Constant-proportion portfolio (${\pi }_t=\gamma V_t$)

A constant-proportion portfolio holds a fixed fraction $\gamma$ of its portfolio wealth in the stock. In the Black-Scholes model with bank rate $\rho$, initial wealth $V_0$, and no consumption, its value isThe stock holding is $\gamma V_t/S_t$ and the remaining value $(1-\gamma )V_t$ is invested in the bank account. The Itô formula verifies $d{V}_t/V_t=[\rho +\gamma (\mu -\rho )]dt+\gamma \sigma d{W}_t$ when the physical stock drift is $\mu$.