A Reduced Basis Method for Parabolic Partial Differential Equations with Parameter Functions and Application to Option Pricing

TL;DR

This paper develops a reduced basis method for parameterized parabolic PDEs, exemplified by the Heston model, enabling efficient calibration and option pricing with variable initial conditions.

Contribution

It introduces a space-time variational formulation and a reduced basis method that handle parameter functions in both coefficients and initial conditions.

Findings

Stable discretization in space and time.

Effective reduced basis method for parameter functions.

Numerical results demonstrating method efficiency.

Abstract

We consider the Heston model as an example of a parameterized parabolic partial differential equation. A space-time variational formulation is derived that allows for parameters in the coefficients (for calibration) as well as choosing the initial condition (for option pricing) as a parameter function. A corresponding discretization in space and time amd initial condition is introduced and shown to be stable. Finally, a Reduced Basis Method (RBM) is introduced that is able to use parameter functions also for the initial condition. Corresponding numerical results are shown.