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TL;DR

The SDE \( dS = \mu S dt + \sigma S dW \) with solution \( S_t = S_0 \exp\left((\mu - \sigma^2/2)t + \sigma W_t\right) \). It models multiplicative random growth and ensures positive prices.

By Valenke Exam Prep Team·Last updated 2026-06-03

Geometric Brownian Motion

The SDE \( dS = \mu S dt + \sigma S dW \) with solution \( S_t = S_0 \exp\left((\mu - \sigma^2/2)t + \sigma W_t\right) \). It models multiplicative random growth and ensures positive prices.

Why it matters for interviews

The standard model for stock prices in Black-Scholes theory. Understanding GBM -- its derivation via Ito's lemma, log-normal distribution, and limitations (no jumps, constant volatility) -- is fundamental to derivatives pricing.

Definition and Mathematical Foundation

The SDE \( dS = \mu S dt + \sigma S dW \) with solution \( S_t = S_0 \exp\left((\mu - \sigma^2/2)t + \sigma W_t\right) \). It models multiplicative random growth and ensures positive prices.

Application in Quantitative Finance

The standard model for stock prices in Black-Scholes theory. Understanding GBM -- its derivation via Ito's lemma, log-normal distribution, and limitations (no jumps, constant volatility) -- is fundamental to derivatives pricing.

Related Concepts

Brownian Motion (Wiener Process)

Related Terms

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Frequently Asked Questions

Why does the drift of log-price have a -sigma^2/2 correction?▼
By Ito's lemma, \( d\ln S = (\mu - \sigma^2/2)dt + \sigma dW \). The \( -\sigma^2/2 \) is the Ito correction from the quadratic variation of Brownian motion. Without it, \( E[S_t] \) would not equal \( S_0 e^{\mu t} \).
What distribution does the stock price follow under GBM?▼
\( \ln(S_t/S_0) \sim N((\mu - \sigma^2/2)t, \sigma^2 t) \), so \( S_t \) is log-normally distributed. This ensures prices stay positive and percentage returns are normally distributed.
What are the main limitations of GBM?▼
Constant volatility (empirically, vol varies), no jumps (markets do jump), thin tails (real returns have fat tails), and independence of returns (empirically, vol clusters). These motivate extensions like Heston, Merton, and GARCH.