Geometric Brownian Motion Simulations
The figure plots simulated paths for the stock price S_{n \Delta t} = S_{0} \exp\left(\left(\mu - \frac{1}{2} \sigma^{2} \right) (T - t) + \sigma \sqrt{\Delta t} \sum_{i=1}^{n} \varepsilon_{i}\right) where n = 1, 2, \ldots, 5000, S_{0} = 100, \mu = 0.20, \sigma = 0.20, \{\varepsilon_{i}\} is a white noise process with unit variance and \Delta t = 10 / 5000 = 0.002 years. The dashed line denotes \operatorname{E}\left( S_{t} \right) = S_{0} e^{\mu t}.