文档介绍：Stochastic Processes, Ito Calculus and
Black-Scholes formula
Tobias Galla
ICTP
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May 18, 2006
Contents
1 Motivation 3
2 Examples 3
Random walk: diffusion equation . . . . . . . . . . . . . . . . . . . . 3
Particles in a graviational field . . . . . . . . . . . . . . . . . . . . . . 4
Langevin dynamics: Brownian particles with a linear drift . . . . . . 5
General rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
Langevin dynamics in statistical mechanics . . . . . . . . . . . . . . . 8
How to put a stochastic differential equation onto puter ? . . . 8
3 Markov processes 10
Chapman-Kolmogorov equation . . . . . . . . . . . . . . . . . . . . . 10
Master equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
Fokker-Planck equation . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1
4 Stochastic differential equations 13
dW 2 = dt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
Ito’s Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
General case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
General SDEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
5 Stochastic integration - Ito Calculus 17
General remarks (section not proof read) . . . . . . . . . . . . . . . . 17
Ito versus Stratonovich SDE (section not proof read) . . . . . . . . . 19
6 Financial derivatives 21
General remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
No free lunch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
7 Pricing of financial derivatives, the Black-Scholes formula 24
The Cox-Ross-Rubinstein-Modell . . . . . . . . . . . . . . . . . . .