XVA

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 xviiList of Figures xxiAcknowledgements xxvCHAPTER 1 Introduction: The Valuation of Derivative Portfolios 11.1 What this book is about 11.2 Prices and Values 41.2.1 Before the Fall... 41.2.2 The Post-Crisis World... 51.3 Trade Economics in Derivative Pricing 61.3.1 The Components of a Price 61.3.2 Risk-Neutral Valuation 81.3.3 Hedging and Management Costs 111.3.4 Credit Risk: CVA/DVA 111.3.5 FVA 131.3.6 Regulatory Capital and KVA 141.4 Post-Crisis Derivative Valuation or How I Learned to Stop Worrying and Love FVA 161.4.1 The FVA Debate and the Assault on Black-Scholes-Merton 161.4.2 Different Values for Different Purposes 191.4.3 Summary: The Valuation Paradigm Shift 211.5 Reading this Book 21PART ONE CVA and DVA: Counterparty Credit Risk and Credit Valuation AdjustmentCHAPTER 2 Introducing Counterparty Risk 252.1 Defining Counterparty Risk 252.1.1 Wrong-way and Right-way Risk 272.2 CVA and DVA: Credit Valuation Adjustment and Debit Valuation Adjustment Defined 272.3 The Default Process 282.3.1 Example Default: The Collapse of Lehman Brothers 302.4 Credit Risk Mitigants 302.4.1 Netting 302.4.2 Collateral/Security 312.4.3 Central Clearing and Margin 342.4.4 Capital 352.4.5 Break Clauses 352.4.6 Buying Protection 37CHAPTER 3 CVA and DVA: Credit and Debit Valuation Adjustment Models 393.1 Introduction 393.1.1 Close-out and CVA 403.2 Unilateral CVA Model 423.2.1 Unilateral CVA by Expectation 423.2.2 Unilateral CVA by Replication 433.3 Bilateral CVA Model: CVA and DVA 483.3.1 Bilateral CVA by Expectation 483.3.2 Bilateral CVA by Replication 503.3.3 DVA and Controversy 533.4 Modelling Dependence between Counterparties 553.4.1 Gaussian Copula Model 553.4.2 Other Copula Models 563.5 Components of a CVA Calculation Engine 573.5.1 Monte Carlo Simulation 573.5.2 Trade Valuation and Approximations 573.5.3 Expected Exposure Calculation 593.5.4 Credit Integration 593.6 Counterparty Level CVA vs. Trade Level CVA 593.6.1 Incremental CVA 603.6.2 Allocated CVA 603.7 Recovery Rate/Loss-Given-Default Assumptions 63CHAPTER 4 CDS and Default Probabilities 654.1 Survival Probabilities and CVA 654.2 Historical versus Implied Survival Probabilities 664.3 Credit Default Swap Valuation 674.3.1 Credit Default Swaps 674.3.2 Premium Leg 694.3.3 Protection Leg 714.3.4 CDS Value and Breakeven Spread 724.4 Bootstrapping the Survival Probability Function 724.4.1 Upfront Payments 744.4.2 Choice of Hazard Rate Function and CVA: CVA Carry 754.4.3 Calibration Problems 764.5 CDS and Capital Relief 774.6 Liquid and Illiquid Counterparties 784.6.1 Mapping to Representative CDS 794.6.2 Mapping to Baskets and Indices 804.6.3 Cross-sectional Maps 81CHAPTER 5 Analytic Models for CVA and DVA 835.1 Analytic CVA Formulae 835.2 Interest Rate Swaps 845.2.1 Unilateral CVA 845.2.2 Bilateral CVA 865.3 Options: Interest Rate Caplets and Floorlets 865.4 FX Forwards 88CHAPTER 6 Modelling Credit Mitigants 916.1 Credit Mitigants 916.2 Close-out Netting 916.3 Break Clauses 936.3.1 Mandatory Break Clauses 936.3.2 Optional Break Clauses 936.4 Variation Margin and CSA Agreements 976.4.1 Simple Model: Modifying the Payout Function 976.4.2 Modelling Collateral Directly 996.4.3 Lookback Method 1016.4.4 Modelling Downgrade Triggers in CSA Agreements 1026.5 Non-financial Security and the Default Waterfall 107CHAPTER 7 Wrong-way and Right-way Risk for CVA 1097.1 Introduction: Wrong-way and Right-way Risks 1097.1.1 Modelling Wrong-way Risk and CVA 1107.2 Distributional Models of Wrong-way/Right-way Risk 1117.2.1 Simple Model: Increased Exposure 1117.2.2 Copula Models 1117.2.3 Linear Models and Discrete Models 1147.3 A Generalised Discrete Approach to Wrong-way Risk 1167.4 Stochastic Credit Models of Wrong-way/Right-way Risk 1187.4.1 Sovereign Wrong-way Risk 1197.5 Wrong-way/Right-way Risk and DVA 119PART TWO FVA: Funding Valuation AdjustmentCHAPTER 8 The Discount Curve 1238.1 Introduction 1238.2 A Single Curve World 1238.3 Curve Interpolation and Smooth Curves 1268.4 Cross-currency Basis 1278.5 Multi-curve and Tenor Basis 1288.6 OIS and CSA Discounting 1298.6.1 OIS as the Risk-free Rate 1298.6.2 OIS and CSA Discounting 1318.6.3 Multi-currency Collateral and the Collateral Option 1348.7 Conclusions: Discounting 138CHAPTER 9 Funding Costs: Funding Valuation Adjustment (FVA) 1399.1 Explaining Funding Costs 1399.1.1 What is FVA? 1399.1.2 General Principle of Funding Costs 1459.2 First Generation FVA: Discount Models 1459.3 Double Counting and DVA 1469.4 Second Generation FVA: Exposure Models 1479.4.1 The Burgard-Kjaer Semi-Replication Model 1489.5 Residual FVA and CSAs 1609.6 Asymmetry 1619.6.1 Case 1: Corporate vs. Bank Asymmetry 1619.6.2 Case 2: Bank vs. Bank Asymmetry 1629.7 Risk Neutrality, Capital and the Modigliani-Miller Theorem 1629.7.1 No Market-wide Risk-neutral Measure 1629.7.2 Consequences 1659.7.3 The Modigliani-Miller Theorem 1659.8 Wrong-way/Right-way Risk and FVA 166CHAPTER 10 Other Sources of Funding Costs: CCPs and MVA 16710.1 Other Sources of Funding Costs 16710.1.1 Central Counterparty Funding Costs 16710.1.2 Bilateral Initial Margin 17010.1.3 Rating Agency Volatility Buffers and Overcollateralisation 17010.1.4 Liquidity Buffers 17010.2 MVA: Margin Valuation Adjustment by Replication 17110.2.1 Semi-replication with no Shortfall on Default 17410.3 Calculating MVA Efficiently 17510.3.1 Sizing the Problem 17510.3.2 Aside: Longstaff-Schwartz for Valuations and Expected Exposures 17610.3.3 Calculating VaR inside a Monte Carlo 17910.3.4 Case Study: Swap Portfolios 18210.3.5 Adapting LSAC to VaR under Delta-Gamma Approximation 18410.4 Conclusions on MVA 184CHAPTER 11 The Funding Curve 18711.1 Sources for the Funding Curve 18711.1.1 Term Funding 18811.1.2 Rolling Funding 18811.2 Internal Funding Curves 18811.2.1 Bank CDS Spread 18811.2.2 Bank Bond Spread 18911.2.3 Bank Bond-CDS Basis 18911.2.4 Bank Treasury Transfer Price 19011.2.5 Funding Strategy Approaches 19011.3 External Funding Curves and Accounting 19111.4 Multi-currency/Multi-asset Funding 192PART THREE KVA: Capital Valuation Adjustment and RegulationCHAPTER 12 Regulation: the Basel II and Basel III Frameworks 19512.1 Introducing the Regulatory Capital Framework 19512.1.1 Economic Capital 19612.1.2 The Development of the Basel Framework 19612.1.3 Pillar I: Capital Types and Choices 20112.2 Market Risk 20212.2.1 Trading Book and Banking Book 20212.2.2 Standardised Method 20212.2.3 Internal Model Method (IMM) 20412.3 Counterparty Credit Risk 20512.3.1 Weight Calculation 20512.3.2 EAD Calculation 20612.3.3 Internal Model Method (IMM) 20812.4 CVA Capital 20912.4.1 Standardised 20912.4.2 Advanced 21112.5 Other Sources of Regulatory Capital 21312.5.1 Incremental Risk Charge (IRC) 21312.5.2 Leverage Ratio 21312.6 Forthcoming Regulation with Pricing Impact 21412.6.1 Fundamental Review of the Trading Book 21412.6.2 Revised Standardised Approach to Credit Risk 21812.6.3 Bilateral Initial Margin 22012.6.4 Prudent Valuation 22012.6.5 EMIR and Frontloading 224CHAPTER 13 KVA: Capital Valuation Adjustment 22713.1 Introduction: Capital Costs in Pricing 22713.1.1 Capital, Funding and Default 22713.2 Extending Semi-replication to Include Capital 22813.3 The Cost of Capital 23213.4 KVA for Market Risk, Counterparty Credit Risk and CVA Regulatory Capital 23213.4.1 Standardised Approaches 23213.4.2 IMM Approaches 23313.5 The Size of KVA 23313.6 Conclusion: KVA 237CHAPTER 14 CVA Risk Warehousing and Tax Valuation Adjustment (TVA) 23914.1 Risk Warehousing XVA 23914.2 Taxation 23914.3 CVA Hedging and Regulatory Capital 24014.4 Warehousing CVA Risk and Double Semi-Replication 240CHAPTER 15 Portfolio KVA and the Leverage Ratio 24715.1 The Need for a Portfolio Level Model 24715.2 Portfolio Level Semi-replication 24815.3 Capital Allocation 25415.3.1 Market Risk 25515.3.2 Counterparty Credit Risk (CCR) 25515.3.3 CVA Capital 25515.3.4 Leverage Ratio 25615.3.5 Capital Allocation and Uniqueness 25715.4 Cost of Capital to the Business 25715.5 Portfolio KVA 25815.6 Calculating Portfolio KVA by Regression 258PART FOUR XVA ImplementationCHAPTER 16 Hybrid Monte Carlo Models for XVA: Building a Model for the Expected-Exposure Engine 26316.1 Introduction 26316.1.1 Implementing XVA 26316.1.2 XVA and Monte Carlo 26316.1.3 XVA and Models 26416.1.4 A Roadmap to XVA Hybrid Monte Carlo 26716.2 Choosing the Calibration: Historical versus Implied 26816.2.1 The Case for Historical Calibration 26816.2.2 The Case for Market Implied Calibration 28116.3 The Choice of Interest Rate Modelling Framework 28516.3.1 Interest Rate Models (for XVA) 28616.3.2 The Heath-Jarrow-Morton (HJM) Framework and Models of the Short Rate 28616.3.3 The Brace-Gaterak-Musiela (BGM) or Market Model Framework 30516.3.4 Choice of Numeraire 31316.3.5 Multi-curve: Tenor and Cross-currency Basis 31416.3.6 Close-out and the Choice of Discount Curve 31816.4 FX and Cross-currency Models 31916.4.1 A Multi-currency Generalised Hull-White Model 32016.4.2 The Triangle Rule and Options on the FX Cross 32216.4.3 Models with FX Volatility Smiles 32416.5 Inflation 32716.5.1 The Jarrow-Yildirim Model (using Hull-White Dynamics) 32716.5.2 Other Approaches 33616.6 Equities 33716.6.1 A Simple Log-normal Model 33716.6.2 Dividends 33916.6.3 Indices and Baskets 33916.6.4 Managing Correlations 34016.6.5 Skew: Local Volatility and Other Models 34016.7 Commodities 34216.7.1 Precious Metals 34216.7.2 Forward-based Commodities 34216.7.3 Electricity and Spark Spreads 34716.8 Credit 34816.8.1 A Simple Gaussian Model 34916.8.2 JCIR++ 35016.8.3 Other Credit Models, Wrong-way Risk Models and Credit Correlation 351CHAPTER 17 Monte Carlo Implementation 35317.1 Introduction 35317.2 Errors in Monte Carlo 35317.2.1 Discretisation Errors 35417.2.2 Random Errors 35717.3 Random Numbers 35917.3.1 Pseudo-random Number Generators 35917.3.2 Quasi-random Number Generators 36417.3.3 Generating Normal Samples 36917.4 Correlation 37217.4.1 Correlation Matrix Regularisation 37217.4.2 Inducing Correlation 37317.5 Path Generation 37517.5.1 Forward Induction 37517.5.2 Backward Induction 375CHAPTER 18 Monte Carlo Variance Reduction and Performance Enhancements 37718.1 Introduction 37718.2 Classic Methods 37718.2.1 Antithetics 37718.2.2 Control Variates 37818.3 Orthogonalisation 37918.4 Portfolio Compression 38118.5 Conclusion: Making it Go Faster! 382CHAPTER 19 Valuation Models for Use with Monte Carlo Exposure Engines 38319.1 Valuation Models 38319.1.1 Consistent or Inconsistent Valuation? 38419.1.2 Performance Constraints 38419.1.3 The Case for XVA Valuation Consistent with Trade Level Valuations 38519.1.4 The Case for Consistent XVA Dynamics 38619.1.5 Simulated Market Data and Valuation Model Compatibility 38719.1.6 Valuation Differences as a KPI 38719.1.7 Scaling 38719.2 Implied Volatility Modelling 38819.2.1 Deterministic Models 38819.2.2 Stochastic Models 38919.3 State Variable-based Valuation Techniques 38919.3.1 Grid Interpolation 39019.3.2 Longstaff-Schwartz 391CHAPTER 20 Building the Technological Infrastructure 39320.1 Introduction 39320.2 System Components 39320.2.1 Input Data 39420.2.2 Calculation 40120.2.3 Reporting 40520.3 Hardware 40520.3.1 CPU 40620.3.2 GPU and GPGPU 40620.3.3 Intel® Xeon PhiTM 40720.3.4 FPGA 40820.3.5 Supercomputers 40820.4 Software 40820.4.1 Roles and Responsibilities 40920.4.2 Development and Project Management Practice 41020.4.3 Language Choice 41520.4.4 CPU Languages 41620.4.5 GPU Languages 41720.4.6 Scripting and Payout Languages 41820.4.7 Distributed Computing and Parallelism 41820.5 Conclusion 421PART FIVE Managing XVACHAPTER 21 Calculating XVA Sensitivities 42521.1 XVA Sensitivities 42521.1.1 Defining the Sensitivities 42521.1.2 Jacobians and Hessians 42621.1.3 Theta, Time Decay and Carry 42721.1.4 The Explain 43121.2 Finite Difference Approximation 43421.2.1 Estimating Sensitivities 43421.2.2 Recalibration? 43521.2.3 Exercise Boundaries and Sensitivities 43621.3 Pathwise Derivatives and Algorithmic Differentiation 43721.3.1 Preliminaries: The Pathwise Method 43821.3.2 Adjoints 44021.3.3 Adjoint Algorithmic Differentiation 44221.3.4 Hybrid Approaches and Longstaff-Schwartz 44321.4 Scenarios and Stress Tests 445CHAPTER 22 Managing XVA 44722.1 Introduction 44722.2 Organisational Design 44822.2.1 Separate XVA Functions 44822.2.2 Central XVA 45122.3 XVA, Treasury and Portfolio Management 45322.3.1 Treasury 45322.3.2 Loan Portfolio Management 45422.4 Active XVA Management 45422.4.1 Market Risks 45522.4.2 Counterparty Credit Risk Hedging 45722.4.3 Hedging DVA? 45822.4.4 Hedging FVA 45922.4.5 Managing and Hedging Capital 45922.4.6 Managing Collateral and MVA 46022.5 Passive XVA Management 46022.6 Internal Charging for XVA 46022.6.1 Payment Structures 46122.6.2 The Charging Process 46122.7 Managing Default and Distress 462PART SIX The FutureCHAPTER 23 The Future of Derivatives? 46523.1 Reflecting on the Years of Change... 46523.2 The Market in the Future 46523.2.1 Products 46623.2.2 CCPs, Clearing and MVA 46623.2.3 Regulation, Capital and KVA 46723.2.4 Computation, Automation and eTrading 46723.2.5 Future Models and Future XVA 468Bibliography 469Index 489