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Pricing of Options on Commodity Futures with Stochastic Term Structures of Convenience Yields and Interest Rates

Published online by Cambridge University Press:  09 June 2010

Kristian R. Miltersen  and
Eduardo S. Schwartz
Kristian R. Miltersen
Affiliation:
Department of Management, School of Business and Economics, Odense University, Campusvej 55, DK-5230 Odense M, Denmark
Eduardo S. Schwartz
Affiliation:
Department of Finance, John E. Anderson Graduate School of Management at UCLA, 110 Westwood Plaza, Box 951481, UCLA, Los Angeles, CA 90095-1481
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Abstract

We develop a model to value options on commodity futures in the presence of stochastic interest rates as well as stochastic convenience yields. In the development of the model, we distinguish between forward and future convenience yields, a distinction that has not been recognized in the literature. Assuming normality of continuously compounded forward interest rates and convenience yields and log-normality of the spot price of the underlying commodity, we obtain closed-form solutions generalizing the Black-Scholes/Merton's formulas. We provide numerical examples with realistic parameter values showing that both the effect of introducing stochastic convenience yields into the model and the effect of having a short time lag between the maturity of a European call option and the underlying futures contract have significant impact on the option prices.

Type
Research Article
Information
Journal of Financial and Quantitative Analysis , Volume 33 , Issue 1 , March 1998 , pp. 33 - 59
Copyright
Copyright © School of Business Administration, University of Washington 1998

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References

Amin, K. I., and Jarrow, R. A.. “Pricing Options on Risky Assets in a Stochastic Interest Rate Economy.” Mathematical Finance, 2 (04, 1992), 217–237.CrossRefGoogle Scholar
Amin, K. I.; Ng, V.; and Pirrong, S. C.. “Valuing Energy Derivatives.” In Managing Energy Price Risk. London, England: Risk Publications and Enron Capital & Trade Resources (1995), 57–70.Google Scholar
Besscmbinder, H.; Coughenour, J. F.; Seguin, P. J.; and Smoller, M. M.. “Mean Reversion in Equilibrium Asset Prices: Evidence from the Futures Term Structure.” Journal of Finance, 50 (01, 1995), 361–375.CrossRefGoogle Scholar
Black, F., and Scholes, M.. “The Pricing of Options and Corporate Liabilities.” Journal of Political Economy, 81 (03, 1973), 637–654.CrossRefGoogle Scholar
Brennan, M. J. “The Price of Convenience and the Valuation of Commodity Contingent Claims.” In Stochastic Models and Option Value. Amsterdam, The Netherlands: Elsevier Science Publishers, B. V., North-Holland (1991), 33–71.Google Scholar
Brenner, R. J., and Jarrow, R. A.. “A Simple Formula for Options on Discount Bonds.” In Advances in Futures and Options Research, vol. 6. Greenwich, CT: JAI Press, Inc. (1993), 45–51.Google Scholar
Carr, P. P., and Jarrow, R. A.. “A Discrete Time Synthesis of Derivative Security Valuation Using a Term Structure of Futures Prices.” In Finance, vol. 9 of Handbooks in Operations Research and Management Science. Amsterdam, The Netherlands: North-Holland Publishing Co. (1995), chapter 7, 225–249.CrossRefGoogle Scholar
Chicago Board of Trade. Commodity Trading Manual. Chicago, IL: Chicago Board of Trade (1989).Google Scholar
Cortazar, G., and Schwartz, E. S.. “The Valuation of Commodity-Contingent Claims.” Journal of Derivatives, (1994), 27–39.CrossRefGoogle Scholar
Cox, J.C.; Ingersoll, J. E. Jr; and Ross, S. A.. “The Relation between Forward Prices and Futures Prices.” Journal of Financial Economics, 9 (1981), 321–346.CrossRefGoogle Scholar
Gibson, R., and Schwartz, E. S.. “Stochastic Convenience Yield and the Pricing of Oil Contingent Claims.” Journal of Finance, 45 (03, 1990), 959–976.CrossRefGoogle Scholar
Harrison, M. J., and Krcps, D. M. “Martingales and Arbitrage in Muluperiod Securities Markets.” Journal of Economic Theory, 20 (1979), 381–408.CrossRefGoogle Scholar
Harrison, M. J., and Pliska, S. R.. “Martingales and Stochastic Integrals in the Theory of Continuous Trading.” Stochastic Processes and Their Applications, 11 (1981), 215–260. Addendum: Harrison and Pliska (1983).CrossRefGoogle Scholar
Harrison, M. J., and Pliska, S. R.. “A Stochastic Calculus Model of Continuous Trading: Complete Markets.’ Stochastic Processes and Their Applications, 15 (1983), 313–316.CrossRefGoogle Scholar
Heath, D.; Jarrow, R. A.; and Morton, A. J.. “Bond Pricing and the Term Structure of Interest Rates: A New Methodology for Contingent Claims Valuation.” Econometrica, 60 (01, 1992), 77–105.CrossRefGoogle Scholar
Hilliard, J. E., and Reis, J.. “Valuation of Commodity Futures and Options under Stochastic Convenience Yields, Interest Rates, and Jump Diffusions in the Spot.” Journal of Financial and Quantitative Analysis, 33 (1998), 61–86.CrossRefGoogle Scholar
Jarrow, R. A., and Turnbull, S. M.. “A Unified Approach for Pricing Contingent Claims on Multiple Term Structures.” Working Paper, Johnson Graduate School of Management, Cornell Univ, Ithaca, NY, forthcoming in Review of Quantitative Finance and Accounting.Google Scholar
Merton, R. C. “Theory of Rational Option Pricing.” Bell Journal of Economics and Management Science, 4 (1973), 141–183. Reprinted in Merton (1990, Chapter 8).Google Scholar
Merton, R. C.Continuous-Time Finance. Padstow, Great Britain: Basil Blackwell Inc. (1990).Google Scholar
Reismann, H. “Movements of the Term Structure of Commodity Futures and Pricing of Commodity Claims.” Working Paper, Faculty of I. E. and Management, Technion-Israel Institute of Technology, Haifa 32000, Israel (1992).Google Scholar
Routledge, B. R.; Scppi, D. J.; and Spatt, C. J.. “Equilibrium Forward Curves for Commodities.” Working Paper, Graduate School of Industrial Administration, Carnegie Mellon Univ., Pittsburgh, PA (1997).Google Scholar
Schwartz, E. S. “The Stochastic Behavior of Commodity Prices: Implications for Valuation and Hedging.” Journal of Finance, 52 (07 1997), 922–973.CrossRefGoogle Scholar