1
Stats243 Summer 2007
Reviews of Martingale
approach for Black-Scholes
Models
¾
Brownian motion
¾
Ito calculus
¾
Change of measure
¾
Martingale representation theorem
¾
Martingale approach for Black-Scholes model
Stats243 Summer 2007
1. Brownian Motion
•
Definition:
The process W = (W
t
: t ¸ 0) is a P-Brownian motion if and only
if
–
(i) W
t
is continuous and W
0
= 0;
–
(ii) The value of W
t
is distributed, under P, as N(0,t);
–
(iii) The increment W
s+t
– W
s
is distributed as N(0,t), under P, and is
independent of F
s
, the history of what the process did up to time s.
•
Odd properties of Brownian motion (BM)
–
BM is continuous but nowhere differentiable.
–
BM could hit any real value, but, with probability one, it will be back
down again to zero.
–
BM is self-similar.
–
BM is also called a
Wiener process
.

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