Geometric Brownian Motion and Its Log-Price Solution

Summary

The document introduces geometric Brownian motion as a model for an asset price whose proportional changes have a constant drift and volatility. It outlines the derivation of the process solution using Itô's lemma: transform the price to its logarithm, obtain a drift-diffusion process, integrate over time, and exponentiate to recover the price. The resulting form expresses price as its initial value multiplied by an exponential involving drift, volatility, elapsed time, and a Brownian motion term.

This derivation is useful for understanding a standard continuous-time model used in finance and for seeing how stochastic calculus handles nonlinear transformations. Its assumptions are restrictive: drift and volatility are constant, and the model does not capture changing regimes or other evolving market conditions. The text provides no empirical fit or asset-specific evidence, so the solution should be read as a mathematical model result rather than a claim that observed prices follow this process.

Key ideas

• Geometric Brownian motion models proportional asset-price changes with drift and diffusion.
• The model assumes constant drift and volatility over time.
• Applying Itô's lemma to the logarithm converts the price process into a drift-diffusion form.
• Integrating the log-price process and exponentiating yields the asset-price solution.
• The derivation is analytical, while its assumptions limit how closely it may describe real markets.

Tags

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.