Standard deviation?

[Kellyb]

Can someone tell me what a standard deviation is with regard to price chart patterns? I came here from investools and I don't recall hearing that before.

[MatthewHaley]

The standard deviation is how you measure the variation of a set of numbers. In an unbiased set of randomly generated numbers from a single source, you should get a series of numbers that will create a bell shaped curve. The larger the variation from the median value, the larger the value of the standard deviation. For simple things that are evenly distributed, one standard deviation would consist of about 2/3rds of all the potential numbers.

If the series were 2,3,3,3,5,6,4,2,5,4 the standard deviation will be very small. For 2,22,67, 4, -3, 44, 23, 16 the standard deviation is larger.

Some people create a probability curve for stock prices, and believe they are independant of support and resistance lines, fibannaccis and other technical indicators. A price outside one standard deviation means it should only happen about 1/3rd of the time. Outside 3 standard deviations should be very rare, only happening less than 5% of the time.

In the think or swim model of trading, probabilities are key, and new trades are set up regularly so that the probabilities are covered through a large series of trades. Many retail traders are following the same methodology. Others are combining the two methods, selling options at a certain probability that is also outside a support or resistance level for example.

The academic papers say the probability numbers are flawed when judged against real history. Since there are very few traders that retire rich, I have to think a pure probabilistic approach has flaws. It works real well with a huge capital base, as the flow of trades ends up working in your favor, but few retail traders are allowed 1000 to 1 margins. The arguement for a probabilistic approach is that the math may not be right, but in general it works for people.

You will also see quotes about a "low probability" trade, which most people define as a 70% or less probability of winning, vs. a high probability trade which provides a 90% cahnce of winning approximately. The price and liquidity and scew differences make it harder for a high probability to win over long periods of time, though any one trade may have a better probability. The probability of winning times the reward shoudl be equal for all trade according the probability models used by think or swim. Since the trading costs are a higher percentage of a high probability trade, the win ratio has to be higher than the probability to make money. This is what drives many investors to combine methods.

Matt

[Karl K]

To expand on Matt's excellent post, here's another way to look at it.

Any random AND normally distributed set of numbers -- numbers that follow a bell curve --means that any given EXAMPLE of that number has a specific chance of falling in a range.

It works very well with large random data sets -- say the height of any adult male person. One standard deviation is 68% of the time. So if you were to pick out any one male person out of a crowd, that person's height 68% of the time would fall between, say 5"3" and 6'0" (I'm making these numbers up, but you'll get the idea). That means if you pick at random 10 males, 7 out of 10 will fall in that range. while 3 out of 10 will fall outside that range.

Two standard deviations is 95% of the time. In this case let's assume the range then is 4'10 to 6'4". So if you took 100 males at random, 95 of them would fall into that range, 5 would fall outside.

Of course this isn't always the case every time -- your selection might not always follow the pattern precisely. But many "samplings" will eventually gravitate to those percentages. The results will be predictable.

How does this relate to options? Well, the Black Scholes pricing model employs standard deviation of a stock's price in the model to price options. And that makes sense from one perspective. If a stock's price is randomly/normally distibuted (actually stocks prices are LOGnormally distributed, since a stock can never be worth negative money), then the probabilites of it falling within a specific price range should be subject to the laws of statistics. A stock (or an index) should fall within a specific price range some percentage of the time -- plus or minus some percentage of its current price.

That's where the volatility numbers come in. Volatility is in fact the variance -- and variance is the standard deviation squared. Of course, the volatilty of a stock and the volatility of an option can be quite different -- that's why we talk about "implied volatility." Implied volatility is the percentage number that solves an option pricing equation for the option price in the market right then. That volatility number can change every second the market is open.

Here's a good rule of thumb. Look at a stock's volatility number and divide it by 16. That will tell you the probability of a one day price move. So if a stock has volatility of 32%, divide by 16 to get 1 standard deviation, in this case 2%. Thus on any given day, the stock has a 68% chance of moving within the range of plus or minus 2%. Two standard deviations means daily move range of plus or minus 4%, an so on. Right now the DOW has a volatility in the mid 20% range. So a daily move of 1.5% -- 150 points -- is simply not all that unusual. It's well within the range of probabilities.

However, stock prices, and the price of its derivatives, don't necessarily follow the normal distribution. Stock prices can sometimes be like knowing the likelihood of picking out a guy whose 6'6" from a random crowd...and sometimes it is a far cry from that.

Think GOOG. GOOG has vols in the mid 40s, which means a 2.5% daily price move is one standard deviation. But GOOG has at times gone up over 10% in one day -- 4 to 5 standard deviations. That SHOULD happen once every 2000 trading days -- i.e., once every 5 years. But guess what, it does happen. So now what happens to volatility?...IT MOVES UP too. It's dynamic.

Bottom line, in the end Standard Deviation (and particularly Bollinger Bands) are useful tools to measure probabilites, especially for indexes, which tend to be what the statiisticians call "mean reverting: (i.e., they tend to come back to their averages). But if you assume that standard deviation will tell the tale all the time, well, you are playing with fire.

[MatthewHaley]

One key sentance in Karl's post is critical to remember.
Any random AND normally distributed set of numbers -- numbers that follow a bell curve --means that any given EXAMPLE of that number has a specific chance of falling in a range.

Unfortunately for probability traders, stock prices are neither normally distributed nor random. That is a key downfall of the probability trades.

Matt

[Karl K]

Although I will say that stock prices, and particularly well diversified indexes, do follow lognormal price distributions.

Of course, they follow such distributions...until such time they don't..

In other words, at a 40% vol, GOOG will tend to move plus or minus 2% on a daily basis. But if it moves 10% in one day, don't be shocked that it was a 5 or 6 standard deviation move, which descriptive statistics tell you should only happen once every 2000 or once every 5000 trading days...or it moves 8 standard deviations, a move that should happen, well, statistically never.

Actually, daily price spikes, understood as standard deviations, are a very useful technical reading. If GOOG spikes 5 standard deviations in one day, what is the likelihood it will spike up ANOTHER 5 standard deviations the next day? Pretty unlikely.

Unusual daily prices spikes present interesting trading opportunities. If GOOG spikes up or down 10% in one day, then selling a spread another 5 standard deviations away is a very appealing trade.

Same thing with a broken wing/lopsided butterfly. If the Dow drops 600 points on Monday -- a 5 standard deviation move down -- setting up a broken wing butterfly another standard deviation or two down the curve, with the flat wing on the upside, becomes a very intriguing possibility.

As Matt pointed out, the probs on TOS should be taken with a dose of skepticism. They give you a good guideline of how your trade might payoff. But all the numbers are very dynamic. Numbers in the market are never static, things can move both for you and against you pretty quickly.

[awtrader]

Thanks for these thoughtful posts guys.

Matt, you say:

You will also see quotes about a "low probability" trade, which most people define as a 70% or less probability of winning, vs. a high probability trade which provides a 90% cahnce of winning approximately. The price and liquidity and scew differences make it harder for a high probability to win over long periods of time, though any one trade may have a better probability. The probability of winning times the reward shoudl be equal for all trade according the probability models used by think or swim. Since the trading costs are a higher percentage of a high probability trade, the win ratio has to be higher than the probability to make money. This is what drives many investors to combine methods.

I have the impression that many of the IC and DD trades on RUT and other indices which are detailed here are probability-based trades. I am thinking in particular of the trading systems that are following posted rules. Am I mistaken in thinking this? Or do they somehow incorporate information from other methods?

If so what other information is incorporated? Do you consider possible pricing patterns or ideas deduced from historical data about a certain security to be external to a probablistic model? For example you just mentioned the historic volatility of the Russell 2000 in another thread.

Or technical analysis - I presume that is clearly beyond a purely stochastic approach?

But in the analyses folks here write up or that I listen to at TOS and the CBOE site there is certainly a lot of talk about probabilities based on the lognormal distribution.

-aw

[awtrader]

Karl -

Another interesting post. Are those high sigma moves what I have seen referred to here as "Black Swan" events? You speculate on trading after them. Have you actually tried it?

And as to the probs on TOS, are you suggesting that they are not the best calculations, or that we should not trust any statistical calculations. I know someone here mentioned changing the "Volatility Strategy" setting when modeling for certain trades. I'm not too sure what that does other than to smooth out the vol curves on the volatility analysis chart. Do those numbers carry over to other calculations too?

-aw

[optiontraderman]

Dont forget the difference between probability of expiring and probability of touching.

Probability of touching is much less but how many traders will sit and watch the RUT or SPX touch their short strike in an IC banking on the probability of expiring being true?

OTM

[MatthewHaley]

Originally Posted by optiontraderman

Dont forget the difference between probability of expiring and probability of touching.

Probability of touching is much less but how many traders will sit and watch the RUT or SPX touch their short strike in an IC banking on the probability of expiring being true?

OTM

Actually, it is the inverse. TOS calculates probability of touching is twice the probability of expiring. So, if you have a 20% probability of expiring, then you have a 40% probability of touching.

Matt

[MatthewHaley]

AWTRADER,

Yes, many trades here are probability derived, though many people also use techncal analysis, like fibbannaccis, horizontal and / or diagonal support and reistance lines, MACD... All of those tools are based on probabilities NOT being true. Probability math requires a random distribution. Trends and resistance lines require the prices to be non-random.

You will read about 3 standard deviation days being extremelly rare, yet you can find hundreds this past year in the financials. Why? Because prices are not randomly distributed. If things were randomly distributed, the market could never rise or fall outside a range. The mere fact that it is up a lot over the past 30 years is proof that the market is not random.

You will often read of people picking a certain probability line, such as 30%, and then looking for the next resistance line. This combination works for many people. And that is great, anything that works I am all for. I have a problem with psuedo math, and the tos calculations are not based on real math. If you notice, since they solve for probability, and use that as the driver, their calculation for delta for ITM strikes is wrong, often it even gets inverted. All so they can have probabilities that are logical. That is troubling. It is also one of the reasons the analyze tab works well with verticals, ICs, straddles and strangles... Basically anything with a same month expiration. It works ok, with near month calendars, and double diagonals, but stinks at longer than that.

I have only my own records, but my sample af a 100 or so stocks over the past 18 months shows little correlation to their probability numbers and the real prices. They don't track it internally, and for most people it doesn't matter, since they have exit rules that gets them out of trades based on the fear that the probabilities are wrong.

There is a huge study on the regression to the mean in the stock market, and the concensus so far is that the regression is to the inflation + GDP growth + capital float differential line. Since this can only be calculated months after the fact, it is interesting to read about, but useless for trading, and can be harmful as human nature is to try to anticipate what that line might be, even without actually knowing the line details.

Anyway, this is well off the path of hte original question.

If probability as calcualted by tos works for you, then great! Since it is a consistently calculated value, it is esy to build rules around the numbers.

I am a boring trend guy, and will stay there. It is a place I know and understand. After many email and face to face exchanges with the Tom's at TOS, I understand exactly what they are doing, and why. I know why the real numbers don't matter when you put on a series of trades. I worry when people follow half their model (use of probabilities for example) but don't lay on additional trades through the 45 days of adding positions for a particular month. Remember, their rules are to be adding trades from 10-4 weeks before expiration and take them off 4-10 days before expiration.

I think the reason a lot of people get destroyed is by putting on trades too close to expiration thinking they are faster to pay off, but that he probabilities will save them. The closer you get to expiration, the less accurate the probabilites have to be. I bet 99% of the people using the probabilites don't know the accuracy drops the closer you get to expiration.

Matt