Cubic Hermite Finite Element Method for Nonlinear Black-Scholes Equation Governing European Options
DOI:
https://doi.org/10.22481/intermaths.v2i2.9481Keywords:
Nonlinear Black-Scholes, Finite Element Method, Crank-Nicolson, Hermite PolynomialsAbstract
A numerical algorithm for solving a generalized Black-Scholes partial differential equation, which arises in European option pricing considering transaction costs is developed. The Crank-Nicolson method is used to discretize in the temporal direction and the Hermite cubic interpolation method to discretize in the spatial direction. The efficiency and accuracy of the proposed method are tested numerically, and the results confirm the theoretical behaviour of the solutions, which is also found to be in good agreement with the exact solution.
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P. P. Boyle and T. Vorst, “Option replication in discrete time with transaction costs”, The Journal of Finance, 47(1), 271–293, 1992. https://doi.org/10.1111/j.1540-6261.1992.tb03986.x
K. Böhmer, “On finite element methods for fully nonlinear elliptic equations of second order”, SIAM Journal on Numerical Analysis, 46(3), 1212–1249, 2008. https://doi.org/10.1137/040621740
D. C. Lesmana and S. Wang, “An upwind finite difference method for a nonlinear Black–Scholes equation governing European option valuation under transaction costs”, Applied Mathematics and Computation, 219(16), 8811–8828, 2013. https://doi.org/10.1016/j.amc.2012.12.077
B. Düring, M. Fournié and A. Jüngel, “High order compact finite difference schemes for a nonlinear Black-Scholes equation”, International Journal of Theoretical and Applied Finance, 6(07), 767–789, 2003. https://doi.org/10.1142/S0219024903002183
J. Topper, Financial engineering with finite elements, vol. 319, John Wiley & Sons, 2005.
S. O. Edeki, O. O. Ugbebor and E. A. Owoloko, “Analytical solutions of the Black–Scholes pricing model for european option valuation via a projected differential transformation method”, Entropy, 17(11), 7510–7521, 2015. https://doi.org/10.3390/e17117510
V. L. Khatskevich, “Some properties of Legendre polynomials and an approximate solution of the Black–Scholes equation governing option pricing”, Differential Equations, 51(9), 1157–1164, 2015. https://doi.org/10.1134/S0012266115090050
M. Bohner, F. H. M. Sanchez and S. Rodriguez, “European Call option pricing by the Adomian decomposition method”, Advances in Dynamical Systems and Applications, 9(1), 75–85, 2014.
F. Black and M. Scholes, “The pricing of options and corporate liabilities”, The Journal of Political Economy, 81(3), 637–654, 1973.
Y.K. Kwok, Mathematical models of financial derivatives, Springer finance. Springer, 1998.
H. E. Leland, “Option pricing and replication with transaction costs”, The journal of finance, 40(5), 1283–1301, 1985. https://doi.org/10.1111/j.1540-6261.1985.tb02383.x
G. Barles and H. M. Soner, “Option pricing with transaction costs and a nonlinear Black-Scholes equation”, Finance and Stochastics, 2(4), 369–397, 1998.
M. Kratka, “No mystery behind the smile”, Risk, 11, 67–71, 1998.
J. Ankudinova and M. Ehrhardt, “On the numerical solution of nonlinear Black–Scholes equations”, Computers & Mathematics with Applications, 56(3), 799–812, 2008. http://doi.org/10.1016/j.camwa.2008.02.005
R. M. Almeida, J. C. Duque, J. Ferreira and R. J. Robalo, “The Crank–Nicolson–Galerkin finite element method for a nonlocal parabolic equation with moving boundaries”, Numerical Methods for Partial Differential Equations, 31(5), 1515–1533, 2015. https://doi.org/10.1002/num.21957
R. M. Almeida, S. N. Antontsev and J. C. Duque, “On the finite element method for a nonlocal degenerate parabolic problem”, Computers & Mathematics with Applications, 73(8), 1724–1740, 2017. https://doi.org/10.1016/j.camwa.2017.02.013
J. C. Duque, R. M. Almeida, S. N. Antontsev and J. Ferreira, “The Euler–Galerkin finite element method for a nonlocal coupled system of reaction–diffusion type”, Journal of Computational and Applied Mathematics, 296, 116–126. https://doi.org/10.1016/j.cam.2015.09.019
H. M. Soner, S. E. Shreve and J. Cvitanic, “There is no nontrivial hedging portfolio for option pricing with transaction costs”, The Annals of Applied Probability, 5(2), 327-355, 1995. https://doi.org/10.1214/aoap/1177004767
W. Yue-Kuen, Mathematical Models of Financial Derivatives, Springer, 1998.
S. Hodges, “Optimal replication of contingent claims under transaction costs”, Review Futures Market, 8, 222–239, 1998.
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