NICEFA Logo
nicefa.
Library
←Module

Solving the SDE: Unveiling the Log-Normal Distribution for Geometric Brownian Motion

Exploring the cinematic intuition of Solving the SDE: Unveiling the Log-Normal Distribution for Geometric Brownian Motion.

Visualizing...

Our institutional research engineers are currently mapping the formal proof for Solving the SDE: Unveiling the Log-Normal Distribution for Geometric Brownian Motion.

Apply for Institutional Early Access →

The Formal Theorem

Let St S_t be a stochastic process satisfying the Geometric Brownian Motion (GBM) SDE given by dSt=μStdt+σStdWt dS_t = \mu S_t dt + \sigma S_t dW_t , where μ∈R \mu \in \mathbb{R} is the drift, σ>0 \sigma > 0 is the volatility, and Wt W_t is a standard Wiener process. Given an initial condition S0>0 S_0 > 0 , the unique solution is:
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 ln⁡(St) \ln(S_t) is normally distributed with mean ln⁡(S0)+(μ−12σ2)t \ln(S_0) + (\mu - \frac{1}{2} \sigma^2) t and variance σ2t \sigma^2 t .

Analytical Intuition.

Imagine a particle moving in a turbulent fluid, not in a linear fashion, but through constant percentage growth. In the deterministic world, dS=μSdt dS = \mu S dt leads to simple exponential growth. However, when we inject the 'jitter' of a Wiener process, the asset price St S_t doesn't just jitter; it multiplicative-compounds its randomness. To tame this, we employ Ito’s Lemma on the transformation f(St)=ln⁡(St) f(S_t) = \ln(S_t) . This 'logarithmic lens' flattens the explosive curvature of the exponential, turning multiplicative shocks into additive ones. The term −12σ2t -\frac{1}{2} \sigma^2 t —often called the convexity adjustment or Ito drift—arises because of the second-order term in the Taylor expansion of the logarithm. It acts as a necessary 'tax' on the drift, reflecting the fact that the volatility of the underlying process drags down the median path compared to the arithmetic mean. Thus, St S_t becomes a log-normal entity, forever trapped above zero, mimicking the observed dynamics of financial markets where assets cannot turn negative, yet exhibit wild, non-linear fluctuations.
CAUTION

Institutional Warning.

Students often struggle with the 'Ito correction' term −12σ2t -\frac{1}{2}\sigma^2 t . They erroneously assume the solution is simply S0eμt+σWt S_0 e^{\mu t + \sigma W_t} , forgetting that the chain rule for stochastic calculus requires accounting for the second-order derivative f′′(S) f''(S) , which does not vanish as it does in standard calculus.

Academic Inquiries.

01

Why does the distribution of St S_t have to be log-normal?

Because ln⁡(St) \ln(S_t) is a linear combination of a deterministic drift and a Brownian motion, both of which are normally distributed. By definition, a variable whose logarithm is normal is log-normal.

02

What is the physical interpretation of the −12σ2 -\frac{1}{2} \sigma^2 term?

It represents the difference between the 'average' path and the 'median' path. Because the exponential function is convex, volatility spreads out the distribution, pulling the median below the mean.

Standardized References.

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

Related Proofs Cluster.

Advanced

Ito's Lemma: The Cornerstone of Stochastic Calculus

Exploring the cinematic intuition of Ito's Lemma: The Cornerstone of Stochastic Calculus.

Advanced

Girsanov's Theorem: Transforming Measures for Risk-Neutral Valuation

Exploring the cinematic intuition of Girsanov's Theorem: Transforming Measures for Risk-Neutral Valuation.

Advanced

Martingales: The Non-Arbitrage Principle in Discounted Asset Prices

Exploring the cinematic intuition of Martingales: The Non-Arbitrage Principle in Discounted Asset Prices.

Advanced

Put-Call Parity: A No-Arbitrage Derivation of Option Relationships

Exploring the cinematic intuition of Put-Call Parity: A No-Arbitrage Derivation of Option Relationships.

Institutional Citation

Reference this proof in your academic research or publications.

NICEFA Visual Mathematics. (2026). Solving the SDE: Unveiling the Log-Normal Distribution for Geometric Brownian Motion: Visual Proof & Intuition. Retrieved from https://nicefa.org/library/advanced-stochastic-processes/solving-the-sde--unveiling-the-log-normal-distribution-for-geometric-brownian-motion

Dominate the Logic.

"Abstract theory is just a movement we haven't seen yet."

Subscribe for Full ProofsEarly Access