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Efficient Discrete Time Jump Process Models in Option Pricing

Published online by Cambridge University Press:  06 April 2009

Edward Omberg
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Abstract

A family of jump process models is derived by applying Gauss-Hermite quadrature to the recursive integration problem presented by a compound option model. The result is jump processes of any order with known efficiency properties in valuing options. In addition, these processes arise in the replication of options over finite periods of time with two or more assets where they again have known efficiency properties. A “sharpened” trinomial process is designed that accounts for the first-derivative discontinuity in option valuation functions at critical exercise points. It is shown to have accuracy superior to that of conventional binomial and trinomial processes and is nearly identical to the trinomial process optimized by Boyle (1988) through trial and error.

Type
Research Article
Information
Journal of Financial and Quantitative Analysis , Volume 23 , Issue 2 , June 1988 , pp. 161 - 174
Copyright
Copyright © School of Business Administration, University of Washington 1988

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References

Abramowitz, M., and Stegun, I., eds. Handbook of Mathematical Functions. Ninth printing, New York: Dover Publications, Inc. (1970).Google Scholar
Black, F., and Scholes, M.. “The Pricing of Options and Corporate Liabilities.” Journal of Political Economy, 81 (05–06 1973), 637–659.CrossRefGoogle Scholar
Blomeyer, E. “An Analytical Approximation for the American Put Price for Options on Stocks with Dividends.” Journal of Financial and Quantitative Analysis, 21 (06 1986), 229–233.CrossRefGoogle Scholar
Boyle, P. “A Lattice Framework for Option Pricing with Two State Variables.” Journal of Financial and Quantitative Analysis, 23 (03 1988), 1–12.CrossRefGoogle Scholar
Brennan, M., and Schwartz, E.. “Finite Difference Methods and Jump Processes Arising in the Pricing of Contingent Claims: A Synthesis.” Journal of Financial and Quantitative Analysis, 13 (09 1978), 461–474.CrossRefGoogle Scholar
Cox, J.; and Ross, S.. “The Valuation of Options for Alternative Stochastic Processes.” Journal of Financial Economics, 3 (01/03 1976), 145–166.CrossRefGoogle Scholar
Cox, J.; Ross, S.; and Rubinstein, M.. “Option Pricing: A Simplified Approach.” Journal of Financial Economics, 1 (09 1979), 229–263.CrossRefGoogle Scholar
Davis, P., and Rabinowitz, P.. Methods of Numerical Integration. 2nd edition, Orlando, FL: Academic Press (1984).Google Scholar
Geske, R. “The Valuation of Corporate Liabilities as Compound Options.” Journal of Financial and Quantitative Analysis, 12 (11 1977), 541–552.CrossRefGoogle Scholar
Geske, R. “A Note on an Analytical Valuation Formula for Unprotected Call Options on Stocks with Known Dividends.” Journal of Financial Economics, 1 (12 1979), 375–380.CrossRefGoogle Scholar
Geske, R., and Johnson, H.. “The American Put Option Valued Analytically.” Journal of Finance, 39 (12 1984), 1511–1524.CrossRefGoogle Scholar
Geske, R., and Shastri, K.. “Valuation by Approximation: A Comparison of Alternative Option Valuation Techniques.” Journal of Financial and Quantitative Analysis, 20 (03 1985), 45–71.CrossRefGoogle Scholar
Jarrow, R., and Rudd, A.. Option Pricing. Homewood, IL: Dow Jones-Irwin (1983).Google Scholar
Johnson, H. “An Analytic Approximation for the American Put Option.” Journal of Financial and Quantitative Analysis, 18 (03 1983), 141–148.CrossRefGoogle Scholar
Omberg, E. “Efficient Discrete-Time Jump Process Models in the Valuation, Hedging and Production of Contingent Claims.” Working Paper, Santa Clara Univ. (04 1986).Google Scholar
Omberg, E. “The Convergence of Binomial Pricing and Compound Option Models.” Journal of Finance, 42 (06 1987a), 463–469.Google Scholar
Omberg, E. “The Valuation of American Put Options with Exponential Exercise Policies.” Advances in Futures and Options Research, 2 (1987b), 117–142.Google Scholar
Parkinson, M. “Option Pricing: The American Put.” Journal of Business, 50 (01 1977), 21–36.CrossRefGoogle Scholar
Rendleman, R. Jr., and Bartter, B.. “Two-State Option Pricing.” Journal of Finance, 34 (12 1979), 1093–1110.Google Scholar
Roll, R. “An Analytic Valuation Formula for Unprotected American Call Options on Stocks with Known Dividends.” Journal of Financial Economics, 5 (11 1977), 251–258.CrossRefGoogle Scholar
Stroud, A., and Secrest, D.. Gaussian Quadrature Formulas. Englewood Cliffs, NJ: Prentice-Hall (1966).Google Scholar
Whaley, R. “On the Valuation of American Call Options on Stocks with Known Dividends.” Journal of Financial Economics, 9 (06 1981), 207–211.CrossRefGoogle Scholar
Whaley, R. “Valuation of American Futures Options: Theory and Empirical Tests.” Journal of Finance, 41 (03 1986), 127–150.CrossRefGoogle Scholar