GBM dynamics
dS_t = mu S_t dt + sigma S_t dW_tBoth expected change and random dispersion scale with the current price.
Foundations
Geometric Brownian Motion (GBM)
A continuous-time random growth model where relative price changes are normal and prices stay strictly positive.
Mental model
GBM models proportional price changes continuously while keeping the modeled price positive.
GBM assumes dS/S = μ dt + σ dW, which means the logarithm of the price follows a Brownian motion with drift; over any horizon, prices are lognormally distributed.
It implies strictly positive prices and a simple relationship between drift, volatility, and the distribution of returns.
Explore the mechanism
How do drift, volatility, and time shape the distribution of a strictly positive modeled price?
Geometric Brownian Motion (GBM) sample paths. Prices stay strictly positive. The drift μ tilts the centre of the fan; the volatility σ controls how wide and skewed the fan becomes.
Horizon snapshot (t = T)
Mean S_T (sim) ≈ 98.90 (theory: 105.13)
Median S_T (sim) ≈ 100.81 (theory: 103.05)
Range S_T (sim) ≈ 65.82 … 118.63
log(S_T / S₀) mean (sim) ≈ -0.022, std (sim) ≈ 0.159
theory: μₗ = 0.030, σₗ = 0.200
Interpretation
The thin grey lines are individual GBM paths. The blue line is the median across paths; the orange dashed line is the expected path S₀ e^{μt}. The shaded band shows the middle 50% of outcomes at each time.
Notice that the mean at T sits above the median: GBM is lognormal, so the distribution of S_T is skewed. In log-space, log(S_T / S₀) is approximately normal with mean (μ − ½σ²)T and std σ√T – exactly what the log-return stats show.
As σ increases, the fan widens dramatically: extreme paths go far above the drift curve, even if the median stays close to it. For control and risk work, this is why GBM is a useful baseline signal model for log-returns, but it underestimates real-world jump risk and fat tails in markets.
UNDERSTAND
What to notice
Its main role is as a tractable baseline for modelling and derivations like Black–Scholes; real markets show jumps, volatility clustering, and fat tails.
Engineering analogy: GBM is a linear stochastic differential equation in log-space — simple enough to solve exactly, but only an approximation of the true system.
The quantitative structure
Notation
S_t- asset price at time t
mu- instantaneous drift
sigma- instantaneous volatility
W_t- Brownian motion
Log-price solution
ln(S_T/S_0) = (mu - 0.5 sigma^2)T + sigma W_TThe logarithmic return is normal under constant parameters.
This model assumes
- Drift and volatility are constant over the modeled horizon.
- Returns have continuous Gaussian innovations and no jumps.
Where it appears
- building a baseline equity-price model
- simulating paths for option pricing
- explaining the Black–Scholes state process
Where the intuition breaks
What this view leaves out
- GBM imposes continuous paths, constant parameters, and lognormal prices, so it misses jumps, volatility clustering, and changing regimes.
Common mistakes
“Normally distributed log returns make extreme moves impossible.”
Not quite.The normal distribution has unbounded tails; the issue is that its tail probabilities and constant volatility can understate observed extremes.
Where this concept sits
Follow the strongest prerequisites, applications, extensions, and contrasts.
Connections for Geometric Brownian Motion (GBM)
Geometric Brownian Motion (GBM)YOU ARE HERE
Built onReturns and Log-ReturnsGBM implies normally distributed log-returns over fixed horizons under its classical assumptions.
Foundation ofBlack–Scholes Option PricingThe classical Black–Scholes model assumes GBM dynamics for the underlying asset.
UsesVolatility (Realised vs Implied)Volatility is the diffusion-scale parameter controlling the dispersion of GBM paths.
SupportsMonte Carlo PricingGBM is a canonical path model used to illustrate risk-neutral Monte Carlo pricing.
SupportsChange of Measure (Girsanov)Girsanov's theorem explains how the drift of diffusion processes such as GBM changes under an equivalent measure.
Built on · Returns and Log-Returns
GBM implies normally distributed log-returns over fixed horizons under its classical assumptions.
Explore concept →