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People use different parameterization schemes to fit the implied volatilities from the market, e.g., SVI. But often times they cannot always fit well, e.g., the "W"-shape before earnings, the VIX vol skew which is not convex, etc.

I was wondering, what's wrong with calibrating implied volatilities with just polynomials?

To be clear,

  • Before fitting the polynomial, we can perform some pre-processing transformation, e.g., parameterize the vol as a function of the log strike, or normalized strike, or delta, etc. Later we can convert it back to the strike space as needed.
  • Adding smoothness, arbitrage, and asymptotic constrains, for example, we could add L1/L2 regularization terms to make the polynomial coefficients small and stable or maybe Kalman Filter based on past values of the coefficients; we could check and remove butterfly arb or calendar arb, etc; we could enforce Roger Lee skew limit; and so on.
Michael
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4 Answers4

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First of all, you don't calibrate IVs. You interpolate and extrapolate IVs in order to calibrate a model.

There is nothing wrong with interpolating and extrapolating with polynomials, as long as no-arbitrage is satisfied. So the challenge is to find a polynomial that can fit market prices reasonably well and satisfy no-arbitrage conditions.

Frido
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Not exactly an implied volatility surface fitting expert but have done some research in that direction.

I think the whole idea about fitting an implied volatility surface is to best fit the IV surface now and how it looks like in the near (and far future). This in itself bears a tradeoff between overfitting and stability. You can't have a IV surface that fits exactly with the current IV surface and stays the same throughout time (therefore giving up stability by asking for a better current fit).

I read a paper recently by Ait-Sahalia and Lo (1998) (abit long ago yes I know), but it talks about something close to what you are asking about, producing a nonparametric estimate of the state price density used to price options (but the SPD here is implied by option prices). The SPD itself is more stable than other SPDs such as the risk-neutral probability distribution produced by the Black-Scholes framework for example, but there are much more things discussed in that paper.

TLDR. You can fit the IV surface with polynomials, in fact if I recall correctly, SVI itself is a nonparametric polynomials fit (correct me if I'm wrong I haven't went deep into the SVI literature). However, there is a tradeoff between completeness of fit (accuracy in the present) and accuracy of prediction (fitting IV surfaces in the future).

KaiSqDist
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    Are you sure vol surface fitting takes potential Future values into account? If so, do you have a reference to a paper where this is discussed? – AKdemy Oct 25 '23 at 01:15
  • @AKdemy you can reference Carr and Wu (2016), specifically under section (2.) "IV Surface: From Near-Term Dynamics to Current Shape". Not an expert on this topic so would love to hear your insights as well! – KaiSqDist Oct 25 '23 at 07:16
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A prior respondent pointed out that you're engaging in both extrapolation and interpolation. However, before diving into interpolation, you might encounter several issues such as:

  • Arbitrage opportunities in prices, which don’t exist under transaction costs,
  • Absence of prices or low liquidity,
  • Gaps in Year to Expiry (YTE),
  • Non-uniform strike prices across different expiration dates.

Even after resolving arbitrage issues, employing cubic interpolation could introduce arbitrage within the Implied Volatility (IV) space.

Furthermore, while fitting a model may simplify the IV surface into certain parameters, there's a chance the model may not accurately represent the IV surface initially.

Focusing solely on the IV surface, one could employ smoothing cubic splines under quadratic programming to adhere to the no-arbitrage condition. This approach tends to yield a near perfect IV surface, in a certain error sense, mirroring the market’s.

For a deeper insight, there’s a recommended article along with a linked paper that elaborates on volatility smoothing algorithms to eliminate arbitrage from volatility surfaces.

lukas kiss
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what is the IV you're trying to get for? IV is simply a parameter that depends on the model you're fitting. You're not so much trying to fit IV as you are inverting market prices to find this model parameter.

If you are using Black-Scholes and you obtain a 'W' shaped IV skew or any other shape based on market prices, then that is a feature of the model itself. If you believe this shape indicates something the model is missing, the issue lies with the model, not the parameter.

If you're looking to bypass the computationally intensive inversion process, then there are various methods to do so, typically involving Taylor or series expansions (e.g. Corrado-Miller).

alps
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  • You are missing the whole point of vol surface creation, it's not simply inverting market prices. See here for a start. If you use commonly used methods like SVI you wouldn't be able to fit a w shaped surface well. See here for general considerations and a simple SVI demo. – AKdemy Oct 25 '23 at 01:13
  • Fair enough. Vol surface creation isn't just about inversion (e.g. when extrapolating or dealing with market displacements taking away from a FMV). My original comment was aimed at how IV serves as a parameter in models like Black-Scholes, which inherently produce certain shapes like the 'W' - for that matter handling noise and trying to figure a better model are two things that contribute to value... but there is no such thing as "fixing the IV W shape" really. No matter how you put it, it's still a parameter into a model that you chose. – alps Oct 25 '23 at 01:26
  • The whole question is how to fit a vol surface when traditional methods do not work well, which w shaped is an example and something voladynamics prides itself for (see link above). So it's all about fitting to a certain shape. – AKdemy Oct 25 '23 at 01:38