## The Problem

You may only have data for options expiring in one month (1M), 2M, 4M and 6M, however you want to know the implied volatility (IV) level at 3M, or at 15 days to expiration (DTE).

## The Solution: Step 1 - ATM IVs

First, a good at-the-money (ATM) IV needs to be determined for each month. This involves the following steps:

- Get good inputs like dividends, interest rates, and solve for the residual rate, that rate that will line up the call and put implied volatilities.
- Calculate initial deltas*.
- Draw a modified cubic spline or other smoothing method through the deltas.
- Calculate deltas again with the spline IVs.
- Interpolate to find the ATM IV at the 50 delta using some type of slope and derivative (skewness and kurtosis). Ignore the call delta strikes > .85 and < .15.

*Deltas are better for basing the skew parameters than say percentages or standard deviations. In our testing, the parameters based on delta performed better and are more consistent and comparable over disparate stock prices and volatility levels. This is part of the ORATS smoothed market values (SMV) system.

## Step 2 - Earnings Adjustments

Now that you have good ATM IVs for each expiration, it is time to adjust the IVs that have earnings associated with them. If this is an underlying without earnings, skip to the next step.

To find a rational term structure, the earnings effect should be removed from the IVs of expirations occurring after earnings announcement. Moreover, expirations with multiple earnings announcements should be adjusted accordingly. There are formulas available online to determine how to calculate the one day effects of earnings.

## Step 3 - Term Structure Considerations

Some type of a term structure equation, and there are some online and in books (ours is proprietary), should be applied to the expirations ATM IVs. The key is that the formula is based upon the square root of time. We set 30 day and 2 year points and draw the best fitting term structure line based on a skewed curve.

For underlyings with many expirations, ORATS isolates the ones with less than 45 days and slopes those IVs separate from the rest of the expirations.

## Step 4 - Putting it all together

Above is the term structure breakdown for A - Agilent on 5/21/19. Notice the EarnAddl and EarnEffect columns that serve to identify the amount of volatility points associated with earnings. Those effects are removed so an ex-earnings IV term structure can be created (unadjVol). In the case above, a 30 day IV of 23.34 and a 2 year IV of 25.80 and an earnings effect of 282% create the term structure that best fits the monthly ATM IVs (vol50). Adding back the earnings effects and comparing the actual IVs (vol50) with the formula IVs (calVol), the lines are graphed in Blue-vol50 and Red (calVol).

Armed with this data, interpolating to find the 15 day or 3 month IV can be accomplished, 15d = 22.7 and 3 month = 25.92.

For more reading on the SMV:

Historical Options Quotes and Greeks

https://blog.orats.com/2018/07/24/term-structure-of-implied-volatility

https://blog.orats.com/implied-volatility-term-structure-and-interpolated-ivs