Estimation of model-free implied volatility is highly dependent upon the extrapolation procedure for non-traded options at extreme out-of-the-money points.
Jiang and Tian (2007) propose that the slope at the lowest/highest moneyness traded point from a cubic spline interpolation be used to extrapolate Black-Scholes implied volatilities.
Carr and Wu (2008) propose that the Black-Scholes implied volatility be held fixed at the level of the lowest/highest moneyness traded point.
Procedures also differ as to whether the extrapolation is done in volatility/strike space (as the papers cited above do) or volatility/delta space (as suggested by Bliss and Panigirtzoglou (2002)).
Which of these procedures leads to the most accurate model-free implied volatilities when the range of available strikes is fairly limited? Are there other extrapolation procedures which may yield better results?
Answer
Well as far as I know it is a really hard but interesting question. Asymptotics of smile in the strike direction is not known in a model free way as far as I know.
I think I can remember that nevertheless you have upper and lower bounds if you know something about the underlying dynamics and especially the first moment of explosion. I can't remember the correct reference I have to look after it when I can. Edit : Here is the reference I was looking for Benaïm, Friz and Lee.
Otherwise, I haven't read the Car Wu article you mention but all that I can tell you is that holding volatilty fixed at the level of highest OTM strike traded option simply doesn't work in the context of interest rates where asymptotics in strike of implied vols are used (and usefull at strikes much higher than higest level of traded interest options), especially in the context of CMS convexity adjustements.
Anyway for the strikes beneath the traded area, the interpolation method question still demands arbitrage free interpolation, to do so I (again) think I can remember that you can find some constraints to be satisfied by any scheme in order for it to be arbitrage free, even if in my experience the resulting arbitrage you get from simpler method are useless because way to narrow. So it is more an intellectual matter (except for some particular cases which usually last for short time window) than a real practical issue.
Edit : Here is a paper with constraints for arbitrage free interpolation methods, by Kahalé but I think there are many other contributions aswell.
By the way, you can extend the question in small time and long time asymptotics aswell where I think it is a really active area of research and a very interesting (and technical) matter.
Best regards.
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