Fitted Finite Volume Method for a Generalized Black-Scholes Equation Transformed on Finite Interval

TL;DR

This paper develops a fitted finite volume method for a generalized Black-Scholes equation on a finite interval, addressing degeneracy at the boundaries and ensuring positivity preservation, with numerical validation.

Contribution

It introduces a novel fitted finite volume element approach for the transformed Black-Scholes equation, handling boundary degeneracy and proving stability and positivity.

Findings

Method is uniquely solvable and positivity-preserving.

Numerical experiments confirm the effectiveness of the approach.

Abstract

A generalized Black-Scholes equation is considered on the semi-axis. It is transformed on the interval (0,1) in order to make the computational domain finite. The new parabolic operator degenerates at the both ends of the interval and we are forced to use the G\"{a}rding inequality rather than the classical coercivity. A fitted finite volume element space approximation is constructed. It is proved that the time \theta-weighted full discretization is uniquely solvable and positivity-preserving. Numerical experiments, performed to illustrate the usefulness of the method, are presented.