# Institute of Mathematics

## Solution of option pricing equations using orthogonal polynomial expansion

#### Falko Baustian, Kateřina Filipová, Jan Pospíšil

###### Received December 13, 2019.   Published online March 29, 2021.

Abstract:  We study both analytic and numerical solutions of option pricing equations using systems of orthogonal polynomials. Using a Galerkin-based method, we solve the parabolic partial differential equation for the Black-Scholes model using Hermite polynomials and for the Heston model using Hermite and Laguerre polynomials. We compare the obtained solutions to existing semi-closed pricing formulas. Special attention is paid to the solution of the Heston model at the boundary with vanishing volatility.
Keywords:  orthogonal polynomial expansion; Hermite polynomial; Laguerre polynomial; Heston model; option pricing
Classification MSC:  33C45, 65M60, 91G20, 91G60

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Affiliations:   Falko Baustian, Department of Mathematics, University of Rostock, Ulmenstr. 69, 18057 Rostock, Germany, e-mail: falko.baustian@uni-rostock.de; Kateřina Filipová, Jan Pospíšil (corresponding author), NTIS - New Technologies for the Information Society, Faculty of Applied Sciences, University of West Bohemia, Univerzitní 2732/8, 301 00 Plzeň, Czech Republic, e-mail: filipovk@ntis.zcu.cz, honik@ntis.zcu.cz

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