# Counterparty Credit Risk, Collateral and Funding With Pricing Cases For All Asset Classes

, by Brigo, Damiano; Morini, Massimo; Pallavicini, Andrea**Note:**Supplemental materials are not guaranteed with Rental or Used book purchases.

- ISBN: 9780470748466 | 047074846X
- Cover: Hardcover
- Copyright: 4/22/2013

**Professor Damiano Brigo** is Chair of Mathematical Finance and co-Head of Group at Imperial College, London. Damiano is also Director of the Capco Research Institute. His previous roles include Gilbart Professor and Head of Group at King's College, Managing Director and Global Head of Quantitative Innovation in Fitch, Head of Credit Models in Banca IMI, Fixed Income Professor at Bocconi University in Milan, and Quantitative Analyst at Banca Intesa. He has worked on quantitative analysis of counterparty risk, interest rates-, FX-, credit- and equity- derivatives, risk management and structured products, and funding costs and collateral modelling. Damiano has published 70+ works in top journals for Mathematical Finance, Systems Theory, Probability and Statistics, with H-index 24 on Scholar, and books for Springer and John Wiley & Sons that became field references in stochastic interest rate and credit modelling. Damiano is Managing Editor of *the International Journal of Theoretical and Applied Finance*, and has been listed as the most cited author in *Risk Magazine* in 2006 and 2010. Damiano obtained a Ph.D. in stochastic filtering with differential geometry in 1996 from the Free University of Amsterdam, following a BSc in Mathematics with honours from the University of Padua.

**Massimo Morini** is Head of Interest Rate and Credit Models and Coordinator of Model Research at IMI Bank of Intesa San Paolo. Massimo is also Professor of Fixed Income at Bocconi University and was a Research Fellow at Cass Business School, City University London. He regularly delivers advanced training in London, New York and worldwide. He has led workshops on credit risk and the financial crisis at major international conferences. He has published papers in journals including *Risk Magazine*, *Mathematical Finance*, and the *Journal of Derivatives*, and is the author of *Understanding and Managing Model Risk: A Practical Guide for Quants, Traders and Validators*. Massimo holds a PhD in Mathematics and an MSc in Economics.

**Andrea Pallavicini** is Head of Equity, FX and Commodity Models at Banca IMI, where he has the responsibility of numerical algorithm's design, financial modelling and research activity. He is also Visiting Professor at the Department of Mathematics of the Imperial College London. Previously, he held positions as Head of Financial Models at Mediobanca and Head of Financial Engineering at Banca Leonardo, he worked also in aerospace industries and financial institutions. He has a Degree in Astrophysics and a Ph.D. in Theoretical and Mathematical Physics from the University of Pavia for his research activity at CERN laboratory in Genève. Over the years he has written books in finance and he published several academic and practitioner-oriented articles in financial modelling, theoretical physics and astrophysics on major peer-reviewed journals. He teaches regularly at professional training courses and at Master and Ph.D. courses in finance at different Universities and private institutions. His main contributions in finance concern interest-rate and credit modelling, counterparty credit risk, and hybrid derivative pricing.

Abbreviations and Notation xxiii

PART I COUNTERPARTY CREDIT RISK, COLLATERAL AND FUNDING

1 Introduction 3

1.1 A Dialogue on CVA 3

1.2 Risk Measurement: Credit VaR 3

1.3 Exposure, CE, PFE, EPE, EE, EAD 5

1.4 Exposure and Credit VaR 7

1.5 Interlude: P and Q 7

1.6 Basel 8

1.7 CVA and Model Dependence 9

1.8 Input and Data Issues on CVA 10

1.9 Emerging Asset Classes: Longevity Risk 11

1.10 CVA and Wrong Way Risk 12

1.11 Basel III: VaR of CVA and Wrong Way Risk 13

1.12 Discrepancies in CVA Valuation: Model Risk and Payoff Risk 14

1.13 Bilateral Counterparty Risk: CVA and DVA 15

1.14 First-to-Default in CVA and DVA 17

1.15 DVA Mark-to-Market and DVA Hedging 18

1.16 Impact of Close-Out in CVA and DVA 19

1.17 Close-Out Contagion 20

1.18 Collateral Modelling in CVA and DVA 21

1.19 Re-Hypothecation 22

1.20 Netting 22

1.21 Funding 23

1.22 Hedging Counterparty Risk: CCDS 25

1.23 Restructuring Counterparty Risk: CVA-CDOs and Margin Lending 26

2 Context 31

2.1 Definition of Default: Six Basic Cases 31

2.2 Definition of Exposures 32

2.3 Definition of Credit Valuation Adjustment (CVA) 35

2.4 Counterparty Risk Mitigants: Netting 37

2.5 Counterparty Risk Mitigants: Collateral 38

2.5.1 The Credit Support Annex (CSA) 39

2.5.2 The ISDA Proposal for a New Standard CSA 40

2.5.3 Collateral Effectiveness as a Mitigant 40

2.6 Funding 41

2.6.1 A First Attack on Funding Cost Modelling 42

2.6.2 The General Funding Theory and its Recursive Nature 42

2.7 Value at Risk (VaR) and Expected Shortfall (ES) of CVA 43

2.8 The Dilemma of Regulators and Basel III 44

3 Modelling the Counterparty Default 47

3.1 Firm Value (or Structural) Models 47

3.1.1 The Geometric Brownian Assumption 47

3.1.2 Merton’s Model 48

3.1.3 Black and Cox’s (1976) Model 50

3.1.4 Credit Default Swaps and Default Probabilities 54

3.1.5 Black and Cox (B&C) Model Calibration to CDS: Problems 55

3.1.6 The AT1P Model 57

3.1.7 A Case Study with AT1P: Lehman Brothers Default History 58

3.1.8 Comments 60

3.1.9 SBTV Model 61

3.1.10 A Case Study with SBTV: Lehman Brothers Default History 62

3.1.11 Comments 64

3.2 Firm Value Models: Hints at the Multiname Picture 64

3.3 Reduced Form (Intensity) Models 65

3.3.1 CDS Calibration and Intensity Models 66

3.3.2 A Simpler Formula for Calibrating Intensity to a Single CDS 70

3.3.3 Stochastic Intensity: The CIR Family 72

3.3.4 The Cox-Ingersoll-Ross Model (CIR) Short-Rate Model for r 72

3.3.5 Time-Inhomogeneous Case: CIR++ Model 74

3.3.6 Stochastic Diffusion Intensity is Not Enough: Adding Jumps. The

JCIR(++) Model 75

3.3.7 The Jump-Diffusion CIR Model (JCIR) 76

3.3.8 Market Incompleteness and Default Unpredictability 78

3.3.9 Further Models 78

3.4 Intensity Models: The Multiname Picture 78

3.4.1 Choice of Variables for the Dependence Structure 78

3.4.2 Firm Value Models? 80

3.4.3 Copula Functions 80

3.4.4 Copula Calibration, CDOs and Criticism of Copula Functions 86

PART II PRICING COUNTERPARTY RISK: UNILATERAL CVA

4 Unilateral CVA and Netting for Interest Rate Products 89

4.1 First Steps towards a CVA Pricing Formula 89

4.1.1 Symmetry versus Asymmetry 90

4.1.2 Modelling the Counterparty Default Process 91

4.2 The Probabilistic Framework 92

4.3 The General Pricing Formula for Unilateral Counterparty Risk 94

4.4 Interest Rate Swap (IRS) Portfolios 97

4.4.1 Counterparty Risk in Single IRS 97

4.4.2 Counterparty Risk in an IRS Portfolio with Netting 100

4.4.3 The Drift Freezing Approximation 102

4.4.4 The Three-Moments Matching Technique 104

4.5 Numerical Tests 106

4.5.1 Case A: IRS with Co-Terminal Payment Dates 106

4.5.2 Case B: IRS with Co-Starting Resetting Date 108

4.5.3 Case C: IRS with First Positive, Then Negative Flow 108

4.5.4 Case D: IRS with First Negative, Then Positive Flows 109

4.5.5 Case E: IRS with First Alternate Flows 113

4.6 Conclusions 120

5 Wrong Way Risk (WWR) for Interest Rates 121

5.1 Modelling Assumptions 122

5.1.1 G2++ Interest Rate Model 122

5.1.2 CIR++ Stochastic Intensity Model 123

5.1.3 CIR++ Model: CDS Calibration 124

5.1.4 Interest Rate/Credit Spread Correlation 126

5.1.5 Adding Jumps to the Credit Spread 126

5.2 Numerical Methods 127

5.2.1 Discretization Scheme 128

5.2.2 Simulating Intensity Jumps 128

5.2.3 “American Monte Carlo” (Pallavicini 2006) 128

5.2.4 Callable Payoffs 128

5.3 Results and Discussion 129

5.3.1 WWR in Single IRS 129

5.3.2 WWR in an IRS Portfolio with Netting 129

5.3.3 WWR in European Swaptions 130

5.3.4 WWR in Bermudan Swaptions 130

5.3.5 WWR in CMS Spread Options 132

5.4 Contingent CDS (CCDS) 132

5.5 Results Interpretation and Conclusions 133

6 Unilateral CVA for Commodities with WWR 135

6.1 Oil Swaps and Counterparty Risk 135

6.2 Modelling Assumptions 137

6.2.1 Commodity Model 137

6.2.2 CIR++ Stochastic-Intensity Model 139

6.3 Forward versus Futures Prices 140

6.3.1 CVA for Commodity Forwards without WWR 141

6.3.2 CVA for Commodity Forwards with WWR 142

6.4 Swaps and Counterparty Risk 142

6.5 UCVA for Commodity Swaps 144

6.5.1 Counterparty Risk from the Payer’s Perspective: The Airline

Computes Counterparty Risk 145

6.5.2 Counterparty Risk from the Receiver’s Perspective: The Bank

Computes Counterparty Risk 148

6.6 Inadequacy of Basel’s WWR Multipliers 148

6.7 Conclusions 151

7 Unilateral CVA for Credit with WWR 153

7.1 Introduction to CDSs with Counterparty Risk 153

7.1.1 The Structure of the Chapter 155

7.2 Modelling Assumptions 155

7.2.1 CIR++ Stochastic-Intensity Model 156

7.2.2 CIR++ Model: CDS Calibration 157

7.3 CDS Options Embedded in CVA Pricing 158

7.4 UCVA for Credit Default Swaps: A Case Study 160

7.4.1 Changing the Copula Parameters 160

7.4.2 Changing the Market Parameters 164

7.5 Conclusions 164

8 Unilateral CVA for Equity with WWR 167

8.1 Counterparty Risk for Equity Without a Full Hybrid Model 167

8.1.1 Calibrating AT1P to the Counterparty’s CDS Data 168

8.1.2 Counterparty Risk in Equity Return Swaps (ERS) 169

8.2 Counterparty Risk with a Hybrid Credit-Equity Structural Model 172

8.2.1 The Credit Model 172

8.2.2 The Equity Model 174

8.2.3 From Barrier Options to Equity Pricing 176

8.2.4 Equity and Equity Options 179

8.3 Model Calibration and Empirical Results 180

8.3.1 BP and FIAT in 2009 181

8.3.2 Uncertainty in Market Expectations 186

8.3.3 Further Results: FIAT in 2008 and BP in 2010 188

8.4 Counterparty Risk and Wrong Way Risk 191

8.4.1 Deterministic Default Barrier 193

8.4.2 Uncertainty on the Default Barrier 198

9 Unilateral CVA for FX 205

9.1 Pricing with Two Currencies: Foundations 206

9.2 Unilateral CVA for a Fixed-Fixed CCS 210

9.2.1 Approximating the Volatility of Cross Currency Swap Rates 216

9.2.2 Parameterization of the FX Correlation 218

9.3 Unilateral CVA for Cross Currency Swaps with Floating Legs 224

9.4 Why a Cross Currency Basis? 226

9.4.1 The Approach of Fujii, Shimada and Takahashi (2010) 227

9.4.2 Collateral Rates versus Risk-Free Rates 228

9.4.3 Consequences of Perfect Collateralization 229

9.5 CVA for CCS in Practice 230

9.5.1 Changing the CCS Moneyness 234

9.5.2 Changing the Volatility 235

9.5.3 Changing the FX Correlations 235

9.6 Novations and the Cost of Liquidity 237

9.6.1 A Synthetic Contingent CDS: The Novation 238

9.6.2 Extending the Approach to the Valuation of Liquidity 241

9.7 Conclusions 243

PART III ADVANCED CREDIT AND FUNDING RISK PRICING

10 New Generation Counterparty and Funding Risk Pricing 247

10.1 Introducing the Advanced Part of the Book 247

10.2 What We Have Seen Before: Unilateral CVA 249

10.2.1 Approximation: Default Bucketing and Independence 250

10.3 Unilateral Debit Valuation Adjustment (UDVA) 250

10.4 Bilateral Risk and DVA 251

10.5 Undesirable Features of DVA 253

10.5.1 Profiting From Own Deteriorating Credit Quality 253

10.5.2 DVA Hedging? 253

10.5.3 DVA: Accounting versus Capital Requirements 254

10.5.4 DVA: Summary and Debate on Realism 255

10.6 Close-Out: Risk-Free or Replacement? 256

10.7 Can We Neglect the First-to-Default Time? 257

10.7.1 A Simplified Formula without First-to-Default: The Case of an

Equity Forward 258

10.8 Payoff Risk 258

10.9 Collateralization, Gap Risk and Re-Hypothecation 259

10.10 Funding Costs 262

10.11 Restructuring Counterparty Risk 263

10.11.1 CVA Volatility: The Wrong Way 263

10.11.2 Floating Margin Lending 264

10.11.3 Global Valuation 265

10.12 Conclusions 266

11 A First Attack on Funding Cost Modelling 269

11.1 The Problem 269

11.2 A Closer Look at Funding and Discounting 271

11.3 The Approach Proposed by Morini and Prampolini (2010) 272

11.3.1 The Borrower’s Case 273

11.3.2 The Lender’s Case 274

11.3.3 The Controversial Role of DVA: The Borrower 275

11.3.4 The Controversial Role of DVA: The Lender 276

11.3.5 Discussion 277

11.4 What Next on Funding? 278

12 Bilateral CVA–DVA and Interest Rate Products 279

12.1 Arbitrage-Free Valuation of Bilateral Counterparty Risk 281

12.1.1 Symmetry versus Asymmetry 285

12.1.2 Worsening of Credit Quality and Positive Mark-to-Market 285

12.2 Modelling Assumptions 286

12.2.1 G2++ Interest Rate Model 286

12.2.2 CIR++ Stochastic Intensity Model 288

12.2.3 Realistic Market Data Set for CDS Options 289

12.3 Numerical Methods 290

12.4 Results and Discussion 291

12.4.1 Bilateral VA in Single IRS 292

12.4.2 Bilateral VA in an IRS Portfolio with Netting 296

12.4.3 Bilateral VA in Exotic Interest Rate Products 301

12.5 Conclusions 302

13 Collateral, Netting, Close-Out and Re-Hypothecation 305

13.1 Trading Under the ISDA Master Agreement 306

13.1.1 Mathematical Setup and CBVA Definition 306

13.1.2 Collateral Delay and Dispute Resolutions 308

13.1.3 Close-Out Netting Rules 308

13.1.4 Collateral Re-Hypothecation 309

13.2 Bilateral CVA Formula under Collateralization 310

13.2.1 Collecting CVA Contributions 310

13.2.2 CBVA General Formula 312

13.2.3 CCVA and CDVA Definitions 312

13.3 Close-Out Amount Evaluation 313

13.4 Special Cases of Collateral-Inclusive Bilateral Credit

Valuation Adjustment 314

13.5 Example of Collateralization Schemes 315

13.5.1 Perfect Collateralization 315

13.5.2 Collateralization Through Margining 316

13.6 Conclusions 316

14 Close-Out and Contagion with Examples of a Simple Payoff 319

14.1 Introduction to Close-Out Modelling and Earlier Work 319

14.1.1 Close-Out Modelling: Context 319

14.1.2 Legal Documentation on Close-Out 320

14.1.3 Literature 320

14.1.4 Risk-Free versus Replacement Close-Out: Practical

Consequences 321

14.2 Classical Unilateral and Bilateral Valuation Adjustments 322

14.3 Bilateral Adjustment and Close-Out: Risk-Free or Replacement? 323

14.4 A Quantitative Analysis and a Numerical Example 323

14.4.1 Contagion Issues 326

14.5 Conclusions 329

15 Bilateral Collateralized CVA and DVA for Rates and Credit 331

15.1 CBVA for Interest Rate Swaps 332

15.1.1 Changing the Margining Frequency 332

15.1.2 Inspecting the Exposure Profiles 334

15.1.3 A Case Where Re-Hypothecation is Worse than No

Collateral at All 335

15.1.4 Changing the Correlation Parameters 336

15.1.5 Changing the Credit Spread Volatility 337

15.2 Modelling Credit Contagion 340

15.2.1 The CDS Price Process 340

15.2.2 Calculation of Survival Probability 341

15.2.3 Modelling Default-Time Dependence 344

15.3 CBVA for Credit Default Swaps 345

15.3.1 Changing the Copula Parameters 345

15.3.2 Inspecting the Contagion Risk 347

15.3.3 Changing the CDS Moneyness 347

15.4 Conclusions 349

16 Including Margining Costs in Collateralized Contracts 351

16.1 Trading Under the ISDA Master Agreement 352

16.1.1 Collateral Accrual Rates 352

16.1.2 Collateral Management and Margining Costs 353

16.2 CBVA General Formula with Margining Costs 355

16.2.1 Perfect Collateralization 356

16.2.2 Futures Contracts 357

16.3 Changing the Collateralization Currency 357

16.3.1 Margining Cost in Foreign Currency 357

16.3.2 Settlement Liquidity Risk 358

16.3.3 Gap Risk in Single-Currency Contracts with

Foreign-Currency Collaterals 359

16.4 Conclusions 359

17 Funding Valuation Adjustment (FVA)? 361

17.1 Dealing with Costs of Funding 361

17.1.1 Central Clearing, CCPs and this Book 362

17.1.2 High Level Features 362

17.1.3 Single-Deal (Micro) vs. Homogeneous (Macro) Funding Models 363

17.1.4 Previous Literature on Funding and Collateral 364

17.1.5 Including FVA along with Credit and Debit Valuation

Adjustment 365

17.1.6 FVA is not DVA 365

17.2 Collateral- and Funding-Inclusive Bilateral Valuation Adjusted Price 366

17.3 Funding Risk and Liquidity Policies 367

17.3.1 Funding, Hedging and Collateralization 367

17.3.2 Liquidity Policies 368

17.4 CBVA Pricing Equation with Funding Costs (CFBVA) 372

17.4.1 Iterative Solution of the CFBVA Pricing Equation 373

17.4.2 Funding Derivative Contracts in a Diffusion Setting 374

17.4.3 Implementing Hedging Strategies via Derivative Markets 377

17.5 Detailed Examples 378

17.5.1 Funding with Collateral 378

17.5.2 Collateralized Contracts Priced by a CCP 379

17.5.3 Dealing with Own Credit Risk: FVA and DVA 380

17.5.4 Deriving Earlier Results on FVA and DVA 381

17.6 Conclusions: FVA and Beyond 382

18 Non-Standard Asset Classes: Longevity Risk 385

18.1 Introduction to Longevity Markets 385

18.1.1 The Longevity Swap Market 385

18.1.2 Longevity Swaps: Collateral and Credit Risk 386

18.1.3 Indexed Longevity Swaps 390

18.1.4 Endogenous Credit Collateral and Funding-Inclusive Swap Rates 390

18.2 Longevity Swaps: The Payoff 391

18.3 Mark-to-Market for Longevity Swaps 394

18.4 Counterparty and Own Default Risk, Collateral and Funding 397

18.5 An Example of Modelling Specification from Biffis et al. (2011) 401

18.6 Discussion of the Results in Biffis et al. (2011) 404

19 Conclusions and Further Work 409

19.1 A Final Dialogue: Models, Regulations, CVA/DVA, Funding and More 409

Bibliography 415

Index 423