Volatility Structures of Forward Rates and the Dynamics of the Term Structure*

For general volatility structures for forward rates, the evolution of interest rates may not be Markovian and the entire path may be necessary to capture the dynamics of the term structure. This chapter identifies conditions on the volatility structure of forward rates that permit the dynamics of the term structure to be represented by a two-dimensional state variable Markov process. The permissible set of volatility structures that accomplishes this goal is shown to be quite large and includes many stochastic structures. In general, analytical characterisation of the terminal distributions of the two state variables is unlikely, and numerical procedures are required to value claims. Efficient simulation algorithms using control variates are developed to price claims against the term structure.