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


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.