Interest Rate Models

In this chapter we illustrate the role of stochastic volatility in the case of interest rate products traded in fixed income markets . Our main example is pricing of bonds when the interest rate is defined in terms of a Vasicek model. As in the previous chapters, we use a two-factor stochastic volatility model and show how one can derive bond price approximations in the regime of separation of time scales. Market bond pricing data are often quoted in terms of the yield curve corresponding to the effective or continously compounded interest rate for the bond as a function of time to maturity. We show how the bond price approximation gives a flexible way of parameterizing this yield curve, also called the term structure of interest rates. The stochastic volatility Vasicek model that we consider here is introduced in Section 12.1 and we carry out the asymptotic expansion for the associated bond price in Section 12.2. The bond price approximation leads to a particular form for the yield curve and we discuss this and calibration issues in Section 12.2.8. The Vasicek example illustrates how our singular and regular perturbation approach easily generalizes to typical problems in the fixed income market. There are many other interest rate products and also interest rate models that can be analyzed in our framework, and we comment on some of these. In Section 12.4 we use the CIR model and in Section 12.3 a quadratic model for the interest rate.