The Black-Scholes equation

The Black-Scholes equation is a parabolic partial differential equation (PDE) for option prices discovered by Black and Scholes in 1973. This discovery revolutionized financial markets in 1990-s. In 1997 Merton and Scholes were awarded the Nobel prize in economics. For any stock price s,

and time t,

the price u for an option expiring at time T satisfies the following Black-Scholes PDE:

where σ(s,t) is the volatility coefficient,

and μ and r are respectively the risk neutral drift and risk free interest which are assumed to be constants.