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B) Dependence on the joint distribution (correlation) of rates measured at various future points in time. American and Bermudan options have logistic pay-offs that depend on the joint distribution of rates and the correlation between them at various time-points. For example, with just two possible exercise days, t1 and t2, Bermudan option pricing will depend on the joint distribution of the term structures. Any exercise decision made at time t1 will depend on the volatility of the underlying measure between t1 and t2. Mathematically, this volatility is a function of 2 marginal volatilities seen at time zero for t1 and t2, correspondingly, as well as the correlation between the two values of the underlying.1
Interestingly enough, changing the mean reversion in a single-factor model can alter the correlation between those two values. Hence, the value of American or Bermudan options can vary even within a single-factor model. Generally, one can draw on this argument and prove that two-factor modeling bound to a given set of European options results in a lower value for an American or Bermudan option.
Valuation Comparison - For the Record
In Appendices A - C we present a comprehensive comparison between
valuation results obtained by different models for various financial
instruments. We reiterate that every model we considered was calibrated
to the same set of swap rates and ATM swaption volatilities. The accuracy
of volatility calibration varied somewhat and certainly contributed
to the pricing results. Since the two inter-rate correlation parameters
employed as inputs do vary historically, we show results for several
input sets. The "100/100" case is the Hull-White model, and
the "8/1" case is actually close to it, too. In each of these
two extreme correlation set-ups, one model's factor is either absent
or dominated by the other. In between, we have true two-factor models
with the "90/70" case being the closest to the 1995-2000 implied
swap rates behavior (see last-month's
article).
The results prove that both fixed-rate MBS and even hybrid ARMs are
valued within 1 bp of OAS, i.e. not much beyond sampling accuracy, regardless
of the number of factors and the inter-rate correlation.
The CATO effect
The above analysis was completed using the AD&Co. OAS system backed
by the most commonly used 4.3.3 family of prepay models. It does not
have the CATO effect and, as such, may not show the role of two-factor
modeling to its full extent. We plan to fully investigate valuation
consequences of CATO under the two-factor modeling when this effect
is finalized in the 5.1 prepay model. According to some street research,
the curve dynamics may contribute up to 6-8 CPR difference between two
positions of >>>
