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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 |
The aim of the paper is to study problem of financial derivatives pricing based on the idea of the Heston model introduced in [
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. |
show all references
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. |
[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 |
[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 |
[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 |
[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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