Review of modern numerical methods for a simple vanilla option pricing problem

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JEL classification

• History of Economic Thought: Quantitative and Mathematical
• Operations Research; Statistical Decision Theory
• Contingent Pricing; Futures Pricing; option pricing

Keywords

Black–Scholes equation, discontinuous Galerkin method, fuzzy transform, option pricing, wavelet method

Abstract

Option pricing is a very attractive issue of financial engineering and optimization. The problem of determining the fair price of an option arises from the assumptions made under a given financial market model. The increasing complexity of these market assumptions contributes to the popularity of the numerical treatment of option valuation. Therefore, the pricing and hedging of plain vanilla options under the Black–Scholes model usually serve as a benchmark for the development of new numerical pricing approaches and methods designed for advanced option pricing models. The objective of the paper is to present and compare the methodological con-cepts for the valuation of simple vanilla options using the relatively modern numerical techniques in this issue which arise from the discontinuous Galerkin method, the wavelet approach and the fuzzy transform technique. A theoretical comparison is accompanied by an empirical study based on the numerical verification of simple vanilla option prices. The resulting numerical schemes represent a particularly effective option pricing tool that enables some features of options that are dependent on the discretization of the computational domain as well as the order of the polynomial approximation to be captured better.