# 11.7 Ho and Lee model within the HJM framework

## 11.7 Ho and Lee model within the HJM framework

By way of an illustrative example, consider the continuous time equivalent of the Ho and Lee [ 27 ] model (studied in Chapter 10) within the above specified framework.

### 11.7.1 The model specifications.

Consider a model with a single Brownian motion and hence single volatility parameter ƒ 1 ( , t, T ) ƒ > 0 where ƒ is a constant. Assume Conditions 1-6 are satisfied with { f (0, T ): T ˆˆ [0, ]} the initial forward rate curve and ( , t ), t ˆˆ [0, ] themarket price of risk corresponding to the single source of randomness. From (11.32) the forward rate process with respect to the equivalent martingale measure is:

and hence the short- term interest rate process is specified as:

Since there are no restrictions placed on the evolution of the short-term or forward interest rates, there exists a positive probability of these rates being negative. To determine the bond price dynamics, substitute (11.38) into (11.2) to give:

From the definition of the forward rate (11.1), we have:

and so:

### 11.7.2 Pricing contingent claims.

Define the following notation:

 C ( t ) - time t value of a European call option on bond P ( t, T ), K - option exercise price, t * - option expiry date, 0 ‰ t ‰ t * ‰ T.

At expiry of the option, its value is:

Making use of (11.36), the time t value of the contingent claim is [18] :

An analysis, similar to that used in the initial formulation of the Black- Scholes valuation formula, yields the contingent claim price to be [19] :

with

and

The above formula is a modification of the Black-Scholes option pricing formula where the required volatility is the standard deviation of instantaneous returns on the forward bond price, that is the standard deviation of returns at time t * of a bond maturing at time T . This may be determined from (11.13), the bond price process within the current framework:

and so the standard deviation of the forward bond price is ƒ ( T ˆ’ t ) ˆ’ ƒ ( t * ˆ’ t ) = ƒ ( T ˆ’ t *).

[18] Here [ ·] denotes the expectation taken with respect to the equivalent martingale measure.

[19] Here N ( ·) is the cumulative normal distribution.

Interest Rate Modelling (Finance and Capital Markets Series)
ISBN: 1403934703
EAN: 2147483647
Year: 2004
Pages: 132