Equity Derivatives and Market Risk Models

The Authors ix

Notation xi

Introduction xiii

Part I. Modelling Framework 1

1. The Black-Scholes Framework 3

1.1 The Black-Scholes equity model 3

1.2 Extentions to Black-Scholes 6

2. Skew Models 13

2.1 Introduction 13

2.2 Volatility surface generation 15

2.3 Volatility smile model 18

2.4 Volatility surface dynamics 20

3. Jump-Diffusion Models 23

3.1 Model Description 23

3.2 Options pricing 25

3.3 Fitting the smile 27

4. Deterministic Volatility Models 31

4.1 Introduction 31

4.2 Calibration techniques 33

4.3 Hedging 36

5. Stochastic Volatility Models 39

5.1 The Hull-White model 39

5.2 The Heston Model 41

5.3 Calibration 43

5.4 Hedging 45

5.5 Introduction to Arch and Garch 46

6. Credit Spread Models 51

6.1 Merton’s model 52

6.2 Structural models 53

6.3 Intensity models 53

6.4 Convertible bonds with credit risk 57

Part II. Numerical Techniques 63

7. Trees 65

7.1 Thich tree 65

7.2 Implied trees 66

7.3 Stochastic trees 69

7.4 Generic tree framework 74

8. Finite Difference 77

8.1 One-dimensional techniques 77

8.2 Path-dependant options 83

8.3 Two-dimensional techniques 85

8.4 Generic finite difference 87

9. Monte Carlo 89

9.1 Local volatility in Monte Carlo 89

9.2 Itô-Taylor expansion 90

9.3 Greeks in Monte Carlo 94

9.4 Generic Monte Carlo framework 102

10. Alternative Approaches 105

10.1 Fourier transforms 105

10.2 Laplace transforms 108

10.3 Path integral 109

Part III. Market Products 111

11. American Options on Multi Assets 113

11.1 Markov chain method 113

11.2 Regression for continuation method 115

11.3 Simulated tree 116

11.4 Stochastic mesh 117

12. Volatility Contracts 119

12.1 Variance swaps 120

12.2 Covariance swaps 124

12.3 Volatility swaps 126

12.4 Volaility options 134

13. Discrete Sampling Options 137

13.1 Barriers 137

13.2 Lookbacks 138

14. Additional Products 143

14.1 Cliquet with smile - analytical approximation 143

14.2 Barrier options with a smile 144

14.3 Passport options 146

Part IV. Risk Management 149

15. Introduction to Risk Management 151

15.2 Credit Risk 152

15.3 Raroc 153

16. Value-at-Risk 157

16.1 The VaR approach 158

16.2 VaR methodologies 164

16.3 Simulated VaR 164

16.4 Analytical VaR 169

16.5 Correlation concepts 174

17. Extreme Value Theory 177

17.1 The domain of attraction 177

17.2 A central limit theorem for maxima 179

17.3 Point process approach 182

17.4 Estimation of the tail distribution 183

17.5 A limit theorem for the excess distribution 186

17.6 The peaks over threshold (POT) method 188

17.7 Dynamic extreme value theory 190

17.8 Multi-day returns 194

17.9 Multivariate EVT 194

17.10 Hill estimation 195

18. Coherent Risk Measures 197

18.1 Axioms for acceptance sets 197

18.2 Correspondence between acceptance sets 198

and risk measures

18.3 Axioms for risk measures 198

18.4 Correspondence between the axioms on 198

acceptance sets and risk measures

18.5 Value-at-risk and expected shortfall 199

18.6 Model-free risk measures 200

18.7 Generalised senarios 201

19. Credit Risk Management 203

19.1 The asset value model 203

19.2 The credit quality migration model 207

19.2 Credit Risk+ 211

Bibliography 219