and the Vasicek-Fong |20~ procedure is used to choose the parameters C(t) so that the analytic solution (5) produces the FNMA term structure of interest rates most consistent with the observed market prices of noncallable FNMA debt obligations.
To value the FNMA ISFDs, I use the analytical procedure described in the preceding section to calibrate the valuation model.
Third, the model was calibrated to the noncallable FNMA debt securities to estimate the parameters C(t).
With the parameters |Kappa~, |Sigma~, C(t), |Theta~(t), and |Lambda~(t) estimated from the observed market prices of Treasury and FNMA bonds, Equation (7) is solved subject to the boundary condition (4) to calculate the estimated value of each FNMA ISFD issue.
The model tends to overprice the FNMA ISFDs slightly.
The options imbedded in the FNMA ISFDs are compound options because the value of each option depends upon whether any of the options corresponding to earlier sinking fund dates were exercised.
In Equation (9a), if the call option is in-the-money, the FNMA retires a fraction |Mathematical Expression Omitted~ of the outstanding bonds, or |Mathematical Expression Omitted~ total principal amount, which exceeds the base sinking fund amount by |Mathematical Expression Omitted~.
00% over the preceding six months so that the call options imbedded in the 1993A, 1993B, 1998A, 1999A, and 1999B FNMA ISFDs were in-the-money, the put options imbedded in those issues were out-of-the-money, and the call and put options imbedded in the 1999C FNMA ISFDs were all at-the-money (the actual sinking fund percentage would equal the base sinking fund percentage if interest rates did not change prior to the next sinking fund payment date).
In Exhibit 5, I have also applied Equation (1) to calculate the price V(bond) of a conventional nonredeemable FNMA note implicit in the observed ISFD price and the calculated option values as of August 30, 1991.
As already noted, the FNMA redesigned the contingent sinking fund when it introduced the ten-year ISFD.
The FNMA issued five-year ISFDs first, and presumably after investors had familiarized themselves with the new security, opted for the ten-year maturity in order to achieve a closer duration matching.
Second, on December 11, 1992, the FNMA issued $150 million principal amount of 6.