Derivation of the Black-Scholes-Merton PDE › Constructing the Hedged Portfolio

The key insight of Black and Scholes was that you can construct a portfolio combining the derivative and the stock that eliminates all randomness. This is the essence of delta hedging: by holding the right proportion of stock against the derivative, you create a riskless portfolio. The possibility of such elimination arises because both the stock and the derivative are driven by the same source of randomness, namely the Wiener process $W_t$. If we can find the right mix, the random fluctuations in one position will exactly offset the random fluctuations in the other.

Consider a portfolio $\Pi$ consisting of:

• Long one derivative (value $V$)
• Short $\Delta$ shares of stock (value $-\Delta S$)

The word "short" here means we have sold shares we don't own, creating a negative position in the stock. This is a standard operation in financial markets, and one of our model assumptions permits it without restriction. The portfolio value is:

$$\Pi = V - \Delta S$$

where:

• $\Pi$: value of the hedged portfolio
• $V$: value of the derivative
• $\Delta$: number of shares of stock held short
• $S$: current stock price

The change in portfolio value over a small time interval captures how the portfolio responds to all the forces acting on its components:

$$d\Pi = dV - \Delta \, dS$$

where:

• $d\Pi$: change in portfolio value
• $dV$: change in derivative value
• $\Delta$: number of shares held short
• $dS$: change in stock price

This expression reflects the fact that when we're short $\Delta$ shares, we lose money when the stock price rises and gain money when it falls. The negative sign captures this inverse relationship.

Substituting our expressions for $dV$ and $dS$, we obtain a detailed view of all the terms affecting the portfolio:

$$d\Pi = \left( \frac{\partial V}{\partial t} + \mu S \frac{\partial V}{\partial S} + \frac{1}{2} \sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} \right) dt + \sigma S \frac{\partial V}{\partial S} dW - \Delta(\mu S \, dt + \sigma S \, dW)$$

where:

• $d\Pi$: change in portfolio value
• $V$: value of the derivative
• $S$: stock price
• $\Delta$: number of shares held short
• $\mu$: expected return of the stock
• $\sigma$: volatility of the stock
• $dt$: small time increment
• $dW$: Wiener process increment

Now we rearrange this expression by collecting the $dt$ and $dW$ terms separately. This organizational step reveals the structure of the portfolio's dynamics:

$$d\Pi = \left( \frac{\partial V}{\partial t} + \mu S \frac{\partial V}{\partial S} + \frac{1}{2} \sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} - \Delta \mu S \right) dt + \left( \sigma S \frac{\partial V}{\partial S} - \Delta \sigma S \right) dW$$

where:

• $d\Pi$: change in portfolio value
• $V$: value of the derivative
• $S$: stock price
• $\Delta$: number of shares held short
• $\mu$: expected return of the stock
• $\sigma$: volatility of the stock
• $dt$: small time increment
• $dW$: Wiener process increment

The coefficient of $dW$ represents the portfolio's exposure to randomness. If this coefficient is nonzero, the portfolio value fluctuates randomly. If we can make this coefficient vanish, the portfolio becomes deterministic over the next instant.

To eliminate the random component, we choose $\Delta$ such that the coefficient of $dW$ equals zero:

$$\begin{aligned} \sigma S \frac{\partial V}{\partial S} - \Delta \sigma S &= 0 \\ \Delta \sigma S &= \sigma S \frac{\partial V}{\partial S} \\ \Delta &= \frac{\partial V}{\partial S} \end{aligned}$$

where:

• $\Delta$: number of shares to short
• $\frac{\partial V}{\partial S}$: derivative's delta

The solution is elegantly simple: we should hold exactly $\frac{\partial V}{\partial S}$ shares short for each unit of derivative we hold long. This quantity, the partial derivative of the option price with respect to the stock price, measures how sensitive the option is to stock price movements. It tells us exactly how many shares we need to offset that sensitivity.

With this choice of $\Delta$, something remarkable happens. The random term vanishes, and the terms involving $\mu$ cancel out as well. Let us trace through this simplification carefully:

$$\begin{aligned} d\Pi &= \left( \frac{\partial V}{\partial t} + \mu S \frac{\partial V}{\partial S} + \frac{1}{2} \sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} - \frac{\partial V}{\partial S} \mu S \right) dt \\ &= \left( \frac{\partial V}{\partial t} + \frac{1}{2} \sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} \right) dt \end{aligned}$$

where:

• $d\Pi$: change in value of the hedged portfolio
• $V$: price of the derivative
• $S$: stock price
• $\sigma$: volatility
• $t$: time
• $dt$: small time increment

The portfolio's change now has no random component, so the portfolio is instantaneously riskless. The expected return $\mu$ of the stock has also vanished from the expression. The portfolio change depends only on the passage of time, the volatility, and the Greeks of the derivative. Because $\mu$ disappears, the derivative price will not depend on how optimistic or pessimistic investors are about the stock's future direction.