After further analysis of my recent post on the 3 SD events, I think there are something more for me to understand the relationship between Bollinger bands and actual probabilities. Considering the 3 days on a roll for the 3SD events of TLT on 8/2/2011, the actual days of 3 SD occurrences were 10. It was not the 8 days that I had posted. It means the actual number of events were much higher than the theoretical value of 7.
Therefore, I searched other related studies available from Internet and found a good post: Standard-Deviation Technicals by Adam Hamilton. The article demonstrated that there were 60% more occurrences of 3 SD events than what the theory suggested for S&P 500 in the long period of its study. The "fat tails" compared with normal distribution were sited as one of the major reasons for the discrepancy.
Another good source for my study is Bolling bands on Wiki. It indicated for 2 SD events, the theory states a 95% of probability of stock prices to stay within the bands while the actual data showed about 88% of probability only.
The following table is obtained from theoretical calculations, and is an extension of my previous study. I added a column "Days/Event" to indicate the number of days required on average for the corresponding event to take place. For example, if we use 2 standard deviations, one event in which the price breaks the bands is likely to happen in about 22 trading days, which corresponds to 1 calendar month.
Table of SD Multiple, Probability, Occurrences
| SD Multiple | Statistical Prob | ITM Probability | Days/Event | Comments |
| 0.5 | 40.0% | 30.00% | 2 | Karen's Adjustment point |
| 1 | 68.3% | 15.87% | 3 | |
| 1.3 | 80.0% | 10.00% | 5 | Karen's short call |
| 1.6 | 90.0% | 5.00% | 10 | Karen's short put |
| 2 | 95.4% | 2.30% | 22 | My short options |
| 3 | 99.7% | 0.13% | 333 |
However, there are known issues in actual stock price distributions and the standard deviation theory. The SD requires a normal distribution as shown in the chart here. But financial products have fat tails due to greed and fear. Secondly, the SD model requires sufficient data sample points. If one uses 20 DMA for Bollinger bands as many traders use as default, it may miss the 2 SD events which require 22 days on average. This results inaccuracy in my opinion which may worth only 2 cents.
To the best of my current knowledge, it would require, at least, 2 times of the days/event of the sample data for the calculation to be valid. In another word, it needs 44 day MA, at least, for a 2SD probability to be close to reality in theory. For the 3SD event, it needs 666 day MA. Lastly, I'm doubtful that stock prices are evenly distributed around any moving prices at all. Even looking at a long term period like 700 days, market may be in a bullish up trend where prices continues to touch or break the upper Bollinger band while making a much smaller number of touches and breaks on the lower band.
In summary, I still like to use Bollinger bands and they should corresponds to some level of probability. However, if one use the associated probability in their trading formula, be careful about the differences between reality and theory. I think there are smart traders that are exploiting these discrepancies.
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