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

The Black-Scholes-Merton partial differential equation is one of the most important results in mathematical finance. Published in 1973 by Fischer Black, Myron Scholes, and Robert Merton, this equation gives a framework for pricing derivative securities by showing that the price of any derivative must satisfy a specific PDE under certain assumptions. What makes this result high is that the equation contains no reference to investor preferences or expected returns: the derivative price depends only on observable quantities and the assumption of no arbitrage.

In this chapter, you will learn how to derive this PDE from first principles using the tools we've developed in previous chapters. The derivation follows a beautiful logical chain: we assume stock prices follow geometric Brownian motion, apply Itô's lemma to find the dynamics of a derivative's price, construct a portfolio that eliminates all randomness through dynamic hedging, and invoke the no-arbitrage principle to obtain the final equation. Understanding this derivation deeply prepares you for the practical applications covered in subsequent chapters, where we'll solve the PDE to obtain explicit pricing formulas.