Geometric Brownian Motion (GBM)

Geometric Brownian Motion (GBM) is a fundamental model widely used to describe the behavior of asset prices over time. This model guarantees that prices will not be negative, which aligns with the real world.

The form of the GBM equation

The asset price S(t) is defined by the stochastic differential equation (SDE):

dS(t)=μS(t)dt+σS(t)dB(t)

Where:

• μ: Average growth rate (drift) — constant
• σ: Volatility — Constant
• B(t): Standard Brownian Motion
• S(t): Asset price at time t

In more complex models, μ and σ may be functions of t, S(t), or even other random processes.

Solving the GBM equation using Ito’s Lemma

Starting from the equation:

S(t)dS(t) = μdt + σdB(t)

To find S(t), consider ln⁡S(t) and use Ito's Lemma with the function ln⁡S(t):

d(lnS(t))=(μ−21​σ2)dt+σdB(t)

which is an Ito drift-diffusion process, then integrate both sides:

lnS(t)=lnS(0)+(μ−21​σ2)t+σB(t)

Raise both sides to the power, you will get:

S(t)=S(0)⋅exp[(μ−21​σ2)t+σB(t)]

Summary

• The GBM model works well for explaining stock prices in a market without dividend payments.
• Guaranteed that S(t) > 0 always.
• It is one of the few cases where an analytical solution to the SDE can be found.

Reference: Geometric Brownian Motion

From https://www.quantstart.com/articles/Geometric-Brownian-Motion/