Derivation of the Black-Scholes-Merton PDE › Alternative Derivation: The Replicating Portfolio

There's an alternative way to derive the BSM PDE that provides additional insight and reinforces the economic intuition. Instead of constructing a hedged portfolio of derivative and stock, we can think about replicating the derivative using a portfolio of stock and risk-free bonds. The idea is that if we can perfectly replicate the derivative's payoff using traded securities, then the derivative must have the same price as the replicating portfolio. Otherwise, arbitrage opportunities would exist.

Suppose we want to replicate a derivative $V(S, t)$ using a portfolio of $\alpha_t$ shares of stock and $\beta_t$ units of risk-free bonds. The portfolio value is:

$$\Pi_t = \alpha_t S_t + \beta_t B_t$$

where:

• $\Pi_t$: value of the replicating portfolio
• $\alpha_t$: shares of stock held
• $\beta_t$: units of risk-free bonds held
• $S_t$: stock price
• $B_t = e^{rt}$: value of a risk-free bond

The quantities $\alpha_t$ and $\beta_t$ can change over time as we continuously adjust the portfolio to maintain replication. This dynamic adjustment is the essence of delta hedging.

For the portfolio to replicate the derivative, we need $\Pi_t = V(S_t, t)$ at all times. The key constraint is that the portfolio must be self-financing: changes in portfolio value come only from price changes, not from adding or removing capital. This means when we rebalance the portfolio, selling some stock to buy bonds or vice versa, the total value must remain unchanged. We can only rearrange what we already own. The self-financing condition is:

$$d\Pi_t = \alpha_t \, dS_t + \beta_t \, dB_t$$

where:

• $d\Pi_t$: change in portfolio value
• $\alpha_t$: shares of stock held
• $dS_t$: change in stock price
• $\beta_t$: units of risk-free bonds held
• $dB_t$: change in bond price

This equation says that the change in portfolio value equals the gains from holding stock plus the gains from holding bonds. There is no term for additional capital injection because the portfolio is self-financing.

Recall that $dB_t = r B_t \, dt$ represents the risk-free bond dynamics: the bond grows deterministically at the risk-free rate. Substituting the dynamics of $S_t$ and $B_t$:

$$\begin{aligned} d\Pi_t &= \alpha_t (\mu S_t \, dt + \sigma S_t \, dW_t) + \beta_t (r B_t \, dt) \\ &= (\alpha_t \mu S_t + \beta_t r B_t) \, dt + \alpha_t \sigma S_t \, dW_t \end{aligned}$$

where:

• $d\Pi_t$: change in the replicating portfolio value
• $\alpha_t$: number of shares of stock
• $\beta_t$: number of units of risk-free bonds
• $\mu$: stock drift
• $r$: risk-free interest rate
• $\sigma$: stock volatility
• $dW_t$: Wiener process increment

For replication, we require $d\Pi_t = dV$. The portfolio must change in exactly the same way as the derivative, both in its deterministic component and its random component. Recall the dynamics of the derivative from Itô's lemma:

$$dV = (\dots) \, dt + \sigma S_t \frac{\partial V}{\partial S} \, dW_t$$

where:

• $dV$: change in derivative price
• $\sigma$: volatility
• $S_t$: stock price
• $dW_t$: Wiener process increment

Matching the coefficients of the random term $dW_t$ gives us the replication condition:

$$\alpha_t \sigma S_t = \sigma S_t \frac{\partial V}{\partial S}$$

where:

• $\alpha_t$: shares of stock
• $\sigma$: volatility
• $S_t$: stock price
• $V$: derivative price

Solving for $\alpha_t$:

$$\alpha_t = \frac{\partial V}{\partial S}$$

where:

• $\alpha_t$: number of shares in the replicating portfolio
• $V$: price of the derivative
• $S$: stock price

This confirms that the delta hedge ratio emerges naturally from replication. To replicate the derivative, we must hold exactly delta shares of stock. The remaining portfolio value goes into bonds: $\beta_t B_t = V - \alpha_t S_t$. Matching the deterministic components and invoking the self-financing constraint leads to the same PDE. The replicating portfolio approach provides a constructive interpretation: it shows that the derivative price satisfies a PDE and explains how to create the derivative synthetically using stock and bonds.