American Institute of Mathematical Sciences

October  2015, 8(5): 913-931. doi: 10.3934/dcdss.2015.8.913

Diamond--cell finite volume scheme for the Heston model

 1 Department of Mathematics, Slovak University of Technology, Radlinského 11, 813 68 Bratislava, Slovak Republic 2 Department of Mathematics, Slovak University of Technology, Radlinskeho 11, 813 68 Bratislava, Slovak Republic

Received  December 2013 Revised  July 2014 Published  July 2015

The objective of this article is to propose a novel numerical scheme for solving the partial differential equation arising in the Heston stochastic volatility model. We discretize the governing advection-diffusion-reaction equation using the finite volume technique. The diffusion tensor is treated by means of the diamond--cell approximation. A theoretical result concerning the existence and uniqueness of the solution to the corresponding system of linear equations is proved. Numerical experiments regarding accuracy and order of convergence are shown.
Citation: Pavol Kútik, Karol Mikula. Diamond--cell finite volume scheme for the Heston model. Discrete and Continuous Dynamical Systems - S, 2015, 8 (5) : 913-931. doi: 10.3934/dcdss.2015.8.913
References:
 [1] L. Andersen, Simple and efficient simulation of the Heston stochastic volatility model, Journal of Computational Finance, 11 (2008), 1-42. [2] F. Black and 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. Cont, Empirical properties of asset returns: Stylized facts and statistical issues, Quantitative Finance, 1 (2001), 223-236. doi: 10.1080/713665670. [4] Y. Coudiere, J. P. Vila and P. Villedieu, Convergence rate of a finite volume scheme for a two-dimensional convection-diffusion problem, M2AN Math. Model. Numer. Anal., 33 (1999), 493-516. doi: 10.1051/m2an:1999149. [5] J. Cox, J. Ingersoll and S. Ross, A theory of the term structure of interest rates, Econometrica, 53 (1985), 385-407. doi: 10.2307/1911242. [6] O. Drblíková and K. Mikula, Convergence analysis of finite volume scheme for nonlinear tensor anisotropic diffusion in image processing, SIAM Journal on Numerical Analysis, 46 (2007), 37-60. doi: 10.1137/070685038. [7] R. Eymard, T. Gallouët and R. Herbin, Finite Volume Methods, in: Handbook Of Numerical Analysis, Elsevier Science B.V., Amsterdam, 2000. [8] G. Fichera, Sulle equazioni differenziali lineari ellittico-paraboliche del secondo ordine, Atti Accad. Naz. Lincei, Mem., Cl. Sci. Fis. Mat. Nat., Sez. I, VIII., 5 (1956), 1-30. [9] P. A. Forsyth, K. R. Vetzal and R. Zvan, A finite element approach to the pricing of discrete lookbacks with stochastic volatility, Applied Mathematical Finance, 6 (1999), 87-106. doi: 10.1080/135048699334564. [10] P. Frolkovič and K. Mikula, High-resolution flux-based level set method, SIAM Journal on Scientific Computing, 29 (2007), 579-597. doi: 10.1137/050646561. [11] J. Gatheral, The Volatility Surface: A Practitioner's Guide, John Wiley & Sons, Inc., New Jersey, 2006. [12] 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. [13] K. J. In't Hout and S. Foulon, ADI finite difference schemes for option pricing in the Heston model with correlation, Internation Journal of Numerical Analysis and Modeling, 7 (2010), 303-320. [14] P. Kútik, Numerical Solution of Partial Differential Equations in Financial Mathematics, PhD. Thesis, Slovak University of Technology, Bratislava, 2014. [15] R. J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge University Press, Cambridge, 2002. doi: 10.1017/CBO9780511791253. [16] R. Merton, Theory of rational option pricing, The Bell Journal of Economics and Management Science, 4 (1973), 141-183. doi: 10.2307/3003143. [17] R. Zvan, P. A. Forsyth and K. R. Vetzal, A finite volume approach for contingent claims valuation, IMA J. Numer. Anal., 21 (2001), 703-731. doi: 10.1093/imanum/21.3.703.

show all references

References:
 [1] L. Andersen, Simple and efficient simulation of the Heston stochastic volatility model, Journal of Computational Finance, 11 (2008), 1-42. [2] F. Black and 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. Cont, Empirical properties of asset returns: Stylized facts and statistical issues, Quantitative Finance, 1 (2001), 223-236. doi: 10.1080/713665670. [4] Y. Coudiere, J. P. Vila and P. Villedieu, Convergence rate of a finite volume scheme for a two-dimensional convection-diffusion problem, M2AN Math. Model. Numer. Anal., 33 (1999), 493-516. doi: 10.1051/m2an:1999149. [5] J. Cox, J. Ingersoll and S. Ross, A theory of the term structure of interest rates, Econometrica, 53 (1985), 385-407. doi: 10.2307/1911242. [6] O. Drblíková and K. Mikula, Convergence analysis of finite volume scheme for nonlinear tensor anisotropic diffusion in image processing, SIAM Journal on Numerical Analysis, 46 (2007), 37-60. doi: 10.1137/070685038. [7] R. Eymard, T. Gallouët and R. Herbin, Finite Volume Methods, in: Handbook Of Numerical Analysis, Elsevier Science B.V., Amsterdam, 2000. [8] G. Fichera, Sulle equazioni differenziali lineari ellittico-paraboliche del secondo ordine, Atti Accad. Naz. Lincei, Mem., Cl. Sci. Fis. Mat. Nat., Sez. I, VIII., 5 (1956), 1-30. [9] P. A. Forsyth, K. R. Vetzal and R. Zvan, A finite element approach to the pricing of discrete lookbacks with stochastic volatility, Applied Mathematical Finance, 6 (1999), 87-106. doi: 10.1080/135048699334564. [10] P. Frolkovič and K. Mikula, High-resolution flux-based level set method, SIAM Journal on Scientific Computing, 29 (2007), 579-597. doi: 10.1137/050646561. [11] J. Gatheral, The Volatility Surface: A Practitioner's Guide, John Wiley & Sons, Inc., New Jersey, 2006. [12] 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. [13] K. J. In't Hout and S. Foulon, ADI finite difference schemes for option pricing in the Heston model with correlation, Internation Journal of Numerical Analysis and Modeling, 7 (2010), 303-320. [14] P. Kútik, Numerical Solution of Partial Differential Equations in Financial Mathematics, PhD. Thesis, Slovak University of Technology, Bratislava, 2014. [15] R. J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge University Press, Cambridge, 2002. doi: 10.1017/CBO9780511791253. [16] R. Merton, Theory of rational option pricing, The Bell Journal of Economics and Management Science, 4 (1973), 141-183. doi: 10.2307/3003143. [17] R. Zvan, P. A. Forsyth and K. R. Vetzal, A finite volume approach for contingent claims valuation, IMA J. Numer. Anal., 21 (2001), 703-731. doi: 10.1093/imanum/21.3.703.
 [1] Hassan Belhadj, Samir Khallouq, Mohamed Rhoudaf. Parallelization of a finite volumes discretization for anisotropic diffusion problems using an improved Schur complement technique. Discrete and Continuous Dynamical Systems - S, 2021, 14 (7) : 2075-2099. doi: 10.3934/dcdss.2020260 [2] Anna Kaźmierczak, Jan Sokolowski, Antoni Zochowski. Drag minimization for the obstacle in compressible flow using shape derivatives and finite volumes. Mathematical Control and Related Fields, 2018, 8 (1) : 89-115. doi: 10.3934/mcrf.2018004 [3] Z. Jackiewicz, B. Zubik-Kowal, B. Basse. Finite-difference and pseudo-spectral methods for the numerical simulations of in vitro human tumor cell population kinetics. Mathematical Biosciences & Engineering, 2009, 6 (3) : 561-572. doi: 10.3934/mbe.2009.6.561 [4] G. Bellettini, Giorgio Fusco, Nicola Guglielmi. A concept of solution and numerical experiments for forward-backward diffusion equations. Discrete and Continuous Dynamical Systems, 2006, 16 (4) : 783-842. doi: 10.3934/dcds.2006.16.783 [5] Christos V. Nikolopoulos, Georgios E. Zouraris. Numerical solution of a non-local elliptic problem modeling a thermistor with a finite element and a finite volume method. Conference Publications, 2007, 2007 (Special) : 768-778. doi: 10.3934/proc.2007.2007.768 [6] Kaouther Bouchama, Yacine Arioua, Abdelkrim Merzougui. The Numerical Solution of the space-time fractional diffusion equation involving the Caputo-Katugampola fractional derivative. Numerical Algebra, Control and Optimization, 2021  doi: 10.3934/naco.2021026 [7] Yones Esmaeelzade Aghdam, Hamid Safdari, Yaqub Azari, Hossein Jafari, Dumitru Baleanu. Numerical investigation of space fractional order diffusion equation by the Chebyshev collocation method of the fourth kind and compact finite difference scheme. Discrete and Continuous Dynamical Systems - S, 2021, 14 (7) : 2025-2039. doi: 10.3934/dcdss.2020402 [8] Yu-Hsien Chang, Guo-Chin Jau. The behavior of the solution for a mathematical model for analysis of the cell cycle. Communications on Pure and Applied Analysis, 2006, 5 (4) : 779-792. doi: 10.3934/cpaa.2006.5.779 [9] Karol Mikula, Róbert Špir, Nadine Peyriéras. Numerical algorithm for tracking cell dynamics in 4D biomedical images. Discrete and Continuous Dynamical Systems - S, 2015, 8 (5) : 953-967. doi: 10.3934/dcdss.2015.8.953 [10] Song Wang. Numerical solution of an obstacle problem with interval coefficients. Numerical Algebra, Control and Optimization, 2020, 10 (1) : 23-38. doi: 10.3934/naco.2019030 [11] Roberto Garrappa, Eleonora Messina, Antonia Vecchio. Effect of perturbation in the numerical solution of fractional differential equations. Discrete and Continuous Dynamical Systems - B, 2018, 23 (7) : 2679-2694. doi: 10.3934/dcdsb.2017188 [12] Kamil Aida-Zade, Jamila Asadova. Numerical solution to optimal control problems of oscillatory processes. Journal of Industrial and Management Optimization, 2021  doi: 10.3934/jimo.2021166 [13] Antoine Gloria Cermics. A direct approach to numerical homogenization in finite elasticity. Networks and Heterogeneous Media, 2006, 1 (1) : 109-141. doi: 10.3934/nhm.2006.1.109 [14] Stefan Klus, Christof Schütte. Towards tensor-based methods for the numerical approximation of the Perron--Frobenius and Koopman operator. Journal of Computational Dynamics, 2016, 3 (2) : 139-161. doi: 10.3934/jcd.2016007 [15] Jinghong Liu, Yinsuo Jia. Gradient superconvergence post-processing of the tensor-product quadratic pentahedral finite element. Discrete and Continuous Dynamical Systems - B, 2015, 20 (2) : 495-504. doi: 10.3934/dcdsb.2015.20.495 [16] Lianzhang Bao, Zhengfang Zhou. Traveling wave solutions for a one dimensional model of cell-to-cell adhesion and diffusion with monostable reaction term. Discrete and Continuous Dynamical Systems - S, 2017, 10 (3) : 395-412. doi: 10.3934/dcdss.2017019 [17] Youngmok Jeon, Eun-Jae Park. Cell boundary element methods for convection-diffusion equations. Communications on Pure and Applied Analysis, 2006, 5 (2) : 309-319. doi: 10.3934/cpaa.2006.5.309 [18] Huan Han. A variational model with fractional-order regularization term arising in registration of diffusion tensor image. Inverse Problems and Imaging, 2018, 12 (6) : 1263-1291. doi: 10.3934/ipi.2018053 [19] Ching-Shan Chou, Yong-Tao Zhang, Rui Zhao, Qing Nie. Numerical methods for stiff reaction-diffusion systems. Discrete and Continuous Dynamical Systems - B, 2007, 7 (3) : 515-525. doi: 10.3934/dcdsb.2007.7.515 [20] Mudassar Imran, Mohamed Ben-Romdhane, Ali R. Ansari, Helmi Temimi. Numerical study of an influenza epidemic dynamical model with diffusion. Discrete and Continuous Dynamical Systems - S, 2020, 13 (10) : 2761-2787. doi: 10.3934/dcdss.2020168

2020 Impact Factor: 2.425