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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?

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Re Jiang and Tian, if your extrapolated volatility is linear (fixed slope), don't you eventually have further-out-of-the-money calls costing more than nearer-out-of-the-money calls? –  barrycarter Oct 25 '11 at 3:55
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2 Answers

up vote 6 down vote accepted

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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My context as well as Carr and Wu's is single stock options. Peter Carr is one of the top names in the field, so I doubt he doesn't know what he's doing. –  Tal Fishman Oct 25 '11 at 1:58
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Hi well I know P. Carr's reputation and there is no doubt that when he asserts such a claim he must have strong evidence for this to be right. I just wanted to say that some caution has to be kept in mind, by using a concrete example for which it is wrong to freeze BS Implied Volatilities for high strikes and very high strikes. Best regards. –  TheBridge Oct 25 '11 at 9:31
    
Thanks for the additional references. I think I will go with a linear extrapolation in vol/delta space with constraints based on arbitrage bounds. –  Tal Fishman Oct 25 '11 at 21:58
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At strikes distant from the forward value, pretending that options have some meaningful implied volatility gets kind of silly. Options really have prices (both bids and offers), and we all just translate that to volatility because doing so provides a convenient normalization.

Just to take one example, discrete price quoting completely obfuscates the volatility value (as opposed to intrinsic value) of deep options. Using the offer price, or even the midpoint price of an exchange traded option automatically sends implied vols to infinity just because no is allowed to offer the option at less than a tick. Is the implied vol really meaningful in that case? Not to me.

Furthermore, since far from the money options are rarely quoted or traded, it might well make more sense to thing of the psychology of their quotes than some model. A trader will quote that option based essentially on what he expects the hedges to cost him, plus a margin corresponding to how much he likes you. With that as the price consideration, what are you willing to infer about implied volatilities?

Now, we all find ourselves using models that end up exploring deep regions of the state space so at some level one ends up making an assumption about deep vols. That usage is by definition in the context of a model. But since you ask for model-free implied volatilities, I have to pose the question: for what purpose?

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"Model-free" in this context typically means that we make no assumption regarding the DGP of the underlying, not that we make no assumptions whatsoever. The purpose is to calculate something like a VIX for individual stock options. But, if you read the papers I linked, you'll see that the VIX methodology is far from perfect, and becomes even more imperfect when applied to single stocks with fewer traded strikes. –  Tal Fishman Oct 25 '11 at 19:12
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\@Brian: I am with you all the way, Brian ;) @Tal: I would suggest you to be not that certain that Brian's concerns are irrelevant and have been carefully taken into account. Actually, I bet Brain has known everything you mentioned. And just an analogy to your comments about Peter Carr in TheBridge's answer: if you carefully check Brian's previous answers in QF.SE, you should doubt that he doesn't know what he's answering. He is obviously an expert/master in this subject. And interestingly, in this industry, proprietary advice is usually much better than the public reference. –  楊祝昇 Oct 25 '11 at 20:54
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@楊祝昇 Brian's concerns are certainly all valid, and I appreciate his pointing them out. As it happens, I have already independently thought about each issue he brings up, and perhaps I should think about them more. Also, I know TheBridge and Brian have both given excellent options-related answers in the past, which is why I thought it would be a good idea to post the question here. I think this is a common problem, so someone should know something about it (in fact, the references TheBridge gave me have already been helpful), but of course I will have to do much work on my own. –  Tal Fishman Oct 25 '11 at 21:55
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@Tal : Your question is legitimate and interesting IMO, if it is for Variance swaps and alikes products purposes, I think there is a stream about pricing bounds on those kind of products. Obloj et al., and Hobson et al. have written about this. But as I am not really well acquainted with those matters I really don't know what they are worth. Best Regards –  TheBridge Oct 26 '11 at 7:02
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Now that I understand it is for something like VIX this comes into clearer focus. (BTW I take no offense whatsoever at your comments above). I still believe the real answer lies in thinking about the deeps the way a trader would, but it is hard to capture that algorithmically. The best automated scheme would be to do what a lot of options shops do...construct a normalized representative "cohort skew" (often cohort=industry) whose wings are largely determined by the most liquidly traded names in the cohort. Then, you un-normalize the cohort skew to obtain the wings of your illiquid names. –  Brian B Oct 26 '11 at 15:31
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