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Quantitative Finance by Matt Davison

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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=mSdt+sSdW

is available in closed form:

St=S0e(m(1/2)s2)t+sWt

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

St=S0e(m(1/2)s2)testZt

where ...

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