An application of finite elements to option pricing

Abstract

This study applied the finite element method (FEM) to pricing options. The FEM estimates the function that satisfies a governing differential equation through the assembly of piecewise continuous functions over the domain of the problem. Two common representations, a variational functional representation, and a weighted residual representation are used in the application of the method. The FEM is a versatile alternative to other popular lattice methods used in option pricing. Advantages include the abilities to directly estimate the Greeks of the option and allow nonuniform mesh construction. As an illustration of the advantages that the FEM offers, the method was used to price European put options and discrete barrier knock‐out put options. © 2001 John Wiley & Sons, Inc. Jrl Fut Mark 21:19–42, 2001

Nonsymmetric Interior Penalty Galerkin Method For Black-Scholes PDEs Modelling European Options

“…The Black-Scholes equation can be converted into a diffusion equation when its coefficients are constant or space-independent. Additionally, the finite element approach has been developed for this model [5,7,13,14].…”

A superconvergent fitted finite volume method for Black–Scholes equations governing European and American option valuation

“…However, numerical solution of Black–Scholes equations becomes increasingly popular in recent years as most option pricing problems, particularly those of early exercise options, are not analytically solvable. Several classic numerical methods have been used for Black–Scholes equations such as lattice techniques , finite differences , and finite elements .…”

An investigation into the application of the finite element method in counting process models

“…The Finite Element method (hereafter abbreviated FEM) has its origins in the mid 1950's, when it was developed as a method for solving elasticity and structural me chanics problems in civil and aeronautical engineering [3,8]. Since then the number of applications using this method has steadily grown and they nowadays include heat transfer, solid and fluid mechanics, wave scattering, mass transfer, data smoothing and option pricing, just to name a few [7,28,4,25,21,1]. The method has been so successful in solving Partial Differential Equations (PDE), that today the term "Finite Element Method" is practically synonymous with the numerical solutions of PDEs.…”

“…In the last test we present in this section, we solved three cases with (M, L) = (25,20), (50,50), and (100,150) on a grid with 31 finite elements giving n = 64 degrees of freedom in the interpolation basis. In each case, we simulated B = 1000 samples and estimated a(s) using GCV and REML.…”