1 Geometric Brownian Motion Definition
A Geometric Brownian Motion \((S_{t})\) has dynamics given by the stochastic differential equation \[ dS_{t}=\mu S_{t}dt+\sigma S_tdW_t, \] where \(W_{t}\) is Brownian Motion and \(\mu, \sigma\) are constants.
2 Solution
To solve the SDE consider the ansatz \(g(S_{t})=\log S_t\). Then by Itô Formula we have that \[ dg(S_t)=d\log(S_t)=\frac{1}{S_{t}}dS_t+\frac12\frac{1}{S_{t}^{2}}dS_t^2=\frac{1}{S_{t}}dS_t+\frac12\sigma^2dt \] If we substitute in our initial expression we obtain \[ \begin{align} d\log(S_{t})=\frac1{S_{t}}(\mu S_tdt+\sigma S_tdW_{t})+\frac12\sigma^2dt=\left(\mu+\frac12\sigma^{2}\right)dt+\sigma dW_t. \end{align} \] Written in integral form we have \[ \begin{align} \log S_{t}&=\log S_0+\left(\mu+\frac12\sigma^{2}\right)t+\sigma W_t\\ \implies S_t&=S_0\exp\left(\left(\mu-\frac{1}{2}\sigma^2\right)t+\sigma W_t\right) \end{align} \] ## Variants of Geometric Brownian Motion ### Mean-Reverting GBM ([[ornstein-uhlenbeck-process]]) Instead of constant drift, the drift term reverts toward a long-term mean: \[ dS_t = \kappa(\mu - \log S_t)S_t dt + \sigma S_t dW_t \] Used for interest rates and commodity prices that have equilibrium levels.
2.1 [[local-volatility-models]]
Volatility is a function of both time and price: \(\sigma = \sigma(t, S_t)\) \[ dS_t = \mu S_t dt + \sigma(t, S_t) S_t dW_t \] Allows fitting to observed option prices more accurately than constant volatility.
2.2 [[stochastic-volatility-models]]
Volatility itself follows a random process (e.g., Heston Model): \[ dS_t = \mu S_t dt + \sqrt{v_t} S_t dW_t^S \] \[ dv_t = \kappa(\theta - v_t) dt + \xi\sqrt{v_t} dW_t^v \] Can capture smile effects and time-varying volatility.
2.3 [[jump-diffusion-models-merton-model]]
Adds random jump component to GBM: \[ dS_t = \mu S_t dt + \sigma S_t dW_t + S_t d\left(\sum_{i=1}^{N_t} (J_i - 1)\right) \] Where \(N_t\) is a Poisson process and \(J_i\) are jump magnitudes. Captures tail risk.
2.4 [[cev-model-constant-elasticity-of-variance]]
Volatility scales with price level: \(\sigma(S_t) = \sigma S_t^{\beta-1}\) \[ dS_t = \mu S_t dt + \sigma S_t^\beta dW_t \] When \(\beta < 1\), exhibits volatility smile observed in options markets.
2.5 [[displaced-diffusion]]
Combines GBM with an Ornstein-Uhlenbeck component for short rates.