Welcome back to my series on "Why Do We Care About Volatility?". Now that we've defined terms, it's time for some math that will show you how volatility really works -- and why it's important in both stocks and options trading.

As a figure, implied volatility tells you how much the marketplace as whole thinks the potential movement will be for the underlying security that the option is based on.  Keep in mind: volatility has no regard for direction. If the implied volatility is a "large" number, the market place thinks the security will make larger price swings in either direction. Similarly, a "small" number implies the underling's price will not move as much.

Now's the time to dust off the old college stats book. Don't get too scared, though -- we're not going to derive the Black/Scholes pricing model or anything too wildly complicated. Once you have the implied volatility for an option, you can actually calculate an expected high side number and an expected low side number for the underlying over the life of the option contract. In other words, you can use implied volatility to see if the marketplace agrees with our outlook for a stock or index. It can be used as both a measure of risk and potential reward. Best of all, volatility is based on statistics, so it's an objective way to test your hunches and manage exit points.

The math starts here. First off, we have to address the biggest assumption made by people who build pricing models -- the log normal distribution of stock and index prices. Check out the bell-shaped curve below, which is the basic normal distribution. A normal distribution of data means that most of the examples in a set of data are close to the "average or mean price", while relatively few examples tend to one extreme or the other.  Again in laymen's terms, most of the time stocks and indexes do nothing and stay close to their current price and only every once and awhile they make an extreme move.

(The difference between a normal and a log normal distribution is the fact that a stock or index can theoretically go to infinity on the upside and only zero on the downside. A log normal distribution accounts for this fact and because of this has a slight upside bias in the distribution.)

Say we have a stock trading at 100 with an implied volatility of 20%. First we'll start with defining standard deviation and how it relates to implied volatility; then we'll reverse the process and show you how standard deviation can give high and low price points for a stock. In subsequent posts we'll evaluate how to use these numbers as you make trading decisions.

Key words for this discussion are standard deviation and annualized. Statistically speaking, implied volatility is a proxy for standard deviation. If we assume a normal price distribution, we can calculate what a one standard deviation move (OSDM) of the stock will be. To do that we first multiply the stock price with the Implied Volatility:

100 x .20 = 20

OSDM = 20 points

The number of points it sums to is a one standard deviation move. This refers to either a move up or down in the stock. So the next step is to add and subtract the OSDM from the current stock price.

100 + 20 = 120

100 - 20 = 80

This gives us the normal expected range of the underlying stock. Standard statistical formulas imply the underlying stock will stay within this range (80 to 120) 68% percent of the time. When discussing a normal distribution most math text books refer to a one, two and three standard deviation move. That means the one standard deviation move will occur at a confidence interval of 68% of all occurrences, a two standard deviation move will occur with a confidence interval of 95% and a three standard deviation move will occur with a confidence interval of 99%.

Now you know how to gauge the underlying's swings over a one-year period. Check back next week to find out how to adjust this annualized figure for options contracts of any length.

Regards,
Brian (OG)

Options involve risk and are not suitable for all investors. Please read Characteristics and Risks of Standardized Options.

While implied volatility represents the consensus of the marketplace as to the future level of stock price volatility there is no guarantee that this forecast will be correct.

68% is derived from statistical theory assuming a normal distribution, or in this case log normal.   A standard deviation in a normal distribution is always this range, 2 SDs is always 95% and 3 SDs 99%.    If you wanted to know more you'd have to look iy up in a prob and stats book. 

Yes, 2 SDs would be 40 on each side and 3 SDs 60.  The larger range pertains to antipated price, not predicted volatility.  For example there is a 99% chance that it will trade between 40 and 160.

Stocktalk is right on with his explanation of standard deviation. Also realize that the real world market does not always behave the way it is supposed to. Example is the 87 crash, the market made about am 8 Standard deviation move and statically there was only a very minuscule probability
of that happening. Before that crash many market makers on the floor used to sell way out of the money options naked for a "tiny" or 1/16 (as it was called back then when options were quoted in fractions). As you can suspect for many years at each expiration month these options would expire worthless and then in one day most of the market markets that implemented the "tiny" strategy went bankrupt. So as long as we realize the unrealistic move does and will happen eventually. We can plan for them by either hedging with other options or having hard stops prices in place and sticking to them.

Regards,

Brian