Simulating stock price dynamics using Geometric Brownian Motion

Thanks to the unpredictability of financial markets, simulating stock prices plays an important role in the valuation of many derivatives, such as options. Due to the aforementioned randomness in price movement, these simulations rely on stochastic differential equations (SDE).

A stochastic process is said to follow the Geometric Brownian Motion (GBM) when it satisfies the following SDE:

Here, we have the following:

• S: Stock price
• μ: The drift coefficient, that is, the average return over a given period or the instantaneous expected return
• σ: The diffusion coefficient, that is, how much volatility is in the drift
• W_t: The Brownian Motion

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 are calculated...