DG method for pricing European options under Merton jump-diffusion model

Jiří Hozman; Tomáš Tichý; Miloslav Vlasák

Applications of Mathematics (2019)

  • Volume: 64, Issue: 5, page 501-530
  • ISSN: 0862-7940

Abstract

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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.

How to cite

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Hozman, Jiří, Tichý, Tomáš, and Vlasák, Miloslav. "DG method for pricing European options under Merton jump-diffusion model." Applications of Mathematics 64.5 (2019): 501-530. <http://eudml.org/doc/294256>.

@article{Hozman2019,
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.},
author = {Hozman, Jiří, Tichý, Tomáš, Vlasák, Miloslav},
journal = {Applications of Mathematics},
keywords = {option pricing; Merton jump-diffusion model; integro-differential equation; discontinuous Galerkin method; semi-implicit discretization; a priori error estimates},
language = {eng},
number = {5},
pages = {501-530},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {DG method for pricing European options under Merton jump-diffusion model},
url = {http://eudml.org/doc/294256},
volume = {64},
year = {2019},
}

TY - JOUR
AU - Hozman, Jiří
AU - Tichý, Tomáš
AU - Vlasák, Miloslav
TI - DG method for pricing European options under Merton jump-diffusion model
JO - Applications of Mathematics
PY - 2019
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 64
IS - 5
SP - 501
EP - 530
AB - 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.
LA - eng
KW - option pricing; Merton jump-diffusion model; integro-differential equation; discontinuous Galerkin method; semi-implicit discretization; a priori error estimates
UR - http://eudml.org/doc/294256
ER -

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