Counterparty Credit Risk, Collateral and Funding
With Pricing Cases For All Asset Classes
Wiley Finance Series
1. Edition March 2013
464 Pages, Hardcover
Professional Book
ISBN:
978-0-470-74846-6
John Wiley & Sons
The book's content is focused on rigorous and advanced quantitative methods for the pricing and hedging of counterparty credit and funding risk. The new general theory that is required for this methodology is developed from scratch, leading to a consistent and comprehensive framework for counterparty credit and funding risk, inclusive of collateral, netting rules, possible debit valuation adjustments, re-hypothecation and closeout rules. The book however also looks at quite practical problems, linking particular models to particular 'concrete' financial situations across asset classes, including interest rates, FX, commodities, equity, credit itself, and the emerging asset class of longevity.
The authors also aim to help quantitative analysts, traders, and anyone else needing to frame and price counterparty credit and funding risk, to develop a 'feel' for applying sophisticated mathematics and stochastic calculus to solve practical problems.
The main models are illustrated from theoretical formulation to final implementation with calibration to market data, always keeping in mind the concrete questions being dealt with. The authors stress that each model is suited to different situations and products, pointing out that there does not exist a single model which is uniformly better than all the others, although the problems originated by counterparty credit and funding risk point in the direction of global valuation.
Finally, proposals for restructuring counterparty credit risk, ranging from contingent credit default swaps to margin lending, are considered.
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
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
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.
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.
