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 as a product of a Standard Normal random variable (rv ∼ N(0,1)) and the square root of the time increment. Another way to say this is that the Brownian increment comes from rv ∼ N(0,t), where t is the time increment. We obtain the Brownian path by taking the cumulative sum of the Brownian increments. 

The SDE has a closed-form solution (only a few SDEs have it):

Here,  is the initial value of the process, which in this case is the initial price of a stock. The preceding equation presents the relationship compared to the initial stock price.

For simulations, we can use the following recursive formula:

Here, Z_i is a Standard Normal random variable and  is the time index. This specification is possible because the increments of W are independent and normally distributed.

In this recipe, we use Monte Carlo methods and the Geometric Brownian Motion to simulate Microsoft's stock prices one month ahead.