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Interest Rate Derivatives in a Duffie and Kan Model with Stochastic Volatility: An Arrow-Debreu Pricing Approach

Posted: 18 Apr 2000  

João Pedro Vidal Nunes

ISCTE Business School

Les Clewlow

Lacima; University of Warwick - Financial Options Research Centre (FORC); University of Technology Sydney (UTS) - School of Finance and Economics

Stewart D. Hodges

University of Warwick - Financial Options Research Centre (FORC)

Abstract

Simple analytical pricing formulae have been derived, by different authors and for several interest rate contingent claims, under the Gaussian Langetieg (1980) model. The purpose of this paper is to use such exact Gaussian solutions in order to obtain approximate analytical pricing formulae under the most general stochastic volatility specification of the Duffie and Kan (1996) model, for several European-style interest rate derivatives, namely for: default-free bonds, FRAs, IRSs, short-term and long-term interest rate futures, European spot and futures options on zero-coupon bonds, interest rate caps and floors, European (conventional and pure) futures options on short-term interest rates, and even for European swaptions.
First, the functional form of an Arrow-Debreu price, under the Gaussian specification of the Duffie and Kan (1996) model, is obtained in a slightly more general form than the one given by Beaglehole and Tenney (1991). Then, and following Chen (1996), each stochastic volatility pricing solution is expressed in terms of one integral with respect to each one of the model's state variables, and another integral with respect to the time-to-maturity of the contingent claim under valuation. Finally, unlike in Chen (1996) and as the original contribution of this paper, all stochastic volatility closed form solutions are simplified into first order approximate pricing formulae that do not involve any integration with respect to the model's factors: only one time-integral is involved, irrespective of the model dimension. Consequently, such approximations will be shown to be much faster than the existing exact numerical solutions, as well as accurate. Moreover, asymptotic error bounds are provided for the proposed approximations.

JEL Classification: G132 G13, C63, E43

Suggested Citation

Nunes, João Pedro Vidal and Clewlow, Les and Hodges, Stewart D., Interest Rate Derivatives in a Duffie and Kan Model with Stochastic Volatility: An Arrow-Debreu Pricing Approach. Review of Derivatives Research, Vol. 3, Pp. 5-66, 1999. Available at SSRN: https://ssrn.com/abstract=221951

João Pedro Vidal Nunes (Contact Author)

ISCTE Business School ( email )

Ed. INDEG/ISCTE
Av. Prof. Anibal Bettencourt
Lisboa, 1600-189
Portugal
+351 21 790 39 32 (Phone)
+351 21 793 87 09 (Fax)

HOME PAGE: http://iscte.pt/~jpvn/

Les Clewlow

University of Warwick - Financial Options Research Centre (FORC)

Coventry CV4 7AL
United Kingdom

HOME PAGE: http://www.wbs.ac.uk/expertise/research_teaching/f

University of Technology Sydney (UTS) - School of Finance and Economics ( email )

Haymarket
Sydney, NSW 2007
Australia

HOME PAGE: http://www.business.uts.edu.au/finance/

Lacima ( email )

London
United Kingdom

HOME PAGE: http://www.lacimagroup.com

Stewart D. Hodges

University of Warwick - Financial Options Research Centre (FORC) ( email )

Warwick Business School
Coventry CV4 7AL
United Kingdom
01203-523606 (Phone)

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