This paper characterizes the arbitrage-free dynamics of interest rates, in the presence of both jumps and diffusion, when the term structure is modeled through simple forward rates (i.e., through discretely compounded forward rates evolving continuously in time) or forward swap rates. Whereas instantaneous continuously compounded rates form the basis of most interest rate models, simply compounded rates and their parameters are more directly observable in practice. We consider very general

more »

... of jump processes, allowing randomness in jump sizes and dependence between jump sizes, jump times, and interest rates. We make explicit how jump and diffusion risk premia enter into the dynamics of simple forward rates. We also formulate reasonably tractable subclasses of models and provide pricing formulas for some derivative securities, including interest rate caps and options on swaps. Through these formulas, we illustrate the effect of jumps on implied volatilities in interest rate derivatives. * Musiela [15], Musiela and Rutkowski [45], and Jamshidian [35] ). The motivation for models based on simple forwards (in contrast to the instantaneous rates traditionally treated in continuous-time models) lies in building a model based on observable quantities. Most market rates are indeed based on simple compounding, so instantaneous continuously compounded rates often represent an idealized approximation to market data. This point is relevant whether one tries to infer model parameters from time-series data or from prices of derivative securities because most derivatives contracts are tied to simple rates. Motivation for including jumps comes from several sources. There is evidence of the importance of jumps in capturing both returns and option prices in other markets, including equity and foreign exchange (Akgiray and Booth [2], Andersen, Benzoni, and Lund [5], Bakshi, Cao, and Chen [8] Bates [10], Bates [11], Jorion [41]) so there would seem to be no a priori reason for ruling out the possibility of jumps in interest rates. Motivating theory in the form of equilibirum asset pricing models with jumps include Naik and Lee [47] and, in the interest rate setting, Ahn and Thompson [1], Attari [6], Das and Foresi [23], Naik and Lee [48], Nietert [49]; the pricing of interest rate derivatives in the presence of jumps is considered in Björk et al. [13], Burnetas and Ritchken [17], Chacko and Das [18], Das [21], Das and Foresi [23], Duffie and Kan [26], Duffie, Pan, and Singleton [27], Jarrow and Madan [38, 39], and Shirakawa [51]. Specific sources of jumps in interest rates, including economic news and moves by central banks, are put forward in Balduzzi, Elton, and Green [9], Babbs and Webber [7], Das [20, 22], El-Jahel, Lindberg, and Perraudin [29], Honoré [33], and Johannes [40]. These studies find compelling empirical evidence for jumps; Das [22] and Johannes [40] note that the kurtosis in short-term interest rates is incompatible with a pure-diffusion model. The possibility of default (as modeled in Duffie and Singleton [28] and Jarrow and Turnbull [37]) provides further motivation for including jumps, though we do not consider credit risk here. Jumps in interest rates can also be used to try to reproduce the patterns in implied volatilities derived from market prices of interest rate derivatives. Although reliable information for away-fromthe-money options traded over-the-counter is not readily available, broker quotes provided through commercial financial information services indicate a very pronounced skew in implied volatilities for Yen-denominated interest rate caps. A similar but less pronounced pattern can be seen in US Dollar caps, and both markets also show a dependence on maturity in implied volatilities (decreasing in Japan, humped in the US). Implied volatilities extracted from interest rate caps are putative parameters of simple forward rates, which again motivates adopting simple forwards as the building blocks of a model. (Similarly, implied volatilities extracted from options on interest rate swaps are putative parameters of forward swap rates.) In special cases of the general framework we develop, interest rate caps or swaptions can