We briefly touched on the idea of partial differential equations in the glossary section of the website, but here we aim to give you a better view of how PDEs work and what are there uses within finance. First, we can define a PDE as

"...an equation relating a function of two or more independent variables and its partial derivatives.."

You may have come across differential equations (DE) in high-school mathematics. Like those, PDes are equations to be solved in which the unknown element is a function, but whereas a standard DE has only 1 variable, PDEs function is of one of several variables. This makes PDEs somewhat trickier to solve than DEs as increased dimensions make increases computations.

In finance, the concept of PDEs are equally as important as applications within engineering and physics as many of the derivatives we encounter require a solution to a PDE. From the simple Black-Scholes (see below) to the pricing of exotics, we must solve the PDEs using the applicable boundary conditions in order to derive prices for these options.

For our plain vanilla European options, we can solve the Black-Scholes-Merton PDE using the boundary condition of a call or a put.

Call:

Put:

(A full derivation of the Black-Scholes-Merton formula is found in Hull's "Options, Futures & Other Derivatives" in chapter 12.)

Useful Resources:

Books:

Hull, J. "Options, Futures & Other Derivatives" (5th Edition) pg. 241-244)

Websites:

The Mathematical Atlas (David Rusin)