ASIAN OPTIONS, THE SUM OF LOGNORMALS, AND THE RECIPROCAL GAMMA DISTRIBUTION

Moshe Milevsky and Steven E. Posner
Additional contact information
Steven E. Posner: Marsh & McLennan Securities Corp., 114 West 47th Street, New York, NY 10036, USA

Chapter 7 in Quantitative Analysis in Financial Markets:Collected Papers of the New York University Mathematical Finance Seminar, 1999, pp 203-218 from World Scientific Publishing Co. Pte. Ltd.

Abstract: AbstractArithmetic 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.

Date: 1999
References: View references in EconPapers View complete reference list from CitEc
Citations:

Downloads: (external link)
https://www.worldscientific.com/doi/pdf/10.1142/9789812812599_0007 (application/pdf)
https://www.worldscientific.com/doi/abs/10.1142/9789812812599_0007 (text/html)
Ebook Access is available upon purchase.

Related works:
Journal Article: Asian Options, the Sum of Lognormals, and the Reciprocal Gamma Distribution (1998)
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:wsi:wschap:9789812812599_0007

Ordering information: This item can be ordered from

Access Statistics for this chapter

More chapters in World Scientific Book Chapters from World Scientific Publishing Co. Pte. Ltd.
Bibliographic data for series maintained by Tai Tone Lim ().