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Stochastic Differential Equations in Finance
Katerina Loizou
Candidate No: 51046, School of Mathematics,
University of Sussex,
United Kingdom
kl202@sussex.ac.uk,
02 September 2010
Contents
1
Introduction 6
2
Probabilities & Random Variables 8
2.1 Random
Sequences . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2 Normal
Random Variables . . . . . . . . . . . . . . . . . . . . 10
2.3
Discrete Random Variables . . . . . . . . . . . . . . . . . . .
. 11
2.4
Continuous Random Variables . . . . . . . . . . . .. . . . . 12
2.5
Expectation, Variance and High Moments . . . . . . . 19
3
Computational Simulations 24
3.1
Generation of Random Numbers . . . . . . . . . . . . . . . . .
24
3.2 Monte Carlo Simulation . . . . . . . . .
. . . . . . . . . . . . 25
4 Random
Processes 27
4.1 Characterisation of Random Processes . . .
. . . . . . . . . . 27
4.2
Gaussian Process . . . . . . . . . . . . . . . . . . . . . . .
. . 29
4.3 Random
Walk . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4.4 Wiener
Process - Brownian Motion . . . . . . . . . . . . . . . 31
5
Deterministic Differential Equations 36
5.1 Taylor
Series . . . . . . . . . . . . . . . . . . . . . . . . . . . .
36
5.2 Euler
Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
5.3 Higher
Order Taylor
Methods . . . . . . . . . . . . . . . . . . 37
5.4 Error
Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
5.5 Runge-Kutta Method . . . . . . . . . . . .
. . . . . . . . . . . 43
5.6 Multistep
Methods . . . . . . . . . . . . . . . . . . . . . . . . 45
5.7
Predictor-Corrector Method . . . . . . . . . . . . . . . . . .
. 46
5.8
Consistency, Convergence and Stability . . . . . . . . . . . .
. 46
5.9
Stability . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . 49
5.10
Absolute Stability . . . . . . . . . . . . . . . . . . . . . .
. . . 49
6
Stochastic Differential Equations 52
6.1
Introduction to Stochastic Differential Equations . . . . . . .
. 52
6.2
Euler-Maruyama Scheme . . . . . . . . . . . . . . . . . . . . .
55
6.3
Convergence of the Euler-Maruyama scheme . . . . . . . . . . 58
6.4
Milstein Approximation . . . . . . . . . . . . . . . . . . . .
. 62
6.5
Convergence of the Milstein Approximation . . . . . . . . . .
65
6.6 Euler
and Milstein Comparison . . . . . . . . . . . . . . . . . 66
7 Examples in
finance 72
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