Valuation Commentary
How to Validate an Interest Rate Model
Part II: Valuation of floaters
By Alex Levin
Part I of "How to Validate an Interest Rate Model"
was featured in the February 2004 Valuation Commentary of The Pipeline
(click here to read). In that article,
we discussed pricing Swaps and bonds. This month, we turn our focus
to the valuation of floaters
Since any MBS system must manage ARMs, it certainly
should be smart enough to figure the index rate in the course of simulations.
Surprisingly, this is not a trivial task at all. Monte-Carlo sampling
simulates only the short rate or, generally speaking, market "factors"
on its own; longer rates remain to be constructed. When using the
Hull-White model, exact analytics can link any-maturity zero rate
to the short rate. Such a convenience is not presented in most non-linear
models, such as the Black-Karasinski model.
In the AD&Co system, this is not a significant issue.
Sampled paths are randomly directed through the lattice nodes where
1-mo, 1-yr, 2-yr, and 10-yr rates have already been pre-computed.
Small imperfections in ARM index forecasting can be caused by the
shortcuts we implemented to boost speed and efficiency:
• AD&Co. uses
a progressively sparse pricing grid. Time steps between two sequential
branching nodes range from 2 months
to 12 months. Hence, index rates falling between the nodes should
be interpolated, leading to slight errors.
• An ARM index (such
as the 3-yr) may differ from the pre-computed set. One can always
ask the library to compute the missing-maturity
rate on a lattice. Alternatively, AD&Co. takes a shortcut and
interpolates using the four rates listed above and forward curves
for all participants of this approximation.
We apply the same approach when running a CMT ARM being valued off
the LIBOR curve, without constructing
an additional lattice of Treasury rates.
Test 3. CMS Valuation of perfect floaters indexed to the pricing
benchmark
The easiest way to validate our rate reconstruction is to run a perfect
floater. Specifically, we set a >>>
