Deriving Forward Vol from Straddle and Put Spread

Assuming we have the fair value of the July straddle on a certain strike, what would be the implied July-Oct forward vol if an order comes in to sell the July-Oct Put Spread (on the same strike) for a certain price?

I'm wondering how one would go about solving this problem, assuming we have complete information on the prices/greeks of the relevant calls and puts, and information on the number of variance days and volatility of each month.

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.

Lets say:

• 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:

• July = 4%
• October = 5%

Vega of the Straddles:

• July = 30
• October = 40

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.

$$\sigma_{t,T}=\sqrt{\frac{T\sigma_{0,T}^2-t\sigma_{0,t}^2}{T-t}}.$$

If the time to expiry of the two options is:

• July = 0.1 (In Years)
• October = 0.4

Then this gives,

$$\sigma_{t,T}=\sqrt{\frac{0.4*5.1^2-0.1*4^2}{0.4-0.1}}$$ =6.3%

Let me know if you need anymore clarity.