viii Contents
4 Derivatives Modeling in Practice ......................................... 43
4.1 Model Applications ................................................. 43
4.2 Calibration .......................................................... 45
4.3 Risk Management................................................... 53
4.4 Model Limitations .................................................. 69
4.5 Testing ............................................................... 73
Bibliography ................................................................. 78
Part II Skew and Smile Techniques
5 Continuous Stochastic Processes .......................................... 81
5.1 The Linear SDE ..................................................... 82
5.2 The Lognormal SDE ................................................ 82
5.3 The Normal SDE.................................................... 84
5.4 The Shifted Lognormal SDE ....................................... 86
5.5 The Quadratic SDE ................................................. 88
5.6 The Ornstein-Uhlenbeck Process .................................. 90
5.7 The Brownian Bridge ............................................... 91
5.8 The CEV Process ................................................... 93
5.9 The Bessel Process.................................................. 98
5.10 Non-Analytic SDEs................................................. 102
Bibliography ................................................................. 105
6 Local Volatility Models..................................................... 107
6.1 ATM Perturbation ................................................... 108
6.2 Dupire’s Equation ................................................... 113
6.3 Short Maturity Expansion .......................................... 114
6.4 Dynamics............................................................ 117
Bibliography ................................................................. 117
7 Stochastic Volatility Models ............................................... 119
7.1 Skew and Smile ..................................................... 120
7.2 Perturbation for Small Volatility of Volatility ..................... 123
7.3 Conditional Expectation Approach ................................ 129
7.4 Fourier Transform Approach ....................................... 130
7.5 Comparison of Methods ............................................ 134
7.6 Relations to Implied and Local Volatility .......................... 135
7.7 Dynamics............................................................ 136
7.8 Local Stochastic Volatility.......................................... 138
Bibliography ................................................................. 138
8L
´
evy Models ................................................................. 139
8.1 L´evy Processes ...................................................... 140
8.2 L´evy-Ito Decomposition............................................ 141
8.3 Stochastic Calculus ................................................. 144
8.4 Examples of L´evy Processes ....................................... 147