Chapter 21

# Simulating Geometric Brownian Motion

In this chapter, we will show how to use the results of Chapter 20 to simulate geometric Brownian motion-based stock prices, first at a single point in time, and then along a whole path.

This is a very important chapter for practical financial modeling.

## 21.1 Simulating GBM Stock Prices at a Single Future Time

In Section 20.3.5.2, we showed, using Ito’s lemma, that the solution of the geometric Brownian motion stochastic differential equation

$dS=m\text{\hspace{0.17em}}Sdt+s\text{\hspace{0.17em}}SdW$

is available in closed form:

${S}_{t}={S}_{0}{e}^{\left(m-(1/2){s}^{2}\right)t+s{W}_{t}}$

Since, if we use only information available at time 0 (at which time W0 = 0, Wt has zero mean, variance t, and is normally distributed), we can write this as

${S}_{t}={S}_{0}{e}^{\left(m-(1/2){s}^{2}\right)t}{e}^{s\sqrt{t}{Z}_{t}}$

where ...