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Asian Options, the Sum of Lognormals, and the Reciprocal Gamma Distribution

Published online by Cambridge University Press:  06 April 2009

Moshe Arye Milevsky  and
Steven E. Posner
Moshe Arye Milevsky
Affiliation:
Schulich School of Business, York University, 4700 Keele Street, Ontario, M3J 1P3, Canada
Steven E. Posner
Affiliation:
Marsh & McLennan Securities Corp., Two World Trade Center, New York, NY 10048
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Abstract

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.

Type
Research Article
Information
Journal of Financial and Quantitative Analysis , Volume 33 , Issue 3 , September 1998 , pp. 409 - 422
Copyright
Copyright © School of Business Administration, University of Washington 1998

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References

Alziary, B.; Decamps, J.; and Koehl, P.. “A P.D.E. Approach to Asian Options: Analytical and Numerical Evidence.” Journal of Banking and Finance, 21 (1997), 613–640.CrossRefGoogle Scholar
Black, R., and Scholes, M. S.. “The Pricing of Options and Corporate Liabilities.” Journal of Political Economy, 81 (1973), 637–653.Google Scholar
Bouaziz, L.; Briys, E.; and Crouhy, M.. “The Pricing of Forward-Start Asian Options.” Journal of Banking and Finance, 18 (1994), 823–839.Google Scholar
Boyle, P. “Options: A Monte Carlo Approach.” Journal of Financial Economics, 4 (1977), 323–338.CrossRefGoogle Scholar
Carverhill, A., and Clewlow, L.. “Flexible Convolution.” RISK, 5 (04 1990), 25–29.Google Scholar
Chacko, G., and Das, S.. “Average Interest.” Working Paper, Harvard Business School (1997).Google Scholar
Corwin, J.; Boyle, P.; and Tan, K.. “Quasi-Monte Carlo Methods in Numerical Finance.” Management Science, 42 (06 1996), 926–938.Google Scholar
Curran, M. “Valuing Asian and Portfolio Options by Conditioning on the Geometric Mean.” Management Science, 40 (12, 1994), 1705–1711.CrossRefGoogle Scholar
DeSchepper, A.; Teunen, M.; and Goovaerts, M.. “An Analytic Inversion of a Laplace Transform Related to Annuities Certain.” Insurance: Mathematics and Economics, 14 (1, 1994), 33–37.Google Scholar
Dewynne, J., and Wilmott, P.. “Asian Options as Linear Complementarity Problems.” Advances in Futures and Options Research, 8 (1995), 145–173.Google Scholar
Dufresne, D. “The Distribution of a Perpetuity with Applications to Risk Theory and Pension Funding.” Scandinavian Actuarial Journal, 9 (1990), 39–79.Google Scholar
Geman, H., and Yor, M.. “Bessel Processes, Asian Options and Perpetuities.” Mathematical Finance, 3 (4, 10 1993), 349–375.Google Scholar
Haykov, J. “A Better Control Variate for Pricing Standard Asian Options.” Journal of Financial Engineering, 2 (1993), 207–216.Google Scholar
Hull, J., and White, A.. “Efficient Procedures for Valuing European and American Path Dependent Options.” Journal of Derivatives, 1 (Fall 1993), 21–31.CrossRefGoogle Scholar
Hull, J. C.Options, Futures and Other Derivatives, Third ed.Englewood Cliffs, NJ: Prentice Hall (1997).Google Scholar
Ju, N. “Fourier Transformation, Martingale, and the Pricing of Average Rate Derivatives.” Ph.D. Thesis, Univ. of California-Berkeley (1997).Google Scholar
Karatzas, I., and Shreve, S. E.. Brownian Motion and Stochastic Calculus, 2nd ed.New York, NY: Springer-Verlag (1992).Google Scholar
Kemna, A., and Vorst, A.. “A Pricing Method for Options Based on Average Values.” Journal of Banking and Finance, 14 (1990), 113–129.Google Scholar
Kramkov, D., and Mordecky, E.. “Integral Options.” Theory of Probability and Its Applications, 39 (06 1994), 162–171.CrossRefGoogle Scholar
Levy, E. “Pricing European Average Rate Currency Options.” Journal of International Money and Finance, 11 (1992), 474–491.CrossRefGoogle Scholar
Levy, E., and Turnbull, S.. “Average Intelligence” RISK, 5 (02 1992), 53–59.Google Scholar
Majumdar, M., and Radner, R.. “Linear Models of Economic Survival under Production Uncertainty.” Economic Theory, 1 (1991), 13–30.CrossRefGoogle Scholar
Merton, R. “An Asymptotic Theory of Growth under Uncertainty.” Review of Economic Studies, 42 (1975), 375–393.Google Scholar
Merton, R. “Theory of Rational Option Pricing.” Bell Journal of Economics and Management Science, 4 (1973), 141–183.Google Scholar
Milevsky, M. A. “Martingales, Scale Functions and Stochastic Life Annuities.” Insurance: Mathematics and Economics (forthcoming 1998).Google Scholar
Neave, E., and Turnbull, S.. “Quick Solutions for Arithmetic Average Options on a Recombining Random Walk.” 4th Actuarial Approach for Financial Risks International Colloquium (1993), 718–739.Google Scholar
Nielsen, J. A., and Sandman, K.. “The Pricing of Asian Options under Stochastic Interest Rates.” Applied Mathematical Finance, 3 (1996), 209–236.CrossRefGoogle Scholar
Posner, S. E., and Milevsky, M. A.. “Valuing Exotic Options by Approximating the SPD with Higher Moments.” Journal of Financial Engineering (forthcoming 1998).Google Scholar
Rogers, L., and Shi, Z.. “The Value of an Asian Option.” Journal of Applied Probability, 32 (1995), 1077–1088.Google Scholar
Turnbull, S., and Wakeman, L.. “A Quick Algorithm for Pricing European Average Options.” Journal of Financial and Quantitative Analysis, 26 (1991), 377–389.CrossRefGoogle Scholar
Vorst, T. “Prices and Hedge Ratios of Average Exchange Rate Options.” International Review of Financial Analysis, 1 (3, 1992), 179–193.Google Scholar
Vorst, T. “Averaging Options.” In The Handbook of Exotic Options, Nelken, I., ed. Homewood, IL: Irwin (1996), 175–199.Google Scholar
Yor, M. “From Planar Brownian Windings to Asian Options.” Insurance: Mathematics and Economics, 13 (1993), 23–34.Google Scholar