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.
\nWith constant parameters, the geometric Brownian motion solution has the form
\n$$S_t=S_0\\exp\\left((\\mu-\\tfrac12\\sigma^2)t+\\sigma W_t\\right).$$
\nThe \\(-\\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.
","contentMarkdown":"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.\n\nWith constant parameters, the geometric Brownian motion solution has the form\n\n$$S_t=S_0\\exp\\left((\\mu-\\tfrac12\\sigma^2)t+\\sigma W_t\\right).$$\n\nThe \\(-\\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.","dataUrl":"https://sharifhsn.dev/api/posts/brownian-motion-risk-models.json","date":"2024-09-26","datePublished":"2024-09-26","description":"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 proportion…","site":"https://sharifhsn.dev","slug":"brownian-motion-risk-models","source":"FE-535 | Risk Management","sourceUrl":null,"tags":["Risk Management","Brownian Motion","Geometric Brownian Motion"],"title":"Brownian Motion and Geometric Growth","url":"https://sharifhsn.dev/blog/brownian-motion-risk-models/","version":"1","wordCount":91}