An Introduction to Financial Mathematics, 2nd Edition

Chapter 12

The Black-Scholes-Merton Model

With the methods of Chapter 11 at our disposal, we are now able to derive the celebrated Black-Scholes formula for the price of a call option. The formula is based on the solution of a partial differential equation arising from a stochastic differential equation that governs the price of the underlying stock $\mathcal{S}$. We assume throughout that the market is arbitrage-free.

12.1  The Stock Price SDE

Let W be a Brownian motion on a probability space $\left(\Omega ,ℱ,\mathbb{P}\right)$. The price S of a single share of $\mathcal{S}$ is assumed to satisfy the SDE

$\frac{dS}{S}=\sigma dW+\mu dt,$

(12.1)

where μ and σ are constants called, respectively, the drift and volatility of the stock. Equation (12.1) asserts that the relative change in the stock price has two ...