A partial differential equation (PDE) is a mathematical equation that involves multiple independent variables, an unknown function that is dependent on those variables, and partial derivatives of the unknown function with respect to the independent variables. PDEs are commonly used to define multidimensional systems in physics and engineering. In quantitative finance, they have a similar purpose and are typically used in the pricing of derivatives. The Black-Scholes equation is an example of a PDE used to describe the evolution of the price of a European option. Some PDEs have exact solutions, but many aren’t easy to solve as they describe complex systems. In such cases, cumbersome numerical methods have to be used. Numerical methods such as the finite difference method, for instance, work by approximating the derivatives in the PDE and then, using a large number of incremental values of the independent variables, calculating the unknown function at each of those values.
PARTIAL DIFFERENTIAL EQUATION (PDE)
