Interest Rate Derivatives: One Factor Spot Rate Models

In this chapter we survey models of interest rate derivatives which take the instantaneous spot interest rate as the underlying factor. The continuous hedging argument is extended so as to model the term structure of interest rates and other interest rate derivative securities. This basic approach is due to Vasicek (J Financ Econ 5:177–188, 1977) and hence we shall often refer to it as the Vasicek approach. By specifying different functional forms for the drift, the diffusion and the market price of risk, we develop three well known spot rate models, namely the Vasicek model, the Hull–White model and the Cox–Ingersoll–Ross model. Then we present a general framework for pricing bond options and we apply this framework to obtain closed form solutions for bond options under the specifications of the Hull–White and the Cox–Ingersoll–Ross model. Finally we discuss the calibration of the Hull–White model to the currently observed yield curve.