## Standard Deviation : An Option Trader’s Perspective

Let’s talk about standard deviation that is a basis for some of the my income trades that I play for my personal portfolio.

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.*

Thanks for exmplaining it simply. Helps understand much better in perespective with Optionstrading.

Are you sure you want to divide the #of days til expiration by 365? Instead, wouldn’t you want to divide by the number of trading days in a year, which would be 255ish?

Reliance on the concept of standard deviation as a measure of investment risk can be much more dangerous than one might think. Prices can and do fall much further than 3 standard deviations from the mean — to the dismay of the “quants” and their Wall Street employers. A good introduction to the notion that stock prices are not normally distributed can be found in “The (Mis)Behavior of Markets” by Benoit Mandelbrot and Richard L. Hudson. You can get an overview of their ideas by using the “search inside this book” function at amazon.com.

Number of days till expiration should be 365 as days till expiration include weekends/non trading days.

So there are various implied volatility numbers for various time frames available. For example, there are 30 day IV, IV60, IV90, etc… What is the volatility figure that you’ve used in your example?

And, in order to come up with 2 or 3 SDs all you need to do is multiply 1 SD by 2 or 3, right? So in your example above, in order to obtain 2 SDs, you need to 6.27 x 2, correct?

Thanks!

…and, also, you have to recalculate (daily?) the SD to adjust for changes in stock price, days till expiration, and daily volatility change? Is there such a thing as daily volatility?

@Vlad- Thank you for your comments.

Volatility figured used is Weighted Volatility. Most often used volatility figure is 30d…some use forward volatility. However, for simplicity, term structure is not factored in the above example.

For volatility, there are various other forms as well e.g. open to close, open to open, close to open and close to close volatility as well. However, not all vol data is easily available. For approximation I just use 30day vol and adjust it after sharp moves in VIX (or once earnings are over).

All said and done, in financial markets, standard deviation is just a concept. The occurrences of 2-3 std deviation(or even larger) price moves are much more than what the original concept implies. So use it with caution 🙂

Profitable Trading, OP

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