Chapter 10

The Art of Term Structure Models: Volatility and Distribution

This chapter continues the presentation of the elements of term structure modeling, focusing on the volatility of interest rates and on models in which rates are not normally distributed.

TIME-DEPENDENT VOLATILITY: MODEL 3

Just as a time-dependent drift may be used to fit many bond or swap rates, a time-dependent volatility function may be used to fit many option prices. A particularly simple model with a time-dependent volatility function might be written as follows:

Unlike the models presented in Chapter 9, the volatility of the short rate in equation (10.1) depends on time. If, for example, the function were such that and , then the volatility of the short rate in one year is 126 basis points per year while the volatility of the short rate in two years is 120 basis points per year.

To illustrate the features of time-dependent volatility, consider the following special case of (10.1) that will be called Model 3:

In (10.2) the volatility of the short rate starts at the constant and then exponentially ...