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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 $t$.

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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 $(\hat{P}_t,0)$.

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