Geometric Brownian Motion

Stochastic modeling of asset prices

An introduction to Geometric Brownian Motion (GBM), the foundation of modern option pricing and Monte Carlo simulation.

1 Abstract

Geometric Brownian Motion (GBM) is a continuous-time stochastic process used to model stock prices. It forms the foundation of the Black-Scholes option pricing model and assumes that prices follow a random walk consistent with the weak-form efficient market hypothesis.

2 Definition

The stochastic differential equation (SDE) for GBM is:

\[ dS = \mu S \, dt + \sigma S \, dW \]

Where:

• \(S\) = asset price
• \(\mu\) = drift (expected return)
• \(\sigma\) = volatility (standard deviation of returns)
• \(dW\) = Wiener process (standard Brownian motion)

3 Discrete Form

For simulation purposes, we use the discrete approximation:

\[ \frac{\Delta S}{S} = \mu \Delta t + \sigma \varepsilon \sqrt{\Delta t} \]

Where \(\varepsilon \sim N(0,1)\) is a standard normal random variable.

The first term (\(\mu \Delta t\)) is the drift — the expected directional movement. The second term (\(\sigma \varepsilon \sqrt{\Delta t}\)) is the shock — random fluctuation scaled by volatility.

4 Analytical Solution

The exact solution to the GBM SDE is:

\[ S(t) = S_0 \exp\left[\left(\mu - \frac{\sigma^2}{2}\right)t + \sigma W(t)\right] \]

This shows that prices are log-normally distributed (always positive), while returns are normally distributed.

5 Simulation (Python)

We simulate 10 possible price paths for a stock over 1 year using historical SPY parameters.

Initial Price: $694.60
Annual Drift (μ): 20.31%
Annual Volatility (σ): 16.25%

	Path 1	Path 2	Path 3	Path 4	Path 5	Path 6	Path 7	Path 8	Path 9	Path 10
2026-12-29	950.467809	984.150600	825.607601	819.367442	981.679130	929.240159	691.905597	952.873151	898.003029	1168.063516
2026-12-30	945.303270	1002.380961	843.255610	813.574988	976.907896	938.159716	689.688553	953.148291	898.913587	1146.197051
2026-12-31	945.881894	996.078814	839.642079	826.309851	965.149830	946.715273	688.240273	951.139008	906.899236	1151.061032
2027-01-01	952.140424	1006.758513	837.430054	821.766310	955.207382	939.457335	709.869273	964.061715	895.116842	1136.439359
2027-01-04	957.570033	1013.177650	842.783514	820.242233	954.700610	943.326454	711.029323	943.117031	893.686838	1127.426848

6 Simulated Price Paths

7 Distribution of Final Prices

8 Conclusion

Geometric Brownian Motion provides a mathematically tractable model for asset prices. While it has limitations (assumes constant volatility and continuous trading), GBM remains fundamental to quantitative finance for option pricing, risk management, and Monte Carlo simulation.