Mathematical Modeling in Finance with Stochastic Processes

6.3. PROPERTIES OF GEOMETRIC BROWNIAN MOTION 207

Outside Readings and Links:

1.

2.

3.

4.

6.3 Properties of Geometric Brownian Mo-

tion

Rating

Mathematically Mature: may contain mathematics beyond calculus with
proofs.

Section Starter Question

For the ordinary differential equation

dx

dt

=rx x(0) =x 0

what is the rate of growth of the solution?

Key Concepts

1. Geometric Brownian Motion is the continuous time stochastic process
   z 0 exp(μt+σW(t)) whereW(t) is standard Brownian Motion.


2. The mean of Geometric Brownian Motion is

z 0 exp(μt+ (1/2)σ^2 t).

3. The variance of Geometric Brownian Motion is

z^20 exp(2μt+σ^2 t)(exp(σ^2 t)−1).