A common issue in the valuation of mortgage backed bonds is the modelling of the delivery option.
Binary method
Traditionally in RIO this has been modelled using a so called binary approach where prepayments are dictated to be zero if the value of the bond is lower than face value, assuming that the borrower who wishes to refinance, exercises the delivery option rather than the prepayment option.
This has a number of advantages. Firstly, it does not introduce any additional parameters which have to be estimated along with the prepayment models. Further, the underlying assumption is that the borrower is rational, given that the prepayment decision has been made. However, a series of tests has shown that this method can yield numerical instabilities, especially when calculating option adjusted spreads and sensitivity measures for at-the-money bonds.
Logit method
To allieviate this problem a new method for handling the delivery option is being introduced. This version uses a logit distribution to appoximate the option to exercise the delivery option and is implemented as a loan group specific adjustment. Two parameters are estimated, denoting the upper and lower boundary on prices, along with the normal parameters of the prepayment model. If the observed value of the bond is below the range, the delivery option is assumed to be exercised in full, if it is above the delivery option is not exercised. In between the upper and lower boundary the value of the logit distribution is multiplied to the prepayment rate, ensuring a smooth transition between exercise of the delivery option and exercise of the prepayment option.
Advantages
This has a number of advantages: First, numerical tests reveal that this method is much more stable than the binary method. The group specific values allow the modelling of the groups differently, so that for example smaller loan groups can be seen as acting less rationally while large firm loans will still be able to exhibit a binary behaviour with respect to exercise of the delivery option. Another attractive feature is that when looking at the time series behaviour of prepayment rates there will be a smooth transition between exercise of delivery option or not, rather than a 0-1 decision as in the binary model, making the time series pattern of sensitivity measures more stable.
Comparison
The figure below shows the development in CPR for two models estimated with the binary method (DMBS02Q4 green lines) and with the logit method (02Q4_M1 red lines). (Labels on bottom axis indicate the first in the given month.)
We see that while the DMBS model exhibits some discontinuities the M1 model exhibits a very smooth behaviour.
Similarly the figure below shows the development in market price, theoretical value and option adjusted duration.
