> [!tldr] > A geometric Brownian motion $(X_{t})$ satisfies $\frac{dX_{t}}{X_{t}}=\mu dt + \sigma dW_{t}$where $\mu, \sigma$ are constants, and $W_{t}$ is a [[2 - Snippets/Brownian Motion|Weiner process]]. ### Solution to the SDE The solution to the SDE is $X_{t} = X_{0} \cdot \exp\left[ \left( \mu-\frac{\sigma^{2}}{2} \right)t + \sigma W_{t} \right],$so the returns $X_{t} / X_{0}$ follows the [[Log-Normal Distribution|log-normal distribution]]. Alternatively, with $Z:= W_{t} / \sqrt{ t } \sim N(0,1)$, $X_{t} = X_{0} \cdot \exp\left[ \left( \mu - \frac{\sigma^{2}}{2} + \sigma \sqrt{ t } \cdot Z\right) \right].$ - Therefore $\mathbb{E}[X_{t} ~|~ X_{0}]=X_{0} e^{t\mu}$, and $\mathrm{Var}(X_{t} ~|~ X_{0})= X_{0}^{2}(e^{\sigma^{2}t}-1)e^{2\mu t}$. > [!exposition] Deriving the solution > Consider $X_{t} = X_{0}\exp[at + bW_{t}]$. Now applying [[Ito's Lemma]] gives $dX_{t}=X_{t}\left( a + \frac{b^{2}}{2} \right)dt + X_{t} \cdot b~dW_{t}.$Equating this with the original SDE gives $a=\mu-\frac{\sigma^{2}}{2}, ~b=\sigma.$ ### Application in Black-Scholes This SDE is used in Black-Scholes model to model stock price movements. In this context, $\mu$ is the **drift** and $\sigma$ the **volatility**. Moreover, if the underlying $(X_{t})$ pays a continuous dividend $q$, then its dynamics can be modeled by $dX_{t} = (\mu-q)~dt + \sigma ~dW_{t}.$