Exercise: From Physical Drift to Risk-Neutral Drift

Prerequisites: Change of Measure

Problem

A stock follows $dS_t=\mu S_t\,dt+\sigma S_t\,dW_t^{\mathbb{P}}$. Let $\theta=(\mu-r)/\sigma$ and $dW_t^{\mathbb{Q}}=dW_t^{\mathbb{P}}+\theta\,dt$. Substitute to show the risk-neutral drift is $r$.

Hint