Risk Latte - Borel’s Law and the Lehman Brothers Bankruptcy

Borel’s Law and the Lehman Brothers Bankruptcy

Rahul Bhattacharya
September 16, 2009

A year ago, Lehman Brothers went bust. Everyone thought the world was going to go to hell in a hand basket. People wrote obituaries and braced for the worst. Many questions were asked, most were answered. For the last one year all kinds of analysis and debate have been going on at all levels.

However, the biggest question remained unanswered. The Queen of England had asked it in a slightly different and a more general manner when she visited the London School of Economics last November. Her question was how come no body – by which she of course meant the economists – saw it coming. The question then remained: how come nobody saw Lehman crumbling and disappearing into the dust of history.

The simple answer was that the probability of that event happening – i.e. Lehman Brothers going bust – was vanishingly small. In fact, it was so small that the event was certain not to happen.

Dick Fuld and his minions, it seems, had already met Emil Borel.

You are driving home from work and around the bend you suddenly see a dinosaur crossing the road. Of course, you are dreaming. How likely is it that you’ll encounter a dinosaur crossing your path the next time you drive home from work? How likely is it that you’ll get a phone call from Angelina Jolie inviting you to have lunch with her at the Beverly Wilshire? How likely is it that John Thain will appear on TV and say “I screwed up”?

Emile Borel was a French mathematician who at the turn of the twentieth century – around 1909 – made the world of probability stand on its head. That was good. If you don’t believe me, then ask Nassim Taleb who will vouch for the Borel-Cantelli lemma. Anyway, Borel made major contributions in the field of probability theory in mathematics but what we today popularly know as the Borel’s law is not one of technical discoveries that he made in the field of mathematics. Borel’s law was simply a comment that the famous mathematician made in some of his non-technical writings in the early 1960s. How did it come to be called the Borel’s law is a bit of mystery but that is of no concern here.

Everything that Borel did – his entire body of work in the field of probability – is closely related to the way probability theory is applied in the field of quantitative finance. In fact, all theorems of modern probability theory make use of something called the Borel Field and Borel-Cantelli lemma, the lemma that Nassim Taleb popularized in his first book, Dynamic Hedging, has applications in applied math and computer science.

Therefore, it is ironical to find that Borel’s law, the non-technical, non-mathematical, rule of thumb kind of a conjecture that Borel made towards the end of his life, should be so totally at odds with what we see in the world of finance and financial markets today.

Everything that we see in the financial markets around us is a refutation of the above mentioned Borel’s law.

Borel’s law simply states that an event which has extremely low probability of happening will not happen. He defined a cosmic scale of 1050(this number is extremely large and is equal to 1 followed by 50 zeros) and postulated that events which have a probability of 1/1050 or smaller will not happen. Since the number 1050 is incomprehensibly large, a probability of 1/1050 is so small that it is almost - but not equal to – zero. What Borel called the Single Law of Chance "phenomena with very small probabilities do not occur"**

So, Lehman blow up was not supposed to occur. Was the probability of Lehman blowing up equal to or less than 1/1050 ?

Who knows; but one thing is certain most senior executives at Lehman in the fall of 2008 subscribed to Borel’s law whether they were cognizant about it or not. The argument was that Lehman was invincible, Lehman was infallible, nothing can go wrong and so on and so forth. It was like Borel’s law was wired all over their heads.

**see Borel (1962), also http://synechism.org/tpop/spae.pdf

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