# Proper orthogonal decomposition for pricing options

## Olivier Pironneau

#### Abstract

ABSTRACT

It was shown in a 2011 paper by R. Cont, N. Lantos and O. Pironneau that the computational time of the Black-Scholes partial differential equation can be dramatically reduced by writing the solution on a small set of basis functions.We show that this is, in fact, a proper orthogonal decomposition method, and, when using some other variables, it is a spectral method. This allows us to find a good preconditioning matrix with which to minimize the ill-conditioned linear system, and even to obtain explicit solutions.