Simulating stock price dynamics using a geometric Brownian motion

Simulating stock prices plays a crucial role in the valuation of many derivatives, most notably options. Due to the randomness in the price movement, these simulations rely on stochastic differential equations (SDEs). A stochastic process is said to follow a geometric Brownian motion (GBM) when it satisfies the following SDE:

Here, we have the following:

• S_t—Stock price
• —The drift coefficient, that is, the average return over a given period or the instantaneous expected return
• —The diffusion coefficient, that is, how much volatility is in the drift
• W_t —The Brownian motion
• d—This symbolizes the change in the variable over the considered time increment, while dt is the change in time

We will not investigate the properties of the Brownian motion in too much depth, as it is outside the scope of this book. Suffice to say, Brownian increments...