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Hi Xavier
> I have to price very long term implied volatility and
> I don't know exaclty how to do that.
To price vols? what do you mean? What kind of options do you need to price?
> From the derivative exchange I know the smile of implied volatility
> for mid term option ( between 30 days and 2 years) and I want to
> deduce the smile for very long term options (for example 8 years).
this is a very long shot. There is no way to deduce the long term smile
without strong assumptions. You might think that extrapolating the smile
you have at e.g. 2yrs might be safe, but this would be an assumption
nevertheless.
> I tried the simple linear extrapolation and the Local Volatility model
> provided in Quantlib but it doesn't give me good results.
All local volatility models can be consistent with the prices (smile)
observed in the market, but they are unreliable for their implied dynamic
(future) evolution of the smile.
If you're pricing products crucially dependent on the dynamic (future)
evolution of the smile (e.g. cliquet) stay away from local vol models.
In my opinion despite their popularity local vol models are pretty useless.
Besides please note that the Gatheral's parametrization of the local vol
surface provided in QuantLib requires as input a Black vol surface C^2 in
the strike and C^1 in the time variable.
> Do you know if any other model exists? Do you know the model used in
> practice by Market Makers or if I can find an article somewhere?
It depends very much on which kind of options you need to price. Market
makers often rely upon dirty and effective practitioner tricks:
unfortunately they do not publish papers about them.
In general:
a) for a clear account of the smile dynamics there is a chapter in Mark
Joshi's book.
b) for an explanation of the limits of local vol models I suggest you
Gatheral's lectures (if you can't find them on the web I can send them to you).
ciao -- Nando
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