Black–Scholes equation

Definition

Let $\sigma > 0$ and $r \ge 0$ be constants and let $T > 0$. The Black–Scholes equation is the parabolic equation for a function $V(S, t)$ with $S > 0$ and $0 \le t < T$: $$\frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} + r S \frac{\partial V}{\partial S} - r V = 0.$$ Here $S$ is the price of the underlying asset, modelled by geometric Brownian motion with volatility $\sigma$. The constant $r$ is the risk-free interest rate, and $V$ is the price of a derivative security. It is solved with a terminal condition $V(S, T) = \Phi(S)$ given by the payoff $\Phi$.