An Arbitrage Theory of the Term Structure of Interest Rates

In the setting of the Heath-Jarrow-Morton model this paper presents sufficient conditions to assure that the stochastic forward rates are strictly positive while maintaining the martingale property of the discounted bond price processes in the case where the stochastic forward rates are described as stochastic differential equations with explicitly state dependent stochastic volatility. Moreover, the stochastic development of the term structure of interest rates is generalized to be described by a class of continuous local martingales instead of Wiener processes. An example showing that this is a true extension of the Heath-Jarrow-Morton model is provided.