handout00_pre - Reviews of Martingale approach for...

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 .