I will assume that you have a set of "sheets" like a market maker. I.e different calendar terms of theoretical prices that your model has spat out.
- July Straddle = 100
- October Straddle = 150
If someone sells the July-Oct Put Calendar, selling Oct, Buying July for 20. If the theoretical value on our sheets is 22. Then that means we have traded for 2 ticks of "edge".
If we are certain of the fair value of our July straddle = 100. Then this means that our October Put is 2 ticks greater than the theoretical value that our model has given us.
Now through the put-call parity, we know that if hedged calls = puts. Hence the October straddle is 4 ticks greater than what our sheets say. Hence October straddle = 154.
You have said that we know the implied volatility of each month and have the option greeks.
Lets say that the implied volatilities are as follows:
Vega of the Straddles:
To calculate the implied volatility of October we need to compare the 4 ticks of edge versus the vega of the October straddle.
4/40 = 0.1% of Implied Volatility
This then means that the October Straddle has just traded at a 5% + 0.1% = 5.1% implied volatility.
Now we have the two implied volatilities and time to expiry, we can calculate the forward vol using the following equation.
If the time to expiry of the two options is:
- July = 0.1 (In Years)
- October = 0.4
Then this gives,
Let me know if you need anymore clarity.