Arc Sine Law – A Trader's Dilemma
Apr 4, 2006
If you are a trader (whatever the asset class or derivatives, doesn’t matter) and if your P & L is totally random (it is a fair game) then your chances of spending 6 months in a year is either "red" (loss) or "black" (profits) is least likely. This may come as a complete surprise to you, if you are just joining the profession of trading derivatives or cash assets but ask an experienced trader and he will tell you this: there is a very high probability that you will either spend 1 month in a year in "black" (or "red") or spend 11 months in a year in "black" (or "red"). This empirical fact follows from the mathematical axiom of Levy's arcsine law and is perhaps the most counterintuitive aspects of Brownian motion.
Levy’s arcsine law in mathematical terms can be stated as* (you can ignore the formula):
If x is time (normally months) that a trader spends in "black" or "red" then the distribution of his P & L is given by the arcsine probability density function of x, where x is bounded between 0 and 1:
This striking arcsine law of Brownian motion also affects the distribution of the maximum and the minimum of a random walk. The distribution of the extrema of a Brownian motion will such that the random walk will hit a maximum very early on or very late in the process where the process is bounded by time, , where . This can be easily seen from the above graph.
Though a trader will very much like to spend six months out of a year in either "red" or "black" so as to maximize the expected value of his book, this is seldom the case. In all likelihood he will end up making a huge profit (or loss) early on in the game or very late in the game and the process will be such that either he will be very lucky so as to make money for 11 months in a year or pretty unlucky so as to lose money for 11 months in a year. This is the curse (or boon) of the arcsine law of random walk.
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