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
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
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