Skip to content
All library documents

Geometric Brownian Motion: Itô’s Correction and the Linear Approximation

Article Quant Q&A · Author: DQd
Read the original

Summary

The document asks how to derive a stock price at a future time from a geometric Brownian motion and why several familiar expressions differ. It compares direct integration of the proportional price change, the exponential solution with a volatility correction, and a first-order linear approximation. The central explanation is that ordinary calculus does not apply directly to Brownian-driven processes: Itô’s lemma adds a correction term when transforming the process through the logarithm.

The discussion identifies the source of the apparent discrepancy, but provides little detail beyond that point. It does not fully derive the corrected solution or explain the distributional interpretation of the Brownian increment. The linear expression is a local approximation and should not be treated as the exact finite-horizon GBM solution. These notes are useful for distinguishing stochastic calculus from ordinary integration, but further derivation is needed to apply the formulas confidently.

Key ideas

  • Itô’s lemma is needed when transforming a stochastic process through a nonlinear function such as the logarithm.
  • The correction term in the exponential GBM solution arises from the process’s volatility.
  • A Brownian increment over a time interval can be represented using a standard normal variable scaled by the square root of elapsed time.
  • The linear change in price is an approximation and differs from the exact exponential solution.

Tags

EquitiesStatisticsVolatility
Full text
# Different Forms of Geometric Brownian Motion


# Different Forms of Geometric Brownian Motion












If the stock price S follows the geometric brownian motion: $$dS=\mu Sdt+\sigma Sdz$$ $$\frac{dS}S=\mu dt+\sigma dz$$

Where $dz=\epsilon\sqrt{dt}$ is a wiener process.

Integrating this to get $S_T$ as a function of $S_0$

$$\int^T_0\frac{1}Sds=\int^T_0\mu dt +\int^T_0\sigma dz$$ $$=ln(S_T/S_0)=\mu(T-0)+\sigma (z_T-z_0)$$

$$S_T=S_0e^{\mu T+\sigma (z_T-z_0)}$$ Why do others allow $(z_T-z_0)$ to be a standard Wiener process?

And why is this not the same as: $$S_T =S_0 e^{(\mu -0.5\sigma^2)T+\sigma \epsilon \sqrt{T}}$$

And what is the difference between the above version, and this version: $$S_T =S_0+dS=S_0+\mu S_0 T+\sigma S_0 \epsilon \sqrt{T}$$?

## Answer by ExIR (score 2)

https://quant.stackexchange.com/a/45648

Your integration of 1/S dS is incorrect for a stochastic process. You must use stochastic calculus. That would give you the adjustment term, somewhat like a convexity adjustment.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.