Black–Scholes equation

Summary

The Black–Scholes equation is the linear parabolic equation $V_t + \frac{1}{2}\sigma^2 S^2 V_{SS} + r S V_S - r V = 0$ for the price $V(S, t)$ of a financial option. It assumes that the price $S$ of the underlying asset follows geometric Brownian motion with volatility $\sigma$. It also assumes that money earns the constant interest rate $r$. It is solved backward in time from the payoff at expiry. A change of variables, starting with $x = \log S$ and reversing time, turns it into the heat equation.