Exponential time integration and Chebychev discretisation schemes for fast pricing of options. (English) Zbl 1151.91546

Tangman, D. Y.; Gopaul, A.; Bhuruth, M.

Appl. Numer. Math. 58, No. 9, 1309-1319 (2008).

Summary: We consider exponential time differencing (ETD) schemes for the numerical pricing of options. Special treatments for the implementation of the boundary conditions that arise in finance are described. We show that only one explicit time step computation gives unconditional second order accuracy for European, Barrier and Butterfly spread options under both Black-Scholes geometric Brownian motion model and Merton’s jump diffusion model with constant coefficients. In comparison, the commonly used Crank-Nicolson scheme is shown to be only conditionally stable due to lack of \(L_{0}\)-stability. Finally, we describe how the use of spectral spatial discretisation based on a Chebychev grid point concentration strategy gives fourth order accurate option prices for both the Black-Scholes and Merton’s jump-diffusion model.

MSC:

91B28	Finance etc. (MSC2000)
60H15	Stochastic partial differential equations (aspects of stochastic analysis)
35A35	Theoretical approximation in context of PDEs
65N99	Numerical methods for partial differential equations, boundary value problems

Keywords:

option pricing; exponential time differencing; spectral methods; jump-diffusion processes; integro-differential equations

Software:

Matlab

References:

[1]	Almendral, A.; Oosterlee, C. W., Numerical valuation of options with jumps in the underlying, Appl. Numer. Math., 53, 1-18 (2005) · Zbl 1117.91028
[2]	Andersen, L.; Andreasen, J., Jump-diffusion processes: Volatility smile fitting and numerical methods for option pricing, Rev. Derivatives Res., 4, 231-262 (2000) · Zbl 1274.91398
[3]	Bergamaschi, L.; Caliari, M.; Vianello, M., Efficient computation of the exponential operator for discrete 2D advection-diffusion equations, Numer. Linear Algebra Appl., 10, 3, 271-289 (2003) · Zbl 1071.65048
[4]	Black, F.; Scholes, M., The pricing of options and other corporate liabilities, J. Pol. Econ., 81, 637-654 (1973) · Zbl 1092.91524
[6]	Briani, M.; La Chioma, C.; Natalini, R., Convergence of numerical schemes for viscosity solutions to integro-differential degenerate parabolic problems arising in financial theory, Numer. Math., 98, 4, 607-646 (2004) · Zbl 1065.65145
[7]	Cont, R.; Tankov, P., Financial Modelling With Jump Processes (2004), Chapman and Hall/CRC Press: Chapman and Hall/CRC Press Boca Raton, FL · Zbl 1052.91043
[8]	Cont, R.; Voltchkova, E., Finite difference methods for option pricing in jump-diffusion and exponential Lévy models, SIAM J. Numer. Anal., 43, 4, 1596-1626 (2005) · Zbl 1101.47059
[9]	d’Halluin, Y.; Forsyth, P. A.; Vetzal, K. R., A penalty method for American options with jump diffusion processes, Numer. Math., 97, 2, 321-352 (2004) · Zbl 1126.91036
[10]	d’Halluin, Y.; Forsyth, P. A.; Vetzal, K. R., Robust numerical methods for contingent claims under jump diffusion processes, IMA J. Numer. Anal., 25, 87-112 (2005) · Zbl 1134.91405
[11]	Eiermann, M.; Ernst, O. G., A restarted Krylov subspace method for the evaluation of matrix functions, SIAM J. Numer. Anal., 44, 6, 2481-2504 (2006) · Zbl 1129.65019
[12]	Giles, M. B.; Carter, R., Convergence analysis of Crank-Nicolson and Rannacher time-marching, J. Comp. Fin., 9, 4 (2006)
[13]	Hochbruck, M.; Lubich, C., On Krylov subspace approximations to the matrix exponential operator, SIAM J. Numer. Anal., 34, 1911-1925 (1997) · Zbl 0888.65032
[14]	Matache, A. M.; von Petersdorff, T.; Schwab, C., Fast deterministic pricing of options on Lévy driven assets, M2AN Math. Model. Numer. Anal., 38, 37-71 (2004) · Zbl 1072.60052
[15]	Merton, R. C., Option pricing when the underlying stocks are discontinuous, J. Financ. Econ., 5, 125-144 (1976) · Zbl 1131.91344
[16]	Moler, C.; Loan, C. V., Nineteen dubious ways to compute the exponential of a matrix, twenty-five years later, SIAM Rev., 45, 1, 3-49 (2003) · Zbl 1030.65029
[18]	Tangman, D. Y.; Gopaul, A.; Bhuruth, M., Numerical pricing of options using high-order compact finite difference schemes, J. Comput. Appl. Math. (2007) · Zbl 1146.91338
[19]	Tavella, D.; Randall, C., Pricing Financial Instruments: The Finite Difference Method (2000), John Wiley and Sons: John Wiley and Sons New York
[20]	Trefethen, L. N., Spectral Methods in Matlab (2000), SIAM: SIAM Philadelphia · Zbl 0953.68643
[21]	Zhu, Y. L.; Wu, X.; Chern, I. L., Derivatives Securities and Difference Methods (2004), Springer: Springer New York