Valuing Options Under Nonlognormality Using Relaxed Lattices

Abstract

Option pricing models based on an underlying lognormal distribution typically exhibit volatility smiles or smirks where the implied volatility varies by strike price. To adequately model the underlying distribution, a less restrictive model is needed. A relaxed binomial model is developed here that can account for the skewness of the underlying distribution and a relaxed trinomial model is developed that can account for the skewness and kurtosis of the underlying distribution. The new model incorporates the usual binomial and trinomial tree models as restricted special cases. Unlike previous flexible tree models, the size and probability of jumps are held constant at each node so only minor modifications in existing code for lattice models are needed to implement the new approach. Also, the new approach allows calculating implied skewness and implied kurtosis. Numerical results show that the relaxed binomial and trinomial tree models developed in this study are at least as accurate as tree models based on lognormality when the true underlying distribution is lognormal and substantially more accurate when the underlying distribution is not lognormal.