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: $690.36
Annual Drift (μ): 20.04%
Annual Volatility (σ): 16.26%

	Path 1	Path 2	Path 3	Path 4	Path 5	Path 6	Path 7	Path 8	Path 9	Path 10
2026-12-28	942.243156	975.659220	818.379473	812.189460	973.207292	921.184044	685.759801	944.629422	890.195564	1158.129769
2026-12-29	937.109047	993.734387	835.876444	806.434512	968.462903	930.022279	683.553162	944.891751	891.088838	1136.420845
2026-12-30	937.672524	987.470959	832.282569	819.057653	956.787281	938.499272	682.109048	942.887816	899.000650	1141.234020
2026-12-31	943.870650	998.054888	830.078999	814.541555	946.913187	931.288626	703.553301	955.697057	887.302443	1126.714014
2027-01-01	949.246350	1004.411879	835.379944	813.020632	946.399802	935.116381	704.695939	934.908629	885.873925	1117.759574

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.