Reduced basis methods for pricing options with the Black-Scholes and Heston model

TL;DR

This paper introduces a reduced basis method for efficiently pricing European and American options under Black-Scholes and Heston models, combining POD-Greedy techniques with error estimation.

Contribution

It develops a novel reduced basis approach with proven reproduction properties and error estimators for option pricing models.

Findings

High approximation accuracy demonstrated in numerical tests

Convergence of the reduced basis method confirmed

Reliable and effective error estimators validated

Abstract

In this paper, we present a reduced basis method for pricing European and American options based on the Black-Scholes and Heston model. To tackle each model numerically, we formulate the problem in terms of a time dependent variational equality or inequality. We apply a suitable reduced basis approach for both types of options. The characteristic ingredients used in the method are a combined POD-Greedy and Angle-Greedy procedure for the construction of the primal and dual reduced spaces. Analytically, we prove the reproduction property of the reduced scheme and derive a posteriori error estimators. Numerical examples are provided, illustrating the approximation quality and convergence of our approach for the different option pricing models. Also, we investigate the reliability and effectivity of the error estimators.