1 Why Perturbation Methods? 1
1.1 Analytic Pricing of Derivatives 1
1.2 In Defence of Perturbation Methods 3
2 Some Representative Case Studies 7
2.1 Quanto CDS Pricing 7
2.2 Wrong-Way Interest Rate Risk 8
2.3 Contingent CDS Pricing and CVA 9
2.4 Analytic Interest Rate Option Pricing 9
2.5 Exposure Scenario Generation 10
2.6 Model Risk 11
2.7 Machine Learning 12
2.8 Incorporating Interest Rate Skew and Smile 13
3 The Mathematical Foundations 15
3.1 The Pricing Equation 15
3.2 Pricing Kernels 18
3.2.1 What Is a Kernel? 18
3.2.2 Kernels in Financial Engineering 20
3.2.3 Why Use Pricing Kernels? 20
3.3 Evolution Operators 21
3.4 Obtaining the Pricing Kernel 25
3.4.1 Example - The Black-Scholes Pricing Kernel 31
3.4.2 Example - Mean-Reverting Diffusion 32
3.5 Convolutions with Gaussian Pricing Kernels 35
3.6 Proofs for Chapter 3 39
3.6.1 Proof of Theorem 3.2 39
3.6.2 Proof of Lemma 3.1 41
4 Hull-White Short Rate Model 43
4.1 Background of Hull-White Model 44
4.2 The Pricing Kernel 45
4.3 Applications 46
4.3.1 Zero Coupon Bond Pricing 46
4.3.2 LIBOR Pricing 47
4.3.3 Caplet Pricing 49
4.3.4 European Swaption Pricing 50
4.3.5 CMS Index Representation 51
4.4 Proof of Theorem 4.1 52
4.4.1 Preliminary Results 52
4.4.2 Turn the Handle! 54
5 Black-Karasinski Short Rate Model 57
5.1 Background of Black-Karasinski Model 57
5.2 The Pricing Kernel 59
5.3 Applications 61
5.3.1 Zero Coupon Bond Pricing 61
5.3.2 Caplet Pricing 63
5.3.3 European Swaption Pricing 67
5.4 Comparison of Results 67
5.5 Proof of Theorem 5.1 71
5.5.1 Preliminary Result 71
5.5.2 Turn the Handle! 72
5.6 Exact Black-Karasinski Pricing Kernel 74
6 Extension to Multi-Factor Modelling 77
6.1 Multi-Factor Pricing Equation 77
6.2 Derivation of Pricing Kernel 81
6.2.1 Preliminaries 81
6.2.2 Full Solution Using Operator Expansion 82
6.3 Exact Expression for Hull-White Model 83
6.4 Asymptotic Expansion for Black-Karasinski Model 87
6.5 Formal Solution for Rates-Credit Hybrid Model 91
7 Rates-Equity Hybrid Modelling 95
7.1 Statement of Problem 95
7.2 Previous Work 96
7.3 The Pricing Kernel 97
7.3.1 Main Result 97
7.4 Vanilla Option Pricing 100
8 Rates-Credit Hybrid Modelling 101
8.1 Background 101
8.1.1 Black-Karasinski as a Credit Model 101
8.1.2 Analytic Pricing of Rates-Credit Hybrid Products 102
8.1.3 Mathematical Definition of the Model 103
8.1.4 Pricing Credit-Contingent Cash Flows 103
8.2 The Pricing Kernel 104
8.3 CDS Pricing 110
8.3.1 Risky Cash Flow Pricing 110
8.3.2 Protection Leg Pricing 113
8.3.3 Defaultable LIBOR Pricing 114
8.3.4 Defaultable Capped LIBOR Pricing 119
8.3.5 Contingent CDS with IR Swap Underlying 120
9 Credit-Equity Hybrid Modelling 127
9.1 Background 127
9.2 Derivation of Credit-Equity Pricing Kernel 129
9.2.1 Pricing Equation 129
9.2.2 Pricing Kernel 130
9.2.3 Asymptotic Expansion 132
9.3 Convertible Bonds 133
9.4 Contingent CDS on Equity Option 136
10 Credit-FX Hybrid Modelling 139
10.1 Background 139
10.2 Credit-FX Pricing Kernel 140
10.3 Quanto CDS 141
10.3.1 Domestic Currency Fixed Flow 141
10.3.2 Foreign Currency Fixed Flow 142
10.3.3 Foreign Currency LIBOR Flow 143
10.3.4 Foreign Currency Notional Protection 143
10.4 Contingent CDS on Cross-Currency Swaps 145
11 Multi-Currency Modelling 151
11.1 Previous Work 151
11.2 Statement of Problem 152
11.3 The Pricing Kernel 153
11.3.1 Main Result 153
11.3.2 Derivation of Multi-Currency Pricing Kernel 157
11.4 Inflation and FX Options 159
12 Rates-Credit-FX Hybrid Modelling 161
12.1 Previous Work 161
12.2 Derivation of Rates-Credit-FX Pricing Kernel 162
12.2.1 Pricing Equation 162
12.2.2 Pricing Kernel 163
12.3 Quanto CDS Revisited 170
12.3.1 Domestic Currency Fixed Flow 170
12.3.2 Foreign Currency Fixed Flow 171
12.3.3 Foreign Currency Notional Protection 174
12.4 CCDS on Cross-Currency Swaps Revisited 175
13 Risk-Free Rates 181
13.1 Background 181
13.2 Hull-White Kernel Extension 183
13.3 Applications 184
13.3.1 Compounded Rates Payment 184
13.3.2 Caplet Pricing 185
13.3.3 European Swaption Pricing 187
13.3.4 Average Rate Options 187
13.4 Black-Karasinski Kernel Extension 189
13.5 Applications 190
13.5.1 Compounded Rates Payment 190
13.5.2 Caplet Pricing 191
13.6 A Note on Term Rates 195
14 Multi-Curve Framework 197
14.1 Background 197
14.2 Stochastic Spreads 200
14.3 Applications 201
14.3.1 LIBOR Pricing 201
14.3.2 LIBOR Caplet Pricing 202
14.3.3 European Swaption Pricing 206
15 Scenario Generation 209
15.1 Overview 209
15.2 Previous Work 210
15.3 Pricing Equation 213
15.4 Hull-White Rates 214
15.4.1 Two-Factor Pricing Kernel 214
15.4.2 m-Factor Extension 217
15.5 Black-Karasinski Rates 218
15.5.1 Two-Factor Pricing Kernel 218
15.5.2 Asymptotic Expansion 219
15.5.3 m-Factor Extension 221
15.5.4 Representative Calculations 222
15.6 Joint Rates-Credit Scenarios 224
16 Model Risk Management Strategies 227
16.1 Introduction 227
16.2 Model Risk Methodology 229
16.2.1 Previous work 229
16.2.2 Proposed framework 232
16.2.3 Calibration to CDS market 234
16.3 Applications 235
16.3.1 Interest rate swap extinguisher 235
16.3.2 Contingent CDS 236
16.4 Conclusions 238
Bibliography 239
Index 247
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