Thursday, May 19, 2011

Backgammon Cultivates Investing Insight

I play backgammon on Yahoo and I just played a game which was instructive to me as an investor. Backgammon, like many games, is driven by the roll of dice, which is as much as you need to know about the game to understand the learning-point of my game. This particular game developed to the point whereby if I were to  roll a 10 I could obtain a substantial advantage. Due to neglect by my opponent the 10-shot opportunity remained on the board for about 8 rolls and when I eventually hit it I was accused of outrageous "luck" owing to the fact, I was told, that the chances of 10 coming up on any given roll was so small. I countered that the large number of attempts he afforded me made it so that I did not consider my luck to be terribly outrageous.

 
The problem of figuring the odds of 10 coming up in the next 8 rolls of a pair of dice is essentially the same as probability of touching calculations. The analysis tab on the ThinkOrSwim trading platform knows from implied volatility what the chances are of a distant stop price randomly coming up in the next day. This is determined by calculating where your stop price falls on a normal distribution curve and summing the area under the curve. If your stop is far way then there is a very high probability that the market will not randomly encounter your stop price on the next roll, er, day.

 
Now probabilities are numbers between 0 and 1 and even if your next-day chances of getting stopped-out are very low they will always be more than zero. Your probability of getting stopped out in the next two days is one minus the product of not getting stopped out tomorrow with the probability of not getting stopped out on the day after. This calculation is easily extended to any number of days in the future by raising the probability of not getting stopped out tomorrow to the power of the desired number of days. This result gets exponentially smaller the more times you repeat the multiplication and therefore the probability of random fluctuations stopping you out becomes a near certainty over time. Try multiplying .9 by itself a number of times on a calculator. How many times does it take for the result to become less than .5? less than .05?

  • There are about 250 trading days in year, and raising almost any fraction to the 250th power is going to be a number very close to zero.
 
 You see you can always find a time period for which the probability of randomly touching almost any price is almost certain. This is because the normal distribution curve never intersects with the zero line - it extends infinitely to the plus and minus and only approaches zero asymptotically. Thus any price you care to name has a non-zero probabililty of occuring randomly for any stock. One minus a small number is still less than one and if you multiply any such near-one number against itself enough times you will always be able to reduce the result to near zero. This just says that if you wait long enough random fluctuations will produce your price. 

This is absurd, of course, because it says that all stocks will encounter a very large range of prices in a relatively short amount of time.

  • The fact that stock prices don't range widely in the universe of numbers proves that stock prices are not the product of a pure random walk.

At some point human beings step in to confine the range of prices. Within the range, random fluctuations abound as the market attempts to discover valuation. Outside of the range, human emotions of fear and greed intervene to push the price in a non-random direction - back towards the mean.

 
As for the backgammon problem, there are three ways to roll a 10: 6/4, 4/6 and 5/5. Thus, the probability of 10 on the next roll is 3 out of 36, or 1/12. Another way to think of this is that there is a 11/12 (0.917) chance of 10 not coming up on the next roll. In the next eight rolls the chance of 10 not coming up is .917 raised to the power 8,  a number around 1/2.

 
So I ask you, was it so outrageous that I should be the beneficiary of such a coin flip!?


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