A study of Bollinger bands and actual probability in stock market

Many traders use Bollinger bands as one of the trading tools. I also use this tool in my current high probability option selling strategy. I've studied standard deviations, in-the-money probability, Bollinger bands in my previous posts:

• Does 3 standard deviation move of an ETF match reality?
• Surviving a test of 3 SD event on TLT bear call spread trades
• Estimating In-The-Money Probability using IV and Standard Deviation

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

Note the days for each event is the number of trading days. ITM probability is the single side probability.

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.

Worthless puts saved overall positions during the fast sell-off in October

As I have been testing water with my new option selling strategy inspired by the Super trader Karen, I started with TBT as described in my September post. Since then, mistakes had been made and some quick profits were earned as shown in the chart below. More importantly, lessons were learnt during the live trades.

In my October post, the TBT trade was going on according to plan. I did not realize that I had the best opportunity to close all the trades to achieve the maximum profit around 9-18 as shown in the chart above. I thought the far OTM puts were expire worthless as TBT was turning into a up trend. However, it did the opposite, and broke down my support level of $54. On 10-7, I settled to take a 50% profit by buying back Oct$53 put. There were no bids for Oct$49p at the time so I had to leave it on (I was not able to close the position as a spread trade) with only 2 weeks to expiration. It was the far OTM, worthless straight put that saved the overall position several days later when TBT crashed around 10-15 along with the stock market. As mentioned in the previous post, that was a test of 3 SD event. The other bull put position Oct$51/46p got crushed as price tumbled from $51.2 to $48 just 2 days before expiration. I decided to close the straight put which become ITM by selling it and the short bull put spread by buying it back, for a small profit on the crash day.

I think it was this small profit that gave me a little bit of psychological boost to follow my trading rule to hold on to the TLT position which was suffering a temporary loss that was over 3 times of potential MAX profit at the time. I did not look at the intraday loss and felt OK since I had a good amount of capital available in the account for possible rolling adjustments.

As happened many times in strong bull market before, the market sell-off was short lived. The market rallied back quickly. I strongly believe that bulls would not die without a big fight. For 5 weeks after the sell-off, market kept advancing. My TBT bear call spread Nov$57/62c (C5S/C5B in chart) was expired worthless last Friday. Now, I'm out of all TBT positions and transitioned fully into TLT trades as planned.

So what are the lessons to be learned from the market and the trades? This is the tough part of the trade review. With the help of the chart that recorded the trades, I think the following points are important for me.

1. After entering a trade for 1 month and with 1 month left for expiration, the trade may be exited for minor profit or loss if the stock prices goes against the position in general.
• I could have exited positions around 8-14 to open new positions.
2. Need to make sure adjustment sizes to be correct so that the potential profits remain the same.
• Made a mistake around 8-15
3. Should take deep profits as the stock starts to change direction around major support/resistance levels
• Around 9-19, TBT pulled back from 4 month resistance which was identified in an earlier post. That was the best exit point.
4. In a weak market, it may be worth to long really cheap ($0.05) OTM puts when key support level is broken. The contract sizes should depend on the long term bullishness of the market.
• Trades on 10-7 was perfect as market broke down the support level.
5. It's OK to follow the rules to adjust around Delta of 0.65.

Surviving a test of 3 SD event on TLT bear call spread trades

When I started to test my new premium selling strategy using TLT in middle to late September as described in my previous post on my positions on TLT & TBT, I had no idea that a 3 Standard Deviation test would soon appear. In fact, the 1st trades near middle of September were easy: 50% profit in 4 days as shown in the diagram below. At that time, FED announced end of QE. I thought TLT would start going down which was totally wrong as TLT continued to rise and shot up far beyond a 3 SD point at an intraday period.

Apparently, my short position of Nov$121/126c was in trouble as soon as it was opened. By Oct. 13, I began to realize that the uptrend of TLT was still intact as it firmly stayed above the $119 resistance level that I had in mind. So I began my adjustments and sold a larger number of bear calls of Nov$125/130c to make up possible deficits of existing losing position. The Delta of the short call strike Nov$121c reached my adjustment level of 0.65. So I rolled up and out the trade. On Oct 14 which was one day before the big 3 SD test, TLT broke up even further away from its upper Bollinger band. I had an opportunity to close the Original Nov$121/125c position and my sell order for Dec$127/132c was filled. I thought the price of TLT was pretty extended on that day. Then the big surprise came as market had a mini-crash and TLT sky-rocketed to test the 3 SD price. As usually, I watched market for less then one hour that day around Wall Street's lunch time. The Delta of my short strike Nov$125c were below 0.65, my adjustment level. Although shocked, I did not make any adjustments for TLT on that day. Well, the price started to move down which was what my positions favor since then. I think I was lucky to exit all the positions with 50% profits about 2 weeks later.

In retrospect, I think there are a few lessons learned from the adverse test for the high probability option selling strategies:

• Don't panic in the market extreme events and use trading rules to guide the reaction
• Trade a strategy only if you feel confident about it
• Keep sufficient capital to fight the adverse events (I'm still working on it)
• The high probability strategy can have large temporary losses (3 x potential profit this time)
• Profits can be earned even if price projections are wrong at times
• Refrain from over-adjustments

I had the thought of making adjustments if TLT reached and stayed above resistance level of $119. Had I done it, I would have sold TLT calls with lower strikes which would gave me larger temporary losses. It would be a bit more difficult to reach the 50% profit with the lower short call strikes. After reviewing my trades here, I think I should stick to the adjustment rule based on the Delta of short strikes, rather than support/resistance levels obtained from technical analysis.

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.