A New
Member of AD&Co: The Two-factor Gaussian Term Structure, Part 2
By Alex Levin
Last month we introduced a two-factor Gaussian model, the latest addition
to AD&Co's suite of term structures. I pledged to touch on an intriguing
and practically important question: what financial instruments are valued
differently when moving from a single-factor view to the two-factor view?
Since a correctly calibrated two-factor model simulates the rate collection
in a much more realistic and accurate fashion than any single-factor model,
it seems at first glance that two- or more-factor modeling may reveal values
and risks way beyond the primitive picture drawn by any single-factor model.
As paradoxical as it may sound, though it's easy to perceive an instrument as mis-valued by a single-factor model, most MBS by types and an absolute majority of them by outstanding volume, will not be valued materially differently if we switch the business regimen to the use of the two-factor model. Whereas some instruments and exotic options certainly require two- or more-factor modeling, we see the most important role of this new model in assessing the interest rate risk, not in finding today's value (or OAS).
Common Perception of
MBS
During my career I have asked a number of practitioners about the "2-factor
versus 1-factor" dilemma. The most common ("collective") answer
was as such: the value of the embedded prepayment option depends on the correlation
between a long rate that drives the prepay speed and the short rate that is
used for discounting (see Kazarian et al [1998], or Belbase [2000]). Hence,
the use of a realistic two-factor model should deflate the prepay option and
increase the value of MBS.
What is Wrong with
the Common Perception?
If the common perception were right, it would affect European options too.
Consider, for example, a European swaption. The exercise is triggered by the
long coupon rate; the discounting is done using an arbitrage-free sequence
of the short rates, yet its price is known and independent of the rate model
selection. Levin [2001] has given a simple argument reminiscent to the classic
Black-Scholes setting that should we model the prepay option as a sequence
of European pay-offs on a single long rate and fix volatility of this rate
beforehand, we won't find material dependence on the model specification for
most MBS. This statement seems surprisingly robust: it holds true even if
we relax stiff assumptions. For example, we may assume that prepay decisions
are based on a "remaining-maturity" rate rather than on a single
tenor.
Since the majority of commonly used practical models simulate prepay speeds as the sequence of European options on the MBS current market rate, volatility selection effectively "freezes" the value of a prepay option. Perhaps, the most important point that many practitioners miss is that model selection should be studied after an option volatility matrix is matched. This leaves much less freedom to deviate from the value given by a single-factor model.
What Makes the Difference?
A) Asymmetric response to curve's twist. Note that so far we bound
the prepay speed to a single rate. What if the prepay option is triggered
by the difference between two points of the curve? If the dependence is linear,
an MBS would lose and gain the same amount from steepening and flattening,
thereby causing no need for a prepay option overhaul. It is only when the
pay-off depends on the curve shape asymmetrically that a curve's random twist
changes the value.
The Curve-At-Origination (CATO) prepay effect can serve a good example. If the yield curve steepens from origination, homeowners who originally elected a fixed-rate mortgage may review their choice and switch to an ARM. If the curve flattens, they will not change their selection. Hence, CATO causes an asymmetric prepayment signal in response to random twist.
Another example of asymmetric pricing can be found in floater and inverse floater IO tranches. Suppose that x denotes the random twist factor; without loss of generality, we can center it on zero by interpreting x as the deviation of the curve's slope from the historical average. Suppose further that the positive value of x corresponds to steepening. The value of a 1% fixed-rate IO will increase with x due to slower prepayments; let us write it down as a + bx (where b > 0), ignoring higher-order terms. The actual floating coupon will change with x as c + dx where d is positive for an inverse floater and negative for a direct floater. Hence, the total value of an IO will be (a + bx)(c + dx). Let us take mathematical expectation of this expression; keeping in mind that the expectation of x is zero, we ultimately get ac + bds2. Therefore, the slope's volatility (s) boosts the value of an inverse IO, which pays a higher rate longer and lower rate shorter: vice versa, it deflates the value of a direct IO floater.
Once the effect of using two-factor model is proven for IO floaters, it can be extrapolated to some other CMO tranches having embedded IO floating components.
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.
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 the term structure, flat and steep. Once in the model,
we expect CATO to cause a small (1-2 basis points, at most) OAS difference
between 1- and 2-factor worlds. The reasoning is as follows:
A two-factor model can both increase and reduce the value of prepay option modeled with CATO. In order to explain this paradox, let us account for the limited contribution that CATO can make in a prepayment forecast. If the market curve is steep, the CATO-related prepay acceleration effect may already be saturated. Hence, the two-factor model presents a greater chance for deceleration via flattening, i.e. the value of a premium MBS can improve. We ran some preliminary versions of a CATO-embedded model for 8/31/2004 when the curve was steep to see a tiny OAS improvement within accuracy of pricing.In contrast, a similar study performed on 2/11/05 (much flatter curve) suggested up to 1.0-1.5 bps OAS tightening when using the two-factor model. Indeed, a flat market curve can steepen, thereby triggering refinancing to ARMs.
Two-factor Risk Management
Using the two-factor
model changes the price of few instruments. However, it simulates the historical
curve's dynamics 3-4 times more accurately than does a single-factor model
(Levin [2001]). Therefore, hedging against rate moves ("principal components")
that stem from the two-factor view may be beneficial for maintaining a market-neutral
position. As we demonstrated in this article, a single-factor pricing model
can do very well, but a single-factor risk assessment that relies solely on
the parallel-shock sensitivity is not a viable practical option. The real
choice remains between hedging principal components coming from a two-factor
model or using a set of "key rates." The former is more elegant
and quick; the latter is more accurate and time-consuming.
References
E. Belbase, "A Lattice Implementation of the Black-Karasinski Rate
Process," Quantitative Perspectives, Andrew Davidson & Co.,
Inc., June 2000.
D. Kazarian, D. Joneja, and W. Huan Ren, "The Lehman Brothers Yield Curve Model: Implication for MBS, Lehman Brothers," MBS and ABS Research, 1998.
A. Levin, A Linearization
Approach to Quasi-affine Modeling Coupon and Swap Rate Term Structures and
Related Derivatives, in M. Avellaneda (ed.), Quantitative Analysis of Financial
Markets, collected papers of the NYU Mathematical Finance Seminar, vol.3,
World Scientific Publishing, 2001, pp. 199-221.
Appendix A. Comparative valuation for fixed-rate TBAs
Appendix B. Comparative valuation for fixed-rate TBA
Appendix C. Comparative valuation for fixed-rate TBAs
