I've been mentioning standard deviations quite often when I write about various options trades. For example, recently I compared iron condors and iron butterflies. Depending on the strikes chosen, it is often possible to place the sold calls of an iron condor a little more than a standard deviation away from the current price of the underlying and the puts, about 1.5 standard deviations. What does that mean?

I'll give a simplistic definition that will probably make some statisticians wince. About 68.2 percent of the time, an underlying's moves will be contained within with a standard deviation either side of its current price. I'll provide a formula for computing a standard deviation range, but first let's just think logically about standard deviation. First, we would expect that a one-day standard deviation will be smaller than a one-month standard-deviation, and it is. Obviously, then, the formula for calculating standard deviation will include a factor that relates to the time the trade will be open.

Also, the more volatile prices are, the more we expect prices to range. Equally obviously, then, the formula must contain some reference to implied volatilities. We use the implied volatilities of the at-the-money calls, usually, to calculate the standard deviation.

One of my articles written way back in 2009 referenced a CBOE webinar presented by CBOE instructor and book author James Bittman along with frequent CBOE presenter Dan Sheridan. That webinar included a formula for calculating a standard deviation.

Formula from Bittman/Sheridan Presentation:

Some traders are confused as to whether to use calendar days in a year or trading days in that "Days per year" portion of the calculation. Bittman and others advise that, most of the time, the differences will be minor. Don't worry about it.

As I originally roughed out this article well ahead of publication, TESLA (TSLA) had had a volatile couple of days. Using think-or-swim's values, let's calculate a standard deviation for TSLA from May 15, 2013, when this article is being roughed out to June expiration, then 37 days away. Please understand that I am not recommending a TSLA trade. I would not recommend a trade in such a heavily shorted stock. At this moment, most shorts have probably been shaken out by the huge move over the last couple of days, but they may well pile back on.

TOS Supplied Snapshot of Part of June Option Chain:

TSLA closed at 84.42 on May 15, the day this article was roughed out and that image snapped. The closest call to an at-the-money (ATM) strike was the 85 call. We see that the implied volatility for the June13 85 strike is 80.42 percent, so 0.8042 will be the figure used in that formula seen above, along with 37 days to expiration. Because I'm not sure which symbols will translate well onto the printed page, I'll be using the oh-so-scientific notation "square root of" rather than the actual square-root symbol.


(84.42 x 0.842 x square root of 37)/square root of 365 = 432.37/19.10 = 22.63.

Remember that the price can range up or down, so this will be a +/- figure. I probably made some rounding-off errors there, too, having been long out of my physics and mathematics classes, but this gives us an estimated range either side of the then-current price.

We can see that if implied volatilities have quadrupled over the last few days, a standard deviation is now two times (square root of 4) times larger than it was before the volatility exploded higher. This intuitively makes sense to us. We expect a volatile underlying to range wider over a period of time than that same underlying when it's in a calm period.

Of course, if we choose an option with more time to expiration, we also expect price to range further. That's intuitive, too.

Of course, calculating standard deviations can be tedious. Many online brokerages calculate them for us. For example, the chart below charts an ATM straddle (long call and long put, not recommended when options are so expensive). The light-colored vertical band marks the prices within a one-standard deviation.

Chart Marking a Standard Deviation:

This chart calculates that TSLA could range to about 116.38 on the upside and 66.98 to the downside by June expiration and still be within a standard deviation of its May 15, 2013 closing price. That's 31.54 to the upside and 17.86 to the downside.

Why the discrepancy with the difference to the upside and the downside and from my roughly calculated standard deviation? TSLA was up after-hours trading, at 92.25 rather than the 84.84 at which it closed. In addition, my hand-eye coordination is compromised these days and I may just have eyeballed the edges of that lighter section incorrectly. In addition, there are those rounding-off errors. A third difference might be in the "days to expiration" calculation. This was taken from TOS's own days to expiration listed on its options montage but that doesn't mean that the days listed there are the same ones employed in the calculations. Days to expiration is sometimes loosely (and incorrectly) taken to mean days until an option stops trading or days until its settlement value is determined, but the expiration for these equity options is actually on Saturday. The days to expiration might differ by a day or so on different listings.

If there are this many problems, why would you use a brokerage's calculation? Remember that most of the differences seen here result from the after-hours price and would sort themselves out at the open the next morning. The trader simulating a trade and trying out different strikes has a quick idea which will provide the most protection within a standard deviation move.

What about the trader who never intends to stay in a trade to expiration but instead wants out the Friday before expiration? That trader can calculate a standard-deviation move for herself, substituting the number of days she intends to be in the trade for the "Days to Expiration" portion. Also, TOS, at least, allows clients to reset the "probability date" and the "probability range." Remember that I said 68.2 percent of the time, prices will be contained within a standard deviation either side of the current price? About 95.4 percent of the time, prices will be contained within two standard deviations either side of the current price. I could input 95 percent instead of the default 68 percent. I'm not certain whether other platforms allow this manipulation of standard deviation calculations, but I know that many do show a standard deviation move over the period until expiration.

When I throw out that "standard deviation" term, I wanted readers who might not yet be familiar with it to understand what I meant. I know some of you could teach me about standard deviations, but this was meant to be a quick and simple explanation.