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.