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March  2021, 14(3): 1181-1195. doi: 10.3934/dcdss.2020226

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

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

* Corresponding authors:Matúš Tibenský

Received  December 2018 Revised  September 2019 Published  December 2019

Fund Project: Authors are supported by grants APVV 15-0522 and VEGA 1/0728/15

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.

Citation: Matúš Tibenský, Angela Handlovičová. Convergence analysis of the discrete duality finite volume scheme for the regularised Heston model. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1181-1195. doi: 10.3934/dcdss.2020226
##### References:

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##### References:
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
 $N_x$ $N_y$ $N_{ts}$ $L_2 D$ $L_2 R, \epsilon = 10^{-2}$ $L_2 R, \epsilon = 10^{-4}$ $L_2 R, \epsilon = 10^{-6}$ [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
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
 $N_x$ $N_y$ $N_{ts}$ $L_2 D$ $L_2 R, \epsilon = 10^{-2}$ $L_2 R, \epsilon = 10^{-4}$ $L_2 R, \epsilon = 10^{-6}$ [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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