Derivation of the Black-Scholes-Merton PDE › Dynamics of the Derivative Price

To find how the derivative price changes over time, we must apply Itô's lemma. This is where the machinery of stochastic calculus becomes essential. In ordinary calculus, if you have a function $f(x)$ and $x$ changes by a small amount $dx$, then $f$ changes by approximately $f'(x) \cdot dx$. But in stochastic calculus, the presence of random fluctuations introduces additional terms that have no analog in the deterministic world.

Recall from our chapter on Itô's Lemma and Stochastic Calculus that for any sufficiently smooth function $V(S, t)$ where $S$ follows a diffusion process, the differential $dV$ has a specific form that accounts for the second-order effects of stochastic calculus. The key insight is that terms involving $(dW)^2$, which would vanish in ordinary calculus because they're second-order small, contribute to first order in stochastic calculus because $(dW)^2 = dt$.

Since $S_t$ follows geometric Brownian motion, Itô's lemma gives:

$$dV = \frac{\partial V}{\partial t} dt + \frac{\partial V}{\partial S} dS + \frac{1}{2} \frac{\partial^2 V}{\partial S^2} (dS)^2$$

where:

• $V(S, t)$: price of the derivative security
• $\frac{\partial V}{\partial t}$: sensitivity of option value to time (Theta)
• $\frac{\partial V}{\partial S}$: sensitivity of option value to stock price (Delta)
• $\frac{\partial^2 V}{\partial S^2}$: curvature of option value with respect to stock price (Gamma)
• $dt$: small time increment
• $dS$: change in stock price

Notice that this formula looks almost like the ordinary chain rule, except for the final term involving the second derivative. This term arises specifically because of the stochastic nature of the price process. In deterministic calculus, this term would be negligibly small, a second-order effect that vanishes in the limit. But in stochastic calculus, the roughness of Brownian motion paths means this term survives and contributes to the dynamics.

We need to evaluate $(dS)^2$ by substituting the expression for $dS$ and applying Itô multiplication rules. This calculation reveals why stochastic calculus differs fundamentally from ordinary calculus:

$$\begin{aligned} (dS)^2 &= (\mu S \, dt + \sigma S \, dW)^2 \\ &= \mu^2 S^2 (dt)^2 + 2\mu\sigma S^2 \, dt \, dW + \sigma^2 S^2 (dW)^2 && \text{(expand square)} \\ &= \sigma^2 S^2 \, dt && \text{(apply } (dt)^2=0, dt \cdot dW=0, (dW)^2=dt \text{)} \end{aligned}$$

where:

• $dS$: change in stock price
• $\mu$: drift coefficient
• $\sigma$: volatility coefficient
• $S$: stock price
• $dt$: small time increment
• $dW$: Wiener process increment

The Itô multiplication rules deserve careful attention because they encode the essence of what makes stochastic calculus different. The rule $(dt)^2 = 0$ is intuitive: when we square an infinitesimally small quantity, we get something even more negligibly small. The rule $dt \cdot dW = 0$ follows from similar reasoning. But the rule $(dW)^2 = dt$ is remarkable: squaring a random increment of order $\sqrt{dt}$ produces something of order $dt$, which is not negligible. This is the mathematical expression of the fact that Brownian motion paths are extremely rough, changing direction infinitely often in any finite time interval.

Substituting $(dS)^2$ and $dS$ back into the expression for $dV$ and grouping terms, we arrive at the complete dynamics of the derivative price:

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

where:

• $dV$: change in the derivative price
• $V(S, t)$: price of the derivative
• $S$: stock price
• $t$: time
• $\mu$: expected return of the stock
• $\sigma$: volatility of the stock
• $dW$: standard Wiener process increment

This equation tells us that the derivative price has both a deterministic component (the $dt$ term) and a random component (the $dW$ term). The deterministic component captures the systematic evolution of the derivative price due to the passage of time, the drift in the stock price, and the convexity effect from the second derivative. The random component, proportional to $dW$, represents the uncertainty in the derivative price arising from the uncertainty in the stock price.

The presence of $dW$ means that holding the derivative alone exposes you to the same source of randomness as holding the stock. Both the stock and the derivative are driven by the same Brownian motion, though with different sensitivities. This shared source of randomness makes the hedging argument that follows possible.