2.9 Properties of the bond price under the CIR model


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:




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

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