# Calculating Standard Deviation for Options Traders

#### CostaKapo

##### Active member
164 12
Let’s first understand what is a standard deviation, also referred to as “sigma”

Standard deviation is the most common measure of statistical dispersion, measuring how widely spread the values in a data set are. If the data points are all close to the mean, then the standard deviation is close to zero. If many data points are far from the mean, then the standard deviation is far from zero. Say it other way, A standard deviation is a unit of measure for volatility, and measures how tightly data is bunched around a mean, or average.

In the option trading world, this may be defined as how tightly stock or index prices are bunched around the current price. Some stocks like KO don’t range too much up or down from the current price. Other stock like AAPL can vary hugely. The standard deviation tells you about the potential percentage move or the dollar move a stock or index might make by a certain date. You can say that for a \$100 stock, the standard deviation is either 10%, or \$10.

The standard deviation is based mainly on an estimate of future stock or index volatility, a future date, and the current stock or index price. It also incorporates interest rates, but to a lesser extent.
•The higher the volatility, the bigger the standard deviation.
•The further the future date is, the bigger the standard deviation.
•The larger the stock price, the bigger the standard deviation.

You may use historical volatility, but in my opinion implied volatility is a better estimate of future volatility. Here is how you can calculate stadard deviation:
1 standard deviation = stock price * volatility * square root of days to expiration/365.

Let’s take an example. With SPY trading at 142.00, and March expiration 53 days away, and a volatility of 11.6%, what is the 1 standard deviation range for the SPY at March’07 expiration? 142.00 * .116 * square root (53/365) = 6.27

The above means 68% of the time, the index will be 142+/-6.27 by Mar’07 expiration. This assumes that stock and index price returns are normally distributed. One standard deviation covers the same percentage number of occurrences regardless of the size of the standard deviation. That is, the \$100 stock with a \$10 standard deviation will be between \$90 and \$110 68% of the time. And a \$20 stock with at \$3 standard deviation will be between \$17 and \$23 68% of the time.

For your information, +1/-1 standard deviation covers 68% of occurrences, +2/-2 standard deviations cover 95% of occurrences, and +3/-3 standard deviations cover 99% of occurrences.
So, how do you use standard deviations to trade? You could create your own trades based on your outlook , risk/reward of the market/Index.

-Concepts are general and lead thoughts are based on conversation in ToS’s trader’s lounge.

Personal note, I divide by trading days in a year 252 instead of calendar days because I trade options.

#### ACstudio

##### Active member
138 14
Excellent post. I might add a couple of points.

For the beginning options trader this may not seem important or even make a lot of sense.

First to simplify the equation just look at the Delta of the option. This will be roughly equivalent to the probability of the option expiring In The Money (ITM). So an option at .30 deltas will have about a 30% chance of expiring ITM or more importantly....a 70% chance of expiring Out Of The Money (OTM) or worthless. These numbers are very efficient in liquid markets.

Also the CBOE uses the full 365 in their equation to calculate volatility/extrinsic value. So when you have a weekend nobody knows when or how they are going to pull the decay. It might be anytime Fri...it might be anytime on Mon...they might do a little now and a little later. Nobody knows but them...I'm not sure they even know.

All of this becomes important when you want to be the seller of premium. And this is where selling premium gives you an edge over buying it.

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