Mathematically Calculating 1SD Expected Move Using a Probability Analysis Chart

As discussed in the "Expected Move" post, the expected movement of a stock can be calculated with the following formula, where S subscript 0 is the stock's current price, IV is implied volatility, and the final term is the square root of days to expiration divided by 365:

Though an intimidating formula at first glance, it provides traders with the approximate range in which a stock's price may travel in a given amount of time or days to expiration (DTE). Note: The image above should say plus or minus preceding the square root sign.

To visualize, let's look at an example. Assume a SPY stock's price on low for the year on Feb 11th 2016 as $182.86, with an implied volatility of 28.14%. To calculate the expected move in 30 days, we substitute "DTE" with actual number of dates and solve for EM. 

The plus or minus range for each expiration will approximately equate to the EM calculations using the mid-price fills of EM spreads and the MMM value. 

So, in the above example, there's a 68% probability (1 SD) that SPY would stay in a range of +/- $14.75 from the current price of $182.86 in 30 days from now. There's a 95% (2 SD) probability that the stock price would stay in a range of +/- $29.49 from the current price of $182.86 in 30 days from now. And so for for 3 SD.

Now if we go for 60 days from now, you'll see that the numbers are as follows:

• 1 SD = +/- $20.86
• 2 SD = +/- $41.71
• 3 SD = +/- $62.57

Geek stuff - ​Putting the Equation Into Perspective:
​The equation itself is a transformation of an inverse parabolic function. What? The most basic parabola is y=x^2. When we take the inverse of this function we arrive at y= +/- SQUAREROOT(x). The inverse function and the EM graph are the same shape! The formula of the EM graph is a transformation of the inverse function. Think of the product of (S0) and (IV) as the coefficient, DTE as the independent variable (x), and EM as the dependent variable (y). See the correlation?