1 Modeling Tools for Financial Options ..................................... 1
1.1 Options ................................................................. 1
1.1.1 The Payoff Function........................................... 2
1.1.2 A Priori Bounds ............................................... 5
1.1.3 Options in the Market ......................................... 6
1.1.4 The Geometry of Options ..................................... 7
1.2 Model of the Financial Market ........................................ 9
1.3 Numerical Methods.................................................... 12
1.3.1 Algorithms..................................................... 13
1.3.2 Discretization .................................................. 14
1.3.3 Effificiency ...................................................... 15
1.4 The Binomial Method ................................................. 16
1.4.1 A Discrete Model.............................................. 16
1.4.2 Derivation of Equations ....................................... 18
1.4.3 Solution of the Equations ..................................... 21
1.4.4 A Basic Algorithm ............................................ 21
1.4.5 Improving the Convergence................................... 24
1.4.6 Sensitivities .................................................... 27
1.4.7 Extensions ..................................................... 29
1.5 Risk-Neutral Valuation ................................................ 30
1.6 Stochastic Processes ................................................... 34
1.6.1 Wiener Process ................................................ 35
1.6.2 Stochastic Integral ............................................. 37
1.7 Diffusion Models ...................................................... 41
1.7.1 Itô Process ..................................................... 41
1.7.2 Geometric Brownian Motion ................................. 44
1.7.3 Risk-Neutral Valuation ........................................ 45
1.7.4 Mean Reversion ............................................... 47
1.7.5 Vector-Valued Stochastic Differential Equations ............ 481.8 Itô Lemma and Applications .......................................... 51
1.8.1 Itô Lemma ..................................................... 51
1.8.2 Consequences for Geometric Brownian Motion ............. 52
1.8.3 Integral Representation........................................ 54
1.8.4 Bermudan Options ............................................ 55
1.8.5 Empirical Tests ................................................ 57
1.9 Jump Models........................................................... 58
1.9.1 Poisson Process................................................ 58
1.9.2 Simulating Jumps ............................................. 60
1.9.3 Jump Diffusion ................................................ 61
1.10 Calibration ............................................................. 63
1.11 Notes and Comments .................................................. 66
1.12 Exercises ............................................................... 71