Risk Latte - Arc Sine Law of Brownian motion and the Mathematics of Luck

Arc Sine Law of Brownian motion and the Mathematics of Luck

Team Latte
June 24, 2012

Why do some traders make a lot of money while a vast majority of them incur great losses, so much so, that some almost break their banks? The recent losses or credit derivatives trades at J.P. Morgan and some of the other legendary losses, such as the LTCM case or the Soc Gen case come to mind. "London Whale", code name for Bruno Iksil, the trader who lost almost $2 billion for J.P. Morgan, they say was simply unlucky this time. He and his team at the CIO’S office of J.P. Morgan made an awful lot of money trading on credit derivatives and other securities over the past 4 to 5 years and so everyone, including perhaps, the CIO and the Chief Risk Officer of the bank, thought that he was a genius who would continue to turn in profits. Alas, his luck ran out. The man on the other side of this trade was Boaz Weinstein, a 38 year old hedge fund manager who runs Saba Capital Management in New York. As Bruno Iksil and J.P. Morgan bled Mr Weinstein pocketed his riches from the trade. One could say, that the Lady Luck smiled on Mr Weinstein.

One may say that Mr Weinstein is a genius (just as many said, Bruno Iksil is one) but it must not be forgotten that during the financial crisis he was part of a team in Deutsche Bank which lost nearly $2 billion on credit derivatives trades. One can say, he was unlucky then.

It's all about luck. Luck plays an important role not only in trading but in many other professionals where the dominance of negative skew can be seen. For example, in a year out of say, 100 Hollywood movies, only 4 or 5 will be a big commercial success, a couple of them will break even and the rest will lose money hand over fist. Out of a 1,000 entrepreneurs struggling with their start-ups only one will be able to produce a Facebook or a Google and the rest will be committed to the dustbin of history. The same goes for book publishing and the music industry.

Anyone who has read Nassim Taleb's Fooled by Randomness would recognize the role of "randomness", the elusive, but omnipresent, Brownian motion that drives the fortunes of traders and banks. In an earlier work, Dynamic Hedging, Taleb refers to the Arc Sine law of Brownian motion though he refrains from giving a detailed mathematical description of this topic. This law states that, contrary to what we might expect from conventional wisdom, in a 12 month period a trader is most likely to spend 11 months in black (profits) and 1 month in red (loss) or the other way around, i.e. 11 months in red (losses) and 1 month in black (profit). Ordinarily, we would think that in a 12 month period on an average a trader will spend 6 months making money and the other 6 losing money. Arc Sine law says that is blatantly wrong!

What the Arc Sine law of Brownian motion (which Taleb calls the Arc Sine Law of Trading) tells us is that a random arithmetic process cannot always be positive or negative for long periods of time. Regardless of how skillful you are, if the underlying process is a random arithmetic process, you are bound to run out of luck.

Indeed, if there is any one mathematical law that ravages our lives – or laces it with gold – in the financial markets, thereby establishing the doctrine of luck, it is the Arc Sine law of Brownian motion. For those in the markets and with a modicum of interest in the theory of finance, this may be the only law worth remembering. For a mathematical description of arc sine law see the excellent book Financial Derivatives, Pricing, Applications and Mathematics by JamilBaz and George Chacko. You can also read the article Arc Sine Law – A Trader’s Dilemma on our website.

Assume that is a stochastic (random) process that follows arithmetic Brownian motion, like, the profit and loss of a trader, and is given by:

The above equation says that the money in a trader’s account can be both positive (profits) or negative (losses).

If, then the probability that the process - the trader’s profit and loss – will hit zero at least once in a time interval is given by:

And, the probability that is always be positive or negative – i.e., the trader consistently makes profits or incurs losses – in the time interval is given by

If you input and in the above formula then you’d get the following result.

This means the probability that in a time interval , where we have approximated the as , denoting a time period of one year, the probability that is always be positive or negative is 0.64%. This is quite a small probability. In other words, it is highly unlikely that over a period of one year a trader will consistently make profits or losses. His skill – or, the lack of it – is completely negated by the arc sine law. Instead of one year, change the time interval to 5 years, i.e. choose . The probability comes out to be 0.13%, even smaller. A trader’s luck has to run out somewhere.

Assuming that the trader started out with a very modest capital (this is an assumption to make the process consistent with ) and built his fortune over time, then over a 5 year period, i.e. the probability that , that is the trader will lose his shirt, is given by

This means that there is a big possibility – nearly certain – that all profits made by the trader will be wiped out in a 5 year period.

Dynamic Hedging by Nassim Taleb, John Wiley & Sons
Financial Derivatives, Pricing, Applications and Mathematics by Jamil Baz & George Chacko, Cambridge University Press

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