Perturbation methods in credit derivatives: strategies for efficient risk management

TABLE OF CONTENTS Preface xi Acknowledgments xv Acronyms xvi Chapter 1 Why Perturbation Methods? 1 1.1 Analytic Pricing of Derivatives 1 1.2 In Defence of Perturbation Methods 3 Chapter 2 Some Representative Case Studies 8 2.1 Quanto CDS Pricing 8 2.2 Wrong-Way Interest Rate Risk 9 2.3 Contingent CDS Pricing and CVA 10 2.4 Analytic Interest Rate Option Pricing 10 2.5 Exposure Scenario Generation 11 2.6 Model Risk 11 2.7 Machine Learning 12 2.8 Incorporating Interest Rate Skew and Smile 13 Chapter 3 The Mathematical Foundations 14 3.1 The Pricing Equation 14 3.2 Pricing Kernels 16 3.2.1 What Is a Kernel? 16 3.2.2 Kernels in Financial Engineering 18 3.2.3 Why Use Pricing Kernels? 19 3.3 Evolution Operators 20 3.3.1 Time-Ordered Exponential 21 3.3.2 Magnus Expansion 22 3.4 Obtaining the Pricing Kernel 23 3.4.1 Duhamel-Dyson Expansion Formula 24 3.4.2 Baker-Campbell-Hausdorff Expansion Formula 24 3.4.3 Exponential Expansion Formula 25 3.4.4 Exponentials of Derivatives 26 3.4.5 Example - The Black-Scholes Pricing Kernel 28 3.4.6 Example - Mean-Reverting Diffusion 30 3.5 Convolutions with Gaussian Pricing Kernels 32 3.6 Proofs for Chapter 3 36 3.6.1 Proof of Theorem 3.2 36 3.6.2 Proof of Lemma 3.1 38 Chapter 4 Hull-White Short-Rate Model 40 4.1 Background of Hull-White Model 41 4.2 The Pricing Kernel 42 4.3 Applications 43 4.3.1 Zero Coupon Bond Pricing 43 4.3.2 LIBOR Pricing 44 4.3.3 Caplet Pricing 45 4.3.4 European Swaption Pricing 47 4.4 Proof of Theorem 4.1 48 4.4.1 Preliminary Results 48 4.4.2 Turn the Handle! 49 Chapter 5 Black-Karasinski Short-Rate Model 52 5.1 Background of Black-Karasinski Model 52 5.2 The Pricing Kernel 54 5.3 Applications 56 5.3.1 Zero Coupon Bond Pricing 56 5.3.2 Caplet Pricing 58 5.3.3 European Swaption Pricing 61 5.4 Comparison of Results 62 5.5 Proof of Theorem 5.1 65 5.5.1 Preliminary Result 65 5.5.2 Turn the Handle! 66 5.6 Exact Black-Karasinski Pricing Kernel 67 Chapter 6 Extension to Multi-Factor Modelling 70 6.1 Multi-Factor Pricing Equation 70 6.2 Derivation of Pricing Kernel 73 6.2.1 Preliminaries 73 6.2.2 Full Solution Using Operator Expansion 74 6.3 Exact Expression for Hull-White Model 75 6.4 Asymptotic Expansion for Black-Karasinski Model 78 6.5 Formal Solution for Rates-Credit Hybrid Model 82 Chapter 7 Rates-Equity Hybrid Modelling 86 7.1 Statement of Problem 86 7.2 Previous Work 86 7.3 The Pricing Kernel 87 7.3.1 Main Result 87 7.4 Vanilla Option Pricing 90 Chapter 8 Rates-Credit Hybrid Modelling 92 8.1 Background 92 8.1.1 Black-Karasinski as a Credit Model 92 8.1.2 Analytic Pricing of Rates-Credit Hybrid Products 93 8.1.3 Mathematical Definition of the Model 94 8.1.4 Pricing Credit-Contingent Cash Flows 94 8.2 The Pricing Kernel 95 8.3 CDS Pricing 101 8.3.1 Risky Cash Flow Pricing 101 8.3.2 Protection Leg Pricing 103 8.3.3 Defaultable LIBOR Pricing 105 8.3.4 Defaultable Capped LIBOR Pricing 110 8.3.5 Contingent CDS with IR Swap Underlying 111 Chapter 9 Credit-Equity Hybrid Modelling 116 9.1 Background 116 9.2 Derivation of Credit-Equity Pricing Kernel 117 9.2.1 Pricing Equation 117 9.2.2 Pricing Kernel 119 9.2.3 Asymptotic Expansion 120 9.3 Convertible Bonds 122 9.4 Contingent CDS on Equity Option 124 Chapter 10 Credit-FX Hybrid Modelling 127 10.1 Background 127 10.2 Credit-FX Pricing Kernel 128 10.3 Quanto CDS 129 10.3.1 Domestic Currency Fixed Flow 129 10.3.2 Foreign Currency Fixed Flow 129 10.3.3 Foreign Currency LIBOR Flow 131 10.3.4 Foreign Currency Notional Protection 131 10.4 Contingent CDS on Cross-Currency Swaps 133 Chapter 11 Multi-Currency Modelling 137 11.1 Previous Work 137 11.2 Statement of Problem 138 11.3 The Pricing Kernel 139 11.3.1 Main Result 139 11.3.2 Derivation of Multi-Currency Pricing Kernel 142 11.4 Inflation and FX Options 144 Chapter 12 Rates-Credit-FX Hybrid Modelling 146 12.1 Previous Work 146 12.2 Derivation of Rates-Credit-FX Pricing Kernel 146 12.2.1 Pricing Equation 146 12.2.2 Pricing Kernel 148 12.3 Quanto CDS Revisited 155 12.3.1 Domestic Currency Fixed Flow 155 12.3.2 Foreign Currency Fixed Flow 155 12.3.3 Foreign Currency Notional Protection 158 12.4 CCDS on Cross-Currency Swaps Revisited 159 Chapter 13 Risk-Free Rates 163 13.1 Background 163 13.2 Hull-White Kernel Extension 165 13.3 Applications 166 13.3.1 Compounded Rates Payment 166 13.3.2 Caplet Pricing 166 13.3.3 European Swaption Pricing 169 13.3.4 Average Rate Options 169 13.4 Black-Karasinski Kernel Extension 170 13.5 Applications 171 13.5.1 Compounded Rates Payment 171 13.5.2 Caplet Pricing 172 13.6 A Note on Term Rates 177 Chapter 14 Multi-Curve Framework 178 14.1 Background 178 14.2 Stochastic Spreads 180 14.3 Applications 182 14.3.1 LIBOR Pricing 182 14.3.2 LIBOR Caplet Pricing 183 14.3.3 European Swaption Pricing 186 Chapter 15 Scenario Generation 187 15.1 Overview 187 15.2 Previous Work 188 15.3 Pricing Equation 190 15.4 Hull-White Rates 192 15.4.1 Two-Factor Pricing Kernel 192 15.4.2 m-Factor Extension 194 15.5 Black-Karasinski Rates 195 15.5.1 Two-Factor Pricing Kernel 195 15.5.2 Asymptotic Expansion 195 15.5.3 m-Factor Extension 198 15.5.4 Representative Calculations 198 15.6 Joint Rates-Credit Scenarios 201 Chapter 16 Model Risk Management Strategies 203 16.1 Introduction 203 16.2 Model Risk Methodology 205 16.2.1 Previous Work 205 16.2.2 Proposed Framework 208 16.2.3 Calibration to CDS Market 209 16.3 Applications 210 16.3.1 Interest Rate Swap Extinguisher 210 16.3.2 Contingent CDS 211 16.4 Conclusions 212 Chapter 17 Machine Learning 213 17.1 Trends in Quantitative Finance Research 213 17.1.1 Some Recent Trends 213 17.1.2 The Arrival of Machine Learning 214 17.2 From Pricing Models to Market Generators 215 17.3 Synergies with Perturbation Methods 217 17.3.1 Asymptotics as Control Variates 217 17.3.2 Data Representation 218 Bibliography 222 Index 229

Perturbation Methods in Credit Derivatives: Strategies for Efficient Risk Management offers an incisive examination of a new approach to pricing credit-contingent financial instruments. Author and experienced financial engineer Dr. Colin Turfus has created an approach that allows model validators to perform rapid benchmarking of risk and pricing models while making the most efficient use possible of computing resources. The book provides innumerable benefits to a wide range of quantitative financial experts attempting to comply with increasingly burdensome regulatory stress-testing requirements, including: Replacing time-consuming Monte Carlo simulations with faster, simpler pricing algorithms for front-office quants Allowing CVA quants to quantify the impact of counterparty risk, including wrong-way correlation risk, more efficiently Developing more efficient algorithms for generating stress scenarios for market risk quants Obtaining more intuitive analytic pricing formulae which offer a clearer intuition of the important relationships among market parameters, modelling assumptions and trade/portfolio characteristics for traders The methods comprehensively taught in Perturbation Methods in Credit Derivatives also apply to CVA/DVA calculations and contingent credit default swap pricing.