## A cubic B-spline collocation method for a numerical solution of the generalized Black-Scholes equation.(English)Zbl 1255.91431

Summary: The uniform cubic B-spline collocation method is implemented to find the numerical solution of the generalized Black-Scholes partial differential equation. We use the horizontal method of lines to discretize the temporal variable and the spatial variable by means of a $$\theta$$-method, $$\theta \in [1/2,1]$$ ($$\theta =1$$ corresponds to the back-ward Euler method and $$\theta =1/2$$ corresponds to the Crank-Nicolson method), and a cubic B-spline collocation method on uniform meshes, respectively. The method corresponding to $$\theta =1$$ is shown to be unconditionally stable and first order accurate with respect to the time variable and second order accurate with respect to the space variable while the method corresponding to $$\theta =1/2$$ is shown to be unconditionally stable and second order accurate with respect to both the variables. Finally, the numerical examples demonstrate the stability and accuracy of the method.

### MSC:

 91G60 Numerical methods (including Monte Carlo methods) 65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs 65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs 91G20 Derivative securities (option pricing, hedging, etc.)
Full Text:

### References:

 [1] Black, F.; Scholes, M., The pricing of options and corporate liabilities, J. Polit. Econ., 81, 637-654 (1973) · Zbl 1092.91524 [2] Merton, R. C., Theory of rational option pricing, Bell J. Econ., 4, 141-183 (1973) · Zbl 1257.91043 [3] Seydel, R., Tools for Computational Finance (2004), Springer: Springer Berlin · Zbl 1042.91049 [4] Tavella, D.; Randall, C., Pricing Financial Instruments: The Finite Difference Method (2000), Wiley: Wiley New York [5] Wilmott, P.; Howison, S.; Dewynne, J., The Mathematics of Financial Derivatives (1997), Cambridge University Press: Cambridge University Press Cambridge [6] Wilmott, P.; Howison, S.; Dewynne, J., Option Pricing: Mathematical Models and Computation (1993), Oxford Financial Press: Oxford Financial Press Oxford · Zbl 0797.60051 [7] Kangro, R.; Nicolaides, R., For field boundary conditions for Black-Scholes equations, SIAM J. Numer. Anal., 38, 1357-1368 (2000) · Zbl 0990.35013 [8] Cox, J. C.; Ross, S.; Rubinstein, M., Option pricing: a simplified approach, J. Fin. Econ., 7, 229-264 (1979) · Zbl 1131.91333 [9] Hull, J. C.; White, A., The use of control variate technique in option pricing, J. Fin. Econ. Quant. Anal., 23, 237-251 (1988) [10] Hull, J. C.; White, A., Hull-White on Derivatives (1996), Risk Publication: Risk Publication London · Zbl 0997.90503 [11] Barles, G.; Daher, Ch.; Romano, M., Convergence of numerical schemes for problems arising in finance theory, Math. Models Mech. Appl. Sci., 5, 125-143 (1995) · Zbl 0822.65056 [12] Barles, G., Convergence of numerical schemes for degenerate parabolic equations arrising in finance theory, (Rogers, L. C.; Tallay, D., Numerical Methods in Finance (1997), Cambridge University Press: Cambridge University Press Cambridge) · Zbl 0898.90015 [13] Courtadon, G., A more accurate finite difference approximation for the valuation of options, J. Fin. Econ. Quant. Anal., 17, 697-703 (1982) [14] Rogers, L. C.G.; Talay, D., Numerical Methods in Finance (1997), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0867.00036 [15] Schwartz, E., The valuation of warrants: implimenting a new approach, J. Fin. Econ., 13, 79-93 (1977) [16] Vazquez, C., An upwind numerical approach for an American and European option pricing model, Appl. Math. Comput., 97, 273-286 (1998) · Zbl 0937.91053 [17] Cen, Z.; Le, A., A robust and accurate finite difference method for a generalized Black-Scholes equation, J. Comput. Appl. Math., 235, 3728-3733 (2011) · Zbl 1214.91130 [18] Wang, S., A novel fitted finite volume method for the Black-Scholes equation governing option pricing, IMA J. Numer. Anal., 24, 699-720 (2004) · Zbl 1147.91332 [19] Luskin, M.; Rannacher, R., On the smoothing property of the Crank-Nicolson schem, Appl. Anal., 14, 117-135 (1982/83) · Zbl 0476.65062 [20] Rannacher, R., Finite element solution of diffusion problems with irragular data, Numer. Math., 43, 309-327 (1984) · Zbl 0512.65082 [21] Friedman, A., Partial Differential Equation of Parabolic Type (1983), Robert E. Krieger Publishing Co.: Robert E. Krieger Publishing Co. Huntington, NY [22] O.A. Ladyzenskaja, V.A. Solonnikov, N.N. Ural’ceva, Linear and Quasilinear Equations of Parabolic Type, in: Amer. Math. Soc. Transl., vol. 23, Providence, RI, 1968.; O.A. Ladyzenskaja, V.A. Solonnikov, N.N. Ural’ceva, Linear and Quasilinear Equations of Parabolic Type, in: Amer. Math. Soc. Transl., vol. 23, Providence, RI, 1968. [23] Cho, C.; Kim, T.; Kwon, Y., Estimation of local valatilities in a generalized Black-Scholes model, Appl. Math. Comput., 162, 1135-1149 (2005) · Zbl 1079.91022 [24] Clavero, C.; Gracia, J. L.; Jorge, J. C., High-order numerical methods for one-dimensional parabolic singularly perturbed problems with regular layers, Numer. Methods Partial Differential Equations, 21, 149-169 (2004) · Zbl 1073.65079 [25] Protter, M. H.; Weinberger, H. F., Maximum Principles in Differential Equations (1984), Springer-Verlag: Springer-Verlag New York · Zbl 0153.13602 [26] Clavero, C.; Jorge, J. C.; Lisbona, F., Uniformly convergent schemes for singular perturbation problems combining alternating directions and exponential fitting techniques, (Miller, J. J.H., Applications of Advanced Computational Methods for Boundary and Interior Layers (1993), Boole: Boole Dublin), 33-52 · Zbl 0791.65064 [27] Gonzalez, C.; Palencia, C., Stability of time-stepping methods for abstract time-dependent parabolic problems, SIAM J. Numer. Anal., 35, 973-989 (1998) · Zbl 0923.65027 [28] Lambert, J. D., Computational Methods in Ordinary Differential Equations (1973), Wiley: Wiley New York · Zbl 0258.65069 [29] Lax, P. D.; Richtmyer, R. D., Survey of the stability of linear finite difference equations, Comm. Pure Appl. Math., 9, 267-293 (1956) · Zbl 0072.08903 [30] Prenter, P. M., Splines and Variational Methods (1975), Joahn Wiley and Sons: Joahn Wiley and Sons New York · Zbl 0344.65044 [31] Varah, J. M., A lower bound for the smallest singular value of a matrix, Linear Algebra Appl., 11, 3-5 (1975) · Zbl 0312.65028 [32] Varga, R. S., Matrix Iterative Analysis (1962), Prentice-Hall: Prentice-Hall Englewood Cliffs, NJ · Zbl 0133.08602 [33] Hall, C. A.; Meyer, W. W., Optimal error bounds for cubic spline interpolation, J. Approx. Theory, 16, 105-122 (1976) · Zbl 0316.41007 [34] Giles, M. B.; Carter, B., Convergence analysis of Crank-Nicolson and Rannacher time-marching, J. Comput. Finance, 9, 89-112 (2006) [35] Pooley, D. M.; Vetzal, K. R.; Forsyth, P. A., Convergence remedies for non-smooth payoffs in option pricing, J. Comput. Finance, 6, 25-40 (2003) [36] Zvan, R.; Forsyth, P. A.; Vetzal, K. R., Negative coeffcients in two factor option pricing models, J. Comput. Finance, 7, 37-73 (2003)
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.