ANDREW GREEN heads CVA/FVA Quantitative Research at Lloyds Banking Group. He leads a team of quantitative analysts and developers who are responsible for the design and implementation of models for derivative valuation adjustments. Andrew and his team also work extensively on the implication of regulatory change on derivatives. Andrew previously headed CVA Quantitative Research at Barclays Capital and during his career, has also worked on models for fixed income and equity derivative products as well as ALM. High performance computing is a central element of XVA model implementation and Andrew has extensive experience of the practical implementation of large scale Monte Carlo simulation models in IT systems. Andrew is a regular conference speaker and has co-authored a number of papers on various topics in XVA. He has a DPhil in Theoretical Physics and a BA in Physics from Oxford University, and Part III of the Mathematics Tripos from Cambridge University.

List of Tables xvii

List of Figures xxi

Acknowledgements xxv

CHAPTER 1 Introduction: The Valuation of Derivative Portfolios 1

1.1 What this book is about 1

1.2 Prices and Values 4

1.2.1 Before the Fall... 4

1.2.2 The Post-Crisis World... 5

1.3 Trade Economics in Derivative Pricing 6

1.3.1 The Components of a Price 6

1.3.2 Risk-Neutral Valuation 8

1.3.3 Hedging and Management Costs 11

1.3.4 Credit Risk: CVA/DVA 11

1.3.5 FVA 13

1.3.6 Regulatory Capital and KVA 14

1.4 Post-Crisis Derivative Valuation or How I Learned to Stop Worrying and Love FVA 16

1.4.1 The FVA Debate and the Assault on Black-Scholes-Merton 16

1.4.2 Different Values for Different Purposes 19

1.4.3 Summary: The Valuation Paradigm Shift 21

1.5 Reading this Book 21

PART ONE CVA and DVA: Counterparty Credit Risk and Credit Valuation Adjustment

CHAPTER 2 Introducing Counterparty Risk 25

2.1 Defining Counterparty Risk 25

2.1.1 Wrong-way and Right-way Risk 27

2.2 CVA and DVA: Credit Valuation Adjustment and Debit Valuation Adjustment Defined 27

2.3 The Default Process 28

2.3.1 Example Default: The Collapse of Lehman Brothers 30

2.4 Credit Risk Mitigants 30

2.4.1 Netting 30

2.4.2 Collateral/Security 31

2.4.3 Central Clearing and Margin 34

2.4.4 Capital 35

2.4.5 Break Clauses 35

2.4.6 Buying Protection 37

CHAPTER 3 CVA and DVA: Credit and Debit Valuation Adjustment Models 39

3.1 Introduction 39

3.1.1 Close-out and CVA 40

3.2 Unilateral CVA Model 42

3.2.1 Unilateral CVA by Expectation 42

3.2.2 Unilateral CVA by Replication 43

3.3 Bilateral CVA Model: CVA and DVA 48

3.3.1 Bilateral CVA by Expectation 48

3.3.2 Bilateral CVA by Replication 50

3.3.3 DVA and Controversy 53

3.4 Modelling Dependence between Counterparties 55

3.4.1 Gaussian Copula Model 55

3.4.2 Other Copula Models 56

3.5 Components of a CVA Calculation Engine 57

3.5.1 Monte Carlo Simulation 57

3.5.2 Trade Valuation and Approximations 57

3.5.3 Expected Exposure Calculation 59

3.5.4 Credit Integration 59

3.6 Counterparty Level CVA vs. Trade Level CVA 59

3.6.1 Incremental CVA 60

3.6.2 Allocated CVA 60

3.7 Recovery Rate/Loss-Given-Default Assumptions 63

CHAPTER 4 CDS and Default Probabilities 65

4.1 Survival Probabilities and CVA 65

4.2 Historical versus Implied Survival Probabilities 66

4.3 Credit Default Swap Valuation 67

4.3.1 Credit Default Swaps 67

4.3.2 Premium Leg 69

4.3.3 Protection Leg 71

4.3.4 CDS Value and Breakeven Spread 72

4.4 Bootstrapping the Survival Probability Function 72

4.4.1 Upfront Payments 74

4.4.2 Choice of Hazard Rate Function and CVA: CVA Carry 75

4.4.3 Calibration Problems 76

4.5 CDS and Capital Relief 77

4.6 Liquid and Illiquid Counterparties 78

4.6.1 Mapping to Representative CDS 79

4.6.2 Mapping to Baskets and Indices 80

4.6.3 Cross-sectional Maps 81

CHAPTER 5 Analytic Models for CVA and DVA 83

5.1 Analytic CVA Formulae 83

5.2 Interest Rate Swaps 84

5.2.1 Unilateral CVA 84

5.2.2 Bilateral CVA 86

5.3 Options: Interest Rate Caplets and Floorlets 86

5.4 FX Forwards 88

CHAPTER 6 Modelling Credit Mitigants 91

6.1 Credit Mitigants 91

6.2 Close-out Netting 91

6.3 Break Clauses 93

6.3.1 Mandatory Break Clauses 93

6.3.2 Optional Break Clauses 93

6.4 Variation Margin and CSA Agreements 97

6.4.1 Simple Model: Modifying the Payout Function 97

6.4.2 Modelling Collateral Directly 99

6.4.3 Lookback Method 101

6.4.4 Modelling Downgrade Triggers in CSA Agreements 102

6.5 Non-financial Security and the Default Waterfall 107

CHAPTER 7 Wrong-way and Right-way Risk for CVA 109

7.1 Introduction: Wrong-way and Right-way Risks 109

7.1.1 Modelling Wrong-way Risk and CVA 110

7.2 Distributional Models of Wrong-way/Right-way Risk 111

7.2.1 Simple Model: Increased Exposure 111

7.2.2 Copula Models 111

7.2.3 Linear Models and Discrete Models 114

7.3 A Generalised Discrete Approach to Wrong-way Risk 116

7.4 Stochastic Credit Models of Wrong-way/Right-way Risk 118

7.4.1 Sovereign Wrong-way Risk 119

7.5 Wrong-way/Right-way Risk and DVA 119

PART TWO FVA: Funding Valuation Adjustment

CHAPTER 8 The Discount Curve 123

8.1 Introduction 123

8.2 A Single Curve World 123

8.3 Curve Interpolation and Smooth Curves 126

8.4 Cross-currency Basis 127

8.5 Multi-curve and Tenor Basis 128

8.6 OIS and CSA Discounting 129

8.6.1 OIS as the Risk-free Rate 129

8.6.2 OIS and CSA Discounting 131

8.6.3 Multi-currency Collateral and the Collateral Option 134

8.7 Conclusions: Discounting 138

CHAPTER 9 Funding Costs: Funding Valuation Adjustment (FVA) 139

9.1 Explaining Funding Costs 139

9.1.1 What is FVA? 139

9.1.2 General Principle of Funding Costs 145

9.2 First Generation FVA: Discount Models 145

9.3 Double Counting and DVA 146

9.4 Second Generation FVA: Exposure Models 147

9.4.1 The Burgard-Kjaer Semi-Replication Model 148

9.5 Residual FVA and CSAs 160

9.6 Asymmetry 161

9.6.1 Case 1: Corporate vs. Bank Asymmetry 161

9.6.2 Case 2: Bank vs. Bank Asymmetry 162

9.7 Risk Neutrality, Capital and the Modigliani-Miller Theorem 162

9.7.1 No Market-wide Risk-neutral Measure 162

9.7.2 Consequences 165

9.7.3 The Modigliani-Miller Theorem 165

9.8 Wrong-way/Right-way Risk and FVA 166

CHAPTER 10 Other Sources of Funding Costs: CCPs and MVA 167

10.1 Other Sources of Funding Costs 167

10.1.1 Central Counterparty Funding Costs 167

10.1.2 Bilateral Initial Margin 170

10.1.3 Rating Agency Volatility Buffers and Overcollateralisation 170

10.1.4 Liquidity Buffers 170

10.2 MVA: Margin Valuation Adjustment by Replication 171

10.2.1 Semi-replication with no Shortfall on Default 174

10.3 Calculating MVA Efficiently 175

10.3.1 Sizing the Problem 175

10.3.2 Aside: Longstaff-Schwartz for Valuations and Expected Exposures 176

10.3.3 Calculating VaR inside a Monte Carlo 179

10.3.4 Case Study: Swap Portfolios 182

10.3.5 Adapting LSAC to VaR under Delta-Gamma Approximation 184

10.4 Conclusions on MVA 184

CHAPTER 11 The Funding Curve 187

11.1 Sources for the Funding Curve 187

11.1.1 Term Funding 188

11.1.2 Rolling Funding 188

11.2 Internal Funding Curves 188

11.2.1 Bank CDS Spread 188

11.2.2 Bank Bond Spread 189

11.2.3 Bank Bond-CDS Basis 189

11.2.4 Bank Treasury Transfer Price 190

11.2.5 Funding Strategy Approaches 190

11.3 External Funding Curves and Accounting 191

11.4 Multi-currency/Multi-asset Funding 192

PART THREE KVA: Capital Valuation Adjustment and Regulation

CHAPTER 12 Regulation: the Basel II and Basel III Frameworks 195

12.1 Introducing the Regulatory Capital Framework 195

12.1.1 Economic Capital 196

12.1.2 The Development of the Basel Framework 196

12.1.3 Pillar I: Capital Types and Choices 201

12.2 Market Risk 202

12.2.1 Trading Book and Banking Book 202

12.2.2 Standardised Method 202

12.2.3 Internal Model Method (IMM) 204

12.3 Counterparty Credit Risk 205

12.3.1 Weight Calculation 205

12.3.2 EAD Calculation 206

12.3.3 Internal Model Method (IMM) 208

12.4 CVA Capital 209

12.4.1 Standardised 209

12.4.2 Advanced 211

12.5 Other Sources of Regulatory Capital 213

12.5.1 Incremental Risk Charge (IRC) 213

12.5.2 Leverage Ratio 213

12.6 Forthcoming Regulation with Pricing Impact 214

12.6.1 Fundamental Review of the Trading Book 214

12.6.2 Revised Standardised Approach to Credit Risk 218

12.6.3 Bilateral Initial Margin 220

12.6.4 Prudent Valuation 220

12.6.5 EMIR and Frontloading 224

CHAPTER 13 KVA: Capital Valuation Adjustment 227

13.1 Introduction: Capital Costs in Pricing 227

13.1.1 Capital, Funding and Default 227

13.2 Extending Semi-replication to Include Capital 228

13.3 The Cost of Capital 232

13.4 KVA for Market Risk, Counterparty Credit Risk and CVA Regulatory Capital 232

13.4.1 Standardised Approaches 232

13.4.2 IMM Approaches 233

13.5 The Size of KVA 233

13.6 Conclusion: KVA 237

CHAPTER 14 CVA Risk Warehousing and Tax Valuation Adjustment (TVA) 239

14.1 Risk Warehousing XVA 239

14.2 Taxation 239

14.3 CVA Hedging and Regulatory Capital 240

14.4 Warehousing CVA Risk and Double Semi-Replication 240

CHAPTER 15 Portfolio KVA and the Leverage Ratio 247

15.1 The Need for a Portfolio Level Model 247

15.2 Portfolio Level Semi-replication 248

15.3 Capital Allocation 254

15.3.1 Market Risk 255

15.3.2 Counterparty Credit Risk (CCR) 255

15.3.3 CVA Capital 255

15.3.4 Leverage Ratio 256

15.3.5 Capital Allocation and Uniqueness 257

15.4 Cost of Capital to the Business 257

15.5 Portfolio KVA 258

15.6 Calculating Portfolio KVA by Regression 258

PART FOUR XVA Implementation

CHAPTER 16 Hybrid Monte Carlo Models for XVA: Building a Model for the...