# 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 & 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. [2] F. Black and and M. Scholes, The pricing of options and corporate liabilities,, The Journal of Political Economy, 81 (1973), 637. doi: 10.1086/260062. [3] R. Cont, Empirical properties of asset returns: Stylized facts and statistical issues,, Quantitative Finance, 1 (2001), 223. 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. 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. 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. doi: 10.1137/070685038. [7] R. Eymard, T. Gallouët and R. Herbin, Finite Volume Methods,, in: Handbook Of Numerical Analysis, (2000). [8] G. Fichera, Sulle equazioni differenziali lineari ellittico-paraboliche del secondo ordine,, Atti Accad. Naz. Lincei, 5 (1956), 1. [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. 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. doi: 10.1137/050646561. [11] J. Gatheral, The Volatility Surface: A Practitioner's Guide,, John Wiley & Sons, (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. 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. [14] P. Kútik, Numerical Solution of Partial Differential Equations in Financial Mathematics,, PhD. Thesis, (2014). [15] R. J. LeVeque, Finite Volume Methods for Hyperbolic Problems,, Cambridge University Press, (2002). doi: 10.1017/CBO9780511791253. [16] R. Merton, Theory of rational option pricing,, The Bell Journal of Economics and Management Science, 4 (1973), 141. 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. doi: 10.1093/imanum/21.3.703.

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##### References:
 [1] L. Andersen, Simple and efficient simulation of the Heston stochastic volatility model,, Journal of Computational Finance, 11 (2008), 1. [2] F. Black and and M. Scholes, The pricing of options and corporate liabilities,, The Journal of Political Economy, 81 (1973), 637. doi: 10.1086/260062. [3] R. Cont, Empirical properties of asset returns: Stylized facts and statistical issues,, Quantitative Finance, 1 (2001), 223. 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. 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. 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. doi: 10.1137/070685038. [7] R. Eymard, T. Gallouët and R. Herbin, Finite Volume Methods,, in: Handbook Of Numerical Analysis, (2000). [8] G. Fichera, Sulle equazioni differenziali lineari ellittico-paraboliche del secondo ordine,, Atti Accad. Naz. Lincei, 5 (1956), 1. [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. 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. doi: 10.1137/050646561. [11] J. Gatheral, The Volatility Surface: A Practitioner's Guide,, John Wiley & Sons, (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. 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. [14] P. Kútik, Numerical Solution of Partial Differential Equations in Financial Mathematics,, PhD. Thesis, (2014). [15] R. J. LeVeque, Finite Volume Methods for Hyperbolic Problems,, Cambridge University Press, (2002). doi: 10.1017/CBO9780511791253. [16] R. Merton, Theory of rational option pricing,, The Bell Journal of Economics and Management Science, 4 (1973), 141. 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. doi: 10.1093/imanum/21.3.703.
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