Generalized trapezoidal formulas for valuing American options

Abstract

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.

Keywords:

• American options
• Black–Scholes equation
• Linear complementarity problem
• Crank–Nicolson
• Generalized trapezoidal formulas