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Michael Brenndoerfer

Derivation of the Black-Scholes-Merton PDE › The Black-Scholes-Merton PDE

Michael BrenndoerferNovember 26, 20253 min read

Quantitative FinanceAll chapters

Part 5 of 15Stochastic Calculus and Pricing Theory

Chapter 29Derivation of the Black-Scholes-Merton PDE › The Black-Scholes-Merton PDE

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Black-Scholes-Merton PDE

Any derivative security $V(S, t)$ whose value depends on a stock following geometric Brownian motion must satisfy:

\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

where:

$V(S, t)$ : price of the derivative security
$S$ : current stock price
$r$ : risk-free interest rate
$\sigma$ : volatility of the underlying stock

This equation has several notable features worth examining carefully, as they reveal deep insights about derivative pricing.

The drift $\mu$ does not appear. The expected return on the stock has completely dropped out of the equation. This means the derivative price is the same whether the stock is expected to grow at 5% per year or 50% per year. At first glance, this seems counterintuitive: shouldn't a call option on a rapidly growing stock be worth more? The resolution lies in understanding that in a no-arbitrage framework, the expected return is already reflected in the current stock price. If investors expected the stock to grow rapidly, they would bid up its price today until the expected return was consistent with its risk. What matters for derivative pricing is volatility, not direction. The randomness, not the trend, determines the option's value.

The equation is a backward parabolic PDE. The equation is similar to the heat equation from physics, but with time running backwards. In physics, the heat equation describes how temperature diffuses forward in time from an initial condition. In finance, we typically know the derivative's value at maturity (the boundary condition) and need to work backwards to find its value today. This backward-in-time structure reflects the forward-looking nature of financial valuation: we discount future payoffs back to the present.

The equation is linear. If $V_1$ and $V_2$ are solutions, then $aV_1 + bV_2$ is also a solution for any constants $a$ and $b$ . This linearity reflects the additivity of derivative portfolios: a portfolio of derivatives can be priced by summing the prices of its components. You can verify this linearity by substituting $aV_1 + bV_2$ into the PDE and using the linearity of partial differentiation.

All derivatives on the same underlying satisfy the same PDE. A call option, a put option, and an exotic option all satisfy the same equation. What distinguishes them is the boundary condition at maturity. The PDE captures the universal dynamics of derivative pricing, while the boundary condition encodes the specific payoff structure. This separation of dynamics from payoffs is what makes the BSM framework so powerful and general.