Pricing Asian options via compound gamma and orthogonal polynomials.

Abstract

Asian Options are options where the payoff depends on the average price of the underlying asset during at least some part of the life of the option. They belong to the so-called path dependent securities and are one of the most difficult options to price and hedge as they do not have closed-form analytical solutions. The main reason for this difficulty is that the payoff depends on the finite sum of correlated lognormal random variables, for which there is no known distribution. Asian options are much cheaper than plain vanilla European options; this makes them very attractive especially for thinly traded assets or commodities such as gold or crude oil, since extreme price manipulation is inhibited. In addition, they are used by company treasurers who want to hedge the risk of the exchange rate of money flowing over a given period. The literature on Asian options has explored a large variety of ways to price arithmetic Asian options, including among others, Monte Carlo simulations, bounds, and analytical approximations. The last class, to which this research belongs to, is the most desirable because most of the other approaches are complex and time consuming. This thesis is along two directions. First, we develop a pricing formula, CG3, based on approximating the probability density function of the sum of correlated lognormal random variables to a compound gamma distribution. The formula is shown to perform notably well over a wide range of parameter values while maintaining its ease of implementation by providing an excellent efficiency-effectiveness trade-off. Second, in an effort to improve the accuracy of our original approximation, we develop a new ad-hoc orthogonal polynomial and introduce a novel approach for pricing Asian options, and possibly other exotics, by matching higher moments via these polynomials. The resulting option formula, CGn, is in closed-from and our numerical results show that it is extremely accurate ranking first in almost all of the tested cases for both continuous and discrete