Where does a Black Swan Come from?
Rahul Bhattacharya September 9, 2012
Where does a Black Swan come from?
A definitive answer cannot be found in Nassim Taleb's book, Black Swan, which formalized the notion of Black Swans; it certainly cannot be found in the myriads of those pseudoscientific research papers, written by extraders and market experts masquerading as mathematicians, or in those best sellers written, once again, by extraders and quants trying to emulate Taleb. The answer can only be found within the probability theory, that is, if we do not thoroughly discredit it first. You cannot trash a branch of mathematics and then use the theories of the same to prove a point, as has become fashionable amongst many practicing traders and tradersturnedbest seller writers.
It is true that if we are talking about Taleb’s Black Swan then we cannot see it coming. Black Swan, the rarest of the rare event has no history of happening, no past data of its occurrence. In simple terms, to paraphrase Taleb, we would be bearing risk without having any idea about what these risks are. We will not have any expectation of a Black Swan and the entire frequentist approach of measuring the probability of a Black Swan event would be rendered useless. This is so because we choose to believe in a Gaussian world, a world of compact probability distributions, where expectations – the mean, or the first moment – and the deviation from that expectation – variance or the volatility – matters the most and influences all the outcomes.
Within the world of financial derivatives, where the Gaussian still rules supreme, there is no way to predict or even prepare for a Black Swan other than to work with fancy and esoteric volatility models that try to capture fat tails. This is, to say the least, an inadequate risk management approach.
Taleb himself realizes this point.He has thoroughly discredited the Gaussian distribution in all his writings and comments, including both his books, Black Swan and Fooled by Randomness. As a matter of fact, in many of his technical papers, he has discussed in detail the drawbacks of the Gaussian distribution and shown how the compactness of a Gaussian distribution is really an illusion. But somehow as far the mathematically modeling a Black Swan is concerned, for him, it has been more about philosophy than mathematics. At best, he has taken a trader’s refuge in asserting the importance of volatility of volatility in explaining the fat tails, the technical term for outliers in a set of observation. In both his bestselling books, even though he tears apart the Gaussian distribution, he does not really venture outside it. This is a bit disappointing but perhaps, understandable.
He is cognizant of the problem though. As a man who has spent decades trading financial derivatives and getting to see the dynamics of the financial markets from very close quarters, he knows that the real problem with risk management is that either you cannot measure the probability of an event at all or, more plausibly, the distribution is utterly pathological which gives infinite variance. Over the years, in many of his writings, including the technical book on options trading, Dynamic Hedging, he has alluded to Power laws, Pareto Levi distribution and all those things that are essentially non Gaussian. In fact, his theory is heavily predicated on the work of the FrenchAmerican Mathematician, Benoit Mandelbrot and the central idea behind Mandelbrot’s thesis: that asset returns follow some kind of a stable Paretian distribution where tails can be really fat and variance can get blown out of proportion. It is obvious from Taleb’s writings that he believes that just by building heteroscedasticity – changing volatility – in the model or incorporating volatility of volatility in the pricing of the financial derivatives is not enough. It is the distribution that is suspect and needs to be thrown out.
However, in the end Taleb has returned to his roots as an options trader. He has settled for a noncompact, deformed and highly adapted Gaussian distribution and explained a Black Swan in purely traders’ parlance. It is not Pareto Levy that blows the variance out of proportion and causes a Black Swan, it is simply the volatility of volatility, the fourth moment of a deformed normal distribution that causes fat tails, so we at the mercy of a trader’s volatility model. For Taleb, Pareto Levi distribution or some kind of a stable Paretian probability distribution might be able to explain the reality to a mathematician, but for a trader – and, in the end, for obvious reasons, his allegiance to the options trading community is the strongest – Pareto Levi is "simply the result of a very high volatility of volatility on very small volatility."
So a rarest of the rare event – a Black Swan – cannot be predicted, for the probability of such an event cannot be measured; or, as I stated above, more likely it is because the volatility is so huge, almost infinite, that makes any prediction meaningless.
I would still say that this entire obsession with volatility of volatility and all these fancy volatility models take us nowhere near a Black Swan. A Black Swan is hidden inside a stable Paretian distribution. If we are looking for one, then all we have to do is to look inside any pathological probability distribution. If we play around with the Cauchy distribution we’d find a flock of these Black Swans appear every now and then.
Reference:
On the Unfortunate Problem of the Nonobservability of the Probability Distribution, Nassim Nicholas Taleb and Avital Pilpel, (2001, 2004), Fooled by Randomness.com.
The Black Swan: The Impact of the Highly Improbable, Nassim Nicholas Taleb, Random House, 2007.
Dynamic Hedging, NassimTaleb, John Wiley & Sons, 1997.
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