Allowing CVA quants to quantify the impact of counterparty risk, includ
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
