Principle:Avhz RustQuant Geometric Brownian Motion
| Knowledge Sources | |
|---|---|
| Domains | Stochastic_Processes, Mathematical_Finance, Monte_Carlo |
| Last Updated | 2026-02-07 20:00 GMT |
Overview
A continuous-time stochastic process where the logarithm of the underlying variable follows a Brownian motion with constant drift and volatility, widely used to model asset prices.
Description
Geometric Brownian Motion (GBM) is the standard model for stock price dynamics in the Black-Scholes framework. Unlike arithmetic Brownian motion, GBM ensures that prices remain strictly positive, which is a key requirement for modeling financial assets.
The model assumes that returns (not prices) are normally distributed, which leads to log-normally distributed prices. GBM is the foundation of:
- Black-Scholes option pricing
- Monte Carlo simulation of asset paths
- Risk-neutral pricing via the equivalent martingale measure
The process is fully characterized by two parameters: drift (mu) and volatility (sigma).
Usage
Use GBM when simulating asset price paths for Monte Carlo pricing, when the underlying is assumed to follow log-normal dynamics. GBM is appropriate for equity and index modeling under standard assumptions (constant volatility, no jumps).
Theoretical Basis
The GBM stochastic differential equation:
where W_t is a standard Brownian motion.
The exact solution is:
Key properties:
- Drift: mu * S_t (proportional to current price)
- Diffusion: sigma * S_t (proportional to current price)
- Distribution: S_T is log-normally distributed
- Expected value: E[S_T] = S_0 * exp(mu * T)
- Variance: Var[S_T] = S_0^2 * exp(2*mu*T) * (exp(sigma^2*T) - 1)
For risk-neutral pricing, mu is replaced by the risk-free rate r.