Section 18.3

Ito's Lemma

Dynamics of functions of Ito processes (logarithms and powers of GBM) and moments of GBM.

Toolkit

Itô’s lemma extends the chain rule to functions of an Itô process, accounting for quadratic variation. If $dX_t = \mu_t\,dt + \sigma_t\,dW_t$ and $f(t,x)$ is $C^{1,2}$ , then

df(t,X_t) = \left(f_t + \mu_t f_x + \tfrac12\sigma_t^2 f_{xx}\right)dt + \sigma_t f_x\,dW_t.

For GBM $dS_t = \mu S_t\,dt + \sigma S_t\,dW_t$ , applying this with $f(x)=\ln x$ gives

d\ln S_t = \left(\mu - \tfrac12\sigma^2\right)dt + \sigma\,dW_t,

so $S_t = S_0\exp\left((\mu-\tfrac12\sigma^2)t + \sigma W_t\right)$ . More generally, for $f(x)=x^p$ ,

d(S_t^p) = S_t^p\left(p\mu + \tfrac12 p(p-1)\sigma^2\right)dt + pS_t^p\sigma\,dW_t,

giving $\mathbb{E}[S_t^p] = S_0^p\exp\left(\left(p\mu+\tfrac12 p(p-1)\sigma^2\right)t\right)$ .

Problems

18.7 Log-Price Dynamics of GBM+3 variations Easy Free