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Reality Swaps

by

Ross M. Miller

Miller Risk Advisors

www.millerrisk.com

March 13, 2006

*[This commentary, fully illustrated with penguins, appears in the
March/April
2006 issue
of *Financial Engineering News*.
This is the raw, unedited version of that
commentary.*]

Perhaps it was not such a good idea to see March of the
Penguins right before going to the 2006 American Finance Association (AFA)
meetings in Boston. Fortunately, many of the world's top academic
financial economists choose to congregate at a giant shopping mall
anchored by three hotels and the Hynes Convention Center to demonstrate
their fecundity, rather than on a barren Antarctic ice shelf.

In past years, the more worthwhile papers at the
meetings have been the theoretical ones—an area where academics possess
a comparative advantage over their industry brethren. Empirical studies,
on the other hand, are more problematic for the academy. Apparently,
generalizing from the presentations that I attended, it is some
fundamental breech of research protocol to interview participants in the
market that one is studying to get an idea of how things "really
work" when constructing their statistical models. I only mention this
because this article is about how markets can be used to reduce model risk
by constructing a bridge between such models and the underlying financial
reality.

My reason for attending the meetings, aside from the
social pull of the financial penguins, was to uncover a paper that I could
write about for this column. Amazingly, it turned out to be the
first paper on the program: "An Economic Motivation for Variance
Contracts," by Nicole Branger and Christian Schlag, both from Goethe
University in Frankfurt.

The standard example of a variance contract is the
CBOE's S&P
500 Three-Month Variance Futures contract. Variance futures are
exactly what the name would indicate: Futures contracts that are settled
based on the observed variance of the rate of return of the underlying
security (or, in this case, index) over the specified timeframe. In the
case of the CBOE's S&P 500 contract, this variance is computed from
daily returns taken over a 3-month period prior to the settlement date.
The resulting variance is then annualized and multiplied by 10,000 to give
a tradable number known as variance points.

These contracts are less-than-user-friendly because
virtually all traders think in terms of volatility (the standard deviation
of returns), and not variance (the square of that number). For example, an
annualized volatility of 12% (or 0.12), when squared becomes a variance of
0.0144. Multiplying by 10,000 produces 144 variance points. The value of a
single futures contract is computed as $50 times the number of variance
points.

Variance contracts move with changes in expectations of
future volatility. If the market thinks the volatility of the S&P 500
will rise from 12% to 13%, then that will, in theory, increase the number
of variance points from 144 to 169, generating a profit of $1,250 ($50 x
25) for each long contract. Over the life of the contract, its price in
variance points can be expected to trend downward as long as no spike in
volatility, such as one generated by a "jump" (especially a
downward one) in the price of the underlying index, materializes.

Given how seemingly unnatural these contracts are, it
should come as no surprise that they have gone untraded on the majority of
trading days since they were introduced on May 18, 2004. It should also
come as no surprise that academics would see fit to write papers
justifying their existence. (Neither author of the aforementioned working
paper appears to have received funding from the CBOE or any other
exchange, and I certainly have not.)

The whole raison d'être for the variance contract has
to be brought into question because the CBOE also offers a vastly more
liquid contract based on the volatility index for the S&P 500,
popularly known as the VIX. This index is computed from an implied
volatility model that uses the prices from an array of S&P index
options to derive a volatility number similar to the one can be backed out
of the Black-Scholes-Merton model. (The precise method for computing the
VIX is quite clever and the details are available at http://www.cboe.com/micro/vix/vixwhite.pdf.)
The VIX in its various incarnations is generally believed to work better
than historical measures of volatility as a predictor of future
volatility.

For all its simplicity and elegance, however, the
volatility index is still subject to prediction error because there is
more to movements in the S&P 500 than can be captured by it. The main
point of the Branger and Schlag paper is that because of these intrinsic
imperfections, as well as those of any other volatility measure derived
from options prices and the underlying S&P index, variance futures
provide traders and investors with something that the VIX does not. That
something is a way to hedge against the model risk that comes from using a
VIX-like model (particularly, one that does not allow for stochastic
volatility or index jumps) to forecast volatility. Unfortunately for the
CBOE, the way that their variance contract is constructed (along with the
tremendous liquidity risk associated with it), makes using the contract to
hedge model risk impractical in all but the most exceptional
circumstances.

If you really want to hedge model risk based on realized
returns, what you need is not a variance contract, but rather an
unexplained variance contract. In other words, something that pays off
according to the underlying model's tracking or prediction error. Such a
contract could be designed so that it is more valuable the greater the
absolute difference between realized volatility and that predicted by the
VIX model. (In typical academic fashion, I leave all implementation
details as "an exercise for the reader.")

This "reality futures" notion is not limited
to the VIX, but can also be applied to any model whose predictions can be
verified in a timely manner. The basic idea can be immediately extended to
forward contracts and anything derived from them. Hence, we can construct
reality swaps, caps, floors, swaptions—you name it. With these financial
instruments, users of a given model can insure themselves against a model
generating estimates that are too high, too low, or both.

Consider, for example, a hopelessly naïve model of
three-month Eurodollar rates that predicts that they will remain constant
indefinitely. The reality, of course, is that Eurodollars do fluctuate,
sometimes by quite a bit. Properly set up, this swap of the
"model" of fixed rates for the "reality" of floating
rates is identical to a plain-vanilla interest rate swap.

Aside from their speculative and academic appeal, these
products can help financial institutions cope with all kinds of model
risk, especially from the use of commercially available models with an
established client base. Models that purport to predict everything from
equity factor loadings to credit spreads could be put to the test on a
daily basis.

Of course, the creation of reality-based financial
products leads to an interesting circularity. As soon a reality contract
gains traction, traders will need models to value those contracts. For
such models to produce any value they would have to be capable of
explaining more of reality that the models whose behavior they were trying
to predict. Model evolution would be accelerated because the models used
to beat the models that underlie reality contracts today could well become
the new benchmarks for tomorrow's model. Global warming and all, the
future could be far more secure for penguins than it is for financial
models.

Copyright 2006 by Miller Risk Advisors and Financial
Engineering News.