make money online | forex made easy

Make Money Online

In the Ho-Lee model, zero-coupon bonds and European options on zero-coupon bonds can be valued analytically. The expression for the price of a zero-coupon bond at time f in terms of the short rate is

Pit, T) = Ait, 7>- r( ' )(r -') (23.14)

See T. S. Y. Ho and S.-B. Lee, "Term Structure Movements and Pricing Interest Rate Contingent Claims," Journal of Finance, 41 (December 1986), 1011-29.

Initial forward curve Time

Figure 23.3 The Ho-Lee model

In these equations, time zero is today. Times t and T are general times in the future with T A t. The equations therefore define the price of a zero-coupon bond at a future time t in terms of the short rate at time t and the prices of bonds today. The latter can be calculated from today's term structure.

For the remainder of this chapter, we will denote the <5r-period interest rate at time t by R(t) or just R. From equation (23.14) we can show (see Problem 23.22) that where L is the principal of the bond, K is its strike price,

1 LP(0,s) oP op P(0,T)K + 2 and

a P = o(s — T)VT

The price of a put option on the bond is

KP{0, T)N(-h + op) - LP(O, s)N{-h)

In practice, we usually compute bond prices in terms of R rather than r and so equation (23.15) is more useful than equation (23.14). Equations (23.15) and (23.16) involve only bond prices at time zero, not partial derivatives of these prices. They demonstrate that we do not require the initial zero curve to be differentiable in applications of the model.

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

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