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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  March 2021 Early access  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 and Continuous Dynamical Systems - S, 2021, 14 (3) : 1181-1195. doi: 10.3934/dcdss.2020226
##### References:
 [1] B. Andreianov, F. Boyer and F. Hubert, Discrete duality finite volume schemes for Leray-Lions type problems on general 2D meshes, Numerical Methods for PDEs, 23 (2007), 145-195.  doi: 10.1002/num.20170. [2] F. Black and M. Scholes, The pricing of options and corporate liabilities, The Journal of Political Economy, 81 (1973), 637-654.  doi: 10.1086/260062. [3] R. Eymard, T. Gallouët and R. Herbin, Finite volume method, Handbook of Numerical Analysis, Handb. Numer. Anal., Ⅶ, North-Holland, Amsterdam, 7 (2000), 713-1020.  doi: 10.1086/phos.67.4.188705. [4] R. Eymard, A. Handlovičová and K. Mikula, Study of a finite volume scheme for regularised mean curvature flow level set equation, IMA Journal on Numerical Analysis, 31 (2011), 813-846.  doi: 10.1093/imanum/drq025. [5] G. Fichera, Sulle equazioni differenziali lineari ellittico-paraboliche del secondo ordine, Atti Accad. Naz. Lincei, Mem., Cl. Sci. Fis. Mat. Nat., 5 (1956), 1-30. [6] A. Handlovičová, Discrete duality finite volume scheme for solving Heston model, Proccedings of ALGORITMY, (2016), 264–274. [7] A. Handlovičová, Stability estimates for discrete duality finite volume scheme for Heston model, Computer Methods in Materials Science, 17 (2017), 101-110. [8] A. Handlovičová and D. Kotorová, Numerical analysis of a semi-implicit discrete duality finite volume scheme for the curvature driven level set equation in 2D, Kybernetika, 49 (2013), 829-854. [9] S. L. Heston, A closed-form solution for options with stochastic volatility with applications to bond and currency options, The Review of Financial Studies, 6 (1993), 327-343.  doi: 10.1093/rfs/6.2.327. [10] P. Kútik, Numerical Solution of Partial Differential Equations and Their Application, Ph.D thesis, Slovak University of Technology in Bratislava, Slovakia, 2013. [11] P. Kútik and K. Mikula, Diamond-cell finite volume scheme for the Heston model, Discrete and Continuous Dynamical Systems, 8 (2015), 913-931.  doi: 10.3934/dcdss.2015.8.913. [12] O. A. Ladyžhenskaya, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, Translations of Mathematical Monographs, Vol. 23 American Mathematical Society, Providence, R.I. 1968. [13] O.A. Oleǐnik and E. V. Radkevič, Second order equations with nonnegative characteristic form, Mathematical Analysis, 1969, Akad. Nauk SSSR Vsesojuzn. Inst. Naučn. i Tehn. Informacii, Moscow, (1971), 7–252.

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##### References:
 [1] B. Andreianov, F. Boyer and F. Hubert, Discrete duality finite volume schemes for Leray-Lions type problems on general 2D meshes, Numerical Methods for PDEs, 23 (2007), 145-195.  doi: 10.1002/num.20170. [2] F. Black and M. Scholes, The pricing of options and corporate liabilities, The Journal of Political Economy, 81 (1973), 637-654.  doi: 10.1086/260062. [3] R. Eymard, T. Gallouët and R. Herbin, Finite volume method, Handbook of Numerical Analysis, Handb. Numer. Anal., Ⅶ, North-Holland, Amsterdam, 7 (2000), 713-1020.  doi: 10.1086/phos.67.4.188705. [4] R. Eymard, A. Handlovičová and K. Mikula, Study of a finite volume scheme for regularised mean curvature flow level set equation, IMA Journal on Numerical Analysis, 31 (2011), 813-846.  doi: 10.1093/imanum/drq025. [5] G. Fichera, Sulle equazioni differenziali lineari ellittico-paraboliche del secondo ordine, Atti Accad. Naz. Lincei, Mem., Cl. Sci. Fis. Mat. Nat., 5 (1956), 1-30. [6] A. Handlovičová, Discrete duality finite volume scheme for solving Heston model, Proccedings of ALGORITMY, (2016), 264–274. [7] A. Handlovičová, Stability estimates for discrete duality finite volume scheme for Heston model, Computer Methods in Materials Science, 17 (2017), 101-110. [8] A. Handlovičová and D. Kotorová, Numerical analysis of a semi-implicit discrete duality finite volume scheme for the curvature driven level set equation in 2D, Kybernetika, 49 (2013), 829-854. [9] S. L. Heston, A closed-form solution for options with stochastic volatility with applications to bond and currency options, The Review of Financial Studies, 6 (1993), 327-343.  doi: 10.1093/rfs/6.2.327. [10] P. Kútik, Numerical Solution of Partial Differential Equations and Their Application, Ph.D thesis, Slovak University of Technology in Bratislava, Slovakia, 2013. [11] P. Kútik and K. Mikula, Diamond-cell finite volume scheme for the Heston model, Discrete and Continuous Dynamical Systems, 8 (2015), 913-931.  doi: 10.3934/dcdss.2015.8.913. [12] O. A. Ladyžhenskaya, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, Translations of Mathematical Monographs, Vol. 23 American Mathematical Society, Providence, R.I. 1968. [13] O.A. Oleǐnik and E. V. Radkevič, Second order equations with nonnegative characteristic form, Mathematical Analysis, 1969, Akad. Nauk SSSR Vsesojuzn. Inst. Naučn. i Tehn. Informacii, Moscow, (1971), 7–252.
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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