ASIAN OPTIONS, THE SUM OF LOGNORMALS, AND THE RECIPROCAL GAMMA DISTRIBUTION
This project was supported by a financial grant from the Social Sciences and Humanities Research Council of Canada and by the York University Research Authority. The authors acknowledge the helpful comments and suggestions of the seminar participants at the Fields Institute at the University of Toronto, the 8th Annual Derivative Securities Conference at Boston University, New York University, and York University, as well as Marco Avellaneda, Peter Carr, George Chacko, Sanjiv Ranjan Das, Aron Gottesman, Dilip Madan, Tom Salisbury, Marti Subrahmanyam, Hans Tuenter, Jonathan Karpoff (the editor), and Robert Jarrow (associate editor and referee). The usual disclaimer applies.
Arithmetic Asian options are difficult to price and hedge as they do not have closed-form analytic solutions. The main theoretical reason for this difficulty is that the payoff depends on the finite sum of correlated lognormal variables, which is not lognormal and for which there is no recognizable probability density function. We use elementary techniques to derive the probability density function of the infinite sum of correlated lognormal random variables and show that it is reciprocal gamma distributed, under suitable parameter restrictions. A random variable is reciprocal gamma distributed if its inverse is gamma distributed. We use this result to approximate the finite sum of correlated lognormal variables and then value arithmetic Asian options using the reciprocal gamma distribution as the state-price density function. We thus obtain a closed-form analytic expression for the value of an arithmetic Asian option, where the cumulative density function of the gamma distribution, G(d) in our formula, plays the exact same role as N(d) in the Black-Scholes/Merton formula. In addition to being theoretically justified and exact in the limit, we compare our method against other algorithms in the literature and show our method is quicker, at least as accurate, and, in our opinion, more intuitive and pedagogically appealing than any previously published result, especially when applied to high yielding currency options.
