# An improvement of the douglas scheme for the Black-Scholes equation

## Keywords:

Black-Scholes, crank-nicolson, douglas scheme, option pricing.## Abstract

A well-known finite difference scheme for the valuation of options from the Black-Sholes equation is the Crank-Nicolson scheme. However, in the case of non-smoothpayoffs, the Crank-Nicolson scheme is known to produce unwanted oscillations forthe computed solution. As an alternative, Douglas scheme is generally recommendedfor better resolution of option price because it has fourth order accuracy in assetderivative. However, as noted by Shaw in his book, both these methods show“potentially nasty behavior when applied to simple option pricing”. We note that boththe Crank-Nicolson scheme and the Douglas scheme use a trapezoidal formula for timeintegration which is known to produce unwanted oscillations in the computed solution.This works since the trapezoidal formula is only A-stable and not L-stable. Chawlaand Evans proposed a new L-stable Simpson rule. We investigate the application ofthis L-stable third order rule for the time integration in the Black-Sholes equationafter it has been semi-discretized in the asset derivative by Numerov discretisation.By numerical experimentation with real option valuation problems, we compare theperformance of this new improved version of Douglas with both Crank-Nicolson andDouglas schemes. We also study the performance of this scheme for the valuation ofthe Greeks.## References

Black, F. & Scholes, M. 1973. The pricing of options and corporate liabilities, The Journal of Political

Economy, 81, 637-659.

Brennan, M.J. & Schwartz, E. S. 1978. Finite difference methods and jump processes arising in the

pricing of contingent claims, A synthesis, Journal of Financial and Quantitative Analysis., 13,

-474.

Chawla, M. M. & Evans, D. J. 2005. A new L-stable Simpson rule for the diffusion equation,

International Journal of Computer Mathematics, 82 , 601-607.

Crank, J. & Nicolson, P. 1947. A practical method for numerical evaluation of solutions of partial

differential equations of the heat-conduction type, Mathematical Proceedings of the Cambridge

Philosophical Society., 43, 50-67.

Geske, R. & Shastri, K. 1985. Valuation by approximation: a comparison of alternative option valuation

techniques, Journal of Financial and Quantitative Analysis., 20, 45-71.

Lambert, J. D. 1991. Numerical methods for ordinary differential systems, John Wiley, New York.

Numerov, B. V. 1924. A method of extrapolation of perturbations, Monthly Notices of the Royal

Astronomical Society 84, 592-601.

Shaw, W. T. 1998. Modelling financial derivatives with mathematica, Cambridge University Press,

Cambridge.

Smith, G. D. 1985. Numerical solution of partial differential equations, 3rd edition, Oxford University

Press, Oxford.

Wilmott, P., Howison, S. & Dewynne, J. 1995. The mathematics of financial derivatives, A Student

Introduction, Cambridge University Press, Cambridge