## 2.8 Bond pricing formula

Allowing the bond price to take the form:

we have ^{ [18] } :

and so the bond price equation (2.18) reduces to:

Since the left-hand side is a function of the short- term interest rate * r * ( * t * ), while the right-hand side is independent of the short-term interest rate, the following two equations must be satisfied:

To solve for * A * ( * t, T * ) and * B * ( * t, T * ) first consider (2.29). This is a Ricatti equation ^{ [19] } with solution * B * ( * t, T * ) = * v * ( * t, T * )/ * u * ( * t, T * )where * v * ( * t, T * ) and * u * ( * t, T * )are solutions to the following system of equations ^{ [20] } :

Set = * T * ˆ’ * t * where * T * is the bond maturity date; then and the above system of equations may be written as:

From (2.30) we have:

substituting into (2.31) gives:

Expressing this in terms of * D * -operators results in a simple quadratic equation:

The roots of this quadratic equation are ( ³ + » ˆ’ )/2 and ( ˆ’³ + » ˆ’ )/2 where and hence the solution may be written as:

where * k * _{ 1 } and * k * _{ 2 } are constants. Since * B * ( * T,T * ) = 0 = * v * (0)/ * u * (0), * v * (0) = 0 and hence * k * _{ 1 } = ˆ’ * k * _{ 2 } . Setting * k * _{ 1 } = 1 and * k * _{ 2 } = ˆ’ 1, * v * ( ) becomes:

Substituting (2.34) and

into (2.32) gives:

Since = * T * ˆ’ * t * , the solution of the Riccati equation is obtained from (2.34) and (2.35) as:

Now consider equation (2.28) with fixed bond maturity * T * , so the bond price is a function of * t * only. Hence:

where and are constants. Substituting * B * ( * t, T * ) from (2.36) into (2.37):

Let * y * = e ^{ ³ ( T ˆ’ s ) } then and . Making this substitution and noting that ( ³ ˆ’ » ˆ’ )( ³ + » + ) = ³ ^{ 2 } ˆ’ ( + » ) ^{ 2 } = 2 ƒ ^{ 2 } , the integral in the above equation becomes:

and hence the solution for * A * ( * t, T * )is:

An analysis similar to that in the derivation of Vasicek's model in Chapter 1 can be applied to determine the bond price dynamics. Since the bond price is a function of the short-term interest rate, Ito's Lemma is used to give:

where

The existence of a factor risk premium * q * ( * r, t * ), implies:

Vasicek assumes a constant market price of risk, while CIR specify:

and hence under the CIR model the bond price process is specified by:

^{ [18] } Where the subscript indicates a partial derivative.

^{ [19] } The general form of the Ricatti equation is:

The solution of this equation can be written as * w * ( * t * ) = * v * ( * t * )/ * u * ( * t * ) where * v * ( * t * ) and * u * ( * t * )are solutions of the associated system of first order linear equations:

For more details on solving the Ricatti equation see [ ** 47 ** ].

^{ [20] } * v * and * u * are functions of * t * and * T * , but * T * is fixed, hence * v * ² ( * t, T * ) and ( * t, T * ) denote the derivative with respect to * t * .

## 2.9 Properties of the bond price under the CIR model

### 2.9.1 Yield-to-maturity .

Since it is market convention to quote bond prices in terms of yield-to-maturity, it is more insightful to examine the behaviour of the yield-to-maturity in the case of a very short and very long time to maturity. Since a zero coupon bond is a pure discount instrument, its price is written as:

where * R * ( * r, t, T * ) is the yield-to-maturity. Equating (2.27) and (2.39) we derive the yield-to-maturity in terms of * A * ( * t, T * ) and * B * ( * t, T * )as:

As * t * ’ * T * , * R * ( * r, t, T * ) ’ * r * , ^{ [21] } since as the bond approaches maturity, it converges to an instrument with instantaneous maturity. Now consider the yield-to- maturity as * T * ’ ˆ . This may be viewed as the yield on a perpetual bond:

Consider ln * A * ( * t, T * )where * A * ( * t, T * ) is given by (2.38):

Now, since

we have:

Also from (2.36):

and hence the yield on a perpetual bond is ^{ [22] } :

Hence, for bonds with increasing maturity, the yield approaches a limit independent of current rate of interest, but proportional to the mean reversion level [ ** 45 ** ].

### 2.9.2 Possible shapes of the term structure.

As for the Vasicek model, the CIR term structure can assume various shapes according to the level of the current interest rate, * r * ( * t * ). See Figure 2.1 below. For , the long-term yield, the term structure is uniformly increasing while for the term structure is uniformly decreasing . For values of * r * ( * t * ) lying between these two extremes the term structure is humped.

Figure 2.1: Possible shapes of the term structure. = 0.3, » = 0, ƒ =0.6, = 0.15

^{ [21] } * e ^{ x } * can be approximated by its power series expansion as:

and hence [ ** 45 ** ]

Therefore as * t * ’ * T * we have:

and also

for small * T * ˆ’ * t * . Hence from (2.27), * P * ( * r, t, T * ) = e ^{ ˆ’ r ( T ˆ’ t ) } .

^{ [22] } Here make use of the following: