Black-Scholes models the dynamics of a financial market containing derivative investment instruments. It is described by the following Partial Differential Equation (PDE)
\[\frac{\partial U}{\partial t} + \frac{1}{2} \sigma^2 S^2 \frac{\partial^2 U}{\partial S^2} + rS \frac{\partial U}{\partial S} - rU = 0\]
The model assumes a constant volatility and risk-free rate. The model can be generalized to remove this restriction.This form of the equation can be use to price options using a local volatility description.
\[\frac{\partial U}{\partial t} + \frac{1}{2} \sigma^2(S,t) S^2 \frac{\partial^2 U}{\partial S^2} + r(t)S \frac{\partial U}{\partial S} - r(t)U = 0\]