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

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.