Why the Black-Scholes Replicating Portfolio Includes Stock
Summary
The document outlines a risk-neutral pricing argument for a European derivative on a dividend-paying stock. Starting from a geometric Brownian motion model, it describes changing to a risk-neutral measure, expressing discounted stock and derivative prices as martingales, and using martingale representation to identify a stock exposure that replicates the derivative’s discounted value.
The central clarification is that matching the derivative’s value at one instant is not enough to establish replication. The portfolio combines cash with a changing stock position derived from the martingale representation; this hedge is constructed so its value continues to track the derivative over time. Holding only cash equal to the derivative’s current value would generally fail to match its next movement. The argument relies on the stated model assumptions and the theoretical replication framework. The document gives no worked payoff example or empirical test, and its conclusion concerns the model rather than practical frictions such as transaction costs or imperfect hedging.
Key ideas
- Risk-neutral valuation expresses a derivative price as a discounted conditional expectation of its payoff.
- Martingale representation identifies the stock exposure needed to track the discounted derivative value.
- The replicating portfolio pairs that stock exposure with a cash position to match the derivative over time.
- A cash-only portfolio can match the derivative’s value now without reproducing its subsequent changes.
- The result rests on the model assumptions and idealized replication argument.
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# The choice of portfolio in the proof of the Black-Scholes formula
# 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:
- 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.
- 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.
- 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 $\mathbb{Q}$.
- 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 $P_t$.
- 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 $t$.
## Answer by Bob Jansen (score 4, accepted)
https://quant.stackexchange.com/a/43410
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 $(\hat{P}_t,0)$.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.