On the rate of convergence of binomial Greeks

Abstract

This study investigates the convergence patterns and the rates of convergence of binomial Greeks for the CRR model and several smooth price convergence models in the literature, including the binomial Black-Scholes (BBS) model of Broadie M and Detemple J (1996), the flexible binomial model (FB) of Tian YS (1999), the smoothed payoff (SPF) approach of Heston S and Zhou G (2000), the GCRR-XPC models of Chung SL and Shih PT (2007), the modified FB-XPC model, and the modified GCRR-FT model. We prove that the rate of convergence of the CRR model for computing deltas and gammas is of order O(1/n), with a quadratic error term relating to the position of the final nodes around the strike price. Moreover, most smooth price convergence models generate deltas and gammas with monotonic and smooth convergence with order O(1/n). Thus, one can apply an extrapolation formula to enhance their accuracy. The numerical results show that placing the strike price at the center of the tree seems to On the Rate of Convergence of Binomial Greeks 563Journal of Futures Markets DOI: 10.1002/fut enhance the accuracy substantially. Among all the binomial models considered in this study, the FB-XPC and the GCRR-XPC model with a two-point extrapolation are the most efficient methods to compute Greeks.

The Binomial CEV Model and the Greeks

“…The graph reveals also that the theta value obtained through the ExtTreeBCEV method seems to converge to the benchmark value monotonically and smoothly. Therefore, the use of the extrapolation formula derived by Chung et al () is able to enhance significantly the accuracy of the theta value, as highlighted by the ExtTreeBCEVR method.…”

“…These results confirm the argument stated by Chung and Shackleton () that the inclusion of nonlinear and smoothly differentiable Black–Scholes functions into the holding value of the final pricing nodes before maturity improves significantly the straight NumDiff method under the GBM framework. Nevertheless, the ExtTreeBBS method is a binomial model with the smooth and monotonic convergence property and, therefore, we may apply the two‐point extrapolation formula derived by Chung et al (). Consequently, the ExtTreeBBSR method clearly offers the best speed‐accuracy trade‐off among the methods tested here under the GBM setup…”

“…However, it is well known that the use of such scheme for computing Greeks (and prices) may be flawed by the nature of the binomial discretization behavior observed in tree methods. See, for instance, Chung and Shackleton (, ), Chung et al (), and Pelsser and Vorst () for details under the geometric Brownian motion (henceforth, GBM) setup.…”

“…However, this is true only when we avoid the use of a Richardson extrapolation technique. Overall, we conclude that the use of an extended tree binomial CEV model possessing the smooth and monotonic convergence property substantially enhances the accuracy of Greeks because we can apply the extrapolation formula suggested by Chung et al ().…”

“…See Chung et al () for thorough discussions on other recent binomial models under the GBM assumption whose binomial option prices converge to the true value monotonically and smoothly.…”