Advanced Modelling in Finance Using Excel and VBA

Chapter 15. Interest Rate Models

This chapter concentrates on the valuation of zero-coupon bonds using an interest rate model. In this approach, changes in the short rate are captured in a stochastic model which generates a term structure of zero-coupon prices. This approach via an interest rate model produces analytic solutions for zero-coupon prices. Jamshidian (1989) has suggested how the zero prices can be used to value options on zero-coupon bonds and additionally options on coupon bonds.

We look at two leading interest rate models, those of Vasicek (1977) and of Cox, Ingersoll and Ross (CIR; Cox et al., 1985). Both models assume that the risk-neutral process for the (instantaneous) short rate r is stochastic, with one source of uncertainty. The stochastic process includes drift and volatility parameters which depend only on the short rate r, and not on time. The short rate model involves a number of variables, and different parameter choices for these variables will lead to different shapes for the term structure generated from the model.

Both of the interest rate models feature 'so-called' mean reversion of the short rate, that is, a tendency for the short rate to drift back to some underlying rate. This is an observed feature of the way interest rates appear to vary. The two models differ in the handling of volatility. We start with Vasicek's model, and then consider the CIR model. Both the accompanying spreadsheets (the Vasicek and the CIR sheets) in the BOND1 workbook have ...