This preview shows pages 1–3. Sign up to view the full content.
1
Stats243 Summer 2007
Reviews of Martingale
approach for BlackScholes
Models
¾
Brownian motion
¾
Ito calculus
¾
Change of measure
¾
Martingale representation theorem
¾
Martingale approach for BlackScholes model
Stats243 Summer 2007
1. Brownian Motion
•
Definition:
The process W = (W
t
: t ¸ 0) is a PBrownian 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 selfsimilar.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document 2
Stats243 Summer 2007
1. Brownian Motion
• Brownian motion with drift
– What is the covariance function?
• Geometric Brownian motion (GBM)
– GBM with drift is a model often used for stock prices.
– Assume that
μ
is a drift factor and
σ
is a noise factor.
– What is the covariance function?
Stats243 Summer 2007
•
Consider functions of Brownian motion, can we establish certain calculus
rules?
•
Recall that, in
This is the end of the preview. Sign up
to
access the rest of the document.
This note was uploaded on 10/30/2011 for the course AMS 517 taught by Professor Xinghaipeng during the Spring '11 term at SUNY Stony Brook.
 Spring '11
 XingHaipeng

Click to edit the document details