Geometric Brownian Motion · dS = μS·dt + σS·dW · Lognormal distribution · Black-Scholes foundation
Simulate multiple stock price trajectories using the stochastic differential equation dS = μS·dt + σS·dW — the model at the heart of quantitative finance and the Black-Scholes options pricing formula.
GBM is the model underlying the Black-Scholes options pricing formula. Each path follows dS = μS·dt + σS·dW where dW is Brownian noise. The logarithm of S is normally distributed — leading to the lognormal distribution of stock prices (Samuelson 1965). μ is the expected annual return; σ is annual volatility. The bold dashed line shows the analytical mean E[S] = S₀·e^(μt), which lies above the median because the lognormal distribution is right-skewed.
Drag μ to positive values for a bull market (most paths end above S₀) or negative for a bear market. Increase σ to see trajectories fan out — more uncertainty, wider lognormal spread. Click "Run New" for a fresh ensemble. The dashed white line shows the analytical mean E[S] = S₀·e^(μt). The histogram below shows the distribution of final prices — green bars are outcomes above S₀, red bars below.
The Black-Scholes formula (Fischer Black, Myron Scholes, Robert Merton — 1973) won the 1997 Nobel Prize in Economics. It assumes GBM with constant μ and σ — in practice σ is not constant (the "volatility smile"), which is why options traders use more sophisticated models such as stochastic volatility (Heston model) or jump-diffusion (Merton jump model). Even so, GBM remains the foundational building block of quantitative finance.