Inferring Forward Volatility from Calendar Put Spreads and Straddles
Summary
The document explains how to infer a later-dated option’s implied volatility from a calendar put spread when the earlier straddle’s fair value is known. It first compares the trade price with model values to estimate the spread’s edge, then uses put-call parity to translate the put mispricing into an adjustment to the later straddle. Dividing that adjustment by the later straddle’s vega gives an approximate implied-volatility change.
With adjusted implied volatilities and times to expiry, the example derives the intervening forward volatility by subtracting earlier total variance from later total variance and annualizing over the remaining period. The worked illustration uses stated straddle prices, vegas, volatilities, and maturities to show the calculation. The method assumes reliable theoretical values, valid put-call parity, and a small enough price change for a vega-based linear approximation; it is an illustrative setup rather than evidence that the resulting volatility will be realized.
Key ideas
- Compare the calendar spread’s execution price with its theoretical value to estimate its pricing edge.
- Use put-call parity to relate the put-spread edge to an adjustment in the later straddle value.
- Approximate the later implied-volatility change by dividing the straddle value adjustment by its vega.
- Calculate forward volatility from the difference in total variance across the two expiries.
- The example relies on accurate model values and a local vega approximation.
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Full text
# Deriving Forward Vol from Straddle and Put Spread
# 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.
## Answer by avava (score 4)
https://quant.stackexchange.com/a/55802
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.