"The deviousness of Black Swans ...".

Although we are in a new trading year, and the recent statistics such as GDP look good, I am far from sure that we are in a stable and upwardly mobile market. In addition I wonder if the past wisdom of "Das Uber President" is still intact, or more lately if it has deserted him in our time of economic need?

As recently as January 20th through to 22nd we had three consecutive days of around one to one and a half standard deviation moves in the DOW. This started me thinking about the probability of such a move. For example if one standard deviation is 68% then the probability of price moving outside this range is 32%. Not to bad, but what is the probability of three days in a row?

Market prices of the DOW as at 28th January 2010.

Under the Central Limit Theorem, price moves should be normally distributed, and as an adjunct to this, the law of large numbers would see the appearance of a greater than one standard deviation is just 32% chance in a year. However as Taleb points out in his various books, we are all being fooled by what constitutes randomness within the normal distribution and the inability of market price to converge to the mean.

Referring to the price movement in the chart above, lets look at calculating the probability of a single standard deviation move and then the combined probability for the three days, 20th, 21st and 22nd of January.

Probability of event.

First lets look at the theory. and set out the proof for a one standard deviation move. 

Proof of methodology.

We know that a 1 standard deviation = 0.68 or 68%, therefore a move greater than 1 standard deviation = 0.32 or 32%. To convert this to find out how many days we take the reciprocal and divide it by the number of trading days in the year, 220, as follows:.

.32/220 = 0.00145 x reciprocal = 687.5 x .32 /220 = 1 year.

Probability of a Black Swan.

Now the daily price moves during January were actually around 1.5 to 2 standard deviations, so lets settle on a 1.5 move for convenience and use the above method to calculate the January Black Swan. 

A 1.5 standard deviation move is calculated as follows: 1-((0.68 + 0.95)/2) = 0.185 or 18.5% chance of happening.

Next we calculate the probability of moving beyond a one Standard deviation move in one day as :

Probability of 1.5 standard deviation in a day 0.185/220= 0.0008409.

Next we need to get a little fancy and multiply the probability of a move for the first, second and third days together to calculate the combined probability of occurrence.

Thus the probability of 3 x 1.5 standard deviation in a row is 0.185/220 x 0.185/219 x 0.185/218. Notice how the number of trading days drops from 220 down by one day for each calculation to account for the remaining trading days in the year. 

Next we take the reciprocal to convert the probability of happening to the range in days that this event might occur.

So we Multiply out the 1.5 standard deviation moves to get combined probability of 3 back to back moves as above and then we divide by the number of trading days remaining , to get trading years :.

In our case the probability of getting three days in a row of greater than a 1.5 standard deviation move is equal to once every 1,394,945 years. 

Release the Swans to do their worst.

But wait, there is more than one swan. If you refer to the chart above you can see from the highlighted sections that over November and January we had a number of instances and that this three day event was not an isolated case. Lets now add the November plunge in as well to see how our probabilities work out.

The earlier moves in previous months can be included by multiplying out the event probability of 18.5% by the remaining number of days in the trading year.

Thus the probability of 5 greater than 1.5 standard deviation moves in a month is now :
0.185/220 x 0.185/219 x 0.185/218 X 0.185/217 x 0.185/216 x 0.185/215. Again notice how we drop the trading days from 220 down to 215 as each trading event happens.

When we convert this very small probability number to years, we get the probability of once every 222,020,900,000,000 years which is longer than the life of the known universe. And remember, this calculation is only taking into account the last two Swans in November and January. What about July, August, September and October ?.  I thought about looking at these, but my eyesight is too dim and my calculator too old to handle the pace.

So forget your endothermic or exothermic reactions, Hell just keeps freezing over and it is only January - It would appear that game theory is no longer a game!.

Trading in reasonable times - Alice are you sure ?.
 

Cheers