Derivation of the Black-Scholes-Merton PDE › Applying the No-Arbitrage Principle

We now have a portfolio with no risk over the next instant. Building on the no-arbitrage principle from our previous chapter on risk-neutral valuation, a riskless portfolio must earn the risk-free rate. The logic is ironclad: if the portfolio earned more than the risk-free rate, you could borrow money at the risk-free rate, invest in the portfolio, and pocket the difference as guaranteed profit. If it earned less, you could short the portfolio, invest the proceeds in the risk-free asset, and again earn guaranteed profit. Either scenario represents arbitrage, a free lunch that cannot persist in a well-functioning market.

The no-arbitrage condition states:

$$d\Pi = r \Pi \, dt$$

where:

• $r$: risk-free interest rate
• $\Pi$: value of the riskless portfolio
• $dt$: time increment

This equation simply says that the return on a riskless investment must equal the risk-free rate. The risk-free rate $r$ represents the return you could earn on a perfectly safe investment, such as a government bond or a bank deposit.

Now we substitute our expressions for $d\Pi$ and $\Pi$ and perform algebraic manipulations to extract the PDE:

$$\begin{aligned} \left( \frac{\partial V}{\partial t} + \frac{1}{2} \sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} \right) dt &= r \left( V - \frac{\partial V}{\partial S} S \right) dt \\ \frac{\partial V}{\partial t} + \frac{1}{2} \sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} &= rV - rS\frac{\partial V}{\partial S} && \text{(divide by } dt \text{)} \\ \frac{\partial V}{\partial t} + rS\frac{\partial V}{\partial S} + \frac{1}{2} \sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} - rV &= 0 && \text{(group terms)} \end{aligned}$$

where:

• $V$: derivative price
• $t$: time
• $S$: stock price
• $r$: risk-free rate
• $\sigma$: volatility

This is the Black-Scholes-Merton partial differential equation. We have arrived at a fundamental constraint that any arbitrage-free derivative price must satisfy. The equation emerged from combining three key ingredients: the assumption that stock prices follow geometric Brownian motion, the application of Itô's lemma to characterize derivative dynamics, and the no-arbitrage condition that riskless portfolios must earn the risk-free rate.