Mathematics for Traders
September 28, 2006
We are really tired of differential and integral equations, Black-Scholes, Markowitz, efficient frontier, vol surface, CPPI, synthetic puts and what not. Maybe it is because of our diminished intellectual capabilities. Every quant these days is a Ph.D. and every trader is a mathematician. We are embarrassed of ourselves. But honestly, how much math do you really need to make money in the markets and live happily ever after?
If you ask us, the following is the entire math (rendition in "plain English") you would need if you are contemplating a life as a trader in a bank or a financial institution.
Math Theorem # 1
A stop loss order is the best put option that you can ever buy. And a corollary of this is that the simplest Portfolio Insurance strategy, and perhaps the most effective, that a trader can implement is a stop loss order.
The conclusion: markets are essentially non-linear and do not acquire steady state; therefore, the only portfolio insurance that works is a stop loss order.
Math Theorem # 2
Life for traders follow the arc sine law of Brownian motion and it is highly likely that in a year a trader will spend either one month or eleven months in the "red" (making a loss) rather than spending six months in "red" and six months in "black".
The conclusion: a single year's results will never reflect the performance of a trader.
Math Theorem # 3
Our man, Keynes, got it right: diversification is counter-productive. And to paraphrase him, a small investment in a large number of companies in which a manager has very little knowledge (research has its limits) and information to reach a good judgment as opposed to a substantial investment in one single company where a manager's information and knowledge is more or less adequate is a losing strategy.
In other words, fund managers don't make money from their skills but rather from the management fees on the assets under management and gross stupidity of the investors. A trader, on the other hand, makes money from his skills.
Math Theorem # 4
Borel-Cantelli lemma for traders states that if there are a very large number of players in the market place then the probability that one of them will produce outstanding return in a year is very high. But the conditional probability that the same player will produce another consecutive outstanding return in year two after generating a outstanding return in year one is, in fact, quite low. Nassim Taleb calls this the "Monkeys on a Typewriter" syndrome.
The conclusion: if you are a trader and have an exceptionally good year then try to extract - if that is possible - as much bonus as you can from your present employer and then jump ship and try to get a hefty signing bonus from your next employer. The fund managers needn't worry about this lemma because whether they make profits or losses they fleece the investor on management fees and anyway 99% of the investors are either stupid or have too much money.
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