Generalized exponential basis for efficient solving of homogeneous diffusion free boundary problems: Russian option pricing

TL;DR

This paper introduces a novel method combining exponential solutions, transmutation operators, and Bessel function series to efficiently solve free boundary problems in diffusion models, demonstrated through Russian option pricing.

Contribution

It develops a new approach using a generalized exponential basis and advanced mathematical tools for solving homogeneous diffusion free boundary problems.

Findings

Accurate valuation of Russian options with different horizons.

Comparison shows improved efficiency over existing methods.

Method effectively handles complex boundary conditions.

Abstract

This paper develops a method for solving free boundary problems for time-homogeneous diffusions. We combine the complete exponential system of solutions for the heat equation, transmutation operators and recently discovered Neumann series of Bessel functions representation for solutions of Sturm-Liouville equations to construct a complete system of solutions for the considered partial differential equations. The conceptual algorithm for the application of the method is presented. The valuation of Russian options with finite horizon is used as a numerical illustration. The solution under different horizons is computed and compared to the results that appear in the literature.