Advanced Modelling in Finance Using Excel and VBA

Chapter 16. Matching the Term Structure

In the previous chapter, the starting point was the continuous stochastic models for the short rate of Vasicek and Cox, Ingersoll and Ross. From these models, analytic solutions were obtained for zero-coupon bond prices and subsequently methods developed for valuing options on bonds. In this chapter by contrast we model the short rate using a discrete binomial tree. By so doing, we can match a given term structure of zero-coupon prices and subsequently derive tree-based values for options on bonds.

In section 14.3, we used an example of an interest rate tree that allowed the matching of zero prices. In this chapter, we describe how such a rate tree can be calculated so that it matches the term structure and the associated volatilities. The price we pay, in computational terms, is that analytic solutions are no longer available and an iterative approach is needed to develop the binomial tree for the short rate.

When building binomial trees for equity options, the starting point involves two known values, the current share price S and the (risk-free) interest rate r, from which a distribution of share prices at the expiry date for the option is generated. The binomial tree is then used to value options, or indeed any pattern of cash flows that can be associated with the nodes in the tree.

With binomial trees for interest rate options, the two known values are the current zero-coupon value P and the cash flow at maturity (equal to 1) from which ...