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 Banca IMI 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 in 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.

Ignition xv

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

2.6 Funding 41

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

3.2 Firm Value Models: Hints at the Multiname Picture 64

3.3 Reduced Form (Intensity) Models 65

3.4 Intensity Models: The Multiname Picture 78

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

4.2 The Probabilistic Framework 92

4.3 The General Pricing Formula for Unilateral Counterparty Risk 94

4.4 Interest Rate Swap (IRS) Portfolios 97

4.5 Numerical Tests 106

4.6 Conclusions 120

5 Wrong Way Risk (WWR) for Interest Rates 121

5.1 Modelling Assumptions 122

5.2 Numerical Methods 127

5.3 Results and Discussion 129

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

6.3 Forward versus Futures Prices 140

6.4 Swaps and Counterparty Risk 142

6.5 UCVA for Commodity Swaps 144

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

7.2 Modelling Assumptions 155

7.3 CDS Options Embedded in CVA Pricing 158

7.4 UCVA for Credit Default Swaps: A Case Study 160

7.5 Conclusions 164

8 Unilateral CVA for Equity with WWR 167

8.1 Counterparty Risk for Equity Without a Full Hybrid Model 167

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

8.3 Model Calibration and Empirical Results 180

8.4 Counterparty Risk and Wrong Way Risk 191

9 Unilateral CVA for FX 205

9.1 Pricing with Two Currencies: Foundations 206

9.2 Unilateral CVA for a Fixed-Fixed CCS 210

9.3 Unilateral CVA for Cross Currency Swaps with Floating Legs 224

9.4 Why a Cross Currency Basis? 226

9.5 CVA for CCS in Practice 230

9.6 Novations and the Cost of Liquidity 237

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

10.3 Unilateral Debit Valuation Adjustment (UDVA) 250

10.4 Bilateral Risk and DVA 251

10.5 Undesirable Features of DVA 253

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

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

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

10.12 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

11.4 What Next on Funding? 278

12 Bilateral CVA-DVA and Interest Rate Products 279

12.1 Arbitrage-Free Valuation of Bilateral Counterparty Risk 281

12.2 Modelling Assumptions 286

12.3 Numerical Methods 290

12.4 Results and Discussion 291

12.5 Conclusions 302

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

13.1 Trading Under the ISDA Master Agreement 306

13.2 Bilateral CVA Formula under Collateralization 310

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

13.6 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

14.2 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

14.5 Conclusions 329

15 Bilateral Collateralized CVA and DVA for Rates and Credit 331

15.1 CBVA for Interest Rate Swaps 332

15.2 Modelling Credit Contagion 340

15.3 CBVA for Credit Default Swaps 345

15.4 Conclusions 349

16 Including Margining Costs in Collateralized Contracts 351

16.1 Trading Under the ISDA Master Agreement 352

16.2 CBVA General Formula with Margining Costs 355

16.3 Changing the Collateralization Currency 357

Foreign-Currency Collaterals 359

16.4 Conclusions 359

17 Funding Valuation Adjustment (FVA)? 361

17.1 Dealing with Costs of Funding 361

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

17.3 Funding Risk and Liquidity Policies 367

17.4 CBVA Pricing Equation with Funding Costs (CFBVA) 372

17.5 Detailed Examples 378

17.6 Conclusions: FVA and Beyond 382

18 Non-Standard Asset Classes: Longevity Risk 385

18.1 Introduction to Longevity Markets 385

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