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This is essentially the same as Black's model for pricing bond options in Section 22.2. The bond price volatility is o(s — T) and the standard deviation of the logarithm of the bond price at time T is o P . As explained in Section 22.3, an interest rate cap or floor can be expressed as a portfolio of options on zero-coupon bonds. It can therefore be valued analytically using the Ho-Lee model. The Ho-Lee model is an analytically tractable no-arbitrage model. It is easy to apply and provides an exact fit to the current term structure of interest rates. One disadvantage of the model is that it gives the user very little flexibility in choosing the volatility structure. The changes in all spot and forward rates during a short period of time have the same standard deviation. A related disadvantage of the model is that it has no mean reversion. Equation (23.12) shows that, regardless of how high or low interest rates are at a particular point in time, the average direction in which interest rates move over the next short period of time is always the same.

THE HULL-WHITE MODEL

In a paper published in 1990, Hull and White explored extensions of the Vasicek model that provide an exact fit to the initial term structure. 9 One version of the extended Vasicek model that they consider is

dr = [0(t) - ar] dt + odz (23.17)

where a and o are constants. This is known as the Hull-White model. It can be characterized as the Ho-Lee model with mean reversion at rate a. Alternatively, it can be characterized as the Vasicek model with a time-dependent reversion level. At time t the short rate reverts to 0{t)/a at rate a. The Ho-Lee model is a particular case of the Hull-White model with a = 0.

The model has the same amount of analytic tractability as Ho-Lee. The function 0(t) can be calculated from the initial term structure:

9(t) = F t (0, t) + aF(0, t) + A -(l - e- 2at ) (23.18)

The last term in this equation is usually fairly small.. If we ignore it, the equation implies that the drift of the process for r at time t is F t (0, t) + a[F(0, t) — r]. This shows that, on average, r follows

See J. Hull and A. White, "Pricing Interest Rate Derivative Securities," Review ofFinancial Studies, 3, no. 4 (1990), 573-92.

Initial forward curve

the slope of the initial instantaneous forward rate curve. When it deviates from that curve, it reverts back to it at rate a. The model is illustrated in Figure 23.4. Bond prices at time t in the Hull-White model are given by

P(t, T) = A(t, T)e' B(t - T)r(t)
Bit, T) =1 ----- e-<

Equations (23.19), (23.20), and (23.21) define the price of a zero-coupon bond at a future time t in terms of the short rate at time f and the prices of bonds today. The latter can be calculated from today's term structure.

As in the case of the Ho-Lee model, it is more relevant to relate P(t, T) to R(t), the <5t-period rate at time f. We can show (see Problem 23.23) that

The price of a put option on the bond is

KP(0, T)N(-h + a P ) - LP(0, s)N(-h) (23.26)

As with the Ho-Lee model, these equations do not require the initial zero curve to be differenti- able.

The price at time zero of a call option that matures at time T on a zero-coupon bond maturing at times is

LP(0,s)N(h)-KP(0,T)N(h-a P ) (23.25)

where L is the principal of the bond, K is its strike price,