Pricing Asian and basket options via Taylor expansion

Nengjiu Ju

Abstract

ABSTRACT

Asian options belong to the so-called path-dependent derivatives. They are among the most difficult to price and hedge, both analytically and numerically. Basket options are even harder to price and hedge because of the large number of state variables. Several approaches have been proposed in the literature, including, among others, Monte Carlo simulations, tree-based methods, partial differential equations, and analytical approximations. The last category is the most appealing because most of the other methods are very complex and slow. Our method belongs to the analytical approximation class. It is based on the observation that, though the weighted average of lognormal variables is no longer lognormal, it can be approximated by a lognormal random variable if the first two moments match the true first two moments. To have a better approximation, we consider the Taylor expansion of the ratio of the characteristic function of the average to that of the approximating log-normal random variable around zero volatility. We include terms up to ó⁶ in the expansion. The resulting option formulas are in closed form. We treat discrete Asian options as a special case of basket options. Formulas for continuous Asian options are obtained from their discrete counterparts. Numerical tests indicate that the formulas are very accurate. Comparisons with all other leading analytical approximations show that our method has performed the best overall in terms of accuracy for both short- and long-maturity options. Furthermore, unlike some other methods, our approximation treats basket (portfolio) and Asian options in a unified way. Lastly, in an Appendix we point out a serious mathematical error in a popular method of pricing Asian options in the literature.