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: $691.66
Annual Drift (μ): 20.38%
Annual Volatility (σ): 16.24%

	Path 1	Path 2	Path 3	Path 4	Path 5	Path 6	Path 7	Path 8	Path 9	Path 10
2026-12-30	947.002138	980.541710	822.666427	816.452200	978.080772	925.864385	689.513508	949.397276	894.759014	1163.661092
2026-12-31	941.862468	998.697418	840.243599	810.686349	973.332931	934.749152	687.307615	949.674244	895.668566	1141.893537
2027-01-01	942.441615	992.425301	836.647761	823.370953	961.627885	943.271466	685.867360	947.676469	903.623463	1146.739972
2027-01-04	948.676636	1003.062601	834.447572	818.848852	951.730608	936.047187	707.410941	960.547418	891.893410	1132.185408
2027-01-05	954.086261	1009.457524	839.781418	817.333672	951.228983	939.902919	708.568518	939.694380	890.472216	1123.215520

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.