# CHAPTER 11

# RISK NEUTRAL TREES AND DERIVATIVE PRICING

The methodology illustrated in Chapter 10 to build a binomial tree has some drawbacks. For instance, there is no guarantee that the probability that are obtained from matching the term structure of interest rates are always between zero and one, as they should be. To make sure that are between these natural boundaries, we sometimes need to decrease the step size Δ appropriately, a relatively cumbersome procedure.

To overcome this problem, the industry practice has moved to a different strategy, namely, the construction of risk neutral trees without any reference to the true interest rate tree. In this section, we review two popular risk neutral tree constructions, in which the risk neutral probabilities are set equal to *p*^{*} = 1/2, and the nodes of the tree are chosen in a way consistent with the prices of interest rate securities. In addition, in this chapter we extend the binomial tree methodology to price a wide variety of interest rate securities, from coupon bonds to standard derivatives, such as caps, floors, and swaptions.

## 11.1 RISK NEUTRAL TREES

In this section we describe two popular interest rate models that are widely used to price and hedge interest rate derivative securities.

### 11.1.1 The Ho-Lee Model

The Ho-Lee model ...