Table of Contents

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