The Term Structure of Simple Forward Rates with Jump Risk

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 traditional interest rate models, simply compounded rates and their parameters are more directly observable in practice and are the basis of recent researchon “market models.” We consider very general types of jump processes, modeled through marked point 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. 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—that is, through discretely compounded forward rates evolving continuously in time—or through forward swap rates. We consider very general types of jump processes (allowing randomness in jump sizes and dependence between jump sizes, jump times, and interest rates) and identify how jump and diffusion risk premia enter into the dynamics of simple forward rates. We also formulate a reasonably tractable subclass of models and provide pricing formulas for some term structure derivatives. Our investigation builds on several strands of research, in particular on the dynamics of instantaneous continuously compounded rates (as in Heath, Jarrow, and Morton 1992), option pricing withjumps (as in Merton 1976), LIBOR and swap rate market models (including Brace, Gatarek, and Musiela 1997; Jamshidian 1997; Miltersen,