Geometric Brownian Motion (GBM)

Diffusion

The foundational model for stock prices. GBM assumes log-normal price distributions with constant drift and volatility. It underlies the Black-Scholes option pricing formula and remains widely used in quantitative finance for its analytical tractability.

Mathematical Formulation

Parameters

SymbolDescriptionConstraint
Asset price at time t
Drift (expected return)
Volatility (diffusion coefficient)
Standard Wiener process (Brownian motion)

Key Assumptions

  • •Continuous price paths (no jumps)
  • •Constant volatility over time
  • •Log-normal price distribution
  • •Independent, normally distributed returns

Reference

Black, F. & Scholes, M. (1973). "The Pricing of Options and Corporate Liabilities." Journal of Political Economy.

Model Hierarchy

Extended by more complex models
More ComplexStandard

Heston

Stochastic Vol
More ComplexStandard

Merton

Jump-Diffusion
More ComplexStandard

VG

Lévy
More ComplexStandard

Dupire

Local Vol

Complexity increases from left to right. More complex models capture additional market phenomena but require more parameters and may be harder to estimate.

Empirical Context (SPY Returns)

How GBM relates to observed return properties

SPY returns exhibit volatility clustering, negative skew, and heavy tails. While GBM captures the mean return and average volatility, it assumes constant volatility and normal returns—failing to reproduce the time-varying volatility and extreme events clearly visible in the data.

What GBM captures

  • Mean return level
  • Overall volatility magnitude

What it cannot capture

  • Volatility clustering
  • Fat tails (excess kurtosis)
  • Negative skewness
  • Extreme tail events
View full empirical analysis of SPY returns

Estimation Data

Select ticker for P-measure estimation

SPY

Parameter Estimation (P-Measure)

Maximum likelihood estimation using SPDR S&P 500 (SPY) daily returns(Jan 2, 2016 to Jan 2, 2026)

999 observations• Computed in 0.6ms

Adequate fit for GBM. Using 999 daily returns for SPDR S&P 500 (SPY), the estimated annualized volatility is 12.9% with an annualized drift of 13.75% (0.0546% daily).

Estimated Parameters

Geometric Brownian Motion under the physical (P) measure

ParameterEstimateStd. Errort-stat95% CISig.
mu(Annualized drift (expected return))
13.75%6.49%2.12[1.04%, 26.47%]*
sigma(Annualized volatility)
12.92%0.29%44.70[12.35%, 13.48%]*
Sig.: * indicates statistical significance at 95% confidence level (|t| > 1.96)

Model Fit Statistics

Log-Likelihood

3,389.56

AIC

-6,775.12

Lower is better

BIC

-6,765.31

Lower is better

Estimation Method

Method: Maximum Likelihood Estimation (MLE)

Model: Daily log returns follow rt = μΔt + σ√Δt εt, where εt ~ N(0,1)

Estimators: Closed-form: μ̂ = mean(r)/Δt, σ̂ = std(r)/√Δt with Δt = 1/252

Convergence: Closed-form MLE solution

Estimation Diagnostics

Visual assessment of model fit and residual properties

Empirical vs Fitted Return Distribution

Histogram of observed returns with fitted normal overlay

-4.27%-2.43%-0.59%1.25%3.09%4.93%Daily Log Return024487397DensityEmpiricalFitted Normal
Empirical Mean:0.0546%
Empirical Std:0.8137%
Skewness:-0.629
Excess Kurtosis:4.678

QQ-Plot: Log Returns vs Normal

Deviations from red line indicate non-normality (fat tails)

-3.37-1.75-0.121.503.13Theoretical Normal Quantiles-5.318-2.4910.3363.1645.991Sample Quantiles (Log Returns)Normal
Lower Tail Deviation:-27.16%
Upper Tail Deviation:-8.20%
Points:999

ACF of Standardized Residuals

Test for remaining serial correlation (should be insignificant under GBM)

01020Lag (days)-0.50.00.51.0
Significant lags (outside 95% CI): 3

Standardized Residuals Over Time

Should oscillate around zero without clustering; ±2 bands shown (95% under normality)

Jan 16Dec 16Dec 17Dec 18Dec 19-202
Observations outside ±2 bounds: 57 (5.7%)(Expected ~5% under normality)

Interpreting the Diagnostics

  • Distribution Plot: The fitted normal (blue) should closely match the empirical histogram; deviations in the tails indicate fat tails
  • QQ-Plot: Points near the red line indicate normality; S-shaped deviations suggest heavier tails than the normal distribution
  • Residual ACF: Significant autocorrelation suggests predictable patterns not captured by GBM
  • Residual Time Series: Standardized residuals should oscillate around zero without persistent trends or volatility clustering

Related Models

Heston

Stochastic Vol

Models volatility as a mean-reverting stochastic process correlated with returns. Captures volatility clustering (high vol follows high vol), leverage...

Compare with Heston →

Merton

Jump-Diffusion

Extends GBM with compound Poisson jumps to capture sudden price movements from earnings announcements, market shocks, or other discontinuous events. T...

Compare with Merton →
View all stochastic models

Run Simulation

Watch the Geometric Brownian Motion (GBM) in action. Adjust parameters and observe how the price path evolves in real-time.

Charts:

Price Path Simulation

Select a model and click Start to begin simulation

Log Returns

Log returns will appear here

Model

Classic log-normal model for asset prices with constant drift and volatility. Underlies the Black-Scholes option pricing formula.

Category: diffusion

Parameters

Starting asset price

Expected annual return (e.g., 0.05 = 5%)

Annual volatility (e.g., 0.2 = 20%)

Simulation time in years

Total simulation steps (more = smoother path, more candles)

Simulation

0.25x

Adjust speed before or during simulation

Status:idle

Disclaimer: These simulations are for educational purposes only. They demonstrate the behavior of mathematical models and should not be used for trading decisions. Real market dynamics are significantly more complex than any single stochastic model can capture.