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Extension to Other Models

The procedure that has just been outlined can be extended to more general models of the form

df(r) = [6(t) - af(r)] dt + adz (23.30)

This family of models has the property that they can fit any term structure. 14 As before, we assume that the <5t-period rate, R, follows the same process as r:

df(R) = [6(t)-af(R)]dt

We start by setting x = f(R), so that

dx = [6(t)-ax]dt + adz

The first stage is to build a tree for a variable x* that follows the same process as x except that 9(t) = 0 and the initial value of x is zero. The procedure here is identical to the procedure already outlined for building a tree such as that in Figure 23.8.

As in Figure 23.9, we then displace the nodes at time iSt by an amount a, to provide an exact fit to the initial term structure. The equations for determining a, and Q t ; inductively are slightly different from those for thef(R) -R case. Q o ,o = 1- Suppose that the 6,-,/s have been determined for i' < m (m > 0). The next step is to determine a m so that the tree correctly prices an (m + \)St zero-coupon bond. Define g as the inverse function of/ so that the <5t-period interest rate at the jth node at time m St is

9(a m +j8x)

The price of a zero-coupon bond maturing at time (m + l)<5t is given by

P m+ i= J2 Qmtp[-g(a m +jSx)St] (23.31)

14 Not all no-arbitrage models have this property. For example, the extended-CIR model, considered by Cox, Ingersoll, and Ross (1985) and Hull and White (1990), which has the form

dr = [9(t) - ar] dt + ojrdz

cannot fit yield curves where the forward rate declines sharply. This is because the process is not well defined when d(t) is negative.

This equation can be solved using a numerical procedure such as Newton-Raphson. The value a 0 of a when m = 0 is f(R(0)). Once a m has been determined, the Q, y for i = m + 1 can be calculated using

where q(k, j) is the probability of moving from node (m, k) to node (m + 1,j) and the summation is taken over all values of k where this is nonzero. Figure 23.10 shows the results of applying the procedure to the model

d ln(r) = [6(t) - a ln(r)] dt + adz when a - 0.22, a = 0.25, St = 0.5, and the zero rates are as in Table 23.1.