Does 3 standard deviation move of an ETF match reality?

It was a fantastical test in the last couple of weeks for the premium selling strategy that I'm working on in both real and paper trades, as the market went extremely turbulent. It will take a while for me to digest the trades to firm up my trading rules.

In preparation to analyze my TLT trades as it ventured above the 3 Standard Deviation (SD) line intraday on 10-15, I studied the historical data. I think 3 SD corresponds to 99.7% probability approximately as I outlined in a post before. If I understand it correctly, there will be a 3 SD event every 1 year 7.5 months which corresponds to 333 (1/0.3%) trading days on average (assuming there are 200 trading days per year).

I'm interested how these numbers matching the reality. So I went through the TLT history using Prophet Chart available from TOS. I created the Bollinger bands with 30 day MA & 3 SD on TLT chart, scrolled through the entire price chart whose data started at 2002-7-26 for over 12 years. I found the follow 3 SD occurrences where prices closed outside the Bollinger bands by visual inspections only.

Number	Date	Days	Up/Low	Days outside 3 SD
1	7/2/2004	0	Up	Just 1 day
2	2/27/2007	970	Up	Just 1 day
3	11/20/2008	632	Up	Just 1 day, but continued to rise
4	5/6/2010	532	Up	Just 1 day
5	8/16/2010	102	Up	Just 1 day
6	8/2/2011	351	Up	3 days
7	3/14/2012	225	Low	Just 1 day
8	4/5/2013	387	Up	Just 1 day

From 2004-7-2 to 2014-10-31, there are 4480 calendar days. That is a period of 12 years and 3 month. I equated it to 2460 trading days roughly. With the 0.3% probability, we should have 7.3 (2460 x 0.3%) occurrences outside 3 SD in the period. In reality, we've had 8 occurrences as shown in the above table. I think this is very surprisingly close to the theory.

I have a couple of other findings that should be recorded here.

• 6 out of 8 times the price came back to Bollinger bands in the next day
• 1 out of 8 times the prices fell below the lower Bollinger band
• I guess there will be more events that the prices fall below the Bollinger band in the next 10 to 20 years.

I'm running out of time today. So I have to present my trade analysis (Lessons from surviving the test of 3 SD events) next time.