Term Structure Movements and Pricing Interest Rate Contingent Claims

Abstract

This paper derives an arbitrage-free interest rate movements model (AR model). This model takes the complete term structure as given and derives the subsequent stochastic movement of the term structure such that the movement is arbitrage free. We then show that the AR model can be used to price interest rate contingent claims relative to the observed complete term structure of interest rates. This paper also studies the behavior and the economics of the model. Our approach can be used to price a broad range of interest rate contingent claims, including bond options and callable bonds. INTEREST RATE OPTIONS, CALLABLE bonds, and floating rate notes are a fewexamples of interest rate contingent claims. They are characterized by their finite lives and their price behavior, which crucially depends 'on the term structure and its stochastic movements. In recent years, with the increase in interest rate volatility and the prevalent use of the contingent claims, the pricing of these securities has become a primary concern in financial research. The purpose of this paper is to present a general methodology to price a broad class of interest rate contingent claims.The crux of the problem in pricing interest rate contingent claims is to model the term structure movements and to relate the movements to the assets' prices.Much academic literature has been devoted to this problem. One earlier attempt is that of Pye [15]. He assumed that the interest rates move according to a (Markov) transition probabilities matrix, and he then used the expectation hypothesis to price the expected cash flow of the asset-in his case, a callable bond (Pye [16]). Recently, investigators have focused more on developing equilibrium models.Cox, Ingersoll, and Ross (CIR) [7] assumed that the short rate follows a meanreverting process. By further assuming that all interest rate contingent claims are priced contingent on only the short rate, using a continuous arbitrage argument they derived an equilibrium pricing model. Brennan and Schwartz (BS) [2] extended the CIR model to incorporate both short and long rates and studied the pricing of a broad range of contingent claims (BS [1, 3]). In these for their helpful comments on the earlier version of the paper. We would also like to thank, in particular, the referee, Eduardo Schwartz, for many of his helpful comments on this paper. We are responsible for the remaining errors. 1011 1012The Journal of Finance approaches, both the term structure and the contingent claims are derived in an equilibrium context.1 This paper proposes an alternative approach to these pricing models. We take the term structure as given and derive the feasible subsequent term structure movements. These movements must satisfy certain constraints to ensure that they are consistent with an equilibrium framework. Specifically, the movements cannot permit arbitrage profit opportunities. We shall call these interest rate movements arbitrage-free rate movements (AR). When the AR movements are determined, the interest rate contingent cl...

Pricing Range Accrual Interest Rate Swap employing LIBOR market models with jump risks

“…By further assuming that all interest rate claims are priced contingent on only the short rate, CIR derive an equilibrium pricing model that relies on a continuous arbitrage argument. Ho and Lee (1986) are the first to incorporate a discount function into the pricing of contingent claims. Given a term structure, their model derives the subsequent stochastic movement of the term structure such that the movement is arbitrage-free.…”

“…However, the short rate movement is assumed to be constant, which results in a failure to derive an interest rate that is always consistent with the market interest rate. Therefore, Hull and White (1990) incorporate the discount function into the Vasicek model and relax the constant short rate assumptions of Ho and Lee (1986) to derive the one-factor equilibrium term structure model, which is capable of determining a short rate process that is consistent with the current term structure of interest rates. Furthermore, Ho and Lee (1986), and Hull and White (1990) apply term structure equilibrium models to the pricing of discount bonds.…”

“…Therefore, Hull and White (1990) incorporate the discount function into the Vasicek model and relax the constant short rate assumptions of Ho and Lee (1986) to derive the one-factor equilibrium term structure model, which is capable of determining a short rate process that is consistent with the current term structure of interest rates. Furthermore, Ho and Lee (1986), and Hull and White (1990) apply term structure equilibrium models to the pricing of discount bonds. In these models, all discount bonds are priced relative to the stochastic short rate such that there are no arbitrage opportunities in their trading.…”

The valuation of American options on bonds

“…For example, Ho and Lee (1986), Black et al (1990), and Jamshidian (1989) all build a process for the evolution of the term structure that is based on a single-factor model. Although Heath et al (1990aHeath et al ( , b, 1992 provide a framework for the pricing of claims using a general multifactor approach to characterize the term structure, the implementation of this methodology using a binomial lattice becomes di cult when the number of factors increases, due to the computational problems associated with building a multidimensional lattice of bond prices or interest rates.…”

“…The latter paper uses a one-factor duration model to generate bond volatilities. The third approach builds a no-arbitrage term structure and was ®rst used by Ho and Lee (1986) and then by Heath et al (1990aHeath et al ( , b, 1992. In this paper, we use a variation of the second of the approaches outlined 16 In the case of a bond, the forward price of the underlying asset for delivery at time t i , p 0Yti , depends upon the coupon-interest payments on the bond.…”