# The choice of portfolio in the proof of the Black-Scholes formula

Consider a stock whose price $$S$$ satisfies $$dS_t=\mu S_tdt+\sigma S_tdW_t$$ for constants $$\mu,\sigma$$ and where $$W$$ is a $$\mathbb{P}$$-Brownian motion. Further assume that the stock pays out dividends continuously at a rate of $$d$$ proportional to the current stock price.

Let $$p_t$$ denote the price at time $$t$$ of a European-style derivative which has a payoff of $$f(S_T)$$ at time $$T$$. In order to determine a formula for $$p_t$$ we essentially carry out the following steps:

1. Use Girsanov's theorem to determine the risk-neutral probability measure $$\mathbb{Q}$$ such that $$\widetilde{W}_t=\left(\frac{\mu+d-r}{\sigma}\right)t+W_t$$ is a $$\mathbb{Q}$$-Brownian motion.
2. Define $$P_t=e^{-r(T-t)}\mathbb{E}_{\mathbb{Q}}[f(S_T)\mid\mathcal{F}_t]$$. Show that both $$\hat{S}_t=e^{-(r-d)t}S_t$$ and $$\hat{P_t}=e^{-rt}P_t$$ are $$\mathbb{Q}$$-martingales.
3. Use the Martingale Representation Theorem to conclude the existence of a predictable process $$A$$ such that $$\hat{P}_t=\hat{P}_0+\int_0^tA_sd\hat{S}_s$$ under $$\mathbb{Q}$$.
4. Construct the portfolio $$(\hat{P}_t-A_t\hat{S}_t,A_te^{dt})$$ which consists of $$\hat{P}_t-A_t\hat{S}_t$$ units of cash and $$A_te^{dt}$$ units of the stock at time $$t$$. The value of this portfolio is $$P_t$$.
5. Since $$P_T=p_T$$ we conclude from the Law of One Price that $$P_t=p_t$$ for all $$0\leq t\leq T$$. In other words, $$p_t=e^{-r(T-t)}\mathbb{E}_{\mathbb{Q}}[f(S_T)\mid\mathcal{F}_t]$$.

After going through the above steps I am wondering why the portfolio needs to be $$(\hat{P}_t-A_t\hat{S}_t,A_te^{dt})$$. It seems like we could simply choose $$(\hat{P}_t,0)$$ as our portfolio and this would still have a value of $$P_t$$ at time $$t$$.

The portfolio $$(\hat{P}_t-A_t\hat{S}_t,A_te^{dt})$$ is chosen because it is a hedging portfolio. That is, unlike $$(\hat{P}_t,0)$$ it will have the same value as the derivative an instant later. This is not generally the case for the portfolio $$(\hat{P}_t,0)$$.