Applications of Mathematics, Vol. 64, No. 5, pp. 501-530, 2019
DG method for pricing European options under Merton jump-diffusion model
Jiří Hozman, Tomáš Tichý, Miloslav Vlasák
Received November 2, 2018. Published online October 2, 2019.
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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Affiliations: Jiří Hozman, Technical University of Liberec, Faculty of Science, Humanities and Education, Department of Mathematics and Didactics of Mathematics, Studentská 1402/2, 461 17 Liberec, Czech Republic, e-mail: jiri.hozman@tul.cz; Tomáš Tichý, VŠB-TU Ostrava, Faculty of Economics, Department of Finance, Sokolská třída 33, 702 00 Ostrava, Czech Republic, e-mail: tomas.tichy@vsb.cz; Miloslav Vlasák, Charles University, Faculty of Mathematics and Physics, Department of Numerical Mathematics, Sokolovská 83, 186 75 Praha 8, Czech Republic, e-mail: vlasak@karlin.mff.cuni.cz
