Multilayer heat equations: application to finance

TL;DR

This paper introduces a Multilayer method for solving one-factor parabolic equations in finance, combining analytical and numerical techniques to improve speed and accuracy over traditional methods like finite differences and Monte Carlo simulations.

Contribution

The paper extends the application of multilayer heat equations from physics to finance, developing efficient algorithms for option pricing and solving the Dupire equation.

Findings

The ML method is faster than finite difference methods.

The ML approach provides higher accuracy in solving PDEs.

Numerical examples demonstrate significant efficiency improvements.

Abstract

In this paper, we develop a Multilayer (ML) method for solving one-factor parabolic equations. Our approach provides a powerful alternative to the well-known finite difference and Monte Carlo methods. We discuss various advantages of this approach, which judiciously combines semi-analytical and numerical techniques and provides a fast and accurate way of finding solutions to the corresponding equations. To introduce the core of the method, we consider multilayer heat equations, known in physics for a relatively long time but never used when solving financial problems. Thus, we expand the analytic machinery of quantitative finance by augmenting it with the ML method. We demonstrate how one can solve various problems of mathematical finance by using our approach. Specifically, we develop efficient algorithms for pricing barrier options for time-dependent one-factor short-rate models, such as Black-Karasinski and Verhulst. Besides, we show how to solve the well-known Dupire equation quickly and accurately. Numerical examples confirm that our approach is considerably more efficient for solving the corresponding partial differential equations than the conventional finite difference method by being much faster and more accurate than the known alternatives.