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The generalization of the Geske–formula for compound options to stochastic interest rates is not trivial–a note

Part of: Stochastic processes

Published online by Cambridge University Press:  14 July 2016

Rüdiger Frey  and
Daniel Sommer
Rüdiger Frey*
Affiliation:
Eidgenössische Technische Hochschule Zürich
Daniel Sommer*
Affiliation:
Universität Bonn
*
∗ Postal address: Dept. of Mathematics, ETH Zurich, ETH Zentrum, CH-8092 Zurich, Switzerland. E-mail address: frey@math.ethz.ch
∗∗ Postal address: Statistische Abteilung, Institut für Wirtschafts- und Gesellschaftswissenschaften, Rechts- und Staatswissenschaftliche Fakultät, Universität Bonn, Adenauerallee 24–42, D-53113 Bonn.
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Abstract

This note refers to the paper by Geman, El-Karoui and Rochet (1995), in which an extension of the Geske-formula for compound options to the case of stochastic interest rates is proposed. We show that such an extension is not possible in general. However, we point out modifications of Geske's original problem in which closed formulas can still be obtained under stochastic interest rates. In particular we consider the case of an option on a future-style option. Moreover, we sketch a numerical solution to Geske's original problem when interest rates are random.

Keywords

Compound optionsstochastic interest ratesfutures-style optionschange of numeraire technique

MSC classification

Secondary: 60G35: Signal detection and filtering

Information

Type
Short Communications
Information
Journal of Applied Probability , Volume 35 , Issue 2 , June 1998 , pp. 501 - 509
Copyright
Copyright © Applied Probability Trust 1998 

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References

Cox, J.C., Ingersoll, J.E., and Ross, S.A. (1981). The relation between forward prices and futures prices. J. Financial Economics 9, 321–346.CrossRefGoogle Scholar
Demange, G., and Rochet, J.-C. (1992). Methodes Mathématiques de la Finance. Economica.Google Scholar
Delbaen, F., and Schachermayer, W. (1995). The no-arbitrage property under a change of numeraire. Stochastics and Stochastis Reports 53, 213–226.Google Scholar
Duffie, D. (1992). Dynamic Asset Pricing Theory. Princeton University Press, Princeton, New Jersey.Google Scholar
El-Karoui, N., Myneni, R., and Viswanathan, R. (1992). Arbitrage pricing and hedging of interest rate claims with state variables: 1, theory. Preprint, University of Paris VI.Google Scholar
El-Karoui, N., Myneni, R., and Viswanathan, R. (1992). Arbitrage pricing and hedging of interest rate claims with state variables: 2, applications. Preprint, University of Paris VI.Google Scholar
El-Karoui, N., and Rochet, J.-C. (1989). A pricing formula for options on coupon bonds. Preprint, University of Paris VI.Google Scholar
Frey, R., and Sommer, D. (1996). A systematic approach to pricing and hedging of international derivatives with interest rate risk. Applied Mathematical Finance 3, 295–317.Google Scholar
Geman, H., El-Karoui, N., and Rochet, J.-C. (1995). Changes of numéraire, changes of probability measure and option pricing. J. Appl. Prob. 32, 443–458.Google Scholar
Geske, R. (1979). The valuation of compound options. J. Financial Economics 7, 63–81.Google Scholar
Heath, D., Jarrow, R., and Morton, A. (1992). Bond pricing and the term structure of interest rates: A new methodology for contingent claim valuation. Econometrica 60, 77–105.Google Scholar
Harrison, J.M., and Pliska, S.R. (1981). Martingales and stochastic integrals in the theory of continuous trading. Stochastic Proc. Appl. 11, 215–260.Google Scholar