Abstract
In this paper, we propose a high accurate method based on non-standard Runge–Kutta (NRK), modified weighted essentially non-oscillatory (MWENO) and grid stretching methods to solve the Black–Scholes equation with discontinuous final condition. For the spatial and temporal discretization of the Black–Scholes equation, the MWENO method and the NRK method are applied, respectively. The MWENO method is a high-order method that prevents the appearance of spurious solutions close to non-smooth points. To achieve the high-order accuracy in non-smooth points as well as smooth points, a grid stretching technique is employed. The accuracy analysis and the CFL stability condition of this hybrid method are presented. The high efficiency of this method for the solution of non-linear Black–Scholes equation is demonstrated numerically. Comparisons are made with the available methods in the literature.
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References
Wilmott P, Howison S, Dewynne J (2002) The Mathematics of financial derivatives. A student introduction. Cambridge University Press, Cambridge
Black F, Scholes M (1973) The pricing of options and corporate liabilities. J Polit Econ 81:637–654
Merton RC (1973) Theory of rational option pricing. Bell J Econ 4:141–183
Forsyth P, Vetzal K, Zvan R (1999) A finite element approach to the pricing of discrete lookbacks with stochastic volatility. Appl Math Finance 6:87–106
Kluge T (2002) Pricing derivatives in stochastic volatility models using the finite difference method. Dipl. thesis, TU Chemnitz
Company R, Navarro E, Pintos JP, Ponsoda E (2008) Numerical solution of linear and nonlinear Black–Scholes option pricing equations. Comput. Math. Appl. 56:813–821
Company R, Jodar L, Pintos J (2012) A consistent stable numerical scheme for a nonlinear option pricing model in illiquid markets. Math. Comput. Simul. 82:1972–1985
Ankudinova J, Ehrhardt M (2008) On the numerical solution of nonlinear Black–Scholes equations. Comput. Math. Appl. 56:799–812
Tangman DY, Gopau A, Bhuruth M (2008) Numerical pricing of options using high-order compact finite difference schemes. J. Comput Appl Math 218:270–280
Dremkova E, Ehrhardt M (2011) A high-order compact method for nonlinear Black–Scholes option pricing equations of American options. Int J Comput Math 88:2782–2797
Bohner M, Zheng Y (2009) On analytical solutions of the Black–Scholes equation. Appl Math Lett 22:309–313
Oosterlee CW, Frisch JC, Gaspar FJ (2004) TVD, WENO and blended BDF discretizations for Asian options. Comput Visual Sci 6:131–138
Hajipour M, Malek A (2012) High accurate NRK and MWENO scheme for nonlinear degenerate parabolic PDEs. Appl Math Model 36:4439–4451
Hajipour M, Malek A (2011) An efficient high order modified WENO scheme for nonlinear parabolic equations. Int J Appl Math 24:443–458
Liu Y-Y, Shu C-W, Zhang M (2011) High order finite difference WENO schemes for nonlinear degenerate parabolic equations. SIAM J Sci Comput 33:939–965
Liu X-D, Osher S, Chan T (1994) Weighted essentially non-oscillatory schemes. J Comput Phys 115:200–212
Jiang G, Shu C-W (1996) Efficient implementation of weighted ENO schemes. J Comput Phys 126:202–228
Balsara D, Shu C-W (2000) Monotonicity preserving weighted essentially non-oscillatory schemes with increasingly high order of accuracy. J Comput Phys 160:405–452
Hidalgo A, Dumbser M (2011) ADER schemes for nonlinear systems of stiff advection-diffusion-reaction equations. J Sci Comput 48:173–189
Shu C-W (2009) High order weighted essentially non-oscillatory schemes for convection dominated problems. SIAM Rev 51(1):82–126
Pedro JC, Banda MK, Sibanda P (2013) On one-dimensional arbitrary high-order WENO schemes for systems of hyperbolic conservation laws. Comput Appl Math. doi:10.1007/s40314-013-0066-y
Jandačka M, Ševčovič D (2005) On the risk-adjusted pricing-methodology-based valuation of vanilla options and explanation of the volatility smile. J Appl Math 2005(3):235–258
Leland HE (1985) Option pricing and replication with transactions costs. J Finance 40:1283–1301
Barles G, Soner HM (1998) Option pricing with transaction costs and a nonlinear Black–Scholes equation. Finance Stoch 2:369–397
Dormand JR, Prince PJ (1980) A family of embedded Runge–Kutta formulae. J Comput Appl Math 6:19–26
Wang R, Spiteri RJ (2007) Linear instability of the fifth-order WENO method. SIAM J Numer Anal 45(5):1871–1901
Chen B, Solis F (1998) Discretizations of nonlinear differential equations using explicit finite order methods. J Comput Appl Math 90:171–183
Mickens RE (1994) Nonstandard finite difference models of differential equations. World Scientific, Singapore
Carpenter MH, Gottlieb D, Abarbanel S, Don W-S (1995) The theoretical accuracy of Runge-Kutta time discretizations for the initial boundary value problem: a study of the boundary error. SIAM J Sci Comput 16:1241–1252
Oosterlee CW, Leentvaar CCW, Huang X (2005) Accurate American option pricing by grid stretching and high-order finite differences, Working papers. Delft University of Technology, the Netherlands, DIAM
Hull JC (1989) Options, futures and other derivatives. Prentice-Hall Int. Inc, London
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Communicated by Eduardo Souza de Cursi.
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Hajipour, M., Malek, A. High accurate modified WENO method for the solution of Black–Scholes equation. Comp. Appl. Math. 34, 125–140 (2015). https://doi.org/10.1007/s40314-013-0108-5
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DOI: https://doi.org/10.1007/s40314-013-0108-5