Can one price accurately by only using vanilla options a derivative that is exposed/sensitive mainly to the forward volatility ?
If it is impossible, why do we hear sometimes "being long a long dated straddle and short a short dated straddle" is being exposed to forward vol ?
Here are some examples :
a) In equity markets :
- pricing a volatility swap starting in 1y and expiring 1y later.
- pricing a forward starting option with the strike determined in 1y as 100% of the spot and expiring in 5y.
b) In rates markets : (FVA swaption) a 1y5y5y Swaption, which is 6y5y swaption with the strike determined in 1y.
In the equity world, a way to express the question is : If we use a sufficiently rich model like Stochastic Local Volatility model (SLV) where the local component of the model is calibrated on vanillas (hence the price of any vanillas will be unique regardless of the choice of the stochastic part). Would our model provide a unique price of the above instruments regardless of the stochastic component choice ?
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