Bhatoo, Omishwary; Peer, Arshad Ahmud Iqbal; Tadmor, Eitan; Tangman, Désiré Yannick; Saib, Aslam Aly El Faidal Conservative third-order central-upwind schemes for option pricing problems. (English) Zbl 1437.91452 Vietnam J. Math. 47, No. 4, 813-833 (2019). Summary: In this paper, we propose the application of third-order semi-discrete central-upwind conservative schemes to option pricing partial differential equations (PDEs). Our method is a high-order extension of the recent efficient second-order “Black-Box” schemes that successfully priced several option pricing problems. We consider the Kurganov-Levy scheme and its extensions, namely the Kurganov-Noelle-Petrova and the Kolb schemes. These “Black-Box” solvers ensure non-oscillatory property and achieve desired accuracy using a third-order central weighted essentially non-oscillatory (CWENO) reconstruction. We compare the schemes using a European test case and observe that the Kolb scheme performs better. We apply the Kolb scheme to one-dimensional butterfly, barrier, American and non-linear options under the Black-Scholes model. Further, we extend the Kurganov-Levy scheme to solve two-dimensional convection-dominated Asian PDE. We also price American options under the constant elasticity of variance (CEV) model, which treats volatility as a stochastic instead of a constant as in Black-Scholes model. Numerical experiments achieve third-order, non-oscillatory and high-resolution solutions. MSC: 91G60 Numerical methods (including Monte Carlo methods) 65M08 Finite volume methods for initial value and initial-boundary value problems involving PDEs 91G20 Derivative securities (option pricing, hedging, etc.) 35Q91 PDEs in connection with game theory, economics, social and behavioral sciences 65D32 Numerical quadrature and cubature formulas Keywords:conservative central-upwind schemes; CWENO reconstruction; Black-Scholes PDEs; nonlinear PDE; two-dimensional PDE; CEV model Software:Ode15s; ode23; ode45; MATLAB ODE suite; CentPack; ode113; Matlab; BENCHOP; ode23s PDFBibTeX XMLCite \textit{O. Bhatoo} et al., Vietnam J. Math. 47, No. 4, 813--833 (2019; Zbl 1437.91452) Full Text: DOI References: [1] Andersen, L.; Andreasen, J., Jump-diffusion processes: volatility smile fitting and numerical methods for option pricing, Rev. Deriv. Res., 4, 231-262 (2000) · Zbl 1274.91398 · doi:10.1023/A:1011354913068 [2] Balbas, J., Tadmor, E.: Centpack: a package of high-resolution central schemes for nonlinear conservation laws and related problems. http://www.cscamm.umd.edu/centpack/ [3] Barraquand, J.; Pudet, T., Pricing of American path-dependent contingent claims, Math. Financ., 6, 17-51 (1996) · Zbl 0919.90005 · doi:10.1111/j.1467-9965.1996.tb00111.x [4] Bhatoo, O.; Peer, Aai; Tadmor, E.; Tangman, Dy; Saib, Aaef, Efficient conservative second-order central-upwind schemes for option-pricing problems, J. Comput. Financ., 22, 71-101 (2019) · doi:10.21314/JCF.2019.363 [5] Black, F.; Scholes, M., The pricing of options and corporate liabilities, J. Politi. Econ., 81, 637-654 (1973) · Zbl 1092.91524 · doi:10.1086/260062 [6] Cox, J.C.: Notes on option pricing I: constant elasticity of variance diffusions. Unpublished note, Stanford University Graduate School of Business (1975) [7] Hajipour, M.; Malek, A., Efficient high-order numerical methods for pricing of options, Comput. Econ., 45, 31-47 (2015) · doi:10.1007/s10614-013-9405-8 [8] Hajipour, M.; Malek, A., High accurate modified WENO method for the solution of Black-Scholes equation, Comput. Appl. Math., 34, 125-140 (2015) · Zbl 1314.91238 · doi:10.1007/s40314-013-0108-5 [9] Jiang, Gs; Levy, D.; Lin, Ct; Osher, S.; Tadmor, E., High-resolution nonoscillatory central schemes with nonstaggered grids for hyperbolic conservation laws, SIAM J. Numer. Anal., 35, 2147-2168 (1998) · Zbl 0920.65053 · doi:10.1137/S0036142997317560 [10] Jiang, Gs; Shu, Cw, Efficient implementation of weighted ENO schemes, J. Comput. Phys., 126, 202-228 (1996) · Zbl 0877.65065 · doi:10.1006/jcph.1996.0130 [11] Kolb, O., On the full and global accuracy of a compact third order WENO scheme, SIAM J. Numer. Anal., 52, 2335-2355 (2014) · Zbl 1408.65062 · doi:10.1137/130947568 [12] Kurganov, A.; Levy, D., A third-order semidiscrete central scheme for conservation laws and convection-diffusion equations, SIAM J. Sci. Comput., 22, 1461-1488 (2000) · Zbl 0979.65077 · doi:10.1137/S1064827599360236 [13] Kurganov, A.; Noelle, S.; Petrova, G., Semi-discrete central-upwind schemes for hyperbolic conservation laws and Hamilton-Jacobi equations, SIAM J. Sci. Comput., 23, 707-740 (2001) · Zbl 0998.65091 · doi:10.1137/S1064827500373413 [14] Kurganov, A.; Tadmor, E., New high-resolution central schemes for nonlinear conservation laws and convection-diffusion equations, J. Comput. Phys., 160, 241-282 (2000) · Zbl 0987.65085 · doi:10.1006/jcph.2000.6459 [15] Leisen, D.; Reimer, M., Binomial models for option valuation - examining and improving convergence, Appl. Math. Financ., 3, 319-346 (1996) · Zbl 1097.91513 · doi:10.1080/13504869600000015 [16] Levy, D.; Puppo, G.; Russo, G., Central WENO schemes for hyperbolic systems of conservation laws, Math. Model. Numer. Anal., 33, 547-571 (1999) · Zbl 0938.65110 · doi:10.1051/m2an:1999152 [17] Levy, D.; Puppo, G.; Russo, G., Compact central WENO schemes for multidimensional conservation laws, SIAM J. Sci. Comput., 22, 656-672 (2000) · Zbl 0967.65089 · doi:10.1137/S1064827599359461 [18] Liu, Xd; Tadmor, E., Third order nonoscillatory central scheme for hyperbolic conservation laws, Numer. Math., 79, 397-425 (1998) · Zbl 0906.65093 · doi:10.1007/s002110050345 [19] Lu, R.; Hsu, Yh, Valuation of standard options under the constant elasticity of variance model, Int. J. Bus. Econ., 4, 157-165 (2005) [20] Medovikov, Aa, High order explicit methods for parabolic equations, BIT Numer Math., 38, 372-390 (1998) · Zbl 0909.65060 · doi:10.1007/BF02512373 [21] Merton, Rc, Theory of rational option pricing, Bell. J. Econ. Manag. Sci., 4, 141-183 (1973) · Zbl 1257.91043 · doi:10.2307/3003143 [22] Oosterlee, Cw; Frisch, Jc; Gaspar, Fj, TVD, WENO and blended BDF discretizations for Asian options, Comput. Visual. Sci., 6, 131-138 (2004) · Zbl 1079.91039 · doi:10.1007/s00791-003-0117-9 [23] Ramírez-Espinoza, Gi; Ehrhardt, M., Conservative and finite volume methods for the convection-dominated pricing problem, Adv. Appl. Math. Mech., 5, 759-790 (2013) · Zbl 1305.65195 · doi:10.4208/aamm.12-m1216 [24] Shampine, Lf; Reichelt, Mw, The MATLAB ODE suite, SIAM J. Sci. Comput., 18, 1-22 (1997) · Zbl 0868.65040 · doi:10.1137/S1064827594276424 [25] Shu, Cw; Osher, S., Efficient implementation of essentially non-oscillatory shock-capturing schemes, J. Comput. Phys., 77, 439-471 (1988) · Zbl 0653.65072 · doi:10.1016/0021-9991(88)90177-5 [26] Von Sydow, L.; Höök, Lj; Larsson, E.; Lindström, E.; Milovanović, S.; Persson, J.; Shcherbakov, V.; Shpolyanskiy, Y.; Sirén, S.; Toivanen, J.; Waldén, J.; Wiktorsson, M.; Levesley, J.; Li, J.; Oosterlee, Cw; Ruijter, Mj; Toropov, A.; Zhao, Y., BENCHOP - the BENCH marking project in option pricing, Int. J. Comput. Math., 92, 2361-2379 (2015) · Zbl 1335.91113 · doi:10.1080/00207160.2015.1072172 [27] Tangman, Dy; Gopaul, A.; Bhuruth, M., A fast high-order finite difference algorithm for pricing American options, J. Comput. Appl. Math., 222, 17-29 (2008) · Zbl 1147.91032 · doi:10.1016/j.cam.2007.10.044 [28] Windcliff, H.; Wang, J.; Forsyth, Pa; Vetzal, Kr, Hedging with a correlated asset: solution of a nonlinear pricing PDE, J. Comput. Appl. Math., 200, 86-115 (2007) · Zbl 1152.91033 · doi:10.1016/j.cam.2005.12.008 [29] Wong, Hy; Zhao, J., An artificial boundary method for American option pricing under the CEV model, SIAM J. Numer. Anal., 46, 2183-2209 (2008) · Zbl 1178.35363 · doi:10.1137/060671541 [30] Zvan, R.; Forsyth, Pa; Vetzal, Kr, Robust numerical methods for PDE models of Asian options, J. Comput. Financ., 1, 39-78 (1997) · doi:10.21314/JCF.1997.006 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.