Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-10T10:58:37.272Z Has data issue: false hasContentIssue false
  • English
  • Français

A Frequency Distribution Method for Valuing Average Options

Published online by Cambridge University Press:  29 August 2014

Edwin H. Neave
Edwin H. Neave*
Affiliation:
School of Business, Queen's University, Kingston, Ontario
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

This paper finds payoff frequency distributions for valuing European and American fixed strike average options on a discrete time, recombining multiplicative binomial asset price process. In comparison to other discrete valuation methods the distributions, obtained analytically from a generating function, greatly reduce the computational requirements needed for accurate valuation. Less data are needed to value geometric than arithmetic averages, but the magnitude of calculations is similar for both instruments. Calculations of order T3 are needed to value European instruments, of order T4 to value their American counterparts. A frequency distribution of a quantity called path sums is used to value geometric average options, and a joint distribution of path sums and realized prices is used to value arithmetic average options. The frequency distributions give an exact value for geometric average, an approximate value for arithmetic average instruments. The method obtains additional information from the generating function to estimate approximation errors relative to the exact binomial solution. If the errors are significant they can be reduced using still further detail from the generating function. Error reduction can be performed selectively to minimize additional calculation.

Keywords

Binomial modelsoptions pricingaverage optionsGaussian binomial coefficientsnumerical methodsg
Type
Articles
Information
ASTIN Bulletin: The Journal of the IAA , Volume 27 , Issue 2 , November 1997 , pp. 173 - 205
Copyright
Copyright © International Actuarial Association 1997

References

REFERENCES

Berman, Gerald, and Fryer, K. D., Introduction to Combinatorics, New York: Addison-Wesley, 1972.Google Scholar
Carverhill, A., and Clewlow, L., 1990, “Flexible Convolution,” Risk 3, 04 25–29.Google Scholar
Chen, Chuang-Chong and Khee-Meng, Koh, Principles and Techniques in Combinatorics, Singapore: World Scientific Publishing, 1992.CrossRefGoogle Scholar
Curran, M., “Beyond Average Intelligence,” Risk, 11 1992, p. 60.Google Scholar
Dixit, Avinash K. and Pindyck, Robert S., Investment Decisions Under Uncertainty, Princeton: Princeton University Press, 1994.CrossRefGoogle Scholar
Feller, William, An Introduction to Probability Theory and its Applications, 2nd Edition, New York: Wiley, 1957.Google Scholar
Geman, Helyette, and Eydeland, A., “Domino Effect,” Risk, 1995, 65–67.Google Scholar
Geman, Helyette, and Yor, Marc, “Bessel Processes, Asian Options, and Perpetuities,” Mathematical Finance 3, 1993, 349–375.CrossRefGoogle Scholar
Goldman, M. Barry, Sosm, Howard B. and Gatto, M., “Path Dependent Options: Buy at the Low, Sell at the High,” Journal of Finance 34, 1979, 1111–1127.Google Scholar
Hoo, Thomas S. Y., “Managing Illiquid Bonds and the Linear Path Space,” Journal of Fixed Income, 06 1992; 80–94.CrossRefGoogle Scholar
Huang, Chi-fu, and Litzenberger, Robert H., Foundations for Financial Economics, New York: North-Holland, 1988.Google Scholar
Hull, John, and White, Alan, “Efficient Procedures for Valuing European and American Path-Dependent Options,” The Journal of Derivatives, Fall 1993, 21–31.CrossRefGoogle Scholar
Jarrow, Robert, and Turnbull, Stuart M., Derivative Securities, Cincinnati: South-Western Publishing Company, 1996.Google Scholar
Kemna, A. and Vorst, Ton “A Pricing Method for Options Based on Average Asset Values,” Journal of Banking and Finance, 1990.CrossRefGoogle Scholar
Levy, Edmond, “A Note on Pricing European Average Options,” Nomura Bank International Working Paper, 1991.Google Scholar
Levy, Edmond, and Turnbull, Stuart M., “Average Intelligence,” Risk Magazine, 1992.Google Scholar
Neave, Edwm H., “Valuing European Average Options on a Recombining Binomial Random Walk,” Actuarial Approach For Financial Risks, vol. 1, pp. 153–180, Third AFIR International Colloquium, Rome, 1993.Google Scholar
Neave, Edwin H., and Stein, Christopher L. “Approximate and Exact Valuation of European Arithmetic Average Spot Price Calls,” Queen's University Working Paper, 1997.Google Scholar
Ritchken, P., Sankarasubramanian, L., and Vijh, A., “The Valuation of Path Dependent Contracts on the Average,” Management Science 39 (10), 1993, 1202–1213.CrossRefGoogle Scholar
Rogers, L. C. G. and Shi, Z., “The Value of an Asian Option,” Journal of Applied Probability, 1995.CrossRefGoogle Scholar
Schwartz, Eduardo S., Review of Dixit and Pindyck, Investment Under Uncertainty, Journal of Finance 49, 1994, 1924–1928.CrossRefGoogle Scholar
Stein, Christopher L., “Computing the Exact Value of European Arithmetic Average Spot Price Calls,” Queen's University Working Paper, 1996.Google Scholar
Tilley, James A., “Valuing American Options in a Path Simulation Model,” Actuarial Approach For Financial Risks, vol. 1, pp. 249–266, Third AFIR International Colloquium, Rome, 1993.Google Scholar
Turnbull, Stuart M. and Wakeman, Lee Macdonald, “A Quick Algorithm for Pricing European Average Options,” Journal of Financial and Quantitative Analysis 26, 1991, 377–389.CrossRefGoogle Scholar
Yor, M., “On Some Exponential Functions of Brownian Motion,” Advances in Applied Probability 24, 1992, 509–531.CrossRefGoogle Scholar
Zhang, P., “Flexible Arithmetic Asian Options,” The Journal of Derivatives, Spring 1995, 53–63.CrossRefGoogle Scholar