Extrapolating Implied Volatility Beyond Traded Option Strikes
Article Quant Q&A · Author: Tal Fishman
Summary
The document examines how to estimate implied volatility for extreme out-of-the-money options when those strikes are not traded, a choice that affects model-free volatility estimates. It describes two proposed strike-space approaches: extend the cubic-spline slope at the edge of observed moneyness, or hold volatility constant at the most distant traded strike. It also notes an alternative of extrapolating in delta space.
The responses do not identify a universally accurate procedure. One emphasizes that model-free tail behavior is poorly determined, discusses bounds tied to assumptions about the underlying, and notes the need for arbitrage-free interpolation constraints. Another cautions that deep options may have unreliable implied volatilities because discrete tick sizes distort prices and quotes may reflect hedging costs and dealer margins. The discussion offers caveats rather than a comparative empirical test, and argues that the appropriate extrapolation depends on the model and intended use.
Key ideas
- Extrapolation choices for untraded option strikes can materially affect model-free implied volatility estimates.
- Proposed methods include extending the edge slope of a cubic spline or holding volatility constant beyond traded strikes.
- Extrapolation can be performed in strike space or delta space.
- Arbitrage-free interpolation imposes constraints, while deep-option quotes and tick sizes can make implied volatility difficult to interpret.
- The discussion provides no empirical ranking of the competing methods.
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# How to extrapolate implied volatility for out of the money options? # How to extrapolate implied volatility for out of the money options? 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 by TheBridge (score 12, accepted) https://quant.stackexchange.com/a/2234 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. ## Answer by Brian B (score 12) https://quant.stackexchange.com/a/2242 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 consider the psychology of their quotes rather 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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