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Be Careful What You Model


Ross M. Miller
Miller Risk Advisors
May 9, 2005

[This commentary appears in the May/June 2005 issue of Financial Engineering News. I was surprised to discover that it was blasted by e-mail to subscribers late in April, so I received some early feedback on it. While most of the feedback was positive, two readers took serious issue with what I had to say. One reader insisted that Fischer Black would never go near CAPM and that I was somehow advocating the inappropriate use of risk-neutrality assumptions. Another reader claims that CAPM indeed gives the expected rate of return for a Black-Scholesian European call option. I have created a numerical valuation spreadsheet to demonstrate to the skeptical that this only works in the degenerate case of a risk-neutral world and that in general the CAPM return for the option is too high. Maybe someone will believe me.]

I grew up watching the television series The Twilight Zone. The standard formula behind its surprise ending was to lull viewers into the believing that the characters inhabit one particular world and then turn the tables on them in the final scene. Rod Serling, the show's creator and narrator, was keenly aware of the thin line that separates belief from reality.

For some time now, the financial markets have inhabited their own twilight zone of falling volatility, shrinking risk premia, and other (as Alan Greenspan would call them) conundrums. I submit for your approval an explanation of what has been going on.

Now that top physics journals are publishing finance papers, it seems reasonable to borrow a trick from that august discipline--the thought experiment. Imagine that you live in that most idyllic of places, Black-Scholes World (BSW). The air is fresh, volatility is both known and constant, markets never close, and so on.

You have in your hand (or on your screen) an at-the-money call option with a year until it expires. Because you are in BSW, you know exactly what that option is worth at the present moment in time. Consider this: What is your expected rate of return on the option between now and the option's expiration in one year?

An easy question, right? Think some more.

The typical profit-and-loss diagram for options, popularly known as the "hockey-stick," assumes that the funds invested in options earn a zero return regardless of the time until expiration. According to this diagram, the absolute return from an option is simply the terminal payoff minus the current cost. The possibility that one might require a positive return to compensate for the opportunity cost of funds tied used to finance the option is either ignored for the sake of pedagogical simplicity or relegated to a footnote.

Zero is clearly the wrong answer, so what about the risk-free rate? That was the nearly unanimous answer to my informal, nonscientific survey and it is what Paul Wilmott appears to be saying (if I understand his notation) on the top of page 34 of the first volume of his magnum opus on quantitative finance.

This answer might be defensible, but it is not what BSW's creators had in mind. A quick perusal of their 1973 blueprint shows that Black-Scholes World was intended as a mere subdivision within a larger CAPM (Capital Asset Pricing Model) World. Fischer Black and Myron Scholes examine the expected rate of return for a call option in their world and use it as the basis of an alternative approach to deriving their famous formula. While Black and Scholes wave their hands a lot, they never present a closed-form solution to the future value of an option over any finite period of time, including the time left until expiration.

The strange world in which every asset earns the risk-free rate of return for the life of the option is not Black-Scholes World, but a universe that I will dub Cox-Ross World (CRW) after the two economists, John Cox and Stephen Ross, who colonized this world in their 1976 Journal of Financial Economics article. (Cox and Ross explicitly refer to their theoretical construct as a "world.") CRW is a degenerate neighborhood of Black-Scholes World in which risk-neutrality rules. What Cox and Ross recommend (and what Black and Scholes allude to in an unpublished early draft of their famous article) is that when you have a messy option it usually pays to visit CRW to find its value.

It is not an exaggeration to attribute much of the revolution in derivatives and financial engineering to the widespread adoption of Cox and Ross's clever trick. An unintended side effect, however, is that in the nearly thirty years since CRW was discovered it has become greatly overpopulated with models lacking proper papers.

Two classes of "undocumented" models now call CRW their home. The first are models that did not meet the requirements to enter BSW and snuck into CRW instead. Most of them are now upstanding members of the community and everyone (save the pickiest economists) is willing to let their illegitimacy slide.

The second class of models is more troubling. These models, rightly or wrongly, got into CRW and never left. You see, the standard visa for getting into CRW does not last long; in fact, it expires the moment that it is granted. The risk-neutral paradise that CRW provides is only valid for a single point in time. As soon as the second hand on the clock moves, all bets are off and the model should be whisked back to a world where people need to be compensated for at least some flavors of risk.

Those who instinctively believe that all option investments, including the at-the-money call in my thought experiment, should earn the risk-free rate of return are trapped in the twilight zone of CRW. In contrast, a well-indoctrinated inhabitant of CAPM World should believe that the expected return on the option will reflect the beta that it inherits from its underlying security, creating a positive risk premium for calls and a negative risk premium for puts. Unlike the beta on the underlying, which is assumed to remain constant, this implied beta will vary over time, so calculating the option's expected return over any noninfinitesimal amount of time is a daunting proposition. The living is much easier in CRW than in the real world.

And this is where the plot twist comes in. What happens to the financial world if enough people (and, more importantly, their models) begin to believe that they reside in CRW? As Madge the Palmolive lady used to say, "You're soaking in it." And for quite a while, I might add.

Take the collapse in spreads on risky debt. In a risk-neutral world, yield spreads are just wide enough to cover the expected capital losses from adverse credit events. While unquestionably much of the tightening over the past few years has come from good news on the credit front, there appears to be more going on--vanishing risk premia.

And then there is the hunt for that elusive alpha that has launched a thousand hedge funds. Because alpha is notoriously difficult to pin down (much less capture) in CAPM World and its environs, much of that hunt has moved to CRW. (You won't find alpha there either, but don't tell anyone.)

It is natural to wonder whether all of this is just another recipe for disaster whipped up in the financial engineers' kitchens. Unfounded assumptions of option replicability (portfolio insurance in 1987) and market liquidity (LTCM in 1998) turned out to have a destabilizing effect on financial markets.

Maybe it will be different this time around. After all, there is nothing inherently unstable about a risk-neutral world, real or imagined. Where the potential instability arises is when everyone wakes up to the fantasy simultaneously. Such a sudden arousal could be triggered by GM, Fannie Mae, or any number of crises that lurk over the horizon.

It all boils down to this: Be careful what you model, you may end up living there. What's more, you never know when the series might be cancelled.

Copyright 2005 by Miller Risk Advisors and Financial Engineering News.