black scholes - Rigorous derivation of $dPi$ for stock with continuous dividend

Suppose we are holding a replicating portfolio $\Pi_t$ of long an option $f(S,t)$ and short some stock, so $$\Pi_t=f(S_t,t)-\Delta_t S_t$$ Suppose the stock follows geometric Brownian motion and pays continuous dividends at rate $q$, so $$d S_t = S_t((\mu-q)dt + \sigma dW_t$$ Naively, $$d\Pi_t = d f(S_t,t)-\Delta_t dS_t$$ However, because the stock pays a dividend, common sense and the literature tell us that $$d\Pi_t=df(S_t,t)-\Delta_t dS_t-\Delta_t S_t dt$$

Question: How do we rigorously arrive at the total derivative for $d\Pi_t$ which includes extra term $-\Delta_t S_t dt$, given that we know $\Pi_t=f(S_t,t)-\Delta_t S_t$, without appeal to common sense i.e., from the equations, without recalling the mechanics of how the stock works? Because, naively, I would not include the extra term, if I just knew the equation defining $\Pi_t$ and nothing about the mechanics of the stock.

Answer

Let $S^0_t = e^{rt}$ be the money market account.

Consider you short a derivative $P_t$ and hedge it with cash and stock. $$\Pi_t = \Delta^0_t S^0_t + \Delta_t S_t - P_t$$ At time $t+dt$, the portfolio is $$\begin{eqnarray*} \Pi_{t+dt} &=& \Delta^0_t S^0_{t+dt} + \Delta_t S_{t+dt} + \Delta_t q_t S_t dt - P_{t+dt} \end{eqnarray*}$$ Where $\Delta_t q_t S_t dt$ corresponds to the dividend received. Then $$\begin{eqnarray*} \Pi_{t+dt} &=& (1+rdt)\Delta^0_t S^0_{t} + \Delta_t S_{t+dt} - P_{t+dt} \\ &=& (1+rdt)( \Pi_t - \Delta_t S_t + P_t ) + \Delta_t S_{t+dt} + \Delta_t q_t S_t - P_{t+dt} \\ &=& (1+rdt)( \Pi_t - \Delta_t S_t + P_t ) + \Delta_t S_{t+dt} + \Delta_t q_t S_t - P_{t+dt} \end{eqnarray*}$$ This can be rewritten $$\begin{eqnarray*} d\Pi_{t}-r\Pi_t &=& \Delta_t (dS_{t} - (r-q)S_tdt) - (dP_{t} - rP_tdt) \end{eqnarray*}$$

Assume the stock price has dynamic $$\frac{dS_t}{S_t} = \mu dt + \sigma dW^\mathbb{P}_t$$ under the real-world measure $\mathbb{P}$ (the drift could even be stochastic here). We search for the price in the form $P_t = P(t,S_t)$. Applying Ito, one finds $$\begin{eqnarray*} dP_{t} &=& \partial_tPdt + \partial_SP dS_t + \frac{1}{2}\sigma^2 S_t^2\partial^2_SP dt \end{eqnarray*}$$ so $$\begin{eqnarray*} d\Pi_{t}-r\Pi_t &=& (\Delta_t - \partial_SP) dS_{t} - \Delta_t S_t (r-q)dt) - \partial_tPdt - \frac{1}{2}\sigma^2 S_t^2\partial^2_SP dt + rP(t,S_t)dt) \end{eqnarray*}$$ In order to kill the stochastic term we should choose $\Delta_t = \partial_SP$. We end up with a risk-less portfolio with PnL $$\begin{eqnarray*} d\Pi_{t}-r\Pi_t &=& -\left(\partial_tP + (r-q)S_t \partial_SP + \frac{1}{2}\sigma^2 S_t^2\partial^2_SP - rP \right)dt \end{eqnarray*}$$ By absence of arbitrage, this has to be 0 otherwise we could make a guaranteed profit without taking any risk. So the right hand side is the diffusion equation $$\begin{eqnarray*} \partial_tP + (r-q)S_t \partial_SP + \frac{1}{2}\sigma^2 S_t^2\partial^2_SP dt = rP \end{eqnarray*}$$ This is a diffusion equation (the Black-Scholes equation). In the case where $P$ pays a single cashflow $P(T,S_T)$ at $T$, the Feynman-Kac theorem ensures that the solution to this PDE can be written as an expectation $$\begin{eqnarray*} P(t,S) &=& \mathbb{E}^\mathbb{Q}_t\left[ e^{-\int_t^T} P(T,Y_T) |Y_t = S\right] \end{eqnarray*}$$ where $(X,\mathbb{Q})$ is any probability space, $W^\mathbb{Q}$ a Brownian motion on it and $Y$ a process satisfying the SDE $$\frac{dY_t}{Y_t} = (r-q) dt + \sigma dW^\mathbb{Q}_t$$ Such a probability $\mathbb{Q}$ is usuallly called the risk-neutral measure and the process $Y$ is usually written $S$. But they are only mathematical constructs that can make computation easier because the real-world drift is irrelevant. The core of the argument is accounting for the PnL of our strategy and absence of arbitrage. The fact that the price does not depend on the drift is due to the fact that it cancels out when we hold the underlying as our hedge.