# SFB 303 Discussion Paper No. B-309

**Author**: Leisen, D. P. J., and M. Reimer

**Title:** Binomial Models for Option Valuation - Examining
and Improving Convergence

**Abstract:** Binomial models, which rebuild the continuous
setup in the limit, serve for approximative valuation of options, especially
where formulas cannot be derived mathematically. Even with the valuation
of European call options distorting irregularities occur. For this case,
sources of convergence patterns are explained. Furthermore, it is proved
order of convergence one for the Cox-Ross-Rubinstein[79] model as well
as for the tree parameter selections of Jarrow and Rudd[83], and Tian[93].
Then, we define new binomial models, where the calculated option prices
converge smoothly to the Black-Scholes solution and remarkably, we even
achieve order of convergence two with much 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.
Approximation of American type options with the new models exhibits order
of convergence one but smaller initial error than previously existing
binomial models.

**Keywords:** binomial model, option valuation, order of convergence,
convergence pattern

**JEL-Classification-Number:** G13

**Creation-Date:** March 1995

**URL:**
../1995/b/bonnsfb309.pdf
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