Convergence analysis of the discrete duality finite volume scheme for the regularised Heston model

Matúš Tibenský; Angela Handlovičová

doi:10.3934/dcdss.2020226

Article Contents

2021, Volume 14, Issue 3: 1181-1195. Doi: 10.3934/dcdss.2020226

This issue Previous Article Comparison of modern heuristics on solving the phase stability testing problem Next Article Stability estimates and a Lagrange-Galerkin scheme for a Navier-Stokes type model of flow in non-homogeneous porous media

Convergence analysis of the discrete duality finite volume scheme for the regularised Heston model

Matúš Tibenský^, and
Angela Handlovičová

Dpt. of Mathematics Slovak University of Technology, Radlinské eho 11,810 05 Bratislava, Slovakia

^* Corresponding authors:Matúš Tibenský
^* Corresponding authors:Matúš Tibenský

Received: December 2018

Revised: September 2019

Early access: December 2019

Published: March 2021

Authors are supported by grants APVV 15-0522 and VEGA 1/0728/15.

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Abstract

The aim of the paper is to study problem of financial derivatives pricing based on the idea of the Heston model introduced in [9]. Following the approach stated in [6] and in [7] we construct the regularised version of the Heston model and the discrete duality finite volume (DDFV) scheme for this model. The numerical analysis is performed for this scheme and stability estimates on the discrete solution and the discrete gradient are obtained. In addition the convergence of the DDFV scheme to the weak solution of the regularised Heston model is proven. The numerical experiments are provided in the end of the paper to test the regularisation parameter impact.

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Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

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Table 1. Results for the regularised and the original DDFV scheme comparison, Experiment Nr. 1

$ N_x $	$ N_y $	$ N_{ts} $	$ L_2 D $	$ L_2 R, \epsilon = 10^{-2} $	$ L_2 R, \epsilon = 10^{-4} $	$ L_2 R, \epsilon = 10^{-6} $
[0.5ex] 20	10	1	0.00318557	0.00329745	0.00318659	0.00318559
40	20	4	0.00206132	0.00211980	0.00206182	0.00206133
80	40	16	0.00151241	0.00156704	0.00151286	0.00151242
160	80	64	0.00125001	0.00130976	0.00125050	0.00125002

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Table 2. Results for the regularised and the original DDFV scheme comparison, Experiment Nr. 2

$ N_x $	$ N_y $	$ N_{ts} $	$ L_2 D $	$ L_2 R, \epsilon = 10^{-2} $	$ L_2 R, \epsilon = 10^{-4} $	$ L_2 R, \epsilon = 10^{-6} $
[0.5ex] 20	10	1	0.00377821	0.00371450	0.00377742	0.00377822
40	20	4	0.00269958	0.00264958	0.00269896	0.00269957
80	40	16	0.00199309	0.00197965	0.00199286	0.00199309
160	80	64	0.00155891	0.00157838	0.00155904	0.00155891

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References

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