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DG Method for Pricing European Options under Merton Jump-Diffusion Model

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Abstract

Under real market conditions, there exist many cases when it is inevitable to adopt numerical approximations of option prices due to non-existence of analytical formulae. Obviously, any numerical technique should be tested for the cases when the analytical solution is well known. The paper is devoted to the discontinuous Galerkin method applied to European option pricing under the Merton jump-diffusion model, when the evolution of the asset prices is driven by a Lévy process with finite activity. The valuation of options under such a model with lognormally distributed jumps requires solving a parabolic partial integro-differential equation which involves both the integrals and the derivatives of the unknown pricing function. The integral term related to jumps leads to new theoretical and numerical issues regarding the solving of the pricing equation in comparison with the standard approach for the Black-Scholes equation. Here we adopt the idea of the relatively modern technique that the integral terms in Merton-type models can be viewed as solutions of proper differential equations, which can be accurately solved in a simple way. For practical purposes of numerical pricing of options in such models we propose a two-stage implicit-explicit scheme arising from the discontinuous piecewise polynomial approximation, i.e., the discontinuous Galerkin method. This solution procedure is accompanied with theoretical results and discussed within the numerical results on reference benchmarks.

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Authors and Affiliations

  1. Faculty of Science, Humanities and Education, Department of Mathematics and Didactics of Mathematics, Technical University of Liberec, Studentská 1402/2, 461 17, Liberec, Czech Republic

    Jiří Hozman

  2. Department of Finance, VŠB-TU Ostrava, Faculty of Economics, Sokolská třída 33, 702 00, Ostrava, Czech Republic

    Tomáš Tichý

  3. Faculty of Mathematics and Physics, Department of Numerical Mathematics, Charles University, Sokolovská 83, 186 75, Praha 8, Czech Republic

    Miloslav Vlasák

Authors
  1. Jiří Hozman
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  2. Tomáš Tichý
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  3. Miloslav Vlasák
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Corresponding authors

Correspondence to Jiří Hozman, Tomáš Tichý or Miloslav Vlasák.

Additional information

The first two authors were supported through the Czech Science Foundation (GA ČR) under project 16-09541S. Furthermore, the second author also acknowledges the support provided within SP2019/5, an SGS research project of VŠB-TU Ostrava. The research of the third author was supported by grant 17-01747S of the Czech Science Foundation; he is a junior member of the university center for mathematical modeling, applied analysis and computational mathematics (MathMAC). The support is greatly acknowledged.

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Hozman, J., Tichý, T. & Vlasák, M. DG Method for Pricing European Options under Merton Jump-Diffusion Model. Appl Math 64, 501–530 (2019). https://doi.org/10.21136/AM.2019.0305-18

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