# CHAPTER 14

# INTEREST RATE MODELS IN CONTINUOUS TIME

In this chapter we move to continuous time. This methodology is by far the most applied in term structure models, and it can be best explained by referring to the Ho-Lee model discussed in Chapter 11. Recall that the Ho-Lee model postulates that from an interest rate *r*_{i, j} in time/node *(i, j)*, the next interest rate on the tree is either of the following, depending on whether there is an upward movement or a downward movement, respectively:

where Δ is the time step, *θ*_{i} is a sequence of constants that depends on time (period) *i*, *σ* is a parameter determining the volatility of interest rates, and “RN” stands for “risk neutral.” Consider the simple case in which *θ*_{i} is constant over periods *i* = 0, 1, 2..., and, for simplicity, assume it is equal to 0, *θ*_{i} = 0. Finally, assume that the interest rate volatility is *σ* = 0.02. Consider now a time interval ...0,1], and let’s divide it into *n* subintervals. Δ = 1/n is then the size of each individual interval. Figure 14.1 plots three paths of interest rates for *n* = 10,100, 1000, that is, with step sizes Δ = 1/10, Δ = 1/100, and Δ = 1/1000. All of these paths start at the same initial interest rate *r*_{0} = 0.06. As can be observed, the interest rate process becomes more and more jagged as Δ gets smaller ...