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Multifactor Generalization of ”Discount-Bond Derivatives on a Recombining Binomial Tree”

By J. Chalupa


The security dynamics described by the Black-Scholes equation with price-dependent variance can be approximated as a damped discrete-time hopping process on a recombining binomial tree. In a previous working paper, such a nonuniform tree was explicitly constructed in terms of the continuous-time variance. The present note outlines how the previous procedure could be extended to multifactor Black-Scholes with price- and timedependent coefficients. The basic idea is to derive new coordinates which give a Black-Scholes equation with all the σ’s equal to unity. In the discretetime tree corresponding to this equation, nodes are uniformly spaced and the hopping probabilities are not constant. When the new coordinates are mapped back onto prices, the ensuing tree is nonuniform. A derivative can be valued with the new coordinates or the original prices. 1 A recent analysis[1] of the Bühler-Käsler discount-bond model examined the security dynamics in the discrete-time formulation. For a price-dependent variance, a recombining binomial tree with the correct continuous-time limiting behavior was explicitly constructed. Left open in that paper was the issue of whether the tree construction procedure could be extended to time-dependent variance and to multifactor continuoustime processes. This note discusses how to go about this. A recent discussion[2] of the finite-difference method vis-à-vis the implied-tree approach[3] stresses the interest of the issue. The present version of this paper is intended to be read in conjunction with Ref. [1]. Equations for the tree structure will be derived, but only a general discussion of the expected solutions will be presented here. In terms of the price variables zi (i = 1;:::;N), the valuation equation of interest for a derivative security f is ∂

Year: 2011
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