Risk Latte - Mathematician and the Trader: Grappling with the Moments of a Gaussian distribution

Mathematician and the Trader: Grappling with the Moments of a Gaussian distribution

Team Latte
May 27, 2011

In 1994 Pierre Boucher graduated from a prestigious Engineering school in France with an advanced degree in mathematics and went to work for a large French bank in Paris. On his first day of job, after the usual orientation programme (equivalent of a boot camp), his manager, the head trader of the G7 swaps desk asked him a question that he still cannot forget.

"What is the difference between a probability theorist and a trader?" the head trader asked. Pierre was clueless so he just kept quiet (also, this was 1994 and the culture in a French bank was very much of subservience to higher intelligence or higher intellect and lots of respect for a senior member of the team).

The head trader shook his head and answered his own question.

"Un mathématiciencroit en la symétried'une distribution gaussienne, pour luices moments singuliersn'existent pas, en fait, le monde du mathématicien se termine avec le second moment de la distribution gaussienne. Voussavezl'écart-type?C'esttout. Iln'y a rien au-delà. D’un autre cote, un trader, doit vivre avec, et surune base quotidienneappréhender, au moins les dix premiers moments de la distribution. Etc’esttrès brutal lorsquevousvousdéplacez au-delà du second moment."

Translated into English, it means:

"For a mathematician odd moments of a Gaussian distribution don’t exist; in fact, a mathematician’s world ends with the second moment of the Gaussian distribution. You know the standard deviation? That’s it. There’s nothing beyond that. A trader on the other hand has to live with, and on a daily basis grapple with, at least the first ten moments of the distribution. And it is very brutal when you move beyond the second moment."

These words would come to haunt Pierre time and again and over the next sixteen years he will witness the decimation of many a trading career due to the third, fifth and the seventh moment of the Normal distribution, so much so that the whole issue of using a Gaussian distribution to model markets would become the biggest joke for him. It would be these beastly higher moments of the distribution that would keep him awake at nights.

However, he had somehow managed to ask the head trader, rather naively, that day, "So how do get rid of these higher moments? By hedging your moves?"

The head trader was incredulous, as if someone had just slapped him on the face.

After a ten second silence he replied, "No, of course not. If you want to get rid of the higher moments of distribution you simply close your book and get a jobat an engineering firm."

So what are these higher moments?

The first four moments of a Normal distribution are the Location of the centre (mean), the width of the distribution (variance), the measure of asymmetry of the distribution (skew) and the measure of the shape of the tails (Kurtosis) respectively. Mathematically, they can be expressed as:

Mean is the first raw moment. The other moments are called central moments, i.e. moments around the mean. The central moment is given by

An elegant way to calculate the higher moments of a Normal distribution is through its moment generating function. The moment generating function is given by the Fourier transform of the probability distribution function (PDF)

If we assume, the distribution to be symmetrical then the third moment, the skew, and all other odd moments vanish. Many probability theorists work with only the even moments of a Normal distribution. All even moments of a Normal distribution are dependent on only one parameter, the standard deviation, .The even central moments of a Normal distribution are given by the general formula:

For example, the 6th moment of the distribution will be given by

And the 8th moment of the distribution will be given by

Any mathematician or even a physicist would dismiss a 6th moment of a Normal distribution. But an option trader will not, especially not if they are trading compound options.


Reference:
- Statistical Methods in High Energy Physics, Michael Schmelling, MPI for Nuclear Physics
- The Handbook of Convertible Bonds, Jan de Spiegeleer and WimSchoutens, (John Wiley & Sons, Ltd., 2011).

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