The XVA of Financial Derivatives: CVA, DVA and FVA Explained Financial Engineering Explained About the series Financial Engineering Explained is a series of concise, practical guides to modern finance, focusing on key, technical areas of risk management and asset pricing. Written for practitioners, researchers and students, the series discusses a range of topics in a non-mathematical but highly intuitive way. Each self-contained volume is dedicated to a specific topic and offers a thorough introduction with all the necessary depth, but without too much technical ballast. Where applicable, theory is illustrated with real world examples, with special attention to the numerical implementation. Series Editor: Wim Schoutens, Department of Mathematics, Catholic University of Leuven. Series Advisory Board: Peter Carr, Executive Director, NYU Mathematical ; Global Head of Market Modeling, Morgan Stanley. Ernst Eberlein, Department of Mathematical Stochastics, University of Freiburg. Matthias Scherer, Chair of , Technische Universität München. Titles in the series: Equity Derivatives Explained, Mohamed Bouzoubaa The Greeks and Hedging Explained, Peter Leoni Smile Pricing Explained, Peter Austing Financial Engineering with Copulas Explained, Matthias Scherer and Jan-Frederik Mai Interest Rates Explained Volume 1, Jörg Kienitz Forthcoming titles: Interest Rates Explained Volume 2, Jörg Kienitz Submissions: Wim Schoutens – [email protected]

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Dongsheng Lu © Dongsheng Lu 2015 All rights reserved. No reproduction, copy or transmission of this publication may be made without written permission. No portion of this publication may be reproduced, copied or transmitted save with written permission or in accordance with the provisions of the Copyright, Designs and Patents Act 1988, or under the terms of any licence permitting limited copying issued by the Copyright Licensing Agency, Saffron House, 6–10 Kirby Street, London EC1N 8TS. Any person who does any unauthorized act in relation to this publication may be liable to criminal prosecution and civil claims for damages. The author has asserted his right to be identified as the author of this work in accordance with the Copyright, Designs and Patents Act 1988. First published 2015 by PALGRAVE MACMILLAN Palgrave Macmillan in the UK is an imprint of Macmillan Publishers Limited, registered in England, company number 785998, of Houndmills, Basingstoke, Hampshire RG21 6XS. Palgrave Macmillan in the US is a division of St Martin’s Press LLC, 175 Fifth Avenue, New York, NY 10010. Palgrave Macmillan is the global academic imprint of the above companies and has companies and representatives throughout the world. Palgrave® and Macmillan® are registered trademarks in the United States, the United Kingdom, Europe and other countries. ISBN 978-1-137-43583-5 ISBN 978-1-137-43584-2 (eBook) DOI 10.105 7/9781137435842 This book is printed on paper suitable for recycling and made from fully managed and sustained forest sources. Logging, pulping and manufacturing processes are expected to conform to the environmental regulations of the country of origin. A catalogue record for this book is available from the British Library. A catalog record for this book is available from the Library of Congress. v

Contents

List of Figures x List of Tables xii Acknowledgements xiii Introduction 1

Part I Introduction to Derivatives Trading 1 Overview of Derivatives Trading 5 1.1 The participants in the and their interactions 5 1.1.1 providers vs. derivative users 5 1.1.2 The dealer-to-customer market and competition 7 1.1.3 The dealer’s business model 9 1.1.4 The dealer-to-dealer market: the broker market 11 1.2 The OTC derivatives trading process 13 1.2.1 Pre-trade 13 1.2.2 Post-trade 14 1.2.3 Risk management and model risk 14 1.3 Regulations and controls 16 1.3.1 External regulations: Dodd–Frank, EMIR and CCAR 17 1.3.2 Basel III 18 1.3.3 Effects of regulations: derivative business changes and unintended consequences 19 Summary 20 2 Legal Aspects and Operations of Derivatives Trading 22 2.1 Legal aspects of derivatives transactions 22 2.1.1 ISDA Master Agreement 23 2.1.2 support annex 23 2.1.3 Confirmations 25 2.1.4 Jurisdiction and enforcement of legal agreements 26 2.2 and operations 26 2.3 Derivatives and adjustments 28 2.4 Treasury, funding channels and 29 Summary 30 vi Contents

Part II Exposition of Various Valuation Adjustments 3 CVA Primer and Credit Default 34 3.1 Risk-free rate, overnight index swap (OIS) rate and LIBOR curves 34 3.1.1 Risk-free rate and OIS discounting 34 3.1.2 Splitting of LIBOR curves 35 3.2 Counterparty 36 3.2.1 Counterparty default and recovery 36 3.2.2 Credit loss and credit exposures 37 3.3 Credit value adjustment 40 3.3.1 Introducing CVA 40 3.3.2 CDS spread, spread and credit default probability 41 3.3.3 Constructing a default probability curve 42 3.3.4 Correlation of market risk exposure and credit default 43 3.3.5 CSA and credit terms affecting CVA calculations 45 3.3.6 Bilateral CVA: DVA 47 3.3.7 DVA: the dilemma 47 3.4 CVA examples 49 3.4.1 An 49 3.4.2 An FX forward example 50 4 FVA Primer: Derivatives Pricing with Funding 52 4.1 The funding market 52 4.1.1 Financial institutions and the 53 4.1.2 Dealer’s business model and funding of derivatives 54 4.2 Collateralization, value of money and differential discounting 55 4.2.1 Credit spreads, funding spreads and notations 56 4.2.2 The value of collateral 57 4.2.3 The value of money: lending to the counterparty 58 4.2.4 The value of money: borrowing money 61 4.2.5 The value of money: summary 61 4.2.6 The equilibrium borrowing/lending rate 61 4.2.7 Collateral asset, repo market and rehypothecation 63 4.2.8 Collateral currency and cross currency market 65 4.2.9 Cross currency swap market 67 4.2.10 Multi-collateral choices and choice options 68 4.2.11 Multi-funding curves, differential discounting and collateral valuation adjustment 70 4.2.12 Collateral operations and replication of collateral values 72 4.2.13 Derivative valuations with collateral 75 4.2.14 Standardized CSA (SCSA) and tri-party optimization 76 4.2.15 Example of fully collateralized derivatives 78 4.3 Funding value adjustment 80 4.3.1 Inseparable principle: Derivative valuation and funding of derivatives 80 4.3.2 Funding for uncollateralized derivatives – From the firm’s economic point of view 81 Contents vii

4.3.3 FBA and DVA: Linking, overlap and double-counting 83 4.3.4 FBA or DVA: Which one to use? 84 4.3.5 Specific funding spread and market funding spread: Which one? 88 4.3.6 “Market price” and “one price” regime 94 4.3.7 FVA asymmetry: Is the no-arbitrage rule broken? 95 4.3.8 Close-out valuation and funding implications 97 4.3.9 Funding policy and construction of funding curve 98 4.3.10 FVA and funding curve dependency on derivative transactions 102 4.4 Derivatives accounting 103 4.4.1 Derivatives accounting with collateralization, CVA and FVA 103 4.4.2 Derivatives valuations, risk neutral measure, replications and deviations 104 4.4.3 Derivatives valuations under collateralization, CVA and FVA 105 4.5 FVA questions and debate 107 4.5.1 Debate point: Existence of FVA 1 107 4.5.2 Debate point: Existence of FVA 2 108 4.5.3 Debate point: FVA as private adjustment 108 4.5.4 Dilemma: Banking book and trading book consistency 109 4.5.5 Discussion: Funding cost for a default risk-free business 112 4.5.6 Discussion: Funding cost for a bond 113 4.5.7 Question: Fair value marking in borrowing and funding cost charge in asset lending 115 4.5.8 Question: Repo of derivatives 116 4.5.9 Question: Supra bond marking and funding spread 117 4.5.10 Question: Differential funding levels with a firm 118 Summary 119 5 Other Valuation Adjustments 121 5.1 Liquidity value adjustment (LVA) 121 5.1.1 Introduction 121 5.1.2 LCR and LVA calculations 122 5.2 RVA: Replacement value adjustment 124 5.2.1 RVA: Special CSA features 124 5.2.2 Collateral event, FVA and LVA 125 5.2.3 RVA: The difficulty 126 5.3 KVA: Capital value adjustment? 127 5.3.1 KVA introduction 127 5.3.2 Accounting, exit pricing and the KVA 129 5.3.3 Calculating KVA 130 5.3.4 KVA management 131 5.4 Summarizing XVAs 132 5.4.1 XVA summary 132 viii Contents

5.4.2 Prudent valuations and additional valuation adjustments (AVA) 133 5.4.3 Integrating XVAs, AVAs and capital 133 5.4.4 Derivative users: Problems and hopes 135 Summary 136

Part III XVA Modeling and Implementation 6 CVA and FVA Modeling and Implementation 141 6.1 General CVA/FVA modeling 141 6.1.1 CVA/FVA Modeling and computations: Complexity 141 6.1.2 General CVA/FVA calculations 142 6.2 Market risk modeling 145 6.2.1 Market risk modeling and instrument valuations 145 6.2.2 Risk neutral measure vs. historical measure 145 6.2.3 Netting and generic time grid 146 6.2.4 Backward Discounting Scheme 148 6.2.5 Backward discounting scheme: Pros and cons 150 6.2.6 Forward simulation scheme 152 6.2.7 Forward simulation scheme: CSA implementation 154 6.2.8 Forward simulation scheme: Pros and cons 155 6.3 Credit risk factor modeling 157 6.3.1 Modeling counterparty credit: Structural and reduced form models 157 6.3.2 Transition matrix and dynamic ratings migration 160 6.3.3 Transitional default calculations 162 6.3.4 Risk neutralization of transition matrix 163 6.3.5 A Simple dynamic credit simulation model 166 6.4 Efficient CVA/FVA calculations 168 6.4.1 Efficient exposure calculations 168 6.4.2 flow compression for linear instruments 172 6.4.3 Efficient incremental and marginal CVA/FVA calculations 173 6.4.4 Greeks computations and distributed computing 174 6.4.5 Adjoint algorithmic differentiation 175 6.5 Other considerations 176 6.5.1 CVA/FVA cross: Default on funded exposure or funding on non-default asset? 176 6.5.2 Netting set and funding set in FVA calculations 177 6.5.3 Mutual put breaks 179 Summary 180

Part IV XVA Risk Management and Hedging 7 CVA and FVA Risk Management 185 7.1 Derivatives risk management primer 185 7.1.1 Derivatives valuations and replication 185 7.1.2 Delta hedging and replication 186 Contents ix

7.1.3 “Scalping” with transaction cost 187 7.1.4 Multiple risk factors 190 7.1.5 Mean variance hedging 194 7.1.6 Specification of model parameters and hedging 195 7.2 Risk managing CVA 196 7.2.1 CVA risk management 196 7.2.2 CVA market risk: Delta, gamma and theta 197 7.2.3 Risk managing credit risk 198 7.2.4 Hedging CVA risk and optimizing capital 202 7.3 XVA desk setup and operations 203 Summary 206 Notes 207 Bibliography 211 Index 213 x

List of Figures

1.1 The derivative market 6 1.2 Illustration of a structured notes issuance and hedging 7 1.3 Illustration of an auction process in customer derivative trading 7 1.4 The derivatives trading processes 13 1.5 External regulations and internal controls for dealers 17 1.6 Changes in a bank’s balance sheet as a result of regulations 19 P.II Illustration of XVA calculation logistics for typical client quoting 32 3.1 FF–Libor spread spikes during the 2007–2008 credit crisis 35 3.2 Collateralized derivative trading 35 3.3 Branching of LIBOR curves 36 3.4 Example PFE and EPE, ENE for an interest rate swap 38 3.5 Example PFE and EPE, ENE for an FX forward contract 39 3.6 Illustration of a CDS transaction 41 3.7 Illustration of wrong-way and right-way exposure 44 4.1 The simplified funding market 53 4.2 A firm’s balance sheet: Borrowing and leverage of financial institutions 54 4.3 Asymmetric funding for dealer’s bilateral derivatives business and hedges 54 4.4 Credit and funding spreads definitions 56 4.5 Transfer pricing of the funding desk 57 4.6 The value of collateral 58 4.7 Schematic diagram for a repo transaction 63 4.8 Constructing funding curves 70 4.9 Differential discounting for collateralized derivatives 72 4.10 Cross currency collateral posting 74 4.11 Reverse repo in collateral operations 74 4.12 CCP and central clearing 77 4.13 Mixture discounting example 79 4.14 Collateralized valuation 81 4.15 Derivative valuation with contingent claims 82 4.16 Asset and liability funding symmetry 83 4.17 Overlapping of DVA and FBA 84 4.18 Centralized funding management 86 List of Figures xi

4.19 Expected and realized funding levels 87 4.20 Overlapping of DVA and FBA with optimized liquidity spread 88 4.21 Overlapping of DVA and FBA with optimized funding spread 88 4.22 Market mechanism: Exit valuation in the market 89 4.23 Illustration of market liquidity and friction 90 4.24 Novation of bilateral derivatives 92 4.25 Intermediation of bilateral derivatives 92 4.26 One asset cash flow in the future 109 4.27 Funding cost for a default risk-free business 112 4.28 Passing funding cost to customers 114 4.29 Repo of derivatives 117 5.1 Illustration of an SPV transaction 124 5.2 Illustration of replacement process 126 5.3 Illustration of funding and capital cost difference 128 5.4 Example for KVA, replication value vs. market exit value 130 5.5 Dynamic replication of derivative value and valuation adjustments 132 5.6 Helping the derivative users 136 6.1 Aggregation of default/funding exposure 142 6.2 Netting of portfolio exposure in Monte Carlo scenarios 143 6.3 General CVA/FVA calculations 144 6.4 Discounting of cash flow payment $1 expected at maturity T 148 6.5 Example ratings distribution P(t,m) for rating m and time t. 151 6.6 Schematic diagram for wrong-way risk calculations 156 6.7 Asset boundaries for different credit ratings 159 6.8 Illustration of transitional defaults 160 6.9 Dynamic simulations of ratings 167 6.10 Illustration of specific trading time grid and generic time grid 170 6.11 Illustration of Brownian bridging of random numbers 170 6.12 Illustration of cash flow compression 172 6.13 Illustration of incremental CVA/FVA calculation for a new trade 174 6.14 Illustration of distributed computing with CPUs and GPUs 175 6.15 Illustration of netting set vs. funding set 178 7.1 An example delta hedging band for a call 188 7.2 Illustration of cross gamma effect 191 7.3 Balancing between CVA P&L hedging and capital optimization 203 7.4 XVA Desk setup and operations 204 xii

List of Tables

1.1 Example of discrete pricing 8 1.2 Example of winning probabilities for A and B 8 1.3 Tabulation of example winnings for A and B 9 2.1 Ratings-based threshold example 24 2.2 Eligible collaterals and haircuts 24 4.1 Summary of funding situations, costs and benefits 62 4.2 Economics of lending and borrowing from SF’s and CP’s point of view. 62 4.3 Example swaption and interest rate details 79 4.4 Example mixture discounting calculations 79 4.5 Example funding spreads implied by S&P index option market 99 4.6 Comparison of funding methodologies 104 4.7 Comparison of implied, realized and non-default spreads 111 4.8 Listing of interest cost terms charged by the lender 115 5.1 Example CSA collateral thresholds 123 6.1 Generic time grid points selection 147 6.2 Example CVA/FVA calculation 149 6.3 Example CVA/FVA calculation 149 6.4 Example mapping function for ratings vs. CDS spreads G(h,m) 160 6.5 S&P’s 1-year corporate transition rates from 1981 to 2011 161 6.6 Propagated rating distribution at 10 years 161 6.7 1-year transition matrix with default through BBB 162 6.8 Propagated 10-year ratings distribution with default through BBB 162 6.9 Probability of default term structure for an example 3-rating system 164 6.10 Optimized transition matrix for the example 3-rating system 164 6.11 Probability of default term structure for an example 8-rating system 165 6.12 The optimized 1-year transition matrix for the 8-rating system 165 6.13 Fitting errors in the 8-rating example case 166 xiii

Acknowledgements

I wish to thank my wife Jiang and my family for their encouragement and tremendous support. I am indebted to Frank Juan for numerous stimulating discussions over the years, and many insightful comments and contributions to this book. I would also like to thank Rob Smith, Chris Wu and Wujiang Lou for helpful suggestions and comments.