Binomial Models for Option Valuation - Examining and Improving Convergence

Dietmar Leisen and Matthias Reimer
Department of Statistics, University of Bonn
Leisen@addi.or.Uni-Bonn.de
Reimer@addi.or.Uni-Bonn.de

Abstract

Binomial models, which rebuild the continuous setup in the limit, serve for approximative valuation of options, especially where formulas cannot be derived mathematically due to properties of the considered option type. Unfortunately, even with the valuation of European call options distorting irregularities occur calculating prices along iteration of tree refinements. For the first time, these convergence patterns in binomial option valuation models are examined and it is proved order of convergence one for the Cox-Ross-Rubinstein [Cox, Ross and Rubinstein1979] model as well as for the tree parameter selections of Jarrow and Rudd [Jarrow and Rudd1983], and Tian [Tian1993]. Then, we define new binomial models, where the calculated option prices converge smoothly to the Black- Scholes solution and we achieve order of convergence two with smaller initial error. Notably, solely the formulas to determine the constant up- and down- factors change. Finally, all tree approaches are compared with respect to speed and accuracy calculating relative root-mean-squared error of approximative option values for a sample of randomly selected parameters across a set of refinements. This approach was used in Broadie and Detemple [Broadie and Detemple1994]. Approximation of American type options with the new models exhibits order of convergence one but smaller initial error than previously existing binomial models.

References

Broadie and Detemple1994

Broadie, M. and Detemple, J., 1994. `American Option Evaluation: New Bounds, Approximations, and a Comparison of Existing Methods', Working Paper, Columbia University, New York.

Cox, Ross and Rubinstein1979

Cox, J., Ross, S.A., and Rubinstein, M., 1979. `Option Pricing: A Simplified Approach', Journal of Financial Economics 7, 229-263.

Jarrow and Rudd1983

Jarrow, R. and Rudd A., 1983. Option Pricing, Homewood, Illinois, 183-188.

Tian1993

Tian, Y., 1993. `A Modified Lattice Approach to Option Pricing', Journal of Futures Markets 13(5), 563-577.