Continuous - Time Short Rate Models

These notes provide an overview of singleand multi-factor models of the short rate. We will begin with a generic single-factor model where the dynamics of rt under the physical measure, P , are given. Following the approach of Vasicek, we then derive the PDE that must be satisfied by derivative security prices. We then use the martingale approach to give an alternative (and familiar) expression for derivative security prices. The consistency of the two approaches is then demonstrated using the Feynman-Kac PDE representation. We show how the Martingale Representation theorem can be used to construct hedging strategies and then discuss some specific single-factor models. These examples include the Vasicek and CIR models, and more generally, affine models. We then conclude with multi-factor models and describe some specific examples.