Brownian Motion and Geometric Growth
- Risk Management
The stochastic-process notes move from a random walk to Brownian motion and then to geometric Brownian motion. Brownian increments are normally distributed with variance proportional to elapsed time, while the geometric model keeps a positive asset level by applying the process to log returns.
With constant parameters, the geometric Brownian motion solution has the form
$$S_t=S_0\exp\left((\mu-\tfrac12\sigma^2)t+\sigma W_t\right).$$
The \(-\tfrac12\sigma^2\) term is the Itô correction. It matters when calibrating a model from observed returns because the mean of the log process is not the same as the mean of the price process.