Implied volatility is the expected magnitude of a stock's future price changes, as implied by the stock's option prices. Implied volatility is represented as an annualized percentage.

Consider the following stocks and their respective option prices (options with 37 days to expiration):

As we can see, both stocks are nearly the same price. However, the same options on each stock have different prices. In the case of UNP, the call and put prices are much higher than PEP's options, which translates to an implied volatility that is higher than PEP. So, instead of looking at option prices all day long, options traders use implied volatility to quickly compare the expected price movements (and therefore, the option prices) of various stocks.

When market participants trade options, they typically do it for one of two reasons:

1) To speculate on movements in the stock price or the stock's option prices (implied volatility).

2) To hedge the risk of an existing position against changes in the stock price.

If market participants are willing to pay a high price for options, then that implies they are expecting significant movements in the stock price or implied volatility. 

Conversely, if market participants aren't willing to pay much for options, then that implies the market is not expecting significant stock price movements.

Since implied volatility represents the overall level of a stock's option prices, implied volatility is just a way to describe the market's expectations for future stock price movements.

Alright, you've learned the basics! In the next sections, you'll learn about what implied volatility represents in terms of probabilities.

As mentioned before, implied volatility represents the expected range for a stock's price over a one year period, based on the current option prices.

More specifically, implied volatility represents the one standard deviation expected price range.

In statistics, a one standard deviation range accounts for approximately 68% of outcomes. As it relates to stock price changes, an 'outcome' is the stock's price at some point in the future.

To calculate the one standard deviation expected range for a stock's price after one year, the following formula can be applied:

To demonstrate what an expected range looks like, consider a stock that's trading for $100 with an IV of 25%:

Based on this graphic, we can see that there's an implied 68% probability that this stock trades between $75 and $125 in a year's time. Now, this doesn't mean that the stock won't trade beyond $125 or below $75, but it does show that the market is pricing in a low probability of such movements.

To take things a step further, multiplying the expected range by two will give us the two standard deviation range:

As you can see, a two standard deviation range encompasses 95% of the expected outcomes. Inversely, this suggests there's only a 5% chance that the stock will be trading below $50 or above $150 in a year.

If we go one step further and multiply the expected range by three, we get a three standard deviation range. In statistics, three standard deviations encompasses 99.7% of the expected outcomes. It is very rare for a stock to experience a three standard deviation move. But, it can (and does) happen!

Next, we'll visualize the difference between two stocks with different implied volatilities.

To compare two stocks trading at different implied volatilities, we'll look at two hypothetical stocks trading for $100. Let's say one stock has an IV of 10%, and the other stock has an IV of 25%. In the following visual, compare each stock's implied probability distribution:

What this visual demonstrates is that low IV stocks are not expected to experience large movements, whereas high IV stocks are expected to experience much larger price fluctuations. More specifically, the implied probability of the 10% IV stock trading below $70 or above $130 in a year is essentially 0%. However, the 25% IV stock has a much higher implied probability of trading below $70 or above $130 in a year.

If we examined out-of-the-money options with the same strike price on each stock, we would find that the 25% IV stock's options are more expensive than the options on the 10% IV stock.

For example, the 70 put or 130 call would be nearly worthless on the 10% IV stock because the implied probability of the stock trading to those strike prices is almost 0%. However, if we looked at the 70 put or 130 call on the 25% IV stock, we'd find that the options have some value because the stock price has a much wider range of expected prices compared to the 10% IV stock.

For one year expected moves, simply multiplying the stock price by implied volatility will do. However, for shorter time frames, the expected range calculation must be adjusted. Here is the formula for calculating a stock's one standard deviation move for any time period:

Note: you can also use trading days to expiration, but you'll have to change the denominator from 365 to 252, as there are 252 trading days in a year.

On a $250 stock with 15% implied volatility, the 30-day one standard deviation move would be:

One thing to note about using this formula is that you should use the implied volatility of the expiration cycle closest to your target time period.

For example, if you're calculating a 5-day expected move, use the IV of the expiration cycle closest to 5 days to expiration. If you're calculating a 180-day expected move, use the IV of a cycle with close to 180 days to expiration.

Why? Because you want to use the implied volatility of the options that match your target time frame. If you use 180-day option prices (implied volatility) for a 3-day expected move calculation, the expected move result will not be accurate.