Generalized trapezoidal formulas for valuing American options

Recently, Chawla et al. [M. J. Brennan and E. S. Schwartz, Finite difference methods and jump processes arising in the pricing of contingent claims: A synthesis, J. Fin. & Quant. Anal., 13 (1978) 461–474] presented time integration schemes for the Black–Scholes equation of option pricing based on the generalized trapezoidal formulas (GTF(α)) introduced in [R. Roll, An analytical formula for unprotected American call options on stocks with known dividends, J. Finan. Econ., 5 (1977) 251–258]. For European options with a nondifferentiable payoff, it was demonstrated that a GTF(α) scheme could provide better approximations for option valuation than the often used Crank–Nicolson (C–N) scheme. In the present article, we extend the application of GTF(α) for valuation of American options. The GTF(1/3) scheme is shown to perform consistently superior to the C–N scheme for American options with a nondifferentiable or discontinuous payoff.

[1]  G. Barone-Adesi,et al.  Efficient Analytic Approximation of American Option Values , 1987 .

[2]  C. Cryer.  The Solution of a Quadratic Programming Problem Using Systematic Overrelaxation , 1971 .

[3]  David J. Evans,et al.  Generalized trapezoidal formulas for the black–scholes equation of option pricing , 2003, Int. J. Comput. Math..

[4]  F. Black,et al.  The Pricing of Options and Corporate Liabilities , 1973, Journal of Political Economy.

[5]  J. Crank,et al.  A practical method for numerical evaluation of solutions of partial differential equations of the heat-conduction type , 1947, Mathematical Proceedings of the Cambridge Philosophical Society.

[6]  Eduardo S. Schwartz,et al.  Finite Difference Methods and Jump Processes Arising in the Pricing of Contingent Claims: A Synthesis , 1977 .

[7]  David J. Evans,et al.  Generalized trapezoidal formulas for parabolic equations , 1999, Int. J. Comput. Math..

[8]  Robert E. Whaley,et al.  On the valuation of American call options on stocks with known dividends , 1981 .

[9]  H. Johnson.  An Analytic Approximation for the American Put Price , 1983, Journal of Financial and Quantitative Analysis.

[10]  R. Roll,et al.  An analytic valuation formula for unprotected American call options on stocks with known dividends , 1977 .

[11]  P. Wilmott,et al.  Option pricing: Mathematical models and computation , 1994 .

[12]  David J. Evans,et al.  A class of generalized trapezoidal formulas for the numerical integration of , 1996, Int. J. Comput. Math..