## 8.1 INTRODUCTION

The geometric Brownian motion is the starting point for most equity derivative models:

The equation above describes the changes *dS*_{t} in the value of the share price and illustrates the dependence of those changes on the expected return *μ* of the stock at its volatility σ. From Equation (7.31) we know that the possible values for *S*_{t} at a future time *t*, starting from the current value *S*_{s} at the current time *s*, are given by:

(8.2)
The random component driving the stock price from *S*_{s} to *S*_{t} is the Brownian motion *W* = {*W*_{t}, *t* ≥ 0}. One factor in the equation above is investor-dependent. Investors will indeed have different values for the drift component *μ* in the equation above. An investor with a bullish mindset will have a very positive value for *μ* opposed to more moderate values for the investors taking a bearish stance on the stock. We now want to calculate the price of a derivative security *P* whose value depends on the future price path followed by the underlying share price *S*. How can we eliminate the different assumptions regarding *μ*? This is where a risk-neutral setting enters on stage, which allows derivative pricing independent from the individual investor’s views. We start illustrating the possibility to hedge away ...