy spread that is appropriate having regard to the rating, industry and region of the counterparty”. LGDMKT is the loss given default based on the (proxy) spread of a counterparty. EEi is the expected exposure and Di is the risk-free discount factor at ti, respectively.

10 The second argument in the max function is an approximate formula for the market implied probability of default, as pointed out by Gregory (2015). That is, the marginal default probability between two times ti−1 and ti can be approximated as:

s t s t PD(t , t ) = exp − i−1 i−1 − exp − i i i−1 i LGD LGD We note that this is an unconditional probability. That is, it is not conditional on that the counterparty has survived until ti−1.

2.6 Default and spread risk

In Section 2.2.3 we noted that one is interested in both knowing the probability of default and the probability of a deterioration in the counterparty’s creditworthiness. We can therefore distinguish between different types of risk:

Default risk: The risk that a scheduled stream of payments is not received. This is perhaps what is typically thought of when thinking about counterparty credit risk.

Recovery risk: The risk that after a default, the amount recovered is less than the amount owed.

Spread risk: The risk that the value of a credit security falls when the market’s perception of the creditworthiness of the obligor changes for the worse, causing a loss to be realised if the security is sold.

Note that even if the typical case of spread risk is that the price of a bond fluctuates as the market’s view of the creditworthiness of the issuer changes, we can also talk about spread risk for CVA. If the is CVA calculated risk-neutrally and marked-to-market daily, the P&L will fluctuate according to changes in the market prices of credit risk (i.e.due to changes in the perception of the counterparty’s credit quality). When considering hedging counterparty credit risk, it is crucial to differentiate between default and spread risk. Default is typically an idiosyncratic event and must be hedged accordingly. That is, if we have some exposure we want to hedge, we must use an instrument with a payoff contingent on the default of the counterparty. Typically, thisis a single-name credit default swap. However, spread risk also has a systematic component, meaning that a so-called cross hedge may be feasible. For example, the market’s perception of outlook for the economy in general may worsen, leading to higher implied default probabilities for all counterparties. This risk could be hedged with a CDS Index contract. However, this means that only the spread and not the default risk is hedged. In Section 2.1.1 we saw that there are two types of financial loss. Default risk is associated with cash losses while spread risk is connected to paper losses. Recall that the credit losses seen during the financial crisis were to two thirds paper losses with the remaining part due to actual defaults.

11 Chapter 3

Introduction to credit modelling

Since the lending and borrowing of money was invented, lenders have been aware of the risk that borrowers might default. For the lender to determine the risk premium required to compensate for the default risk and to manage the credit risk properly, it is crucial for the lender to know the probability of the default event. We will also refer to the survival probability, which is simply the complement. The term credit spread1 is used more or less interchangeably with default probability, the reason being that one can get from one to the other. This chapter begins with an introduction to the two families of default models. In Section 3.2, the difference between the real-world and risk-neutral measure is investigated. The chapter ends with an overview of previous research on proxy modelling.

3.1 Credit models

In essence, a default (or credit) model is a way to model the default time τ for an entity. Models for defaults fall into two general families: firm value (structural) and reduced form (intensity based) models. There are also hybrid models that combine features from the two main families.

3.1.1 Firm value models Firm value, or structural models, are models in which the survival of a company is linked to its ability to repay its debt. This family of models is based on the work of Merton (1974). In its simplest form, one assumes that a firm issues debt to finance its activities. The debt is to be repaid at maturity T . If the firm is not able to pay back its debt tothe bondholders at T , one says that there has been a default. In this setting, default can only happen at time T . To make the model more realistic, one can also allow for default before T . Now, the default time is the first time the firm value passes a barrier form above. That barrier can either be constant, time-varying or stochastic. This family of models is called first passage time models. The fundamental assumption of the standard firm value models is that the value of the firm is driven by a geometric Brownian motion (GBM). The GBM is used forequity modelling in the Black-Scholes framework. In these models, one takes the value of the

1One should note that ”credit spread” can be defined in many ways, as demonstrated in the aptly named chapter ”A plethora of credit spreads” in Kenyon and Stamm (2012).

12 firm to be the sum of the firm’s equity and debt. For Merton style models, regular option pricing techniques can be used. For first passage time models, one needs to employ barrier option pricing theory. An implicit assumption is that the firm value is observable at any time.2 Thus, unlike intensity models (as we will see), the default process is completely monitored based on the default free market information and default does not come as a surprise. In basic versions of structural models, the number of parameters is few and thus the model cannot be exactly calibrated to data such as CDS quotes for different maturities (Brigo and Mercurio, 2006). O’Kane (2008) points out that firm value models are typically not applied in the risk-neutral framework and rarely used for pricing of credit derivatives. Finally, one can note that the firm value family is closer to equity modelling than interest rate modelling. Whereas we will see in the coming sections there are a lot of analogies between reduced form models and interest rate models.

3.1.2 Reduced form models In reduced form models (also known as intensity models), the time of default of an entity is described by an exogenous jump process. The default time τ is the first jump of a Poisson process. The Poisson process can either have a deterministic (constant or time-varying), or stochastic intensity. In this family of models, the default is not triggered by market observables. Instead it is triggered by a component independent of all default free market information. This has two important implications: • Monitoring the default free market (stock prices, interest rates, etc) does not give information about the default process.

• There is no economic reason behind default. As we will see in Chapter 5, the central quantity in reduced form models is the intensity or hazard rate which can be interpreted as the default probability per unit of time, conditional on survival until that point. We will also see that we can interpret the intensity as an instantaneous credit spread, to be added to the short rate. This means that this family of models is well suited for the modelling of credit spreads and is easy to calibrate to credit default swap or bond data. The models are flexible enough to fit the prices of the calibration instruments (bonds, CDSs, etc.). Furthermore, the models allow for capturing the essential risks: default risk and spread risk. One can also model the co-dependence between credit risk and interest rate risk when using the reduced form approach. Since the default time is driven by an external source of randomness (recall that the jump process is given exogenously), independent of all the default free market information, the reduced form models are incomplete. Having the exogenous component means that the default event is unpredictable. In the Merton model described above, the default time is predictable, which means that in the limit dt → 0 we can know with certainty whether default will occur or not in the next interval of time dt. This is not the case in the reduced form approach, where we only know the probability of default in the next interval. Put differently: In reduced form models default always comes as a surprise. This also makes it possible to have non-zero instantaneous credit spread.3 This is in line with the empirical observation that very short-dated credit instruments have a

2In reality, that is not the case (e.g. financial statements are released quarterly). 3Here meaning the credit spread for a very short time to maturity.

13 non-zero credit spread even as the remaining maturity goes to zero (O’Kane, 2008). The explanation for this is that the market adds a premium to capture the uncertainty of the current health of a credit, in turn meaning that the probability of a sudden default is non-zero. This risk is called the jump to default risk. The above features are generally considered to be both favourable and realistic (Brigo and Mercurio, 2006). We will also see in Chapter 5 that reduced form default modelling is quite similar to interest rate modelling. When valuing credit default swaps, the standard approach is to use reduced form models, see O’Kane (2008) and Green (2015). Also, reduced form models are the most commonly used to extract implied default probabilities from market quotes (Brigo and Mercurio, 2006).

3.2 P versus Q There are two methods of estimating the default probability (Kenyon and Stamm, 2012):

• Observe historical credit events for entities4 of the same credit quality. Then, use some statistical procedure to arrive at the estimates. The estimated probabilities are often called real-world probabilities, or the probability of default under the physical measure P. • Use market quotations on credit risky instruments and try to extract the market expectations. Probabilities estimated this way are thus market implied and are commonly referred to as risk-neutral probabilities or the probability of default under the risk-neutral measure Q. This section will explore both of these approaches in some detail.

3.2.1 Real-world default probabilities Real-world estimates are often used when the intention is the reflect the ”true” reality of the economy and to avoid noise such as the market price of risk or supply and demand effects (Gregory, 2015). Real-world default probabilities are often estimated from historical default data for different credit ratings. This type of PD modelling has been usedfor a long time for credit risk assessment, both in internal economic capital models and for regulatory capital in the Basel framework (Green, 2015). The model can either be based on historical defaults experienced by an institution or by external data from rating agencies. Rating agencies calculate default statistics by constructing databases of issuers and monitor the rating and default behaviour of the bonds issued over time. The issuers are grouped into cohorts, where the issuers have the same initial rating. The rating agency then tracks the performance of the issuers in every cohort and can thus calculate the default rate. New cohorts are regularly added and thus it is possible to calculate time-averaged default rates. Thus, one obtains average cumulative default rates for issuers with different ratings. One can also calculate rating migration matrices. The historical default rates can be used as default probabilities. The question is whether historical default rates are a good predictor of future default rates. If one uses time-averaged data, the present state of the economy is not reflected in the estimated probabilities. Furthermore, any variation within a rating category is not reflected.

4Assuming that one has a view of the credit quality of the entity in question

14 A notable product within real-world modelling is Moody’s KMV, which provides expected default frequencies for issuers. It is a firm value type of model calibrated using historical data yielding real-world probabilities. O’Kane (2008) remarks that historical data is not used by market participants for the pricing of derivatives - it is primarily a method for calibrating risk models. When pricing credit risky instruments one needs to be in a setting where one can hedge out risk in these assets. That setting is the risk-neutral framework.

3.2.2 Risk-neutral default probabilities Risk-neutral probabilities are derived from prices observed in the market. One can use prices on credit risky instruments such as bonds or credit default swaps. Typically, one uses CDS quotes together with a reduced form model, as outlined in Section 3.1.2. The market practice is that the valuation of xVAs in general and CVA in particular is done under the Q-measure. This is in line with the pricing and valuation of derivatives, where market participants usually calibrate risk-neutral models and only use real-world models for non-observable parameters. However, this has not always been the case. Gregory (2015) points out that in the past, banks commonly used real-world probabilities when quantifying CVA, because at that point CVA was not universally considered to be included in the fair value of an asset. Moving from P- to Q-probabilities means that the interpretation of CVA changes from ”an actuarial reserve” to ”a market price of credit risk”. The main reasons for using risk-neutral default probabilities for CVA are (Green, 2015): • CVA is a valuation adjustment to a portfolio of derivatives to account for the credit risk of the counterparty. The derivatives themselves are valued in the risk-neutral framework. To be consistent, CVA should be valued using the same principle. • When CVA is valued risk-neutrally, a bank can hedge CVA volatility and default risk by buying the same CDS contracts as used when estimating the default probabilities. • In the Basel III framework CDS spreads are used to calculate the CVA regulatory capital for banks with IMM (internal model method) approval. If banks hedge their CVA risk the required capital is decreased. Furthermore, accounting standards (IFRS 13) require using market observable data when possible, implying the use of Q-probabilities. In total, it is becoming increasingly difficult to defend the use of P-probabilities to auditors and regulators. Since CDSs are traded instruments, the implied default probabilities are at least in theory reflecting all available information. In comparison, credit ratings are onlyup- dated infrequently and will often lag behind the real-time deterioration of an entity’s credit quality (Ruiz, 2015). Using credit ratings and historical default rates is also a backward-looking approach. Hence, using Q-probabilities might seem preferable even without considered accounting and regulatory requirements. However, as pointed out by Gregory (2015), CDS spread contains a large non-default component that might create problems.

3.3 The credit risk premium

We introduced P- and Q-probabilities above. We saw that the methodologies used to compute these differ considerably. It is thus natural to ask if there is any difference

15 Risk premium Risk-neutral default probability

Real-world Default risk default probability

Figure 3.1: The difference between real-world and risk-neutral default probabilities. between the two. Gregory (2015) argues that one would expect risk-neutral probabilities to be higher than real-world probabilities, since investors are risk averse and demand a premium for accepting this risk. The risk-neutral default probability would thus consist of two things: the real, underlying default risk5 and a risk premium as illustrated in Figure 3.1 The risk premium is called the credit risk premium. O’Kane (2008) decomposes the risk premium into three parts: default risk, volatility risk and liquidity risk premium. The credit risk premium has been investigated empirically. Typically, the premium is quantified as the ratio between risk-neutral and real-world default intensities, which some authors call the coverage ratio. This is the number of times the risk-neutral spread ”covers” the real-world (or actuarial) spread. Berndt et al. (2005) derive Q-intensities from CDS prices and P-intensities from Moody’s KMV expected default frequencies. They find that the risk-neutral intensities are roughly twice as large as their real-world counterparts. Heynderickx et al. (2016) extract real-world intensities from rating transition matrices obtained from rating agencies. Risk-neutral intensities are derived from CDS quotes on European corporates. The authors find that the coverage ratio is significantly larger than one. As an example, the coverage ratio for A-rated companies in the post-crisis regime is 7.86. It is also reported that the coverage ratio is significantly larger post-crisis (2013-2014) than before the financial crisis (i.e. prior to 2008).

3.4 Shortage of liquidity problem

As explained in Section 3.2.2 there are number of strong arguments for using risk-neutral default probabilities when calculating CVA. It is also market practice. We have seen that the typical way of obtaining Q-probabilities is to use CDS quotes and a reduced form model. However, in reality banks may have thousands of counterparties. For many of these there are no credit risky instruments to derive risk-neutral default probabilities from. In particular, the CDS market in the Nordic region is limited. That is, there are many ”illiquid credits” for which there is no direct way of obtaining Q-probabilities. This issue is referred to by some authors as the shortage of liquidity problem. This leaves a bank with two options:

5This is called the actuarial spread by O’Kane (2008)

16 1. Derive Q-probabilities using some proxy method 2. Use both Q- and P-probabilities, depending on whether the counterparty is liquid or not There is no standard method for determining a risk-neutral probability of default curve for a given illiquid counterparty. However, even if a proxy method is used some complications arise. A fundamental concept in the risk-neutral valuation framework is the existence of a hedge. Clearly, if we needed a proxy method to determine the Q-probabilities, there will exist no such hedge. Gregory (2015) argues that this is problematic, since CVA is generally larger and more volatile when using Q- compared to P-probabilities and there is no hedging instrument to manage that volatility. Hammarlid and Leniec (2018) point out that the absence of an instrument to hedge away the default risk for most counterparties implies that the market is incomplete. That the market is incomplete means that there is no unique equivalent martingale measure (Björk, 2009). Using a minimum variance hedging argument they conclude that one should actually use P-probabilities in the absence of liquid CDSs for a counterparty. They also suggest a method to make ”maximal use of market information” thus fulfilling the requirements set by IFRS 13. The method amounts to first deriving proxy Q-probabilities and transforming them into their P-counterparts. The proposal to use P instead of Q is in line with what Green (2015) suggests in the following quote: Most counterparties do not have a liquid CDS contract associated with them [...]. Given that the default risk cannot be hedged in such cases, risk ware- housing is inevitable and this leads directly to incomplete markets and the physical measure.

Gregory (2015) emphasises that regulators view Q-probabilities as a central component in CVA calculations and that they are also fundamental in determining exit prices in fair value accounting. Auditors and regulators generally do not allow deviating from the usage of Q-probabilities, even in the case of illiquid credits. As an example, Commission Delegated Regulation (EU) No 526/2014 (2014) states: The application of the advanced method to the determination of own funds requirements for Credit Valuation Adjustment (CVA) risk may involve counterparties for which no Credit Default Swap (CDS) spread is available. Where this is the case, institutions should use a spread that is appropriate having regard to the rating, industry and region of the counterparty. This implies that the arguments by Green (2015) and Hammarlid and Leniec (2018) are more or less academic. The conclusion is that banks needs to derive risk-neutral default probabilities for all counterparties they wish to calculate CVA for.

3.5 Proposed proxy methods

In the previous section, we introduced the shortage of liquidity problem. In this section, we will look at various types of frameworks that have been suggested for the proxy modelling of default probabilities. Sometimes, this is referred to as a ”curve mapping procedure”, due to the fact that some of the methods simply amount to ”mapping” the illiquid name to some proxy curve. The term structure of default probabilities constitutes a probability of default curve. A survival probability curve is the complement. We will refer to both default and survival

17 curves as credit curves. The term proxy curve is used for a credit curve not derived using quoted instruments for the entity in question. A majority of the methods listed below uses CDS spreads in some way. Gregory (2015) comments that there are some technical issues with using CDSs to derive probabilities, but that the CDS market is still believed by regulators and auditors (but perhaps not by all banks) to render to best market implied price for credit risk. We also note that some of the models provides a way of obtaining a proxy CDS spread. Deriving the credit curve from CDS spreads is a straightforward process, as we will see.

3.5.1 Single-name An obvious way to model the default probability is to simply use the CDS quotes from a similar, liquid entity. Gregory (2015) argues that this can be suitable for a subsidiary when the parent company has CDSs traded with sufficient liquidity. Green (2015) explains that the approach can be extended to simply choose a liquid entity that is similar in terms of creditworthiness, rating and sector. We refer to this class of models as single-name since the proxy spread (or probability) is generated using information from a single entity. Brummelhuis and Luo (2017) suggest how to systematically choose the best proxy entity. They employ different machine learning algorithms to, for an illiquid name, find the most ”similar” name amongst the liquid names. The authors report that the top performing algorithms are found in the families of neural networks, support vector ma- chines and bagged trees. The algorithms are trained on a data set containing real-world default probabilities, implied and historical stock volatilities, for different tenors. The performance is evaluated by measuring prediction accuracy using a cross-validation tech- nique. The authors make no effort to investigate whether using single-name proxies is appropriate or not. Green (2015) points out that using single-name proxies has some advantages, primarily the fact that it is obvious what contract to use when hedging CVA. In principle, the credit risk component of the CVA could be completely hedged, giving zero P&L volatility from fluctuations in the market perception of credit risk. However, considerable basis risk persists. If the illiquid entity defaults, there is no protection payment triggered in the CDS bought on the liquid entity. I.e. one can hedge the spread risk, but not the default risk using proxies. It should be noted that this issue prevails for all proxy methods.

3.5.2 Index CDS indices are contracts on benchmark indices in categories of creditworthiness, and sometimes region and sector. The most common are the CDX series covering North America, and iTraxx covering the rest of the world. These contracts are in general traded with greater liquidity than single-name CDSs. As an example, iTraxx Europe Main consists of the 125 most actively traded investment grade names. The CDS index contract can be compared to holding single-name CDS on each of the underlying names. Gregory (2015) and Green (2015) both suggest that that illiquid counterparties can be mapped to CDS indices. Green (2015) emphasises that as for single-name proxies, hedging is obvious (just trade the indices), but naturally, only spread and not default risk can be hedged.

18 3.5.3 Intersection The intersection (or bucket) method stems from an EBA consultation paper (EBA, 2012). To fulfil the requirement that an institution “shall use a proxy spread that is appropriate having regard to the rating, industry and region of the counterparty”, the idea is to use the average spread from the liquid names with the same characteristics as the illiquid name. The proxy spread is in the intersection method simply the arithmetic mean: N 1 X Sproxy = S i N j j=1 where j ∈ {1,...,N} indexes the liquid names of the same rating, sector and region as the entity i. Compared to the using CDS indices, this method provides greater granularity. Only a limited number CDS indices exist and generally the indices are not defined for a combination of region, industry and rating at the same time. To clarify: both a sub-investment grade index and an industrials index exist, but there is no index for sub-investment grade industrials. With the intersection method, any bucket can be defined, as long as there is at least one liquid name with the wanted characteristics. Hedging in this case becomes more complicated compared to the single-name and index approaches. One could use the components of each bucket, but it may turn out to be expensive and inefficient (Green, 2015).

3.5.4 Cross-section (Nomura) In response to the intersection method, the so-called cross-section method was proposed by Chourdakis et al. (2013). This model is also known as the Nomura model. The authors note that the number of liquid names is simply not enough to give robust estimates for all possible buckets. That is, some buckets will only contain very few or even no liquid names. This implies that the proxy spreads will be very sensitive to e.g. rating migrations (when entities enter or leave a bucket). They suggest a model claimed to produce more robust and stable results. The authors propose to model the CDS spread as a product of five factors: • Global • Rating • Region • Industry sector • Seniority Hence, for an entity i: proxy Si = MglobMsctr(i)Mrgn(i)Mrtng(i)Msnty(i)

Note that e.g. Mrtng(i) denotes the factor value for a specific rating, i.e. BB. Thus, the model would read for the senior debt of an AA-rated European Financial obligor: proxy S = MglobMFin MEur MAAMSen We see that the factors in the model will take different values depending on the category. If one uses the Standard & Poor’s rating scheme, the rating factor takes 19 different values. That is, for each factor a number of parameters needs to be estimated. The total number of parameters is the sum of the number of categories per factor. Chourdakis et al. (2013) suggests that the model is calibrated as follows:

19 • Enumerate all the parameters. If we have 11 sectors, 7 regions, 7 ratings, 2 seniorities and the global factor (as in Chourdakis et al. (2013)), the total number of parameters m is 28. The parameters are indexed by j. • Enumerate all the liquid names: i ∈ {1, . . . , n}

• Define the matrix A = [Aij], where the rows correspond to liquid entities and the columns to the parameters. An entity within the financial sector will have a one in the financial column and zeroes in the other sector columns. A is therefore a matrix of dummy variables. proxy • Let yi = ln Si and xi = ln Mj Then the model can be written as: n X yi = Aijxj j=1

market We have n liquid entities for which we have a quoted spread Si . The xj can be proxy chosen as the values that makes the proxy spreads Si as close to the quoted spreads market Si as possible. This can be for example be achieved by performing a standard linear OLS regression. When the factors Mj have been obtained, calculating the proxy spread for an illiquid entity is just a matter of multiplication. The key assumption of the method is that there is a single multiplicative factor for e.g. all AA entities, and that the factor is independent of the sector, region and seniority of the entities. This means that when calibrating the MAA factor, one leverages the information from AA entities across all sectors/regions/seniorities, meaning that each factor is represented by a relatively large number of names. We want to point out that if one uses the 28 parameters as above, the number of possible combinations is 1078, giving 1078 proxy curves. This method has become popular and is presented in widely known textbooks on xVA: Gregory (2015), Green (2015) and Ruiz (2015). Furthermore, Gregory (2015) notes that Markit (a large vendor of financial data and related services) has built a proprietary CDS curve product based on the cross-section method. Sourabh, Hofer, and Kandhai (2018) extends the model further, adding equity returns and volatilities as factors. They claim to find significant improvements over both the intersection and cross-section methods. Hedging becomes even more complex in this case compared to the intersection method. Since a greater number of liquid names contribute to each proxy curve, it is implied that a larger number of instruments may be needed to construct a hedge.

3.5.5 Third party As noted in the previous section, there are vendors that provide curves for different categories of entities based on proprietary models. However, if the constituent entities are not disclosed it is no obvious how to choose hedging instruments.

3.5.6 Others Deutsch and Beinker (2019) outline a method to produce synthetic CDS quotes. The idea is to decompose the spread into three components:

ssynth = sEL + sUL + sL,

20 where sEL denotes the expected loss, sUL the unexpected loss and sL the liquidity premium. The expected loss component would be estimated using historical defaults and rating data (and does thus reflect real-world probabilities). The unexpected loss is ba- sically a premium accounting for the cost of holding capital against unexpected losses. The liquidity premium represents an additional surcharge often present in markets with low liquidity. The authors acknowledge that sUL in particular is difficult to calibrate. An additional weakness is that sEL is based on rating data and is thus inherently slow in responding to changed perceptions of an entity’s creditworthiness. With this method, hedging becomes even more complex, since one is not using any CDSs when constructing the proxy curve.

21 Chapter 4

Interest rate modelling

In this chapter, we will recall some important definitions and properties regarding the modelling of interest rates. Thus, we will also introduce the notation used going forward. As we will see later, credit modelling in intensity models bear much resemblance with standard reduced form modelling. For a comprehensive overview of interest rate modelling, see Brigo and Mercurio (2006). We will also need some of the concepts introduced here when valuing credit default swaps in Chapter 6.

4.1 Bank account and discount factor

Let us first consider the bank account (or money market account). This account isthe representation of a risk-free investment, where the investment accrues continuously at the risk-free interest rate.

Definition 4.1 (Bank account). Let B(t) be the value of the bank account at time t ≥ 0. Assume B(0) = 1 and let the bank account evolve according to the differential equation:

dB(t) = rtB(t)dt, B(0) = 1 where rt is a (possibly stochastic) function of time. Consequently,

Z t B(t) = exp rsds 0

This definition tells us what a unit investment at time 0 is worth at a future time t. rt is the instantaneous interest rate at which the bank account grows. This rate is usually referred to as the short rate. It is natural to ask, what is one unit of currency at time T worth at an earlier time t? The answer to this question is the discount factor, which is the number of currency units at t ”equivalent” to one unit of currency at T .

Definition 4.2 (Stochastic discount factor). The stochastic discount factor between two times t and T , t ≤ T is given by:

B(t) Z T D(t, T ) = = exp − rsds . B(T ) t

22 4.2 Zero bonds and zero rates

We will now introduce a fundamental building block in interest rate theory.

Definition 4.3 (Zero-coupon bond). A zero-coupon bond maturing at T is a contract paying the holder one unit of currency at T , with no intermediate coupon payments. The contract value at an earlier time t is denoted Z (t, T ). Furthermore, Z (T,T ) = 1 for all T ≥ 0.

Now, what is the relationship between the discount factor and the price of a zero- coupon bond? It is important to realise the difference between an ”equivalent currency amount” and a ”price of a contract”. If the short rate r is deterministic, it follows that D is deterministic and consequently D(t, T ) = Z (t, T ). But if rates are stochastic D(t, T ) is a random variable at t, which is dependent on the evolution of r between t and T . However, Z (t, T ) being the price of a contract must be known at t. Under a particular probability measure,1 Z (t, T ) is actually the expectation of D(t, T ). We need to consider how to define the length of intervals of time. Let us say thatwe want to define the amount of time between the present time t and some maturity time T . Defining the amount of time as the difference T −t clearly only makes sense if t and T are real numbers corresponding to two time instants. If t and T refer to two dates, we need to define a measure, converting the distance between the two dates to a real number. We do this by means of a year fraction:

Definition 4.4 (Year fraction). The amount of time between two dates t and T is determined by the time measure δ (t, T ), usually called a year fraction. The year fraction is associated with a day-count convention.

As an example, the year fraction according to the day-count convention ACT/360 is calculated as follows:

• Let D1 and D2 be two dates.

• Let Days(D1,D2) denote the actual number of days between D1 and D2.

• The year fraction between D1 and D2 is δ (D1,D2) = Days(D1,D2)/360.

Zero-coupon bond prices are fundamental quantities in interest rate modelling. All rates can be defined in terms of ZCB prices. To go from ZCB prices to interest rates,or vice versa, we need to know the day-count convention and the compounding type. We will now look at two different rates, both defined using ZCB prices.

Definition 4.5 (Continuously compounded zero rate). The continuously compounded zero rate (or spot rate) at time t, for maturity T is the constant rate for which the amount Z (t, T ) accrues (continuously) to yield one unit of currency at T :

Z (t, T ) eR(t,T )δ(t,T ) = 1 ln Z (t, T ) R(t, T ) = − δ (t, T ) 1The equivalent (risk-neutral) martingale measure.

23 Definition 4.6 (Simply compounded zero rate). The simply compounded zero rate at t for maturity T is the constant rate for which the amount Z (t, T ) accrues to yield one unit of currency at T , where the accrual occurs proportional to time:

Z (t, T ) (1 + L(t, T )δ (t, T )) = 1 1 − Z (t, T ) L(t, T ) = δ (t, T ) Z (t, T )

The notation L(t, T ) is often used since the LIBOR rates are simply compounded. Other common compounding methods are k-times-per-year and annual compounding. The reader is referred to Brigo and Mercurio (2006) for definitions of these.

4.3 Forward rates

We will know continue by looking at forward rates. A forward rate is simply a rate that can be locked in today, for an investment over a future time period. Thus, a forward rate is characterised by three dates: the time t for which the rate is prevailing, the period start T and period end S. Forward rates are set consistently with the current discount factor term structure.

Definition 4.7 (Simply compounded forward rate). The simply compounded forward rate at t, for the period starting at T and ending at S, where t < T < S is defined by 1 Z (t, T ) F (t; T,S) := − 1 . δ (T,S) Z (t, S)

We are also interested in the forward rate for an infinitesimal period [T,S]. This quantity is the instantaneous forward rate.

Definition 4.8 (Instantaneous forward rate). The continuously compounded instantaneous forward rate prevailing at t for maturity T > t is defined as ∂ ln Z (t, T ) f(t, T ) := lim F (t; T,S) = − , S→T + ∂T from which it follows Z T Z (t, T ) = exp − f(t, u)du . t

24 Chapter 5

Reduced form credit modelling

In Chapter 3 credit modelling was introduced. This chapter will consider reduced form modelling in greater detail. First default and survival probabilities are defined. Then, we introduce the Poisson process, the random process underlying reduced form models. In Section 5.3 we introduce some prototypical instruments, that we can use to price derivatives such as credit default swaps (the subject of Chapter 6). The chapter ends with some remarks on the similarities between interest rate and reduced form credit modelling.

5.1 Default and survival probabilities

One might of course have an intuitive notion of what probability of default is. We will now try to define it rigorously. In the rest of this thesis, we will mostly study survival probabilities rather default probabilities, since the survival probability is the quantity most natural to model in intensity models. We will work on a filtered probability space, (Ω, F, (Ft)t≥0, P), where Ω is the sample space, i.e. the set of all possible outcomes. F is a σ-algebra whose elements are the events 1 we want to consider, (Ft)t≥0 is the filtration representing the information structure. P is a probability measure assigning probabilities to the events in F. Note that were not using P or Q in this section and the formulas below are agnostic to the choice of measure. Our task is to model the arrival risk of a credit event. Thus, we need to model an unknown, random point in time τ ∈ R+. If we also want to model events that might never occur, we need to include ∞ and thus τ ∈ R+ ∪ ∞. However, this is not enough. The random time τ needs to be linked to the way information is revealed to us through the filtration (Ft)t≥0. In particular, we want τ (the time of some event), to have the property that at the time of the event it is known that the event has occurred. This leads us to the definition of a stopping time:

Definition 5.1 (Stopping time). Let τ be a random variable defined on the filtered probability space (Ω, F, (Ft)t≥0, P). Then, τ is a stopping time if:

{τ ≤ t} ∈ Ft, for all t ≥ 0.

This simply means that, for any time t, we can given the information available at t say whether or not τ has occurred. Or put differently, we can observe the event at the time

1 In particular, Ft is the information available at t. Also, random variables whose realisation is known at t are said to be measurable w.r.t. Ft.

25 it occurs.2 To represent a stopping time as a stochastic process, we define the indicator process, that jumps from zero to one at the stopping time:

Nτ (t) := 1{τ≤t}.

In our setting of credit modelling, we will use the default indicator function (the previous equation) and the survival indicator function (one minus the default indicator function). We can now proceed to define the survival and default probabilities.

Definition 5.2 (Survival probability). Let τ be a stopping time denoting the time of default, then Q (t) = P [τ > t] = E 1{τ>t} is the cumulative survival probability.

We will also use the notation Q(t, T ), which is the probability of surviving over the interval [t, T ]. Since the default probability is one minus the survival probability, it follows that:

Definition 5.3 (Default probability). Let τ be a stopping time denoting the time of default, then F (t) = P [τ ≤ t] = E 1{τ≤t} is the cumulative default probability.

Sometimes, we want to analyse the default risk over some future interval of time.

Definition 5.4 (Conditional survival probability). The conditional survival probability over the interval [T1,T2] as seen from t is

Q (t, T2) Q (t, T1,T2) = (5.1) Q (t, T1)

The conditional survival probability is thus the probability of surviving over the interval [T1,T2] as seen from t, given survival (no default) until T1. Equation (5.1) can easily be proved using Bayes’ rule for conditional probabilities, noting that surviving to T1 and T2 is the same as surviving to T2. Naturally, it follows from Equation (5.1) that:

Q (t, T2) = Q (t, T1) Q (t, T1,T2) which means that surviving to T2 is the same as surviving until T1 multiplied by the probability of surviving over [T1,T2], given survival until T1.

5.2 The Poisson process

In the family of intensity or reduced form credit models, the default of an entity is modelled as the first jump of a Poisson process, which we will introduce below. We will consider the cases of constant, deterministic time-varying and stochastic time-varying intensity. In particular, we will study the distribution of the first jump time.

2An example of a random time that is not a stopping time is the maximum outdoor temperature during a year. Only at the end of the year one can conclude which day had the highest temperature.

26 5.2.1 Constant intensity A stochastic process representing the number of occurrences of some event over a time interval is called a counting process. Formally:

Definition 5.5. A process {Nt, t ≥ 0} is a counting process if:

1. Nt ≥ 0, N0 = 0

2. Nt is an integer

3. If s ≤ t then Ns ≤ Nt

The process N starts at level 0 and stays at that level until some random time T1 where the process jumps to the new level 1. The process stays at this level, until the next jump at the random time T2 and so on. We refer to the times {Tn; n = 0, 1, 2,...} as the jump times of N. An important counting process is the Poisson process. It can be seen as the jump process analogue of the Brownian motion, since it has independent increments.

Definition 5.6 (Poisson Process). A counting process {Nt, t ≥ 0} is a homogeneous Poisson process with intensity λ > 0 if:

1. N0 = 0

2. The increments Ntk − Ntk−1 are independent for all partitions (i.e. non-overlapping intervals)

3. Nt − Ns ∼ Poisson(λ(t − s)) for t > s As we have indicated above, we will model the default as the first jump of a Poisson process. Thus, we are interested in the time of the first jump. Let τ denote the first jump time. We have: −λt P [τ > t] = P [Nt = 0] = e . (5.2) We see the survival probability has the same structure as a discount factor, where the intensity plays the role of the short rate. Recall that with deterministic and constant rates, we have: Z (0, t) = e−rt. Furthermore, we have: P [τ ∈ (t, t + dt] ∩ τ > t] P [τ ∈ (t, t + dt]] P [τ ∈ (t, t + dt]|τ > t] = = P [τ > t] P [τ > t] P [τ > t] − P [τ > t + dt] e−λt − e−λ(t+dt) = = P [τ > t] e−λt = 1 − e−λdt ≈ λdt

Hence, we see that the probability of default in a small interval dt, given survival until t, is given by λdt. Furthermore, the probability of defaulting in the interval [t, s] is P [s < τ ≤ t] = e−λs − e−λt ≈ λ(t − s) where the approximation is good only for t close to s. We see that the probability of default in the interval is only dependent of the length of the interval and is not dependent on how far away in time the start of the interval is. This stems from the fact that the distribution of the arrival times of the Poisson process possess the memoryless property.

27 5.2.2 Time-varying and deterministic intensity Thus far, we have considered a Poisson process with constant intensity. We will now extend the analysis to allow a time-varying (but deterministic) intensity. Let λ(t) be a positive and (right-) continuous function. Define the quantity Z t Γ(s, t) := λ(u)du, s which we call the cumulative intensity, or hazard function. Now, the time inhomogeneous Poisson process can be defined in the same manner as Definition 5.6 but replacing the third condition with Nt − Ns ∼ Poisson(Γ(s, t)) (Durrett, 2016). If let Mt be a standard Poisson process (i.e. it has intensity one), then for a time inhomogeneous Poisson process Nt it holds that:

Nt = MΓ(t). This means that a time inhomogeneous Poisson process is a time-shifted standard Poisson process. From this relationship, it follows that if N jumps first at t then M jumps first at Γ(t). Since we know that M is a standard Poisson process, we also know that the first jump time is a standard exponential random variable: Γ(t) =: ξ ∼ Exponential(1)

By inverting this, we get an expression for the first jump time: τ = Γ−1(ξ), where ξ is standard exponential. This is a well-known fact of the Poisson process: Trans- forming the time of the first jump of a Poisson process according to its own hazard function gives a standard exponential random variable, independent of all previous processes in the probability space (Brigo and Mercurio, 2006). Now, we have the time inhomogeneous analogue to Equation (5.2):

−Γ(t) − R t λ(u)du P [τ > t] = P [Γ(τ) > Γ(t)] = P [ξ > Γ(t)] = e = e 0 .

Again, we see that this expression is very similar to a discount factor, but with the short rate being replaced by the intensity. Indeed, with deterministic and time-varying short rate, we have: − R t r(s)ds Z (0, t) = e 0 Differentiating the cumulative distribution function yields the density:

∂ ∂ − R t λ(u)du P [τ ≤ t] = − P [τ > t] = λ(t)e 0 . ∂t ∂t Thus, for small dt: − R t λ(u)du P [τ ∈ [t, t + dt)] = λ(t)e 0 dt. Using this, we also have: P [τ ∈ [t, t + dt)] P [τ ∈ [t, t + dt)|τ ≥ t] = = λ(t)dt. P [τ ≥ t] Which means that for arbitrarily small dt, the probability of defaulting in the coming dt years given survival until t is λ(t)dt. In Appendix A.1 we arrive at the same results from a different starting point.

28 5.2.3 Time-varying and stochastic intensity To further generalise the modelling, we now allow the intensity to be stochastic. This type of process is called a Cox process. We assume the intensity to be a Ft-adapted and right continuous and strictly positive process which we denote λt. The cumulative intensity or hazard process is thus: Z t Λ(t) = λsds. 0

That λt is Ft-adapted means that given Ft, the default-free market information up to t, we know the realisation of λ from 0 to t. This also means that the randomness in the intensity is driven by the default free market. As in the deterministic case, we can transform the time of the first jump:

Λ(τ) = ξ ∼ Exponential(1) with ξ being independent of Ft. This means that the time of default is given by

τ = Λ−1(ξ)

In this setting, we have two sources of randomness: the intensity λ and the standard exponential ξ. For this reason, Cox processes are also called doubly stochastic Poisson processes. If we condition λ on Ft (which includes the paths of λ) we obtain the same Poisson structure as if the intensity were deterministic. That is:

P [τ ∈ [t, t + dt)|τ ≥ t, Ft] = λtdt.

The interpretation of this, given t = ”now” is that the probability of default in the (small) dt interval, given survival and the default-free market information so far, is λtdt. We will now derive the survival probability: Z t P [τ ≥ t] = P [Λ(τ) ≥ Λ(t)] = P ξ ≥ λsds 0 t Z h R t i − λsds = E P ξ ≥ λsds Ft = E e 0 0

Also, in this setting, we recognise the similarity with a discount factor. Compare with the stochastic discount factor in Definition 4.2.

5.2.4 Summary We have now looked at three different versions of the Poisson process. We have seen that using Poisson processes (which have exponentially distributed jump times) we end up with expressions for the survival probability that can be interpreted as discount factors, and the default intensities as credit spreads. In Appendix A.2 we take a look at an approach where one does not use the Poisson process, but one instead defines quantities implied from defaultable and risk-free zero- coupon bonds. We arrive at similar expression as for the Poisson approach. When it comes to practical modelling, the constant intensity process is of little use. Deterministic time-varying intensities can be used to capture the term structure of CDS

29 1

0 T 0 τ T (a) No default before T (b) Default (at τ) before T

Figure 5.1: Risky ZCB paying one unit of currency if no default, zero otherwise spreads, which we will see later. Stochastic intensities can be used to model credit spread volatility, something that is beyond the scope of this thesis. We saw that there is a random variable ξ, driving the default process. What is important is that ξ is an external source of randomness, independent of all default-free market information. This makes reduced form models incomplete.

5.3 Building blocks

In this section we will introduce some building blocks fundamental to the valuation of credit risky assets. In particular, these building blocks lets us construct a valuation model for credit default swaps. Since we are intending to value traded instruments we are working under the risk-neutral measure Q. Recall that our target is to obtain risk-neutral default probabilities. First we will take a look at the general structure and payoffs. Then we will look at the pricing of these using the tools outlined in the previous section. Then, we make some simplifying assumptions and consider the pricing expressions again.

5.3.1 Introducing the building blocks Risky zero-coupon bond (no recovery) Consider a zero-coupon bond with a face value of one unit of currency maturing at T . Let us further assume that the issuer of this bond can default and in case this happens, we receive nothing (zero recovery). The payoff for this instrument is shown in Figure 5.1. The present value under the risk-neutral measure is the discounted expected payoff:

h R T i ˆ Q − rsds Z (0,T ) = E e 0 1{τ>T } (5.3) where rs is the short rate. We use ˆ· to denote credit risky quantities. To price this bond, we need to know the probability that τ < T , but we do not need to know τ. Equation (5.3) is a generic formula that does not assume any model for the default time. It also allows for co-dependence between interest rates and default times.

Unit payment at default We will now consider another basic credit structure. This is an instrument that pays one unit of currency at the time of default τ, if default occurs before the maturity T . We will see that this instrument is useful when considering accrual payments in credit default swaps. The payoff is depicted in Figure 5.2. The price of this instrument is givenby:

h R τ i Q − rsds U (0,T ) = E e 0 1{τ≤T }

30 1

0 T 0 τ T (a) No default before T (b) Default (at τ) before T

Figure 5.2: Instrument paying one unit of currency if default before T , zero otherwise

5.3.2 Pricing We will derive the pricing formulas in the most general model considered in this thesis: the reduced form model with stochastic intensity. Model the short rate rs and the intensity λs as potentially correlated stochastic processes. Starting from Equation (5.3), using the law of iterated expectations and condi- tioning on the default-free market information FT , we obtain h R T i ˆ Q − rsds Z (0,T ) =E e 0 1{τ>T } (5.4) h h − R T r ds ii Q Q 0 s =E E e 1{τ>T } FT .

If we look at the inner expectation, we see that since FT is the default-free market information until T , the short rate rs is measurable with respect to FT for 0 ≤ s ≤ T . Furthermore, since the intensity process λt is Ft-adapted, we have: h − R T r ds i − R T r ds − R T (r +λ )ds Q 0 s 0 s Q 0 s s E e 1{τ>T } FT = e E 1{τ>T } FT = e Substituting this back into Equation (5.4), we have:

h − R T (r +λ )dsi Zˆ (0,T ) = EQ e 0 s s (5.5) This last equation clearly shows why we can interpret the intensity as credit spread. It is a quantity added to the short rate and is therefore a spread over the rate. We can now turn to the instrument paying one unit of currency at default, if default occurs before the maturity T . Determining the value of this instrument is a bit trickier than for the risky ZCB. For non-zero interest rates, we need the full probability density distribution for the default time and not just the cumulative distribution. Using the same line of reasoning as above and integrating over the density of default times until T gives (O’Kane, 2008): T Z R t Q − (rs+λs)ds U (0,T ) = E λte 0 dt (5.6) 0

5.3.3 Pricing under simplifying assumptions To progress in our analysis, we will now introduce two simplifying assumptions common in practice. Using either of these assumptions, we can further simplify the expressions in Equations (5.5) and (5.6). Assumption 5.1. Assume that the hazard rate and the risk-free short rate are independent. Assumption 5.2. Assume that the hazard rate is deterministic. Of course, Assumption 5.2 implies that the hazard rate and the risk-free short rate are independent.

31 Independence Assumption 5.1 lets us separate the price of the zero recovery ZCB into two factors. Note that we still allow the hazard rate to be stochastic.

h − R T r +λ dsi Zˆ (0,T ) = EQ e 0 s s

h − R T r dsi h − R T λ dsi = EQ e 0 s EQ e 0 s (5.7) = Z (0,T ) Q (0,T ) .

We see that the price of the risky ZCB with no recovery is the product of the risk-free ZCB price and the survival probability. For the price of a unit payment at default we have T Z R t Q − rs+λsds U (0,T ) = E λte 0 dt 0 T Z h R t i h R t i Q − rsds Q − λsds = E e 0 E λte 0 dt (5.8) 0 Z T = − Z (0, s) dQ (0, s) . 0 The major benefit of the independence assumption is that we can price both the building blocks if we know the term structures of Z (0,T ) and Q (0,T ).

Deterministic hazard rate If now assume that the hazard rate is deterministic, the pricing formulas simplify even further. The price of the risky ZCB becomes

− R T λ(s)ds Zˆ (0,T ) = Z (0,T ) e 0 , and the price of a unit payment at default is

Z T − R t λ(s)ds U (0,T ) = Z (0, t) λ(t)e 0 dt. 0

5.4 Interest rate versus credit modelling

As seen in Section 5.2.3, the expression for the survival probability with stochastic intensity is very similar to the definition of the stochastic discount factor. This is not theonly similarity between interest rate and credit modelling. Indeed, the similarities are deeper, allowing us to use the ”interest rate toolbox” for credit modelling, as suggested by Brigo and Mercurio (2006). In Table 5.1, some of the analogies are listed.

32 Interest rate Credit Discount curve Survival curve Short rate rs Hazard rate (intensity) λs No-arbitrage rs > 0 No-arbitrage λs > 0 h − R t r dsi h − R t λ dsi Discount factor Z(t) = E e 0 s Survival probability Q(t) = E e 0 s Zero rate z(t) = −(1/t) ln Z(t) Zero default rate z(t) = −(1/t) ln Q(t) Forward rate f(t) = −Z(t)−1∂Z(t)/∂t Forward default rate h(t) = −Q(t)−1∂Q(t)/∂t

Table 5.1: Analogies between interest rate and credit modelling. Adapted from O’Kane (2008).

33 Chapter 6

Credit Default Swaps

Credit Default Swaps (CDS) are the most important instrument for hedging and trading default risk (Deutsch and Beinker, 2019). It is also the most liquid single-name credit derivative and is in turn a building block for other credit derivatives. In this chapter, we will first look at the structure of the credit default swap. Then we will look at a valuation model for CDSs, which will be used when calibrating credit curves.

6.1 Basic structure

The workings of the CDS is tightly coupled with legal terms. Even the most fundamental concepts, such as default, must be properly defined. To minimise ambiguity and legal risk, the market has adopted a set of definitions determined by ISDA. The standardisation of the credit derivatives market is seen as one of the main reasons for the significant liquidity in the market (O’Kane, 2008). One of the effects of the ISDA definitions is that the market has developed astandard CDS contract. The standard CDS is a contract with a set of features viewed as the most common. In this chapter, we will describe CDSs of the standard form. The CDS is more of an insurance policy than a swap. The buyer of the swap buys insurance against default for a specific issuer or company, referred to as the reference entity or name. The protection buyer regularly pays a fixed coupon (the CDS premium or CDS spread) to the protection seller. Buying (going long) protection means going short credit risk and vice versa. The basic structure is depicted in Figure 6.1. The premium leg refers to the stream of cash flows from the protection buyer tothe seller. The protection leg is the payment from the protection seller to the buyer, given that default happens during the lifetime of the swap. If the reference entity defaults, the CDS ends and the buyer receives a compensation payment. This payment can either be a fixed, pre-determined amount (for a Digital CDS) or (more commonly) the LGD amount of some reference bond. Note that there is no need

Premium leg Protection buyer Protection seller Protection leg

Figure 6.1: Basic structure of a credit default swap

34 (1 − R)

Protection leg

t t t t τ t0 1 2 3 ··· n tN S0δτ S δ S δ S δ S δ 0 1 0 2 0 3 0 n Premium leg

Figure 6.2: The cash flows of a CDS, from the perspective of the protection buyer. for the protection buyer to actually own the underlying asset (as in traditional insurance), meaning that a CDS can be bought for speculative purposes.

6.1.1 Big and small bang In 2009 the standard CDS market conventions changed, following two protocols introduced by ISDA. These protocols are known as the big bang and small bang protocols. The purpose was to streamline the CDS market and thus paving the way for central clearing (Green, 2015). The main change of interest for this thesis is that the protocols introduced fixed spreads (coupons). In the European market, single-name CDS trades with fixed spreads of 25bp, 100bp, 500bp or 1000bp. To make a new contract fair at initiation, the value is adjusted with an upfront payment in either direction. Note that it is possible to convert this type of quotation (expressed as the coupon and the upfront payment) to the old type with a ”floating” spread.

6.2 Valuation

In this section we will introduce a valuation model for credit default swaps based on O’Kane (2008). This model will be used for curve calibration. We will leverage results and notation introduced in previous chapters. The valuation model assumes independence of interest rates and the default time, meaning that the pricing formulas can be expressed in terms of a risk-free discount curve and survival curve. O’Kane (2008) emphasises that this using this assumption is market practice. We consider only new contracts, i.e. there is no accrued premium to consider. This is sufficient, since we are only interested in a valuation model to be used in curve construction, for which we will only have new contracts (that is what is quoted). We will use the following notation: t The effective date of the CDS T The CDS maturity date tn The premium payment dates, n ∈ {1,...,N}. Typically, tN = T S0 The contractual spread agreed at time 0 R The expected recovery rate δi Shorthand notation for δ (ti−1, ti) (i.e. the day count fraction) We also have Z (t, T ), Q (t, T ), δ (t, T ) as before. The cash flows are shown in Figure 6.2. We will now consider the valuation of the two legs separately.

35 6.2.1 The premium leg We split the valuation of the premium leg into two parts. First, there are the regular, scheduled, coupon payments which are paid conditional on the survival of the reference entity. Secondly, if there is a credit event, there is a payment of accrued interest for the period from the last premium payment until the default. In Section 5.3 we introduced some of the building blocks we need now. Recall the price of a defaultable ZCB, under the assumption of independence between rates and default probabilities (Equation (5.7)):

h R T i ˆ Q − rsds Z (t, T ) = E e t · 1{τ>T } = Z (t, T ) Q (t, T ) .

Let S0 denote the CDS spread. The value of the scheduled premium payments becomes: N X S0 δ (tn−1, tn) Q (t, tn) Z (t, tn) , (6.1) n=1 which is the sum of the premium payments, weighted by the probability of surviving until the payment date and discounted to today. Now we need to consider the accrued interest payment. If there is a default during the protection period of the CDS, the protection buyer pays the interest that has accrued from the last scheduled premium payment to the default date. This amount is thus contingent on the default time. The value at valuation date t, for a unit payment paid at default in the infinitesimal interval [s, s + ds] is given by Z (t, s)(−dQ (t, s)) , which is obtained by differentiating Equation (5.8). The accrual payment to be made depends on when in the premium period default occurs. If the default occurs at tn−1 < τ < tn, then the protection buyer needs to pay:

S0δ (tn−1, τ) .

We can now state the expected present value at t of the accrued premium in the n:th period: Z tn S0 δ (tn−1, s) Z (t, s)(−dQ (t, s)) . tn−1 Summing over all periods gives the expected present value of the accrued premium:

N X Z tn S0 δ (tn−1, s) Z (t, s)(−dQ (t, s)) . (6.2) n=1 tn−1 Combining Equations (6.1) and (6.2) gives the value of the premium leg:

" N X Vpremium(t) =S0 δ (tn−1, tn) Q (t, tn) Z (t, tn) n=1 (6.3) N # X Z tn + δ (tn−1, s) Z (t, s)(−dQ (t, s)) n=1 tn−1 For calibration purposes, we might want to avoid the potentially computationally expensive integral in the accrual term. If we assume that on average, default will occur

36 midway through a premium period, the accrued premium at default is S0δ (tn−1, tn) /2. Furthermore, the probability of defaulting in the n:th period is Q (t, tn−1) − Q (t, tn). Let us also take the discount factor associated with the end point of the period. Then we have: Z tn 1 δ (tn−1, s) Z (t, s)(−dQ (t, s)) ≈ δ (tn−1, tn) Z (t, tn)(Q (t, tn−1) − Q (t, tn)) . (6.4) tn−1 2

Plugging Equation (6.4) into Equation (6.3) and simplifying gives:

N S0 X V (t) = δ (t , t ) Z (t, t )[Q (t, t ) + Q (t, t )] . (6.5) premium 2 n−1 n n n−1 n n=1

6.2.2 The protection leg The protection leg is the payment of the par minus the recovery rate following a default. The quantity paid is uncertain in reality, but here we will take it as constant. In Sec- tion 5.3 we looked at an instrument paying one unit of currency at default. Recalling Equation (5.8) we have:

Z T Vprotection(t) = (1 − R) Z (t, s)(−dQ (t, s)) t The integral can be performed numerically by discretising the interval [t, T ] into K intervals of equal lengths. The intervals have length = (T − t)/K. Consequently, we have

K X Vprotection(t) = (1 − R) Z (t, k)(Q (t, (k − 1)) − Q (t, k)) . (6.6) k=1 O’Kane (2008) points out that we need a very large K to achieve acceptable accuracy. However, by taking the average of the lower and upper bounds of Equation (6.6), acceptable accuracy can be attained with weekly or even monthly time steps. The present value of the protection leg then becomes

K 1 − R X V (t) = [Z (t, (k − 1)) + Z (t, k)] [Q (t, (k − 1)) − Q (t, k)] . (6.7) protection 2 k=1

6.2.3 Full value and breakeven spread The total value for a long protection position, with unit face value is:

VCDS(t) = Vprotection(t) − Vpremium(t) (6.8) with the value of the legs given by Equations (6.5) and (6.7). The breakeven spread is the spread S0 making the present value zero at initiation. That is, the breakeven spread is the value of S0 making VCDS(t) = 0. Therefore, we have

(1 − R) PK [Z (t, (k − 1)) + Z (t, k)] [Q (t, (k − 1)) − Q (t, k)] S = k=1 . 0 PN n=1 δ (tn−1, tn) Z (t, tn)[Q (t, tn−1) + Q (t, tn)]

37 Chapter 7

Model

In this chapter we introduce the two models examined in the thesis. Recall that our goal is to obtain proxy credit curves for entities without liquidly traded CDSs. First, we outline a method for obtaining a credit curve from a set of CDS quotes. Then, we present the proxy models: the Nomura model and a proposal for an altered model.

7.1 Constructing the survival curve

Constructing the survival curve, i.e. the graph T −→ Q(0,T ), is in essence a process of constructing the full term structure of survival probabilities from a finite number of CDS quotes. In theory there are an infinite number of ways of doing this. Thus, to beable to choose a method, we must first consider which properties are desirable for a survival curve.

7.1.1 Desirable properties There are a number of properties generally considered desirable for both interest rate and credit curves. Note that we use credit curve interchangeably with survival curve.

Re-pricing of calibration instruments: We want the curve to be able to re-price the provided instruments exactly. This means that solving for the break-even spread using the curve yields the quoted spread.

Sensible inter- and extrapolation: Since one only have quotes for tenors with considerable gaps between them, the method should give sensible values for times in between the quoted tenors. Furthermore, the curve should also provide sensible values for times after the longest quoted tenor.

Localness: One wants the curve the be local. This refers to how the curve behaves when the input quotes are perturbed. That is, if one shifts the quote for the 5Y tenor, one wants the curve to shift only in the neighbouring region. For curves possessing this property, it is easier to understand model implied hedges and P&L movements.

Smoothness: One usually wants the curve to be smooth, both in the discount factor, zero rate and forward rate sense. Usually there is a trade-off between localness and smoothness.

38 Absence of arbitrage: The curve construction method should yield a curve free from arbitrage.

Speed: The speed of the curve construction is especially important in credit modelling. One may need to build one curve per counterparty, meaning that several hundreds or even thousands of curves needs to be calibrated. Thus, there is a need for the curve construction method to be fast.

7.1.2 Interpolation As explained in the beginning of this section, there are many ways to construct a curve from a limited set of calibration instruments. An important aspect to consider is the interpolation scheme. Usually the curve construction relies on defining a ”skeleton” of points to interpolate between. These points are sometimes called the pillars of the curve. These will correspond to the maturities of the calibration instruments. That is, if we use CDS maturing in 1,2,3,5,7 and 10 years, our pillars will be placed at these times. There are two important choices to make regarding the interpolation: What to interpolate on (the interpolation quantity) and the interpolation method. Different combinations will yield different curve properties. For an overview see Hagan and West (2008), which is set in the interest rate world but applies to intensity-based credit curves as well.

The chosen scheme A common choice of interpolation scheme is to interpolate linearly on the logarithm of the survival probability. This approach is taken by O’Kane (2008) and Green (2015) and is a standard choice in credit modelling. This choice has a nice property as we will see. ∗ The standard formula for linear interpolation for a time t ∈ [tn−1, tn] is:

∗ ∗ (tn − t )f(tn−1) + (t − tn−1)f(tn) f(t∗) = . (7.1) tn − tn−1

Another interpolation method, particularly common in interest rate modelling, is spline interpolation. Cubic splines for example produce very smooth curves and in particular, the forward rate curve is smooth. Since one is primarily interested in the survival probability and not e.g. the three-month default rate (cf. discount factor and three-month forward rate), using splines is not necessary. Recall that if the hazard rate is taken to be deterministic the survival probability is given by Z t Q (t) = e−Γ(t) = exp − h(s)ds . 0 In the last equation, we introduced h(t). This is the hazard rate or instantaneous forward default rate. We change the notation from λ(t) to emphasise that this is a hazard rate extracted from market quotes. If we set the interpolation quantity to

f(t) = − ln Q (t) , (7.2) it follows that Z t f(t) = h(s)ds. 0

39 We notice that this implies that df(t) = h(t). dt Hence, using Equation (7.1):

∗ ∗ df(t ) f(tn) − f(tn−1) h(t ) = ∗ = (7.3) dt tn − tn−1

The main takeaway of the last equation is that the interpolated intensity h(t∗) is independent of the time t∗. This means that interpolating linearly on the log of the survival probability is equivalent to assuming that the hazard rate h(t) is piecewise constant. Let Ti denote the pillar dates (i.e. CDS maturities). Define the hazard rate segments hi as h(t) = hi for t ∈ [Ti−1,Ti).

Furthermore, let β(t) be the index of the first Ti larger than t. Then we have the following expression for the cumulative hazard rate (hazard function):

β(t)−1 Z t X Γ(t) = h(s)ds = (Ti − Ti−1)hi + (t − Tβ(t)−1)hβ(t) 0 i=1

For clarification, we list a few examples:

−h1t Q(t) = e for t ∈ [0,T1)

−(h1(T1−T0)+h2(t−T1)) Q(t) = e for t ∈ [T1,T2)

−(h1(T1−T0)+h2(T2−T1)+h3(t−T2)) Q(t) = e for t ∈ [T2,T3)

For the pillar dates we have:

j Z Tj X Γj := h(s)ds = (Ti − Ti−1)hi 0 i=1

With the notation used the hazard rate segment hi will be associated with the CDS maturing at Ti. Note that this does not mean that hi is determined solely by the Ti-CDS, hi will depend on all CDSs maturing before Ti. From the above, we can also conclude that for t ∈ (Tn−1,Tn):

−hn(t−Tn−1) Q(t) = Q(Tn−1)e

We also have (by plugging Equation (7.2) into Equation (7.3)) for t ∈ (Tn−1,Tn):

1 Q(T ) h(t) = ln n−1 . Tn − Tn−1 Q(Tn)

7.1.3 Bootstrap Bootstrapping refers to the iterative process to find a curve that re-prices the calibration instruments. Starting from the shortest dated instrument, one finds the value1 to be added to the curve that re-prices the calibration instrument currently considered. That is, for each instrument we solve for one parameter determining how to extend the curve

40 Algorithm 1: Credit curve bootstrap Input : n CDS with spreads S1,...,Sn and maturities T1,...,Tn. An expected recovery rate R. Output: A survival curve consisting of times 0,T1,...,Tn associated with values 1,Q(T1),...,Q(Tn). Survival probabilities for intermediate times are interpolated according to a chosen scheme.

Initialise the curve with Q(T0 = 0) = 1.0 for i = 1 to n do For CDS i with spread Si and maturity Ti, solve for the value of Q(Ti) that gives the CDS zero present value using Equation (6.8). Any intermediate survival probabilities (for t < Ti) needed are interpolated given the interpolation scheme and the already determined values Q(T0),...,Q(Ti−1). When Q(Ti) is found, add it to the curve.

to the next maturity point. This process is described in Algorithm 1. Descriptions of the bootstrap method is found in e.g. Schönbucher (2003) and O’Kane (2008). Solving for the survival probability is done numerically using a one-dimensional solver. In this thesis, Brent’s method is used. We will use the bootstrapping algorithm to get the survival curve T → Q(0,T ) both for liquid names and to get proxy curves. Each survival probability added to the curve will depend on the survival probabilities already fitted. This is because valuing a CDS requires knowledge of the survival probability at all times until maturity, not just at the maturity. The intermediate survival probabilities are interpolated from the survival probabilities already calibrated, for shorter dated instruments. Thus, the bootstrap process ensures that all calibration instruments are re-priced and that the pricing is consistent. With consistent pricing we mean that a cash flow two years into the future has the same present value, independent ofwhether that cash flows belongs to e.g. a 3Y or 5Y CDS.

7.2 Proxy models

We will compare two models, or rather two algorithms, for constructing credit curves for illiquid entities. These curves are what we will call proxy curves. In Section 3.5.4 we introduced the Nomura model, also known as the cross-section method. We will call it the Nomura model going forward. This model is common in practice and fulfils the regulatory requirements that the proxy spread should reflect region, sector and rating. Thus, it is appropriate to examine further. We note that the original paper (Chourdakis et al., 2013) in essence describes a method to obtain a proxy CDS spread for a single tenor. This means that one needs to extend this model to be able to obtain default (or survival) probability curves. Green (2015) points out that the typical implementation is to run the regression model for each tenor separately to obtain the proxy spreads associated with each tenor. Then one uses a standard bootstrapping algorithm to obtain the proxy curves, where one typically makes the standard modelling choice of piecewise constant hazard rates.

1For credit curves the survival probability and for interest rate curves the discount factor.

41 Recall that in the Nomura model the proxy spread is given by:2

proxy Si = MglobMsctr(i)Mrgn(i)Mrtng(i), (7.4) meaning that the proxy spread is given as the product of four factors. Chourdakis et al. (2013) provides no argument to why using a multiplicative instead of an additive model. Thus, we will also study the additive formulation:

proxy Si = Mglob + Msctr(i) + Mrgn(i) + Mrtng(i). (7.5)

One should note that each factor takes different values depending on the properties of the entity. That is, each factor is described by a number of parameters. As an example: There is a parameter for rating BBB, belonging to the rating factor. The model parameters are estimated using OLS linear regression using CDS quotes for liquid entities. We argue that this method has two weaknesses:

1. The regression model is ill-posed.

2. Calculating the proxy spread for each tenor independently means that important information about the term structure is lost, potentially leading to calibration issues.

The first point refers to the fact that the Nomura regression model is subject tothe so called dummy variable trap. The regression model includes both a constant term (i.e. the global factor) and an exhaustive set of dummies (meaning that there is one dummy variable for each category in each dimension). Hence, multicollinearity occurs, and the equation system does not have a unique solution.3 There are two options to overcome this problem: exclude the constant term or drop one dummy variable per dimension. The latter means that one needs to define a reference level, towards which the regression coefficients should be compared. In this thesis, we opt for choosing a reference category in each factor, thereby introducing a reference level. We choose not to implement the standard Nomura model (i.e. as described in Chourdakis et al. (2013)), since the ill-posed regression model leads to numerical errors when implemented, meaning that the proxy curves cannot be produced. Together with the additional steps to obtain full proxy curves we can now outline our implementation of the Nomura model in Algorithm 2. The description is for the multiplicative model (Equation (7.4)), but the alterations needed for the additive version (Equation (7.5)) should be obvious. In Line 2 of the algorithm description a dummy variable matrix is created. A sample matrix is shown in Table 7.1. We will now comment to the second point above. We see that the Nomura model handles each tenor separately. For each tenor, we derive a proxy spread. Then a proxy curve can be constructed using the proxy spreads for a combination of properties. The proxy spread is a kind of average of all spreads for that tenor. We suspect that there is important term structure information encoded in the prices for a single entity and that that information is lost when the regression in performed on a tenor-by-tenor basis. Hence, we propose an alternative model. The main idea is to try to keep the term structure information from the liquid entities when creating proxy curves. To accomplish this, we change the modelling quantity from the CDS spread to the hazard rate. We

2Here we have dropped the Seniority factor, since it is not analyzed in the thesis. We only consider senior unsecured debt. 3Solving the equation system involves calculating the inverse of a matrix. With perfect multicollinearity the matrix is singular and can thus not be determined. However, some numerical implementations will fail to recognise that the matrix is singular and provide a solution anyways. Thus, one needs to be careful.

42 Algorithm 2: Nomura proxy method Input : N CDSs with spreads S1,...,SN and maturities T1,...,TN for M liquid entities. Output: Proxy survival curves for all combinations of Region, Sector and Rating.

1 Construct a discount curve Decide on a reference level to be used in the regression. Enumerate the remaining parameters p ∈ {1,...,P }. for Tenor t =∈ {1,...,T } do | Create a vector y = [ln(S1),..., ln(Sn)] holding the logarithm of the quoted spreads for all CDSs with tenor t. 2 Create a matrix A of dummy variables. Rows correspond to entities and columns to parameters. | Using OLS regression, find the optimal β = [x1, . . . , xP ] for the model y = Aβ. Now, Mp = exp(xp) foreach combination of parameters c do ∗ Calculate a proxy spread Sc = MglobMrgn(c)Msctr(c)Mrtng(c) ∗ Create a proxy CDS with tenor t and spread Sc 3 Append proxy the instrument to a list

foreach combination c of factors do Collect the proxy instruments from the list updated in Line 3 with the properties c. Bootstrap the credit curve using Algorithm 1 for the proxy instruments using the discount curve generated in Line 1. return the proxy curves

Global Region Sector Rating

Ticker Global Africa Asia E.Eur India Lat.Amer MiddleEast N.Amer Basic Materials Consumer Goods Consumer Services Energy Government Healthcare Industrials Technology Telecom Services Utilities A BBB BB B CCC AAUK 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 ABX 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 ACAFP 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ACCOR 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 ADCB 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 ADECGRO 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 ADLEREA 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 AEGON 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 AEP 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 AES 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0

Table 7.1: Sample dummy variable matrix

43 extract the hazard rate curve on an entity-by-entity basis for the liquid names, and then perform the regression on the hazard rates. Thereafter, proxy credit curves are easily constructed from proxy hazard rates. To clarify the point regarding keeping the term structure data: Recall that when a credit curve is bootstrapped, each point4 in the curve is dependent on the points preceding it. This means that the hazard rate at the 10Y point is dependent actually dependent on the quoted spread for the 5Y CDS (and any other with CDS maturity shorter than 10Y). In the altered model, the proxy hazard rate for time t and an entity i is given by:

proxy hi (t) = Mglob(t)Msctr(i)(t)Mrgn(i)(t)Mrtng(i)(t). (7.6)

This expression is indeed similar to Equation (7.4), with the main difference that the left-hand side is now a hazard rate. We will also examine an additive version:

proxy hi (t) = Mglob(t) + Msctr(i)(t) + Mrgn(i)(t) + Mrtng(i)(t). (7.7)

The hazard rate and the CDS spread are related. One can show, using simplifying assumptions, that (Brigo and Mercurio, 2006):

S = γ(1 − R). (7.8)

This expression relates the spread for a single CDS to a constant hazard rate γ. R is the recovery rate. Equation (7.8) shows that the CDS spread and the hazard rate are related with a linear relationship. Thus, the proposed alternative model in Equation (7.6) is not totally unfounded. When implementing the alternative model, we need to choose a number of dates for which to calculate the proxy hazard rate. Since CDSs are quoted for a fixed set of tenors, with fixed start and maturity dates (the ”CDS dates”) it is natural to choose thesedates as the model dates. Furthermore, since we opted for a piecewise constant interpolation scheme, there is no extra information obtained by choosing the dates differently. The implementation of the alternative model is outlined in Algorithm 3. Again, the description is for the multiplicative case (Equation (7.6)), but the amendments needed for the additive model (Equation (7.7)) are self-explanatory. We use the same terminology as for the Nomura model. In particular: the values of the model factors are given by a number of parameters. The dummy variable matrix here is also of the type displayed in Table 7.1.

4I.e. the survival probability at a pillar date

44 Algorithm 3: Alternative proxy method Input : N CDSs with spreads S1,...,SN and maturities T1,...,TN for M liquid entities. Output: Proxy survival curves for all combinations of Region, Sector and Rating.

1 Construct a discount curve Decide on a reference level to be used in the regression. Enumerate the remaining factors p ∈ {1,...,P }. for Entity e ∈ {1,...,M} do Bootstrap the survival curve Qe(t) using Algorithm 1 and the discount curve constructed in Line 1. if Qe(t) contains arbitrage (i.e. negative intensity) then Discard the curve and continue with next entity e From Qe(t), obtain the hazard rate function he(t). Extract the hazard rate at discrete points in time: he = [h(t0), h(t1), . . . , h(tn)] | Construct the matrix H = [h1,..., hM ] . Each row corresponds to an entity and each column to a date. foreach Column hc in H do Create the vector y = ln(hc), holding the logarithm of the hazard rate. Create a matrix A of dummy variables. Rows correspond to entities and columns to factors. | Using OLS regression, find the optimal β = [x1, . . . , xP ] for the model y = Aβ. Now, Mp = exp(xp) Store the parameters Mp foreach combination of factors c do foreach point in time t do ∗ Calculate the proxy hazard rate hc = MglobMrgn(c)Msctr(c)Mrtng(c) ∗ Construct a survival curve Qc(t) from the proxy hazard rates hc(t). return the proxy curves

45 Chapter 8

Data

This chapter introduces the data used in the thesis. This data is what the results in the next chapter are based on.

8.1 Credit Default Swap quotes

The most important data for this thesis are CDS quotes alongside with information about the reference names. The data set we use contains the following information for the reference entities: • Ticker • Short name • Region • Sector • Rating For each entity, we have CDS quotes (spreads) for different combinations of: • Currency • Restructuring clause • Seniority (tier) • Term (tenor) In Section 6.1.1, we explained that for the last ten years, CDSs have been traded at fixed coupons (e.g. 500bp) but with an upfront payment. In the data set we have available the quotes have already been converted and are expressed in the old form, i.e. simply as a spread. Before using the data in any model, we ensure that each entity is only included once (but for multiple tenors). That is, the data is filtered to only include USD quotes, for senior unsecured debt and only for a single restructuring clause. The restructuring clause used varies, since what clause is mostly used varies by region. The data set we have available changes over time. The first data is per end of April 2016. Until May 2018, we have monthly data for a smaller set of entities. From May 2018 and onwards, the data includes a larger set of reference names. We have daily data from mid-August 2019 and going forward. For a date in the period with the larger data set, we end up with roughly 600 entities after the filtering described above has been applied. How these entities are distributed over sector, region and rating is shown in Table 8.1.

46 Sector Region AA A BBB BB B CCC Basic Materials Europe 1 2 2 7 4 7 Lat.Amer 6 N.Amer 1 4 2 4 6 5 Consumer Goods Europe 5 8 8 8 6 Lat.Amer 1 N.Amer 1 2 6 6 10 8 Consumer Services Africa 1 Asia 1 Europe 2 4 7 5 9 14 Lat.Amer 1 N.Amer 1 6 11 3 16 16 Energy E.Eur 1 2 Europe 1 3 3 1 Lat.Amer 2 MiddleEast 2 N.Amer 1 1 11 9 Financials Asia 1 E.Eur 5 Europe 1 14 6 18 5 3 India 2 Lat.Amer 2 MiddleEast 2 1 1 N.Amer 4 4 2 10 6 8 Government Africa 1 2 Asia 1 2 2 2 E.Eur 1 3 1 4 1 Lat.Amer 4 5 MiddleEast 1 3 1 1 Healthcare Europe 2 2 1 1 3 MiddleEast 1 N.Amer 7 2 1 3 2 Industrials Asia 1 Europe 1 4 10 13 10 10 Lat.Amer 1 MiddleEast 1 N.Amer 4 6 8 4 9 6 Technology Europe 1 2 2 1 N.Amer 2 1 5 4 5 2 Telecom Services E.Eur 1 Europe 2 6 8 6 3 Lat.Amer 1 MiddleEast 1 N.Amer 2 4 Utilities Africa 1 Asia 1 Continued on next page

47 Sector Region AA A BBB BB B CCC Europe 1 8 3 5 1 MiddleEast 2 1 N.Amer 5 3 1 3 1

Table 8.1: Summary of CDS data per March 31st 2020

We note that some combinations do not have any entities available. That is, there are no liquid quotes available for Asian BBB-rated Consumer Services companies. However, the proxy models studied allow us to calculate a proxy curve for this combination.1

8.2 Interest rates

In this thesis we make no effort to model interest rates in any accurate way. Recall that the CDS valuation model (Section 6.2) requires a discount curve. We set the (continuously compounded) interest rate as constant 1%.

1Note that this is an important difference compared to the intersection model described in Section 3.5.3 The intersection will not give a proxy spread/curve if there is no quote available for a particular bucket.

48 Chapter 9

Results and discussion

In this chapter we present and discuss the results. We investigate the two proxy models presented in Chapter 7 using the data described in Chapter 8.

9.1 Initial results

We introduced two different models for obtaining proxy credit curves together withde- scriptions of their implementation in Section 7.2. The first question we set out to investigate was if there was really any difference in the outcome. As we have noted above, having a factor-based model gives a large number of possible proxy curves. Comparing every single proxy curve is unfeasible and thus we limit ourselves to a few samples. Using data per the 31st of March 2020 we obtain proxy curves using the multiplicative versions of the two proxy models. In Figure 9.1 we display the outcome for two proxy curves and one liquid curve. The proxy curves pertain to European industrials entities rated BBB. The liquid name is Atlas Copco AB (which shares properties with the proxy curves). The markers on the survival probability curves mark the pillar dates (i.e. CDS maturities). The differences between the two proxy curves are minor, in particular when looking at the survival probability. The difference in hazard rate is somewhat more pronounced. Recall that the survival probability is a function of the integral of the hazard rate. Thus, it is natural that a small difference in the hazard rate does not yield a material impact on the survival probability. Survival probabilities and absolute differences for a number of tenors are listed in Table 9.1.

9.2 Additive models

As discussed in Section 7.2, Chourdakis et al. (2013) chooses a multiplicative model for the proxy spread. However, no argument for that choice is provided. Thus, we try a corresponding but additive version of the model (Equation (7.5)). We also test an additive version of the alternate model (Equation (7.7)). However, when starting to produce the proxy curves we quickly run into problems. We often get into cases where the curve fails to build. Furthermore, we encounter multiple instances where the obtained curves are degenerate in the sense that they violate a no- arbitrage constraint. A summary is presented in Table 9.2. The no-arbitrage constraint is discussed in Section 9.4.

49 1.0 0.025

0.9 0.020

0.8 0.015

0.7 Hazard rate 0.010 Survival probability 0.005 0.6

2020 2024 2028 2032 2036 2040 2044 2048 2052 2020 2024 2028 2032 2036 2040 2044 2048 2052

NOMURA_EURO_INDU_BBB ALTRNTV_EURO_INDU_BBB ATCOA

Figure 9.1: Survival curve and hazard rates for the proxy curve for BBB-rated European Industrial companies and the liquid entity Atlas Copco AB. Data per March 31st 2020.

Tenor Nomura Alternative Difference 0D 100.00% 100.00% 0.00% 6M 99.86% 99.85% -0.00% 1Y 99.70% 99.70% 0.00% 2Y 99.11% 99.12% 0.01% 3Y 98.18% 98.20% 0.02% 4Y 96.97% 97.01% 0.05% 5Y 95.43% 95.51% 0.09% 7Y 91.82% 91.94% 0.13% 10Y 86.69% 86.84% 0.16% 15Y 79.82% 79.52% -0.31% 20Y 73.07% 72.71% -0.36% 30Y 61.40% 60.86% -0.54%

Table 9.1: Survival probabilities for European industrial companies rated BBB according to the two models. Data per March 31st 2020.

Model Version Built curves Valid curves Nomura Multiplicative 528 467 Nomura Additive 329 208 Alternative Multiplicative 528 528 Alternative Additive 528 246

Table 9.2: Statistics for proxy credit curves built with different versions of the models. Data per March 31st 2020.

50 1.0 0.40

0.8 0.35 0.30

0.6 0.25

0.20 0.4

Hazard rate 0.15

Survival probability 0.2 0.10

0.05 0.0

2016 2020 2024 2028 2032 2036 2040 2044 2048 2016 2020 2024 2028 2032 2036 2040 2044 2048

NOMURA_EURO_BAMA_CCC ALTRNTV_EURO_BAMA_CCC

Figure 9.2: The Nomura method failing. Data per July 29th 2016.

We will now comment on why these issues occur. With an additive model, we do not ensure that the CDS spread (in the Nomura case) or the hazard rate (in the alternative case) are positive. Recall that in the multiplicative case, we estimate the model parameters by first applying a logarithmic transformation of the model. After the regression coefficients have been determined, the model parameters are given as e raised to the co- efficient. This ensures that each model parameter is non-negative and thus theproxy spread (or hazard rate) is non-negative by construction. Therefore, we can conclude that the multiplicative models are preferable over their additive counterparts. Hence, we will from now on only consider the multiplicative versions of the models.

9.3 The Nomura model collapses

In Section 7.2 we hypothesised that the Nomura model discards important term structure information encoded in the CDS prices and that this may cause issues when constructing the proxy curves. We will now look at a situation where the Nomura model fails to produce a proxy curve, but our alternative model succeeds to do it. In Figure 9.2 we show a case where the Nomura method is failing to construct a curve. When the bootstrap procedure tries to find a survival probability such that the 30Y CDS is re-priced, it fails to do so. I.e. there is no survival probability in the range [0, 1] allowing the 30Y CDS to be repriced from the curve. In contrast, we see that the alternative model handles this case without issues and produces a survival curve stretching to 30Y. In the right subplot of Figure 9.2, we see a distinct difference in the hazard rate curves. We see that the Nomura curve has a large hump between 2026 and 2031, but the alternative model produces a considerably smoother curve. However, when running the model for the full set of liquid names (as we do when the data set becomes larger), the Nomura model only rarely breaks down. That said, we argue that our alternative model is theoretically sound and more robust than the Nomura model in its original formulation.

51 9.4 Arbitrage in the curve

The survival curve should be arbitrage free, meaning that the survival probabilities should be monotonically decreasing with the time horizon: ∂Q(0, t) ≤ 0 for all 0 < t < ∞. ∂t This constraint is equivalent to requiring that the hazard rate (instantaneous forward default rate) is positive: h(t) ≥ 0 for all 0 < t < ∞. (9.1) It is of course natural that we require that the survival curve is monotonically decreasing, and it is consistent with intuition. The probability of surviving the coming five years should naturally be less than the probability of surviving the coming four years. Furthermore, since the intensity can be interpreted as a probability per unit of time, it is natural to require that this quantity is non-negative. Having arbitrage in the CDS curve would mean that one can buy protection for a longer time period [0, t] compared to [0, s] for the same value of the protection leg. We want to point out that the bootstrapping procedure used (Algorithm 1) does not check whether this constraint holds. The procedure is naive in the sense that it tries to find a survival probability Q(0, t) in the range [0, 1]. A more elaborate bootstrapping procedure could indeed check that value of the survival probability to be added to the curve is indeed lower than the previous one. Now, we need to check whether Equation (9.1) holds or not. For all generated proxy and liquid survival curves, we check if there are any negative hazard rates. Since we have assumed a piecewise constant hazard rate it suffices to check the value of the hazard rate at the pillar dates. We find that arbitrage constraint is violated only rarely for the liquid curves.For the Nomura method it happens quite frequently. We find violating cases for many days per month. Most often it occurs for proxy curves concerning the lowest rating category (CCC). We find no cases of this issue for the alternative method. Again, see Table9.2. In Figure 9.3 we plot one instance of a proxy curve with negative hazard rates. The Nomura survival curve exhibits a clear upward slope in one interval. We see that the alternative method does not violate the arbitrage constraint. Now, it is of course natural to ask why these issues only arise for the Nomura model. Since in the Nomura model, the proxy spread is determined on a tenor-by-tenor basis, there is no guarantee that the credit curve will not violate the arbitrage constraint. O’Kane (2008) provides an approximate arbitrage constraint for the spread. For spreads Sm−1 and Sm associated with maturity dates Tm−1 and Tm, the constraint becomes:

Tm−1 Sm ' Sm−1 . Tm This last equation should make it clear that with the Nomura model, we cannot be certain that the proxy spreads obtained will give us a valid credit curve. The next question is then: Why do we not run into the same issues with the alternative model? Recall that the main difference between the Nomura model and our model is that the modelling quantity is changed from the CDS spread to the hazard rate. In the multiplicative case we will, by constructing, obtain non-negative hazard rates. Thus, the no-arbitrage condition Equation (9.1) is always fulfilled. This means that our alternative model will always, by construction, yield valid proxy credit curves.

52 0.14 1.0 0.12

0.8 0.10

0.08 0.6 0.06

0.4 0.04 Hazard rate 0.02 Survival probability 0.2 0.00

−0.02 0.0 2020 2024 2028 2032 2036 2040 2044 2048 2052 2020 2024 2028 2032 2036 2040 2044 2048 2052

NOMURA_EURO_HEAL_CCC ALTRNTV_EURO_HEAL_CCC

Figure 9.3: The Nomura method resulting in negative hazard rates. Data per March 31st 2020.

9.5 Large differences

Having seen that the Nomura method can both break down completely and produce survival curves with arbitrage, it is natural to ask: Can it happen that we get strangely looking (but valid) curves with the Nomura method? We investigated the proxy curves generated, looking for cases where there is a large difference in the hazard rate (but the curves are still valid in the sense thattheyare monotonically decreasing). Indeed, we find such cases. Two examples are shown in Figure 9.4. We see that the Nomura curves are valid in the sense that the hazard rate is always positive. However, particularly in Figure 9.4b the Nomura curve exhibits a large dent in the hazard rate that our alternative curve does not have. We also see that the difference in survival probability is quite noticeable. Recall that CVA as defined in Equation (1.1) is linear in default probability. Thus, using one curve instead of the other could have a material effect. It should be pointed out that there are also cases where the two methods perform similarly, e.g. as shown in Figure 9.1.

9.6 Rating monotonicity

One of the factors included in the modelling is rating. We expect that a better rating implies a lower probability of default. If this holds for all ratings included in the model, we say that model is monotonic in rating. This is a property that we want the model to have. Thus, this is tested for the Nomura model and our alternative model. We plot the five-year probability of default in Figure 9.5 for European Financial for the different ratings. We see that the lines lie in the expected order, and we can thus conclude that both models are monotonic in rating (at least for this time period and combination of region and industry).

53 1.0 0.08

0.07 0.8 0.06

0.6 0.05

0.04

0.4 Hazard rate 0.03 Survival probability

0.2 0.02

2020 2024 2028 2032 2036 2040 2044 2048 2052 2020 2024 2028 2032 2036 2040 2044 2048 2052

NOMURA_NOAM_ENRG_B ALTRNTV_NOAM_ENRG_B

(a) Proxy curves for North America, Energy, B.

1.0 0.175

0.150 0.8 0.125

0.6 0.100

0.075 0.4 Hazard rate 0.050 Survival probability 0.2 0.025

0.000 2020 2024 2028 2032 2036 2040 2044 2048 2052 2020 2024 2028 2032 2036 2040 2044 2048 2052

NOMURA_EEUR_TECH_CCC ALTRNTV_EEUR_TECH_CCC

(b) Proxy curves for Eastern Europe, Technology, CCC.

Figure 9.4: Large differences between the Nomura and the alternative method. Data per the 31st March 2020.

54 Nomura Alternative 1.0

0.9 AA A 0.8 BBB BB 0.7 B CCC 0.6 AA A 0.5 BBB

5Y survival probability BB 0.4 B CCC 2017 2018 2019 2020 2017 2018 2019 2020

AA A BBB BB B CCC

Figure 9.5: 5Y survival probabilities for European corporates in the Financial industry with different ratings according to the two proxy models. Computed using monthly data before August 2019, daily data thereafter.

Other proxy curves and tenors were also investigated. We found some cases where the Nomura model does not exhibit rating monotonicity. This likely due to the behaviour discussed in Section 9.5.

9.7 Volatility

We also investigated how do the two models behave over time. In Figure 9.6 we have plotted the 15Y survival probability for two proxy curves. The left plot in Figure 9.6 depicts the situation for European financial entities with rating A. We see that the two models perform similarly. It is interesting to note the sudden drop (and then recovery) in implied survival probability from mid-February to April. This reflects the turmoil in the financial markets due to the covid-19 pandemic. In the right in Figure 9.6 we see the 15Y survival probabilities for North American consumer goods entities, rated CCC. We see that the Nomura model exhibits a substan- tially higher volatility compared to the alternative model. This reinforces our argument that the alternative model is a more robust choice.

55 EURO_FINA_A NOAM_COGO_CCC 0.86 0.35

0.85 0.30 0.84 0.25 0.83 0.20 0.82 0.15 0.81

0.10 15Y survival0 probability .80

0.79 0.05 Sep Oct Nov Dec Jan Feb Mar Apr Sep Oct Nov Dec Jan Feb Mar Apr 2020 2020 Nomura Alternative

Figure 9.6: 15-year survival probabilities over time for the two proxy models.

56 Chapter 10

Conclusion

In this thesis, we set out to investigate viable modelling frameworks for obtaining proxy credit curves to use for entities for which there are no CDSs liquidly traded. Put differently: We searched for a solution to the shortage of liquidity problem. We saw that a number of suggestions on how to solve the problem have been presented previously. Importantly, there are regulatory requirements to consider. In essence, a proxy credit curve (or spread) should take into account region, sector and rating. A model common in practice fulfilling the regulatory requirements is the so-called Nomura model. Thus,we chose to investigate that model further. The Nomura approach is to model the proxy CDS spread multiplicatively, as a product of five factors. The model parameters are estimated using linear regression andCDS quotes for liquid entities. We argued that there are two fundamental problems with the Nomura model:

• The regression model is ill-posed

• The model discards important term structure information encoded in the CDS spreads

Hence, we proposed an altered model in which the modelling quantity is changed from the CDS spread to the hazard rate. Hazard rate curves are extracted on an entity-by- entity basis for the liquid entities. After the model parameters have been estimated using linear regression, proxy hazard rates and thus proxy credit curves can be obtained. We saw that the alternative model seems to perform better than the Nomura model. The alternative model does not yield degenerate curves, as the Nomura model does in many instances. This stems from the fact that since we model the hazard rate using a multiplicative model, we will always obtain non-negative hazard rates, by construction. This is not the case for the Nomura model. To summarise: We argue that the alternative model proposed is theoretically sound and more robust compared to the Nomura model.

10.1 Further research

In this thesis, we have not questioned whether it is appropriate or not to use risk-neutral default probabilities when there are no liquid CDS. As we have seen, there are clear regulatory requirements for the universal use of risk-neutral default probabilities. However, this view has been challenged e.g. by Green (2015) and Hammarlid and Leniec (2018). It would be of interest to investigate this further.

57 Section 3.5 discussed a number of proxy models. We saw that in some cases, hedging of CVA becomes quite complicated. In particular, that is the case for the type of cross- sectional approach we have explored in this thesis. An area for further research would thus be to investigate the hedging implications in greater detail. This is motivated by that it may be beneficial for banks to hedge their CVA. We saw that there are instances where the credit curves obtained using the alternative model differs significantly from when using the Nomura model. Thus, one may proceed to quantify the differences systematically.

58 References

Basel Committee on Banking Supervision (June 1, 2011). Basel Committee finalises capital treatment for bilateral counterparty credit risk. url: https://www.bis.org/press/ p110601.pdf. — (Dec. 15, 2019). MAR50: Credit valuation adjustment framework. url: https://www. bis.org/basel_framework/chapter/MAR/50.htm?inforce=20191215. Berndt, A. et al. (2005). “Measuring default risk premia from default swap rates and EDFs”. In: BIS Working Papers 173. Björk, T. (2009). Arbitrage theory in continuous time. 3rd ed.. Oxford New York: Oxford University Press. isbn: 978-0199574742. Brigo, D. and F. Mercurio (2006). Interest Rate Models — Theory and Practice: With Smile, Inflation and Credit. Second Edition. Springer Finance. Springer Berlin Hei- delberg. isbn: 978-3-540-22149-4. Brigo, D., M. Morini, and A. Pallavicini (2013). Counterparty credit risk, collateral and funding: with pricing cases for all asset classes. Vol. 478. Wiley finance series. Wiley. Brummelhuis, R. and Z. Luo (2017). CDS Rate Construction Methods by Machine Learn- ing Techniques. arXiv: 1705.06899 [q-fin.ST]. Chourdakis, K. et al. (2013). A cross-section across CVA. url: http://www.nomura. com/resources/europe/pdfs/cva-cross-section.pdf. Commission Delegated Regulation (EU) No 526/2014 (2014). Supplementing Regulation (EU) No 575/2013 of the European Parliament and of the Council with regard to regulatory technical standards for determining proxy spread and limited smaller portfolios for credit valuation adjustment risk. url: http://data.europa.eu/eli/reg_del/ 2014/526/oj. Deutsch, H.-P. and M. W. Beinker (2019). Derivatives and Internal Models: Modern Risk Management. Finance and Capital Market Series. Palgrave Macmillan. isbn: 978-3- 030-22898-9. Durrett, R. (2016). Essentials of Stochastic Processes. 3rd ed. 2016.. Springer Texts in Statistics. isbn: 3-319-45614-8. European Banking Authority (2012). Draft Regulatory Technical Standards for credit valuation adjustment risk on the determination of a proxy spread and the specification of a limited number of smaller portfolios. Consultation Paper. EBA/CP/2012/09. url: https://eba.europa.eu/file/31471. Green, A. (2015). XVA: Credit, Funding and Capital Valuation Adjustments. The Wiley Finance Series. Wiley. isbn: 9781118556788. Gregory, J. (2015). The xVA Challenge: Counterparty Credit Risk, Funding, Collateral and Capital. The Wiley Finance Series. Wiley. isbn: 9781119109419. Hagan, P. S. and G. West (2008). “Methods for constructing a yield curve”. In: Wilmott Magazine, May, pp. 70–81.

59 Hammarlid, O. and M. Leniec (2018). Credit Value Adjustment for Counterparties with Illiquid CDS. arXiv: 1806.07667 [q-fin.MF]. Heynderickx, W. et al. (2016). “The relationship between risk-neutral and actual default probabilities: the credit risk premium”. In: Applied Economics 48.42, pp. 4066–4081. Hull, J. C. (2012). Options futures and other derivatives. 8th. Pearson Education. Kenyon, C. and R. Stamm (2012). Discounting, Libor, CVA and Funding: Interest Rate and Credit Pricing. Palgrave Macmillan. Merton, R. C. (1974). “On the Pricing of Corporate Debt: The Risk Structure of Interest Rates”. In: The Journal of Finance 29.2, pp. 449–470. O’Kane, D. (2008). Modelling single-name and multi-name credit derivatives. John Wiley & Sons. Regulation (EU) No 575/2013 of the European Parliament and of the Council (2013). Prudential requirements for credit institutions and investment firms and amending Regulation (EU) No 648/2012. url: http://data.europa.eu/eli/reg/2013/575/ oj. Ruiz, I. (2015). XVA Desks - A New Era for Risk Management: Understanding, Building and Managing Counterparty, Funding and Capital Risk. Palgrave Macmillan. Schönbucher, P. J. (2003). Credit derivatives pricing models: Models, Pricing and Imple- mentation. John Wiley & Sons. Sourabh, S., M. Hofer, and D. Kandhai (2018). “Liquidity risk in derivatives valuation: an improved credit proxy method”. In: Quantitative Finance 18.3, pp. 467–481.

60 Appendix A

Alternative characterisations

In this appendix, we present two alternative characterisations of the reduced form models.

A.1 Hazard rate

Let F (t) be the cumulative default probability and Q (t) the cumulative survival probability. Now, let us define the hazard rate:

Definition A.1 (Hazard rate). The hazard rate λ(t) is defined such that the probability of default in the interval (t, t+∆t] is λ(t)∆t, conditional on survival to t. (For infinitesimal ∆t)

Now we will look at λ(t) fits into the definitions of F (t) and Q (t). Starting from the probability of default in (t, t + ∆t], conditional on survival to t: P [t < τ ≤ t + ∆t] P [τ ≥ t|t < τ ≤ t + ∆t] P [t < τ ≤ t + ∆t|τ ≥ t] = P [τ ≥ t] P [t < τ ≤ t + ∆t] = P [τ ≥ t] F (t + ∆t) − F (t) = Q (t) Q (t) − Q (t + ∆t) = Q (t)

By definition, this expression should equal λ(t)∆t: Q (t) − Q (t + ∆t) = λ(t)∆t Q (t) Q (t + ∆t) − Q (t) = −λ(t)Q (t) ∆t

Letting ∆t → 0, we obtain dQ (t) = −λ(t)Q (t) dt By integrating, we get the formula for the survival probability in the intensity modelling framework: − R t λ(u)du Q (t) = e 0

61 We note that this expression is very similar to the expression of a discount factor, for a deterministic and time-varying short rate. Furthermore, we have that the default probability is − R t λ(u)du F (t) = 1 − e 0 , and the default probability density is

− R t λ(u)du f(t) = λ(t)e 0 = λ(t)Q (t) . This also means that the hazard rate is f(t) F 0(t) λ(t) = = . Q (t) 1 − F (t)

A.2 Implied quantities

In this section, we will derive some implied quantities for default modelling. This section is largely based on Schönbucher (2003). The fundamental building block is the zero-coupon bond (ZCB). Assume that we have prices of these for all maturities T > t. We will use two versions of the zero-coupon bond: Z (t, T ) = Price at t of default-free ZCB paying 1 at t which is the regular zero-coupon bond, as defined in Definition 4.3. We will now also introduce a defaultable version of the ZCB, which only pays the notional if there is no default before T : Zˆ (t, T ) = Price at t of default-free ZCB paying 1 at t Under the spot martingale measure, the price of a contingent claim is the ex Recall the risk neutral valuation formula (Björk, 2009). For a T claim X, the price at t is given by h − R T r ds i Q t s Π(t; X) = E e X Ft where Q is a martingale measure with the bank account as numeraire. This simply means that the price of a contingent claim is given by the discounted expected payoff. For the default-free ZCB we have: h − R T r ds i Z (t, T ) = EQ e t s · 1 .

The defaultable ZCB only pays 1 if the obligor survives until T , meaning that the payoff at T is given by: ( 1 if default after T .(τ > T ), 1{τ>T } = 0 if default before T .(τ ≤ T ) which leads us to the price: h R T i ˆ Q − rsds Z (t, T ) = E e t · 1{τ>T } .

Now, to be able the derive a relationship between Z and Zˆ, we will introduce an important assumption. Namely that that the default-free interest rate dynamics and the default time are independent. (I.e. independence between Z (t, T ) and τ). Thus, we have: h R T i h R T i ˆ Q − rsds Q − rsds Q Z (t, T ) = E e t · 1{τ>T } = E e t E 1{τ>T }

Q = Z (t, T ) E 1{τ>T } = Z (t, T ) Q (t, T )

62 where we introduced Q (t, T ), denoting the implied probability of survival over [t, T ]. The independence assumption was used in the second equality and the definition of the default-free ZCB in the third. Thus, we have:

Zˆ (t, T ) Q (t, T ) = (A.1) Z (t, T )

The conditional survival probability was defined in Definition 5.4. We want to consider conditional survival and default probabilities, per time interval length. The conditional survival probability per unit of time: 1 1 Q (t, T, T + ∆t) = (1 − F (t, T, T + ∆t)) ∆t ∆t We will now introduce a new quantity, the implied hazard rate, in the same spirit as the forward rate (see Definition 4.7)

Definition A.2 (Implied discrete hazard rate). The discrete hazard rate over [T,T + ∆t] prevailing at t is: 1 Q (t, T ) 1 F (t, T, T + ∆t) H(t; T,T + ∆t) := − 1 = ∆t Q (t, T + ∆t) ∆t Q (t, T, T + ∆t)

Which can be justified at follows: Let the discrete hazard rate be defined analogously to a simply compounding interest rate (see Definition 4.6): 1 Q (t, T ) = . 1 + H(t, T )(T − t)

Recall that: Q (t, T1) Q (t; T1,T2) = Q (t, T2) Hence,

Q (t, T2) Q (t; T1,T2) = Q (t, T1) 1 Q (t, T ) = 2 1 + H(T1,T2)(T2 − T1) Q (t, T1) 1 Q (t, T1) H(T1,T2) = − 1 T2 − T1 Q (t, T2)

Letting ∆t → 0, we have the continuous counterpart:

Definition A.3 (Implied continuous hazard rate). The continuous hazard rate over T prevailing at t is: ∂ 1 ∂ h(t, T ) := lim H(t; T,T + ∆t) = − ln Q (t, T ) = − Q (t, T ) (A.2) ∆t→0 ∂T Q (t, T ) ∂T

A.2.1 Relation to forward rates Let us now define defaultable versions of the forward rates defined in Definitions 4.7 and 4.8. From now on, F denotes a forward rate and not the default probability.

63 The defaultable simply compounded forward rate: ! 1 Zˆ (t, T1) Fˆ(t, T1,T2) = − 1 , T1 − T1 Zˆ (t, T2) and the defaultable instantaneous continuously compounded forward rate: ∂ fˆ(t, T ) = lim Fˆ(t; T,T + ∆t) = − ln Zˆ (t, T ) . ∆t→0 ∂T

Now, if we plug Equation (A.1) into Equation (A.2):

∂ ∂ Zˆ (t, T ) h(t, T ) = − ln S(t, T ) = − ln ∂T ∂T Z (t, T ) ∂ ∂ = − ln Zˆ (t, T ) + ln Z (t, T ) ∂T ∂T = fˆ(t, T ) − f(t, T )

Hence, we have an interpretation of the implied hazard rate: It is the spread of the defaultable over the default-free forward rates. For the discrete (simply compounded) hazard rate, one can prove a similar result: 1 H(t; T1,T2) = Fˆ(t; T1,T2) − F (t; T1,T2) 1 + F (t; T1,T2)(T2 − T1) with the interpretation that implied discrete hazard rate is the spread of defaultable over default-free forward rates, discounted by the default-free forward rate.

TRITA SCI-GRU 2020:071