XVA: Credit, Funding and Capital Valuation Adjustments

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 Expected-Exposure Engine 263
16.1 Introduction 263
16.1.1 Implementing XVA 263
16.1.2 XVA and Monte Carlo 263
16.1.3 XVA and Models 264
16.1.4 A Roadmap to XVA Hybrid Monte Carlo 267
16.2 Choosing the Calibration: Historical versus Implied 268
16.2.1 The Case for Historical Calibration 268
16.2.2 The Case for Market Implied Calibration 281
16.3 The Choice of Interest Rate Modelling Framework 285
16.3.1 Interest Rate Models (for XVA) 286
16.3.2 The Heath-Jarrow-Morton (HJM) Framework and Models of the Short Rate 286
16.3.3 The Brace-Gaterak-Musiela (BGM) or Market Model Framework 305
16.3.4 Choice of Numeraire 313
16.3.5 Multi-curve: Tenor and Cross-currency Basis 314
16.3.6 Close-out and the Choice of Discount Curve 318
16.4 FX and Cross-currency Models 319
16.4.1 A Multi-currency Generalised Hull-White Model 320
16.4.2 The Triangle Rule and Options on the FX Cross 322
16.4.3 Models with FX Volatility Smiles 324
16.5 Inflation 327
16.5.1 The Jarrow-Yildirim Model (using Hull-White Dynamics) 327
16.5.2 Other Approaches 336
16.6 Equities 337
16.6.1 A Simple Log-normal Model 337
16.6.2 Dividends 339
16.6.3 Indices and Baskets 339
16.6.4 Managing Correlations 340
16.6.5 Skew: Local Volatility and Other Models 340
16.7 Commodities 342
16.7.1 Precious Metals 342
16.7.2 Forward-based Commodities 342
16.7.3 Electricity and Spark Spreads 347
16.8 Credit 348
16.8.1 A Simple Gaussian Model 349
16.8.2 JCIR++ 350
16.8.3 Other Credit Models, Wrong-way Risk Models and Credit Correlation 351

CHAPTER 17 Monte Carlo Implementation 353
17.1 Introduction 353
17.2 Errors in Monte Carlo 353
17.2.1 Discretisation Errors 354
17.2.2 Random Errors 357
17.3 Random Numbers 359
17.3.1 Pseudo-random Number Generators 359
17.3.2 Quasi-random Number Generators 364
17.3.3 Generating Normal Samples 369
17.4 Correlation 372
17.4.1 Correlation Matrix Regularisation 372
17.4.2 Inducing Correlation 373
17.5 Path Generation 375
17.5.1 Forward Induction 375
17.5.2 Backward Induction 375

CHAPTER 18 Monte Carlo Variance Reduction and Performance Enhancements 377
18.1 Introduction 377
18.2 Classic Methods 377
18.2.1 Antithetics 377
18.2.2 Control Variates 378
18.3 Orthogonalisation 379
18.4 Portfolio Compression 381
18.5 Conclusion: Making it Go Faster! 382

CHAPTER 19 Valuation Models for Use with Monte Carlo Exposure Engines 383
19.1 Valuation Models 383
19.1.1 Consistent or Inconsistent Valuation? 384
19.1.2 Performance Constraints 384
19.1.3 The Case for XVA Valuation Consistent with Trade Level Valuations 385
19.1.4 The Case for Consistent XVA Dynamics 386
19.1.5 Simulated Market Data and Valuation Model Compatibility 387
19.1.6 Valuation Differences as a KPI 387
19.1.7 Scaling 387
19.2 Implied Volatility Modelling 388
19.2.1 Deterministic Models 388
19.2.2 Stochastic Models 389
19.3 State Variable-based Valuation Techniques 389
19.3.1 Grid Interpolation 390
19.3.2 Longstaff-Schwartz 391

CHAPTER 20 Building the Technological Infrastructure 393
20.1 Introduction 393
20.2 System Components 393
20.2.1 Input Data 394
20.2.2 Calculation 401
20.2.3 Reporting 405
20.3 Hardware 405
20.3.1 CPU 406
20.3.2 GPU and GPGPU 406
20.3.3 Intel Xeon PhiTM 407
20.3.4 FPGA 408
20.3.5 Supercomputers 408
20.4 Software 408
20.4.1 Roles and Responsibilities 409
20.4.2 Development and Project Management Practice 410
20.4.3 Language Choice 415
20.4.4 CPU Languages 416
20.4.5 GPU Languages 417
20.4.6 Scripting and Payout Languages 418
20.4.7 Distributed Computing and Parallelism 418
20.5 Conclusion 421

PART FIVE Managing XVA

CHAPTER 21 Calculating XVA Sensitivities 425
21.1 XVA Sensitivities 425
21.1.1 Defining the Sensitivities 425
21.1.2 Jacobians and Hessians 426
21.1.3 Theta, Time Decay and Carry 427
21.1.4 The Explain 431
21.2 Finite Difference Approximation 434
21.2.1 Estimating Sensitivities 434
21.2.2 Recalibration? 435
21.2.3 Exercise Boundaries and Sensitivities 436
21.3 Pathwise Derivatives and Algorithmic Differentiation 437
21.3.1 Preliminaries: The Pathwise Method 438
21.3.2 Adjoints 440
21.3.3 Adjoint Algorithmic Differentiation 442
21.3.4 Hybrid Approaches and Longstaff-Schwartz 443
21.4 Scenarios and Stress Tests 445

CHAPTER 22 Managing XVA 447
22.1 Introduction 447
22.2 Organisational Design 448
22.2.1 Separate XVA Functions 448
22.2.2 Central XVA 451
22.3 XVA, Treasury and Portfolio Management 453
22.3.1 Treasury 453
22.3.2 Loan Portfolio Management 454
22.4 Active XVA Management 454
22.4.1 Market Risks 455
22.4.2 Counterparty Credit Risk Hedging 457
22.4.3 Hedging DVA? 458
22.4.4 Hedging FVA 459
22.4.5 Managing and Hedging Capital 459
22.4.6 Managing Collateral and MVA 460
22.5 Passive XVA Management 460
22.6 Internal Charging for XVA 460
22.6.1 Payment Structures 461
22.6.2 The Charging Process 461
22.7 Managing Default and Distress 462

PART SIX The Future

CHAPTER 23 The Future of Derivatives? 465
23.1 Reflecting on the Years of Change... 465
23.2 The Market in the Future 465
23.2.1 Products 466
23.2.2 CCPs, Clearing and MVA 466
23.2.3 Regulation, Capital and KVA 467
23.2.4 Computation, Automation and eTrading 467
23.2.5 Future Models and Future XVA 468

Bibliography 469
Index 489