Removing the Correlation Term in Option Pricing HestonModel: Numerical Analysis and Computing

[EN] This paper deals with the numerical solution of option pricing stochastic volatility model described by a time-dependent, twodimensional convection-diffusion reaction equation. Firstly, the mixed spatial derivative of the partial differential equation (PDE) is removed bymeans of the classical technique for reduction of second-order linear partial differential equations to canonical form. An explicit difference scheme with positive coefficients and only five-point computational stencil is constructed. The boundary conditions are adapted to the boundaries of the rhomboid transformed numerical domain. Consistency of the scheme with the PDE is shown and stepsize discretization conditions in order to guarantee stability are established. Illustrative numerical examples are included.This work has been partially supported by the European Union in the FP7-PEOPLE-2012-ITN Program under Grant Agreement no. 304617 (FP7 Marie Curie Action, Project Multi-ITN STRIKE-Novel Methods in Computational Finance) and by the Spanish MEYC Grant DPI2010-20891-C02-01.Company Rossi, R.; Jódar Sánchez, LA.; El-Fakharany, M.; Casabán Bartual, MC. (2013). Removing the Correlation Term in Option Pricing HestonModel: Numerical Analysis and Computing. Abstract and Applied Analysis. 2013:1-11. https://doi.org/10.1155/2013/246724S1112013HULL, J., & WHITE, A. (1987). The Pricing of Options on Assets with Stochastic Volatilities. The Journal of Finance, 42(2), 281-300. doi:10.1111/j.1540-6261.1987.tb02568.xHeston, S. L. (1993). A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options. Review of Financial Studies, 6(2), 327-343. doi:10.1093/rfs/6.2.327Pascucci, A. (2011). PDE and Martingale Methods in Option Pricing. Bocconi & Springer Series. doi:10.1007/978-88-470-1781-8Benhamou, E., Gobet, E., & Miri, M. (2010). Time Dependent Heston Model. SIAM Journal on Financial Mathematics, 1(1), 289-325. doi:10.1137/090753814Hilber, N., Matache, A.-M., & Schwab, C. (2005). Sparse wavelet methods for option pricing under stochastic volatility. The Journal of Computational Finance, 8(4), 1-42. doi:10.21314/jcf.2005.131Zhu, W., & Kopriva, D. A. (2009). A Spectral Element Approximation to Price European Options with One Asset and Stochastic Volatility. Journal of Scientific Computing, 42(3), 426-446. doi:10.1007/s10915-009-9333-xClarke, N., & Parrott, K. (1999). Multigrid for American option pricing with stochastic volatility. Applied Mathematical Finance, 6(3), 177-195. doi:10.1080/135048699334528Düring, B., & Fournié, M. (2012). High-order compact finite difference scheme for option pricing in stochastic volatility models. Journal of Computational and Applied Mathematics, 236(17), 4462-4473. doi:10.1016/j.cam.2012.04.017Zvan, R., Forsyth, P., & Vetzal, K. (2003). Negative coefficients in two-factor option pricing models. The Journal of Computational Finance, 7(1), 37-73. doi:10.21314/jcf.2003.096Company, R., Jódar, L., & Pintos, J.-R. (2009). Consistent stable difference schemes for nonlinear Black-Scholes equations modelling option pricing with transaction costs. ESAIM: Mathematical Modelling and Numerical Analysis, 43(6), 1045-1061. doi:10.1051/m2an/2009014Company, R., Jódar, L., & Pintos, J.-R. (2010). Numerical analysis and computing for option pricing models in illiquid markets. Mathematical and Computer Modelling, 52(7-8), 1066-1073. doi:10.1016/j.mcm.2010.02.037Casabán, M.-C., Company, R., Jódar, L., & Pintos, J.-R. (2011). Numerical analysis and computing of a non-arbitrage liquidity model with observable parameters for derivatives. Computers & Mathematics with Applications, 61(8), 1951-1956. doi:10.1016/j.camwa.2010.08.009Kangro, R., & Nicolaides, R. (2000). Far Field Boundary Conditions for Black--Scholes Equations. SIAM Journal on Numerical Analysis, 38(4), 1357-1368. doi:10.1137/s0036142999355921EHRHARDT, M., & MICKENS, R. E. (2008). A FAST, STABLE AND ACCURATE NUMERICAL METHOD FOR THE BLACK–SCHOLES EQUATION OF AMERICAN OPTIONS. International Journal of Theoretical and Applied Finance, 11(05), 471-501. doi:10.1142/s021902490800489