The Mathematics of the Black‑Scholes Equation

The Black‑Scholes equation (1973) is a partial differential equation that prices financial options: ∂V/∂t + ½σ²S²∂²V/∂S² + rS∂V/∂S – rV = 0. It transformed finance, enabling the growth of derivatives markets. The solution gives the fair price of a European call option. The equation is similar to the heat equation with convection. The 1997 Nobel Prize in Economics was awarded to Scholes and Merton (Black had died). The equation assumes continuous trading, no arbitrage, and log‑normal stock returns. In reality, markets deviate – but the model is still ubiquitous.