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The Log-Normal Distribution of Geometric Brownian Motion

Exploring the cinematic intuition of The Log-Normal Distribution of Geometric Brownian Motion.

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The Formal Theorem

Let St S_t be a stochastic process satisfying the Stochastic Differential Equation (SDE) defined by dSt=μStdt+σStdWt dS_t = \mu S_t dt + \sigma S_t dW_t , where μ \mu is the drift, σ \sigma is the volatility, and Wt W_t is a standard Wiener process. Given an initial value S0>0 S_0 > 0 , the solution at time t t is given by the expression:
St=S0exp⁡((μ−12σ2)t+σWt) S_t = S_0 \exp\left( \left( \mu - \frac{1}{2}\sigma^2 \right)t + \sigma W_t \right)
Consequently, the random variable St S_t follows a log-normal distribution such that ln⁡(St)∼N(ln⁡(S0)+(μ−12σ2)t,σ2t) \ln(S_t) \sim \mathcal{N}\left( \ln(S_0) + (\mu - \frac{1}{2}\sigma^2)t, \sigma^2 t \right) .

Analytical Intuition.

Imagine the trajectory of a stock price as a vessel navigating a turbulent sea. While the underlying force of the market provides a steady 'drift' μ \mu , the unpredictable waves of news and investor sentiment introduce 'volatility' σ \sigma . If we were to apply simple additive noise, the price could drop below zero, which is financially nonsensical. Instead, Geometric Brownian Motion (GBM) uses multiplicative growth. By taking the logarithm, we transform this complex multiplicative process into a simple sum of normally distributed increments—the classic Random Walk. The term −12σ2 -\frac{1}{2}\sigma^2 is the 'Ito correction' or 'convexity adjustment'; it accounts for the fact that volatility drags down the median value of the process over time. Thus, even if the average return is positive, the most likely path the price takes—the median—is skewed by this variance. We are witnessing the beautiful transition from raw, jagged Brownian motion to the smooth, skewed bell curve of the log-normal distribution, ensuring that prices remain strictly positive, reflecting the reality of limited liability.
CAUTION

Institutional Warning.

Students often struggle with the −12σ2 -\frac{1}{2}\sigma^2 term. It arises from Ito’s Lemma because dSt dS_t is not a standard derivative; the second-order term (dWt)2≈dt (dW_t)^2 \approx dt cannot be ignored, pulling the distribution toward zero compared to a naive exponentiation of the drift.

Academic Inquiries.

01

Why does GBM guarantee a positive price?

Because the solution involves an exponential function exp⁡(… ) \exp(\dots) , and the range of the exponential function is strictly (0,∞) (0, \infty) for any real-valued argument.

02

What is the physical significance of the Ito correction?

It represents the difference between the arithmetic mean (the expected value of St S_t ) and the geometric mean (the median), stemming from the non-linear nature of the transformation of the Wiener process.

Standardized References.

  • Definitive Institutional SourceØksendal, B., 'Stochastic Differential Equations: An Introduction with Applications'.

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Institutional Citation

Reference this proof in your academic research or publications.

NICEFA Visual Mathematics. (2026). The Log-Normal Distribution of Geometric Brownian Motion: Visual Proof & Intuition. Retrieved from https://nicefa.org/library/advanced-stochastic-processes/the-log-normal-distribution-of-geometric-brownian-motion

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