Chebyshev polynomial approximation to approximate partial differential equations

1. Introduction

The study of partial differential equations (PDE) is a fundamental topic in applied
mathematics. In fact, PDEs are fundamental in many applications in physics, natural
science and finance. For example, in finance PDEs arise in arbitrage based asset models.
The widely cited Black and Scholes PDE, that each European option must satisfy, within
an arbitrage free market, it is a canonical example.

In the specific example, (i.e. Black and Scholes model above) the PDE has a
specific analytical solution, however, in many other interesting cases in finance as well as
other fields, closed form solutions are very difficult to obtain. Therefore, in these cases
researchers rely on various numerical methods to obtain a solution. The study of these
numerical methods represent the area of Computational Partial Differential Equations.

The most simple applicable algorithm to approximate PDEs rely on the concept of
discretisation. That is, replacing the PDE of interest by a finite dimensional problem.
However, replacing the PDE by a discrete model is not trivial at all and generally the
choice of the finite dimensional model to be used depends on the properties behind the
mathematical model itself.

The development of high speed computers has made easier to find accurate
solutions to PDEs in a very efficient manner, even in most extreme cases of very large
systems of PDEs. In this study we show how using polynomial methods to approximate
PDEs. We shall only be focusing on second-order linear PDEs, although it would also be
interesting to evaluate this methodology when dealing with non-linear types of PDEs. We
leave this on the agenda for future research.

Crack-Nicolson (CN) implicit schemes are amongst the most widely used methods in
these cases. However, the effectiveness of the applications of these schemes rely on the
choice of the time steps used, and the latter, very often, depend on the problem we are
facing. Also CN methods, in general, suffer from poor convergence.

The method suggested in this paper combines polynomial interpolation to
approximate the PDE characterising the option pricing problem, and, given our specific
applications, we use Monte Carlo method to solve the boundary condition for the PDE.

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