2.8 Bond pricing formula


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.

click to expand
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: