You are scrolling through the maze of options data and stumble upon the following tidbit of data on a given asset :

Call Option A - 1 month volatility = 22% ( 30 days )

Call Option B - 2 month volatility = 25% ( 60 days)

You want to buy one of these options but you are not sure which one is cheap and which one is expensive.

Interesting exercise. Let’s read on!

To understand which is cheap or expensive, one must understand the concept of forward volatility.

Forward volatility is a measure of implied volatility of an option over a period of time in the future.

In the above example , let’s assume both options have the same strike price.

Remember that variance is additive, and that variance is equal to volatility ^2.

Annualized variance of option A = 0.22^2

Daily variance of option A = 0.22^2/365

30 day variance of option A = 0.22^2/365*30=0.0039

Similarly, 60 day variance of option B = 0.25^2/365*60 = 0.0102

So the 30 day forward variance = 60 day variance of option B - 30 day variance of option A = 0.0062

Now we need to convert this back into annualized volatility. First, we calculate the annualized variance which is 0.0062/30*365 = 0.0766

Therefore, the annualized implied forward volatility = sqrt(0.0766) = 27.67%

Let’s summarize:

**Option A volatility = 22% ( 1 month )**

**Option B volatility = 25 % ( 2 months )**

**Forward vol from 1 month to 2 month = 27.67%**

Now here is an important concept:

**If there is no significant news expected between the 1st month and 2nd month, forward volatility from month 1 to month 2 should be same as the implied volatility of option A (1 month).**

In light of the above, forward volatility seems high (assuming all else equal i.e. no significant volatility premium for news).

Therefore, as a trade you could short option B and long option A (recall that they have the same strike prices).

Have a look at the following volatility curve for the FX pair USDINR as an example:

Notice how different tenors have different implied volatilities. A series of forward volatilities can be bootstrapped from the curve.

In the next piece of this series, I will teach you how to bootstrap a given volatility curve from basic parameters and introduce the concept of vol carry. Stay tuned, Options Nerds out!

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