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 xvAbbreviations and Notation xxiiiPART I COUNTERPARTY CREDIT RISK, COLLATERAL AND FUNDING1 Introduction 31.1 A Dialogue on CVA 31.2 Risk Measurement: Credit VaR 31.3 Exposure, CE, PFE, EPE, EE, EAD 51.4 Exposure and Credit VaR 71.5 Interlude: P and Q 71.6 Basel 81.7 CVA and Model Dependence 91.8 Input and Data Issues on CVA 101.9 Emerging Asset Classes: Longevity Risk 111.10 CVA and Wrong Way Risk 121.11 Basel III: VaR of CVA and Wrong Way Risk 131.12 Discrepancies in CVA Valuation: Model Risk and Payoff Risk 141.13 Bilateral Counterparty Risk: CVA and DVA 151.14 First-to-Default in CVA and DVA 171.15 DVA Mark-to-Market and DVA Hedging 181.16 Impact of Close-Out in CVA and DVA 191.17 Close-Out Contagion 201.18 Collateral Modelling in CVA and DVA 211.19 Re-Hypothecation 221.20 Netting 221.21 Funding 231.22 Hedging Counterparty Risk: CCDS 251.23 Restructuring Counterparty Risk: CVA-CDOs and Margin Lending 262 Context 312.1 Definition of Default: Six Basic Cases 312.2 Definition of Exposures 322.3 Definition of Credit Valuation Adjustment (CVA) 352.4 Counterparty Risk Mitigants: Netting 372.5 Counterparty Risk Mitigants: Collateral 382.5.1 The Credit Support Annex (CSA) 392.5.2 The ISDA Proposal for a New Standard CSA 402.5.3 Collateral Effectiveness as a Mitigant 402.6 Funding 412.6.1 A First Attack on Funding Cost Modelling 422.6.2 The General Funding Theory and its Recursive Nature 422.7 Value at Risk (VaR) and Expected Shortfall (ES) of CVA 432.8 The Dilemma of Regulators and Basel III 443 Modelling the Counterparty Default 473.1 Firm Value (or Structural) Models 473.1.1 The Geometric Brownian Assumption 473.1.2 Merton’s Model 483.1.3 Black and Cox’s (1976) Model 503.1.4 Credit Default Swaps and Default Probabilities 543.1.5 Black and Cox (B&C) Model Calibration to CDS: Problems 553.1.6 The AT1P Model 573.1.7 A Case Study with AT1P: Lehman Brothers Default History 583.1.8 Comments 603.1.9 SBTV Model 613.1.10 A Case Study with SBTV: Lehman Brothers Default History 623.1.11 Comments 643.2 Firm Value Models: Hints at the Multiname Picture 643.3 Reduced Form (Intensity) Models 653.3.1 CDS Calibration and Intensity Models 663.3.2 A Simpler Formula for Calibrating Intensity to a Single CDS 703.3.3 Stochastic Intensity: The CIR Family 723.3.4 The Cox-Ingersoll-Ross Model (CIR) Short-Rate Model for r 723.3.5 Time-Inhomogeneous Case: CIR++ Model 743.3.6 Stochastic Diffusion Intensity is Not Enough: Adding Jumps. The JCIR(++) Model 753.3.7 The Jump-Diffusion CIR Model (JCIR) 763.3.8 Market Incompleteness and Default Unpredictability 783.3.9 Further Models 783.4 Intensity Models: The Multiname Picture 783.4.1 Choice of Variables for the Dependence Structure 783.4.2 Firm Value Models? 803.4.3 Copula Functions 803.4.4 Copula Calibration, CDOs and Criticism of Copula Functions 86PART II PRICING COUNTERPARTY RISK: UNILATERAL CVA4 Unilateral CVA and Netting for Interest Rate Products 894.1 First Steps towards a CVA Pricing Formula 894.1.1 Symmetry versus Asymmetry 904.1.2 Modelling the Counterparty Default Process 914.2 The Probabilistic Framework 924.3 The General Pricing Formula for Unilateral Counterparty Risk 944.4 Interest Rate Swap (IRS) Portfolios 974.4.1 Counterparty Risk in Single IRS 974.4.2 Counterparty Risk in an IRS Portfolio with Netting 1004.4.3 The Drift Freezing Approximation 1024.4.4 The Three-Moments Matching Technique 1044.5 Numerical Tests 1064.5.1 Case A: IRS with Co-Terminal Payment Dates 1064.5.2 Case B: IRS with Co-Starting Resetting Date 1084.5.3 Case C: IRS with First Positive, Then Negative Flow 1084.5.4 Case D: IRS with First Negative, Then Positive Flows 1094.5.5 Case E: IRS with First Alternate Flows 1134.6 Conclusions 1205 Wrong Way Risk (WWR) for Interest Rates 1215.1 Modelling Assumptions 1225.1.1 G2++ Interest Rate Model 1225.1.2 CIR++ Stochastic Intensity Model 1235.1.3 CIR++ Model: CDS Calibration 1245.1.4 Interest Rate/Credit Spread Correlation 1265.1.5 Adding Jumps to the Credit Spread 1265.2 Numerical Methods 1275.2.1 Discretization Scheme 1285.2.2 Simulating Intensity Jumps 1285.2.3 “American Monte Carlo” (Pallavicini 2006) 1285.2.4 Callable Payoffs 1285.3 Results and Discussion 1295.3.1 WWR in Single IRS 1295.3.2 WWR in an IRS Portfolio with Netting 1295.3.3 WWR in European Swaptions 1305.3.4 WWR in Bermudan Swaptions 1305.3.5 WWR in CMS Spread Options 1325.4 Contingent CDS (CCDS) 1325.5 Results Interpretation and Conclusions 1336 Unilateral CVA for Commodities with WWR 1356.1 Oil Swaps and Counterparty Risk 1356.2 Modelling Assumptions 1376.2.1 Commodity Model 1376.2.2 CIR++ Stochastic-Intensity Model 1396.3 Forward versus Futures Prices 1406.3.1 CVA for Commodity Forwards without WWR 1416.3.2 CVA for Commodity Forwards with WWR 1426.4 Swaps and Counterparty Risk 1426.5 UCVA for Commodity Swaps 1446.5.1 Counterparty Risk from the Payer’s Perspective: The Airline Computes Counterparty Risk 1456.5.2 Counterparty Risk from the Receiver’s Perspective: The Bank Computes Counterparty Risk 1486.6 Inadequacy of Basel’s WWR Multipliers 1486.7 Conclusions 1517 Unilateral CVA for Credit with WWR 1537.1 Introduction to CDSs with Counterparty Risk 1537.1.1 The Structure of the Chapter 1557.2 Modelling Assumptions 1557.2.1 CIR++ Stochastic-Intensity Model 1567.2.2 CIR++ Model: CDS Calibration 1577.3 CDS Options Embedded in CVA Pricing 1587.4 UCVA for Credit Default Swaps: A Case Study 1607.4.1 Changing the Copula Parameters 1607.4.2 Changing the Market Parameters 1647.5 Conclusions 1648 Unilateral CVA for Equity with WWR 1678.1 Counterparty Risk for Equity Without a Full Hybrid Model 1678.1.1 Calibrating AT1P to the Counterparty’s CDS Data 1688.1.2 Counterparty Risk in Equity Return Swaps (ERS) 1698.2 Counterparty Risk with a Hybrid Credit-Equity Structural Model 1728.2.1 The Credit Model 1728.2.2 The Equity Model 1748.2.3 From Barrier Options to Equity Pricing 1768.2.4 Equity and Equity Options 1798.3 Model Calibration and Empirical Results 1808.3.1 BP and FIAT in 2009 1818.3.2 Uncertainty in Market Expectations 1868.3.3 Further Results: FIAT in 2008 and BP in 2010 1888.4 Counterparty Risk and Wrong Way Risk 1918.4.1 Deterministic Default Barrier 1938.4.2 Uncertainty on the Default Barrier 1989 Unilateral CVA for FX 2059.1 Pricing with Two Currencies: Foundations 2069.2 Unilateral CVA for a Fixed-Fixed CCS 2109.2.1 Approximating the Volatility of Cross Currency Swap Rates 2169.2.2 Parameterization of the FX Correlation 2189.3 Unilateral CVA for Cross Currency Swaps with Floating Legs 2249.4 Why a Cross Currency Basis? 2269.4.1 The Approach of Fujii, Shimada and Takahashi (2010) 2279.4.2 Collateral Rates versus Risk-Free Rates 2289.4.3 Consequences of Perfect Collateralization 2299.5 CVA for CCS in Practice 2309.5.1 Changing the CCS Moneyness 2349.5.2 Changing the Volatility 2359.5.3 Changing the FX Correlations 2359.6 Novations and the Cost of Liquidity 2379.6.1 A Synthetic Contingent CDS: The Novation 2389.6.2 Extending the Approach to the Valuation of Liquidity 2419.7 Conclusions 243PART III ADVANCED CREDIT AND FUNDING RISK PRICING10 New Generation Counterparty and Funding Risk Pricing 24710.1 Introducing the Advanced Part of the Book 24710.2 What We Have Seen Before: Unilateral CVA 24910.2.1 Approximation: Default Bucketing and Independence 25010.3 Unilateral Debit Valuation Adjustment (UDVA) 25010.4 Bilateral Risk and DVA 25110.5 Undesirable Features of DVA 25310.5.1 Profiting From Own Deteriorating Credit Quality 25310.5.2 DVA Hedging? 25310.5.3 DVA: Accounting versus Capital Requirements 25410.5.4 DVA: Summary and Debate on Realism 25510.6 Close-Out: Risk-Free or Replacement? 25610.7 Can We Neglect the First-to-Default Time? 25710.7.1 A Simplified Formula without First-to-Default: The Case of an Equity Forward 25810.8 Payoff Risk 25810.9 Collateralization, Gap Risk and Re-Hypothecation 25910.10 Funding Costs 26210.11 Restructuring Counterparty Risk 26310.11.1 CVA Volatility: The Wrong Way 26310.11.2 Floating Margin Lending 26410.11.3 Global Valuation 26510.12 Conclusions 26611 A First Attack on Funding Cost Modelling 26911.1 The Problem 26911.2 A Closer Look at Funding and Discounting 27111.3 The Approach Proposed by Morini and Prampolini (2010) 27211.3.1 The Borrower’s Case 27311.3.2 The Lender’s Case 27411.3.3 The Controversial Role of DVA: The Borrower 27511.3.4 The Controversial Role of DVA: The Lender 27611.3.5 Discussion 27711.4 What Next on Funding? 27812 Bilateral CVA–DVA and Interest Rate Products 27912.1 Arbitrage-Free Valuation of Bilateral Counterparty Risk 28112.1.1 Symmetry versus Asymmetry 28512.1.2 Worsening of Credit Quality and Positive Mark-to-Market 28512.2 Modelling Assumptions 28612.2.1 G2++ Interest Rate Model 28612.2.2 CIR++ Stochastic Intensity Model 28812.2.3 Realistic Market Data Set for CDS Options 28912.3 Numerical Methods 29012.4 Results and Discussion 29112.4.1 Bilateral VA in Single IRS 29212.4.2 Bilateral VA in an IRS Portfolio with Netting 29612.4.3 Bilateral VA in Exotic Interest Rate Products 30112.5 Conclusions 30213 Collateral, Netting, Close-Out and Re-Hypothecation 30513.1 Trading Under the ISDA Master Agreement 30613.1.1 Mathematical Setup and CBVA Definition 30613.1.2 Collateral Delay and Dispute Resolutions 30813.1.3 Close-Out Netting Rules 30813.1.4 Collateral Re-Hypothecation 30913.2 Bilateral CVA Formula under Collateralization 31013.2.1 Collecting CVA Contributions 31013.2.2 CBVA General Formula 31213.2.3 CCVA and CDVA Definitions 31213.3 Close-Out Amount Evaluation 31313.4 Special Cases of Collateral-Inclusive Bilateral Credit Valuation Adjustment 31413.5 Example of Collateralization Schemes 31513.5.1 Perfect Collateralization 31513.5.2 Collateralization Through Margining 31613.6 Conclusions 31614 Close-Out and Contagion with Examples of a Simple Payoff 31914.1 Introduction to Close-Out Modelling and Earlier Work 31914.1.1 Close-Out Modelling: Context 31914.1.2 Legal Documentation on Close-Out 32014.1.3 Literature 32014.1.4 Risk-Free versus Replacement Close-Out: Practical Consequences 32114.2 Classical Unilateral and Bilateral Valuation Adjustments 32214.3 Bilateral Adjustment and Close-Out: Risk-Free or Replacement? 32314.4 A Quantitative Analysis and a Numerical Example 32314.4.1 Contagion Issues 32614.5 Conclusions 32915 Bilateral Collateralized CVA and DVA for Rates and Credit 33115.1 CBVA for Interest Rate Swaps 33215.1.1 Changing the Margining Frequency 33215.1.2 Inspecting the Exposure Profiles 33415.1.3 A Case Where Re-Hypothecation is Worse than No Collateral at All 33515.1.4 Changing the Correlation Parameters 33615.1.5 Changing the Credit Spread Volatility 33715.2 Modelling Credit Contagion 34015.2.1 The CDS Price Process 34015.2.2 Calculation of Survival Probability 34115.2.3 Modelling Default-Time Dependence 34415.3 CBVA for Credit Default Swaps 34515.3.1 Changing the Copula Parameters 34515.3.2 Inspecting the Contagion Risk 34715.3.3 Changing the CDS Moneyness 34715.4 Conclusions 34916 Including Margining Costs in Collateralized Contracts 35116.1 Trading Under the ISDA Master Agreement 35216.1.1 Collateral Accrual Rates 35216.1.2 Collateral Management and Margining Costs 35316.2 CBVA General Formula with Margining Costs 35516.2.1 Perfect Collateralization 35616.2.2 Futures Contracts 35716.3 Changing the Collateralization Currency 35716.3.1 Margining Cost in Foreign Currency 35716.3.2 Settlement Liquidity Risk 35816.3.3 Gap Risk in Single-Currency Contracts with Foreign-Currency Collaterals 35916.4 Conclusions 35917 Funding Valuation Adjustment (FVA)? 36117.1 Dealing with Costs of Funding 36117.1.1 Central Clearing, CCPs and this Book 36217.1.2 High Level Features 36217.1.3 Single-Deal (Micro) vs. Homogeneous (Macro) Funding Models 36317.1.4 Previous Literature on Funding and Collateral 36417.1.5 Including FVA along with Credit and Debit Valuation Adjustment 36517.1.6 FVA is not DVA 36517.2 Collateral- and Funding-Inclusive Bilateral Valuation Adjusted Price 36617.3 Funding Risk and Liquidity Policies 36717.3.1 Funding, Hedging and Collateralization 36717.3.2 Liquidity Policies 36817.4 CBVA Pricing Equation with Funding Costs (CFBVA) 37217.4.1 Iterative Solution of the CFBVA Pricing Equation 37317.4.2 Funding Derivative Contracts in a Diffusion Setting 37417.4.3 Implementing Hedging Strategies via Derivative Markets 37717.5 Detailed Examples 37817.5.1 Funding with Collateral 37817.5.2 Collateralized Contracts Priced by a CCP 37917.5.3 Dealing with Own Credit Risk: FVA and DVA 38017.5.4 Deriving Earlier Results on FVA and DVA 38117.6 Conclusions: FVA and Beyond 38218 Non-Standard Asset Classes: Longevity Risk 38518.1 Introduction to Longevity Markets 38518.1.1 The Longevity Swap Market 38518.1.2 Longevity Swaps: Collateral and Credit Risk 38618.1.3 Indexed Longevity Swaps 39018.1.4 Endogenous Credit Collateral and Funding-Inclusive Swap Rates 39018.2 Longevity Swaps: The Payoff P39118.3 Mark-to-Market for Longevity Swaps 39418.4 Counterparty and Own Default Risk, Collateral and Funding 39718.5 An Example of Modelling Specification from Biffis et al. (2011) 40118.6 Discussion of the Results in Biffis et al. (2011) 40419 Conclusions and Further Work 40919.1 A Final Dialogue: Models, Regulations, CVA/DVA, Funding and More 409Bibliography 415Index 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 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 xvAbbreviations and Notation xxiiiPART I COUNTERPARTY CREDIT RISK, COLLATERAL AND FUNDING1 Introduction 31.1 A Dialogue on CVA 31.2 Risk Measurement: Credit VaR 31.3 Exposure, CE, PFE, EPE, EE, EAD 51.4 Exposure and Credit VaR 71.5 Interlude: P and Q 71.6 Basel 81.7 CVA and Model Dependence 91.8 Input and Data Issues on CVA 101.9 Emerging Asset Classes: Longevity Risk 111.10 CVA and Wrong Way Risk 121.11 Basel III: VaR of CVA and Wrong Way Risk 131.12 Discrepancies in CVA Valuation: Model Risk and Payoff Risk 141.13 Bilateral Counterparty Risk: CVA and DVA 151.14 First-to-Default in CVA and DVA 171.15 DVA Mark-to-Market and DVA Hedging 181.16 Impact of Close-Out in CVA and DVA 191.17 Close-Out Contagion 201.18 Collateral Modelling in CVA and DVA 211.19 Re-Hypothecation 221.20 Netting 221.21 Funding 231.22 Hedging Counterparty Risk: CCDS 251.23 Restructuring Counterparty Risk: CVA-CDOs and Margin Lending 262 Context 312.1 Definition of Default: Six Basic Cases 312.2 Definition of Exposures 322.3 Definition of Credit Valuation Adjustment (CVA) 352.4 Counterparty Risk Mitigants: Netting 372.5 Counterparty Risk Mitigants: Collateral 382.5.1 The Credit Support Annex (CSA) 392.5.2 The ISDA Proposal for a New Standard CSA 402.5.3 Collateral Effectiveness as a Mitigant 402.6 Funding 412.6.1 A First Attack on Funding Cost Modelling 422.6.2 The General Funding Theory and its Recursive Nature 422.7 Value at Risk (VaR) and Expected Shortfall (ES) of CVA 432.8 The Dilemma of Regulators and Basel III 443 Modelling the Counterparty Default 473.1 Firm Value (or Structural) Models 473.1.1 The Geometric Brownian Assumption 473.1.2 Merton’s Model 483.1.3 Black and Cox’s (1976) Model 503.1.4 Credit Default Swaps and Default Probabilities 543.1.5 Black and Cox (B&C) Model Calibration to CDS: Problems 553.1.6 The AT1P Model 573.1.7 A Case Study with AT1P: Lehman Brothers Default History 583.1.8 Comments 603.1.9 SBTV Model 613.1.10 A Case Study with SBTV: Lehman Brothers Default History 623.1.11 Comments 643.2 Firm Value Models: Hints at the Multiname Picture 643.3 Reduced Form (Intensity) Models 653.3.1 CDS Calibration and Intensity Models 663.3.2 A Simpler Formula for Calibrating Intensity to a Single CDS 703.3.3 Stochastic Intensity: The CIR Family 723.3.4 The Cox-Ingersoll-Ross Model (CIR) Short-Rate Model for r 723.3.5 Time-Inhomogeneous Case: CIR++ Model 743.3.6 Stochastic Diffusion Intensity is Not Enough: Adding Jumps. The JCIR(++) Model 753.3.7 The Jump-Diffusion CIR Model (JCIR) 763.3.8 Market Incompleteness and Default Unpredictability 783.3.9 Further Models 783.4 Intensity Models: The Multiname Picture 783.4.1 Choice of Variables for the Dependence Structure 783.4.2 Firm Value Models? 803.4.3 Copula Functions 803.4.4 Copula Calibration, CDOs and Criticism of Copula Functions 86PART II PRICING COUNTERPARTY RISK: UNILATERAL CVA4 Unilateral CVA and Netting for Interest Rate Products 894.1 First Steps towards a CVA Pricing Formula 894.1.1 Symmetry versus Asymmetry 904.1.2 Modelling the Counterparty Default Process 914.2 The Probabilistic Framework 924.3 The General Pricing Formula for Unilateral Counterparty Risk 944.4 Interest Rate Swap (IRS) Portfolios 974.4.1 Counterparty Risk in Single IRS 974.4.2 Counterparty Risk in an IRS Portfolio with Netting 1004.4.3 The Drift Freezing Approximation 1024.4.4 The Three-Moments Matching Technique 1044.5 Numerical Tests 1064.5.1 Case A: IRS with Co-Terminal Payment Dates 1064.5.2 Case B: IRS with Co-Starting Resetting Date 1084.5.3 Case C: IRS with First Positive, Then Negative Flow 1084.5.4 Case D: IRS with First Negative, Then Positive Flows 1094.5.5 Case E: IRS with First Alternate Flows 1134.6 Conclusions 1205 Wrong Way Risk (WWR) for Interest Rates 1215.1 Modelling Assumptions 1225.1.1 G2++ Interest Rate Model 1225.1.2 CIR++ Stochastic Intensity Model 1235.1.3 CIR++ Model: CDS Calibration 1245.1.4 Interest Rate/Credit Spread Correlation 1265.1.5 Adding Jumps to the Credit Spread 1265.2 Numerical Methods 1275.2.1 Discretization Scheme 1285.2.2 Simulating Intensity Jumps 1285.2.3 “American Monte Carlo” (Pallavicini 2006) 1285.2.4 Callable Payoffs 1285.3 Results and Discussion 1295.3.1 WWR in Single IRS 1295.3.2 WWR in an IRS Portfolio with Netting 1295.3.3 WWR in European Swaptions 1305.3.4 WWR in Bermudan Swaptions 1305.3.5 WWR in CMS Spread Options 1325.4 Contingent CDS (CCDS) 1325.5 Results Interpretation and Conclusions 1336 Unilateral CVA for Commodities with WWR 1356.1 Oil Swaps and Counterparty Risk 1356.2 Modelling Assumptions 1376.2.1 Commodity Model 1376.2.2 CIR++ Stochastic-Intensity Model 1396.3 Forward versus Futures Prices 1406.3.1 CVA for Commodity Forwards without WWR 1416.3.2 CVA for Commodity Forwards with WWR 1426.4 Swaps and Counterparty Risk 1426.5 UCVA for Commodity Swaps 1446.5.1 Counterparty Risk from the Payer’s Perspective: The Airline Computes Counterparty Risk 1456.5.2 Counterparty Risk from the Receiver’s Perspective: The Bank Computes Counterparty Risk 1486.6 Inadequacy of Basel’s WWR Multipliers 1486.7 Conclusions 1517 Unilateral CVA for Credit with WWR 1537.1 Introduction to CDSs with Counterparty Risk 1537.1.1 The Structure of the Chapter 1557.2 Modelling Assumptions 1557.2.1 CIR++ Stochastic-Intensity Model 1567.2.2 CIR++ Model: CDS Calibration 1577.3 CDS Options Embedded in CVA Pricing 1587.4 UCVA for Credit Default Swaps: A Case Study 1607.4.1 Changing the Copula Parameters 1607.4.2 Changing the Market Parameters 1647.5 Conclusions 1648 Unilateral CVA for Equity with WWR 1678.1 Counterparty Risk for Equity Without a Full Hybrid Model 1678.1.1 Calibrating AT1P to the Counterparty’s CDS Data 1688.1.2 Counterparty Risk in Equity Return Swaps (ERS) 1698.2 Counterparty Risk with a Hybrid Credit-Equity Structural Model 1728.2.1 The Credit Model 1728.2.2 The Equity Model 1748.2.3 From Barrier Options to Equity Pricing 1768.2.4 Equity and Equity Options 1798.3 Model Calibration and Empirical Results 1808.3.1 BP and FIAT in 2009 1818.3.2 Uncertainty in Market Expectations 1868.3.3 Further Results: FIAT in 2008 and BP in 2010 1888.4 Counterparty Risk and Wrong Way Risk 1918.4.1 Deterministic Default Barrier 1938.4.2 Uncertainty on the Default Barrier 1989 Unilateral CVA for FX 2059.1 Pricing with Two Currencies: Foundations 2069.2 Unilateral CVA for a Fixed-Fixed CCS 2109.2.1 Approximating the Volatility of Cross Currency Swap Rates 2169.2.2 Parameterization of the FX Correlation 2189.3 Unilateral CVA for Cross Currency Swaps with Floating Legs 2249.4 Why a Cross Currency Basis? 2269.4.1 The Approach of Fujii, Shimada and Takahashi (2010) 2279.4.2 Collateral Rates versus Risk-Free Rates 2289.4.3 Consequences of Perfect Collateralization 2299.5 CVA for CCS in Practice 2309.5.1 Changing the CCS Moneyness 2349.5.2 Changing the Volatility 2359.5.3 Changing the FX Correlations 2359.6 Novations and the Cost of Liquidity 2379.6.1 A Synthetic Contingent CDS: The Novation 2389.6.2 Extending the Approach to the Valuation of Liquidity 2419.7 Conclusions 243PART III ADVANCED CREDIT AND FUNDING RISK PRICING10 New Generation Counterparty and Funding Risk Pricing 24710.1 Introducing the Advanced Part of the Book 24710.2 What We Have Seen Before: Unilateral CVA 24910.2.1 Approximation: Default Bucketing and Independence 25010.3 Unilateral Debit Valuation Adjustment (UDVA) 25010.4 Bilateral Risk and DVA 25110.5 Undesirable Features of DVA 25310.5.1 Profiting From Own Deteriorating Credit Quality 25310.5.2 DVA Hedging? 25310.5.3 DVA: Accounting versus Capital Requirements 25410.5.4 DVA: Summary and Debate on Realism 25510.6 Close-Out: Risk-Free or Replacement? 25610.7 Can We Neglect the First-to-Default Time? 25710.7.1 A Simplified Formula without First-to-Default: The Case of an Equity Forward 25810.8 Payoff Risk 25810.9 Collateralization, Gap Risk and Re-Hypothecation 25910.10 Funding Costs 26210.11 Restructuring Counterparty Risk 26310.11.1 CVA Volatility: The Wrong Way 26310.11.2 Floating Margin Lending 26410.11.3 Global Valuation 26510.12 Conclusions 26611 A First Attack on Funding Cost Modelling 26911.1 The Problem 26911.2 A Closer Look at Funding and Discounting 27111.3 The Approach Proposed by Morini and Prampolini (2010) 27211.3.1 The Borrower’s Case 27311.3.2 The Lender’s Case 27411.3.3 The Controversial Role of DVA: The Borrower 27511.3.4 The Controversial Role of DVA: The Lender 27611.3.5 Discussion 27711.4 What Next on Funding? 27812 Bilateral CVA–DVA and Interest Rate Products 27912.1 Arbitrage-Free Valuation of Bilateral Counterparty Risk 28112.1.1 Symmetry versus Asymmetry 28512.1.2 Worsening of Credit Quality and Positive Mark-to-Market 28512.2 Modelling Assumptions 28612.2.1 G2++ Interest Rate Model 28612.2.2 CIR++ Stochastic Intensity Model 28812.2.3 Realistic Market Data Set for CDS Options 28912.3 Numerical Methods 29012.4 Results and Discussion 29112.4.1 Bilateral VA in Single IRS 29212.4.2 Bilateral VA in an IRS Portfolio with Netting 29612.4.3 Bilateral VA in Exotic Interest Rate Products 30112.5 Conclusions 30213 Collateral, Netting, Close-Out and Re-Hypothecation 30513.1 Trading Under the ISDA Master Agreement 30613.1.1 Mathematical Setup and CBVA Definition 30613.1.2 Collateral Delay and Dispute Resolutions 30813.1.3 Close-Out Netting Rules 30813.1.4 Collateral Re-Hypothecation 30913.2 Bilateral CVA Formula under Collateralization 31013.2.1 Collecting CVA Contributions 31013.2.2 CBVA General Formula 31213.2.3 CCVA and CDVA Definitions 31213.3 Close-Out Amount Evaluation 31313.4 Special Cases of Collateral-Inclusive Bilateral Credit Valuation Adjustment 31413.5 Example of Collateralization Schemes 31513.5.1 Perfect Collateralization 31513.5.2 Collateralization Through Margining 31613.6 Conclusions 31614 Close-Out and Contagion with Examples of a Simple Payoff 31914.1 Introduction to Close-Out Modelling and Earlier Work 31914.1.1 Close-Out Modelling: Context 31914.1.2 Legal Documentation on Close-Out 32014.1.3 Literature 32014.1.4 Risk-Free versus Replacement Close-Out: Practical Consequences 32114.2 Classical Unilateral and Bilateral Valuation Adjustments 32214.3 Bilateral Adjustment and Close-Out: Risk-Free or Replacement? 32314.4 A Quantitative Analysis and a Numerical Example 32314.4.1 Contagion Issues 32614.5 Conclusions 32915 Bilateral Collateralized CVA and DVA for Rates and Credit 33115.1 CBVA for Interest Rate Swaps 33215.1.1 Changing the Margining Frequency 33215.1.2 Inspecting the Exposure Profiles 33415.1.3 A Case Where Re-Hypothecation is Worse than No Collateral at All 33515.1.4 Changing the Correlation Parameters 33615.1.5 Changing the Credit Spread Volatility 33715.2 Modelling Credit Contagion 34015.2.1 The CDS Price Process 34015.2.2 Calculation of Survival Probability 34115.2.3 Modelling Default-Time Dependence 34415.3 CBVA for Credit Default Swaps 34515.3.1 Changing the Copula Parameters 34515.3.2 Inspecting the Contagion Risk 34715.3.3 Changing the CDS Moneyness 34715.4 Conclusions 34916 Including Margining Costs in Collateralized Contracts 35116.1 Trading Under the ISDA Master Agreement 35216.1.1 Collateral Accrual Rates 35216.1.2 Collateral Management and Margining Costs 35316.2 CBVA General Formula with Margining Costs 35516.2.1 Perfect Collateralization 35616.2.2 Futures Contracts 35716.3 Changing the Collateralization Currency 35716.3.1 Margining Cost in Foreign Currency 35716.3.2 Settlement Liquidity Risk 35816.3.3 Gap Risk in Single-Currency Contracts with Foreign-Currency Collaterals 35916.4 Conclusions 35917 Funding Valuation Adjustment (FVA)? 36117.1 Dealing with Costs of Funding 36117.1.1 Central Clearing, CCPs and this Book 36217.1.2 High Level Features 36217.1.3 Single-Deal (Micro) vs. Homogeneous (Macro) Funding Models 36317.1.4 Previous Literature on Funding and Collateral 36417.1.5 Including FVA along with Credit and Debit Valuation Adjustment 36517.1.6 FVA is not DVA 36517.2 Collateral- and Funding-Inclusive Bilateral Valuation Adjusted Price 36617.3 Funding Risk and Liquidity Policies 36717.3.1 Funding, Hedging and Collateralization 36717.3.2 Liquidity Policies 36817.4 CBVA Pricing Equation with Funding Costs (CFBVA) 37217.4.1 Iterative Solution of the CFBVA Pricing Equation 37317.4.2 Funding Derivative Contracts in a Diffusion Setting 37417.4.3 Implementing Hedging Strategies via Derivative Markets 37717.5 Detailed Examples 37817.5.1 Funding with Collateral 37817.5.2 Collateralized Contracts Priced by a CCP 37917.5.3 Dealing with Own Credit Risk: FVA and DVA 38017.5.4 Deriving Earlier Results on FVA and DVA 38117.6 Conclusions: FVA and Beyond 38218 Non-Standard Asset Classes: Longevity Risk 38518.1 Introduction to Longevity Markets 38518.1.1 The Longevity Swap Market 38518.1.2 Longevity Swaps: Collateral and Credit Risk 38618.1.3 Indexed Longevity Swaps 39018.1.4 Endogenous Credit Collateral and Funding-Inclusive Swap Rates 39018.2 Longevity Swaps: The Payoff P39118.3 Mark-to-Market for Longevity Swaps 39418.4 Counterparty and Own Default Risk, Collateral and Funding 39718.5 An Example of Modelling Specification from Biffis et al. (2011) 40118.6 Discussion of the Results in Biffis et al. (2011) 40419 Conclusions and Further Work 40919.1 A Final Dialogue: Models, Regulations, CVA/DVA, Funding and More 409Bibliography 415Index 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 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 xvAbbreviations and Notation xxiiiPART I COUNTERPARTY CREDIT RISK, COLLATERAL AND FUNDING1 Introduction 31.1 A Dialogue on CVA 31.2 Risk Measurement: Credit VaR 31.3 Exposure, CE, PFE, EPE, EE, EAD 51.4 Exposure and Credit VaR 71.5 Interlude: P and Q 71.6 Basel 81.7 CVA and Model Dependence 91.8 Input and Data Issues on CVA 101.9 Emerging Asset Classes: Longevity Risk 111.10 CVA and Wrong Way Risk 121.11 Basel III: VaR of CVA and Wrong Way Risk 131.12 Discrepancies in CVA Valuation: Model Risk and Payoff Risk 141.13 Bilateral Counterparty Risk: CVA and DVA 151.14 First-to-Default in CVA and DVA 171.15 DVA Mark-to-Market and DVA Hedging 181.16 Impact of Close-Out in CVA and DVA 191.17 Close-Out Contagion 201.18 Collateral Modelling in CVA and DVA 211.19 Re-Hypothecation 221.20 Netting 221.21 Funding 231.22 Hedging Counterparty Risk: CCDS 251.23 Restructuring Counterparty Risk: CVA-CDOs and Margin Lending 262 Context 312.1 Definition of Default: Six Basic Cases 312.2 Definition of Exposures 322.3 Definition of Credit Valuation Adjustment (CVA) 352.4 Counterparty Risk Mitigants: Netting 372.5 Counterparty Risk Mitigants: Collateral 382.5.1 The Credit Support Annex (CSA) 392.5.2 The ISDA Proposal for a New Standard CSA 402.5.3 Collateral Effectiveness as a Mitigant 402.6 Funding 412.6.1 A First Attack on Funding Cost Modelling 422.6.2 The General Funding Theory and its Recursive Nature 422.7 Value at Risk (VaR) and Expected Shortfall (ES) of CVA 432.8 The Dilemma of Regulators and Basel III 443 Modelling the Counterparty Default 473.1 Firm Value (or Structural) Models 473.1.1 The Geometric Brownian Assumption 473.1.2 Merton’s Model 483.1.3 Black and Cox’s (1976) Model 503.1.4 Credit Default Swaps and Default Probabilities 543.1.5 Black and Cox (B&C) Model Calibration to CDS: Problems 553.1.6 The AT1P Model 573.1.7 A Case Study with AT1P: Lehman Brothers Default History 583.1.8 Comments 603.1.9 SBTV Model 613.1.10 A Case Study with SBTV: Lehman Brothers Default History 623.1.11 Comments 643.2 Firm Value Models: Hints at the Multiname Picture 643.3 Reduced Form (Intensity) Models 653.3.1 CDS Calibration and Intensity Models 663.3.2 A Simpler Formula for Calibrating Intensity to a Single CDS 703.3.3 Stochastic Intensity: The CIR Family 723.3.4 The Cox-Ingersoll-Ross Model (CIR) Short-Rate Model for r 723.3.5 Time-Inhomogeneous Case: CIR++ Model 743.3.6 Stochastic Diffusion Intensity is Not Enough: Adding Jumps. The JCIR(++) Model 753.3.7 The Jump-Diffusion CIR Model (JCIR) 763.3.8 Market Incompleteness and Default Unpredictability 783.3.9 Further Models 783.4 Intensity Models: The Multiname Picture 783.4.1 Choice of Variables for the Dependence Structure 783.4.2 Firm Value Models? 803.4.3 Copula Functions 803.4.4 Copula Calibration, CDOs and Criticism of Copula Functions 86PART II PRICING COUNTERPARTY RISK: UNILATERAL CVA4 Unilateral CVA and Netting for Interest Rate Products 894.1 First Steps towards a CVA Pricing Formula 894.1.1 Symmetry versus Asymmetry 904.1.2 Modelling the Counterparty Default Process 914.2 The Probabilistic Framework 924.3 The General Pricing Formula for Unilateral Counterparty Risk 944.4 Interest Rate Swap (IRS) Portfolios 974.4.1 Counterparty Risk in Single IRS 974.4.2 Counterparty Risk in an IRS Portfolio with Netting 1004.4.3 The Drift Freezing Approximation 1024.4.4 The Three-Moments Matching Technique 1044.5 Numerical Tests 1064.5.1 Case A: IRS with Co-Terminal Payment Dates 1064.5.2 Case B: IRS with Co-Starting Resetting Date 1084.5.3 Case C: IRS with First Positive, Then Negative Flow 1084.5.4 Case D: IRS with First Negative, Then Positive Flows 1094.5.5 Case E: IRS with First Alternate Flows 1134.6 Conclusions 1205 Wrong Way Risk (WWR) for Interest Rates 1215.1 Modelling Assumptions 1225.1.1 G2++ Interest Rate Model 1225.1.2 CIR++ Stochastic Intensity Model 1235.1.3 CIR++ Model: CDS Calibration 1245.1.4 Interest Rate/Credit Spread Correlation 1265.1.5 Adding Jumps to the Credit Spread 1265.2 Numerical Methods 1275.2.1 Discretization Scheme 1285.2.2 Simulating Intensity Jumps 1285.2.3 “American Monte Carlo” (Pallavicini 2006) 1285.2.4 Callable Payoffs 1285.3 Results and Discussion 1295.3.1 WWR in Single IRS 1295.3.2 WWR in an IRS Portfolio with Netting 1295.3.3 WWR in European Swaptions 1305.3.4 WWR in Bermudan Swaptions 1305.3.5 WWR in CMS Spread Options 1325.4 Contingent CDS (CCDS) 1325.5 Results Interpretation and Conclusions 1336 Unilateral CVA for Commodities with WWR 1356.1 Oil Swaps and Counterparty Risk 1356.2 Modelling Assumptions 1376.2.1 Commodity Model 1376.2.2 CIR++ Stochastic-Intensity Model 1396.3 Forward versus Futures Prices 1406.3.1 CVA for Commodity Forwards without WWR 1416.3.2 CVA for Commodity Forwards with WWR 1426.4 Swaps and Counterparty Risk 1426.5 UCVA for Commodity Swaps 1446.5.1 Counterparty Risk from the Payer’s Perspective: The Airline Computes Counterparty Risk 1456.5.2 Counterparty Risk from the Receiver’s Perspective: The Bank Computes Counterparty Risk 1486.6 Inadequacy of Basel’s WWR Multipliers 1486.7 Conclusions 1517 Unilateral CVA for Credit with WWR 1537.1 Introduction to CDSs with Counterparty Risk 1537.1.1 The Structure of the Chapter 1557.2 Modelling Assumptions 1557.2.1 CIR++ Stochastic-Intensity Model 1567.2.2 CIR++ Model: CDS Calibration 1577.3 CDS Options Embedded in CVA Pricing 1587.4 UCVA for Credit Default Swaps: A Case Study 1607.4.1 Changing the Copula Parameters 1607.4.2 Changing the Market Parameters 1647.5 Conclusions 1648 Unilateral CVA for Equity with WWR 1678.1 Counterparty Risk for Equity Without a Full Hybrid Model 1678.1.1 Calibrating AT1P to the Counterparty’s CDS Data 1688.1.2 Counterparty Risk in Equity Return Swaps (ERS) 1698.2 Counterparty Risk with a Hybrid Credit-Equity Structural Model 1728.2.1 The Credit Model 1728.2.2 The Equity Model 1748.2.3 From Barrier Options to Equity Pricing 1768.2.4 Equity and Equity Options 1798.3 Model Calibration and Empirical Results 1808.3.1 BP and FIAT in 2009 1818.3.2 Uncertainty in Market Expectations 1868.3.3 Further Results: FIAT in 2008 and BP in 2010 1888.4 Counterparty Risk and Wrong Way Risk 1918.4.1 Deterministic Default Barrier 1938.4.2 Uncertainty on the Default Barrier 1989 Unilateral CVA for FX 2059.1 Pricing with Two Currencies: Foundations 2069.2 Unilateral CVA for a Fixed-Fixed CCS 2109.2.1 Approximating the Volatility of Cross Currency Swap Rates 2169.2.2 Parameterization of the FX Correlation 2189.3 Unilateral CVA for Cross Currency Swaps with Floating Legs 2249.4 Why a Cross Currency Basis? 2269.4.1 The Approach of Fujii, Shimada and Takahashi (2010) 2279.4.2 Collateral Rates versus Risk-Free Rates 2289.4.3 Consequences of Perfect Collateralization 2299.5 CVA for CCS in Practice 2309.5.1 Changing the CCS Moneyness 2349.5.2 Changing the Volatility 2359.5.3 Changing the FX Correlations 2359.6 Novations and the Cost of Liquidity 2379.6.1 A Synthetic Contingent CDS: The Novation 2389.6.2 Extending the Approach to the Valuation of Liquidity 2419.7 Conclusions 243PART III ADVANCED CREDIT AND FUNDING RISK PRICING10 New Generation Counterparty and Funding Risk Pricing 24710.1 Introducing the Advanced Part of the Book 24710.2 What We Have Seen Before: Unilateral CVA 24910.2.1 Approximation: Default Bucketing and Independence 25010.3 Unilateral Debit Valuation Adjustment (UDVA) 25010.4 Bilateral Risk and DVA 25110.5 Undesirable Features of DVA 25310.5.1 Profiting From Own Deteriorating Credit Quality 25310.5.2 DVA Hedging? 25310.5.3 DVA: Accounting versus Capital Requirements 25410.5.4 DVA: Summary and Debate on Realism 25510.6 Close-Out: Risk-Free or Replacement? 25610.7 Can We Neglect the First-to-Default Time? 25710.7.1 A Simplified Formula without First-to-Default: The Case of an Equity Forward 25810.8 Payoff Risk 25810.9 Collateralization, Gap Risk and Re-Hypothecation 25910.10 Funding Costs 26210.11 Restructuring Counterparty Risk 26310.11.1 CVA Volatility: The Wrong Way 26310.11.2 Floating Margin Lending 26410.11.3 Global Valuation 26510.12 Conclusions 26611 A First Attack on Funding Cost Modelling 26911.1 The Problem 26911.2 A Closer Look at Funding and Discounting 27111.3 The Approach Proposed by Morini and Prampolini (2010) 27211.3.1 The Borrower’s Case 27311.3.2 The Lender’s Case 27411.3.3 The Controversial Role of DVA: The Borrower 27511.3.4 The Controversial Role of DVA: The Lender 27611.3.5 Discussion 27711.4 What Next on Funding? 27812 Bilateral CVA–DVA and Interest Rate Products 27912.1 Arbitrage-Free Valuation of Bilateral Counterparty Risk 28112.1.1 Symmetry versus Asymmetry 28512.1.2 Worsening of Credit Quality and Positive Mark-to-Market 28512.2 Modelling Assumptions 28612.2.1 G2++ Interest Rate Model 28612.2.2 CIR++ Stochastic Intensity Model 28812.2.3 Realistic Market Data Set for CDS Options 28912.3 Numerical Methods 29012.4 Results and Discussion 29112.4.1 Bilateral VA in Single IRS 29212.4.2 Bilateral VA in an IRS Portfolio with Netting 29612.4.3 Bilateral VA in Exotic Interest Rate Products 30112.5 Conclusions 30213 Collateral, Netting, Close-Out and Re-Hypothecation 30513.1 Trading Under the ISDA Master Agreement 30613.1.1 Mathematical Setup and CBVA Definition 30613.1.2 Collateral Delay and Dispute Resolutions 30813.1.3 Close-Out Netting Rules 30813.1.4 Collateral Re-Hypothecation 30913.2 Bilateral CVA Formula under Collateralization 31013.2.1 Collecting CVA Contributions 31013.2.2 CBVA General Formula 31213.2.3 CCVA and CDVA Definitions 31213.3 Close-Out Amount Evaluation 31313.4 Special Cases of Collateral-Inclusive Bilateral Credit Valuation Adjustment 31413.5 Example of Collateralization Schemes 31513.5.1 Perfect Collateralization 31513.5.2 Collateralization Through Margining 31613.6 Conclusions 31614 Close-Out and Contagion with Examples of a Simple Payoff 31914.1 Introduction to Close-Out Modelling and Earlier Work 31914.1.1 Close-Out Modelling: Context 31914.1.2 Legal Documentation on Close-Out 32014.1.3 Literature 32014.1.4 Risk-Free versus Replacement Close-Out: Practical Consequences 32114.2 Classical Unilateral and Bilateral Valuation Adjustments 32214.3 Bilateral Adjustment and Close-Out: Risk-Free or Replacement? 32314.4 A Quantitative Analysis and a Numerical Example 32314.4.1 Contagion Issues 32614.5 Conclusions 32915 Bilateral Collateralized CVA and DVA for Rates and Credit 33115.1 CBVA for Interest Rate Swaps 33215.1.1 Changing the Margining Frequency 33215.1.2 Inspecting the Exposure Profiles 33415.1.3 A Case Where Re-Hypothecation is Worse than No Collateral at All 33515.1.4 Changing the Correlation Parameters 33615.1.5 Changing the Credit Spread Volatility 33715.2 Modelling Credit Contagion 34015.2.1 The CDS Price Process 34015.2.2 Calculation of Survival Probability 34115.2.3 Modelling Default-Time Dependence 34415.3 CBVA for Credit Default Swaps 34515.3.1 Changing the Copula Parameters 34515.3.2 Inspecting the Contagion Risk 34715.3.3 Changing the CDS Moneyness 34715.4 Conclusions 34916 Including Margining Costs in Collateralized Contracts 35116.1 Trading Under the ISDA Master Agreement 35216.1.1 Collateral Accrual Rates 35216.1.2 Collateral Management and Margining Costs 35316.2 CBVA General Formula with Margining Costs 35516.2.1 Perfect Collateralization 35616.2.2 Futures Contracts 35716.3 Changing the Collateralization Currency 35716.3.1 Margining Cost in Foreign Currency 35716.3.2 Settlement Liquidity Risk 35816.3.3 Gap Risk in Single-Currency Contracts with Foreign-Currency Collaterals 35916.4 Conclusions 35917 Funding Valuation Adjustment (FVA)? 36117.1 Dealing with Costs of Funding 36117.1.1 Central Clearing, CCPs and this Book 36217.1.2 High Level Features 36217.1.3 Single-Deal (Micro) vs. Homogeneous (Macro) Funding Models 36317.1.4 Previous Literature on Funding and Collateral 36417.1.5 Including FVA along with Credit and Debit Valuation Adjustment 36517.1.6 FVA is not DVA 36517.2 Collateral- and Funding-Inclusive Bilateral Valuation Adjusted Price 36617.3 Funding Risk and Liquidity Policies 36717.3.1 Funding, Hedging and Collateralization 36717.3.2 Liquidity Policies 36817.4 CBVA Pricing Equation with Funding Costs (CFBVA) 37217.4.1 Iterative Solution of the CFBVA Pricing Equation 37317.4.2 Funding Derivative Contracts in a Diffusion Setting 37417.4.3 Implementing Hedging Strategies via Derivative Markets 37717.5 Detailed Examples 37817.5.1 Funding with Collateral 37817.5.2 Collateralized Contracts Priced by a CCP 37917.5.3 Dealing with Own Credit Risk: FVA and DVA 38017.5.4 Deriving Earlier Results on FVA and DVA 38117.6 Conclusions: FVA and Beyond 38218 Non-Standard Asset Classes: Longevity Risk 38518.1 Introduction to Longevity Markets 38518.1.1 The Longevity Swap Market 38518.1.2 Longevity Swaps: Collateral and Credit Risk 38618.1.3 Indexed Longevity Swaps 39018.1.4 Endogenous Credit Collateral and Funding-Inclusive Swap Rates 39018.2 Longevity Swaps: The Payoff P39118.3 Mark-to-Market for Longevity Swaps 39418.4 Counterparty and Own Default Risk, Collateral and Funding 39718.5 An Example of Modelling Specification from Biffis et al. (2011) 40118.6 Discussion of the Results in Biffis et al. (2011) 40419 Conclusions and Further Work 40919.1 A Final Dialogue: Models, Regulations, CVA/DVA, Funding and More 409Bibliography 415Index 423