Abstract
In a thorough study of binomial trees, Joshi introduced the split tree as a two-phase binomial tree designed to minimize oscillations, and demonstrated empirically its outstanding performance when applied to pricing American put options. Here we introduce a “flexible” version of Joshi’s tree, and develop the corresponding convergence theory in the European case: we find a closed form formula for the coefficients of 1/n and 1/n3/2 in the expansion of the error. Then we define several optimized versions of the tree, and find closed form formulae for the parameters of these optimal variants. In a numerical study, we found that in the American case, an optimized variant of the tree significantly improved the performance of Joshi’s original split tree.
Highlights
The following provides another expression for Splitτ (S0 ) which we will use for defining optimal split trees
With K = 100, r = 0.1, σ = 0.25, T = 1, we calculated the values of B, estimated by (42), for all integer values of S0k k CTBS (S0) such that 0.5 ≤ P BS, that is, for S0 = 86, 87, . . . , 140, where P BS is the price of the option in the Black–Scholes model estimated using Joshi’s original split tree with classical
In this paper we introduced a flexible version of Joshi’s original split tree and we developed the corresponding convergence theory in the European case
Summary
There is a vast collection of literature describing numerical methods for evaluating options. In addition to the intellectual curiosity of understanding how tree models converge to their limits (which is part of the important study of random sums of random variables), the interest in tree methods for pricing security derivatives is motivated by those cases where no simple closed form formula exists This is the case for American options, for which explicit values for the coefficients ci (n) in the expansion of the error c1 ( n ) c2 ( n ). Our numerical results suggest that one of our optimal split trees is capable of significantly improving the accuracy of the convergence of Joshi’s original split tree for the American put We explain how this increased accuracy translates in a measure of increased speed. Our numerical result suggests that one of our optimal split trees could be significantly faster than Joshi’s original tree
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
