COLIN TURFUS, PHD., works in Global Model Validation and Governance at Deutsche Bank. For the last fifteen years, he has been a financial engineer, mainly analysing model risk for credit derivatives and hybrids. He specialises in the application of perturbation methods to risk management, finding efficient analytic methods for computing prices and risk measures. He also taught courses on C++ and Financial Engineering at City, University of London for seven years. Prior to that, Colin worked as a developer consultant in the mobile phone industry after an extended period in academia, teaching applied mathematics and researching in fluid dynamics and turbulent dispersion.??

Preface xiii

Acknowledgments xvii

Acronyms xix

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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