32 Modelling Future Stock Prices Using Geometric Brownian Motion: An Introduction

Peter McQuire

32.1 Introduction

In this chapter we develop, starting with the simple random walk, one of the most widely used models for projecting future share prices – Geometric Brownian Motion (“GBM”). To see where we are aiming, we have set out below, with no explanation at this point, the discrete time approximation of GBM, where upper S Subscript t is the stock price at time t:

(32.1)upper S Subscript t plus normal upper Delta t Baseline almost-equals upper S Subscript t Baseline plus mu upper S Subscript t Baseline normal upper Delta t plus sigma upper S Subscript t Baseline StartRoot normal upper Delta t EndRoot epsilon

which is exact as normal upper Delta t right-arrow 0. Using the rules of stochastic calculus, we arrive at the exact solution for the price upper S Subscript t at time t:

(32.2)upper S Subscript t Baseline equals upper S 0 left-parenthesis e Superscript left-parenthesis mu minus StartFraction sigma squared Over 2 EndFraction right-parenthesis t plus sigma StartRoot t EndRoot epsilon Baseline right-parenthesis

where mu is the expected rate of return per unit time from the stock, is the standard deviation of the returns from the ...

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