Communications in Mathematical Finance

Extended Block Backward Differentiation Formula for the Valuation of Options

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• Abstract

  In this paper, an extended Block Backward Differentiation Formula (BBDF) is proposed for the valuation of options on a non-dividend-paying stock. The development of the method is facilitated by the availability of a Continuous Backward Differentiation Formula (CBDF) that is defined for all values of the independent variable on the range of interest. Hence, discrete schemes which are recovered from the CBDF as by-products are combined to form a BBDF, which is then applied on the entire region as a single block matrix equation by solving a system resulting from the semi-discretization of the Black-Scholes model. The stability and convergence of the BBDF are discussed. It is demonstrated that the American put values are obtained by incorporating an additional equation that generates values for the early exercise boundary which are used to ensure that the put option will be optimal. The performance of the method is tested on some numerical examples.