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Solution of option pricing equations using orthogonal polynomial expansion

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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.

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Acknowledgements

This work is based on the Master’s thesis Filipová [19] titled Solution of option pricing equations using orthogonal polynomial expansion that was written by Kateřina Filipová and supervised by Jan Pospíšil. The thesis was also advised by Falko Baustian during the two months internship of Kateřina Filipová at the University of Rostock.

Our sincere gratitude goes to Prof. Peter Takáč from the University of Rostock, who introduced us to the problem and provided us with valuable suggestions and insightful criticism, and to both anonymous referees for their valuable comments and extensive suggestions.

Computational resources were provided by the CESNET LM2015042 and the CERIT Scientific Cloud LM2015085, provided under the programme “Projects of Large Research, Development, and Innovations Infrastructures”.

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Authors and Affiliations

  1. Department of Mathematics, University of Rostock, Ulmenstr. 69, 18057, Rostock, Germany

    Falko Baustian

  2. NTIS - New Technologies for the Information Society, Faculty of Applied Sciences, University of West Bohemia, Univerzitní 2732/8, 301 00, Plzeň, Czech Republic

    Kateřina Filipová & Jan Pospíšil

Authors
  1. Falko Baustian
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  2. Kateřina Filipová
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  3. Jan Pospíšil
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Corresponding author

Correspondence to Jan Pospíšil.

Additional information

The work was partially supported by the Czech Science Foundation (GACR) grant no. GA18-16680S “Rough models of fractional stochastic volatility”.

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Baustian, F., Filipová, K. & Pospíšil, J. Solution of option pricing equations using orthogonal polynomial expansion. Appl Math 66, 553–582 (2021). https://doi.org/10.21136/AM.2021.0361-19

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