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jkg77
Posts: 6
Joined: September 5th, 2020, 8:46 am

Re: Interpolation of a Implied Volatility Surface

I watch many markets simultaneously so the automatic calibration on the smoothing parameter is kinda important. It's fine to have the calibrated smoothing parameter a bit off from my judgement but at least it should be close (i.e. fit everything within bid/ask)

Well, I give an objective function that I like at eqn (18) of my paper that I cited. Perhaps you could choose the smoothing parameter to minimize that and see if you like the results. Now, $C^{model}_i$ would be the Fengler fits as a function of the smoothing parameter. Note C stands for the out-of-the-money puts or calls, puts on the downside strikes and calls on the upside. But, you likely need more careful cost-of-carrys to use that formula.
If the objective is to minimize the difference between Fengler and the markets, wouldn't this result in the best fit being no smoothing at all?

Alan
Posts: 10387
Joined: December 19th, 2001, 4:01 am
Location: California
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Re: Interpolation of a Implied Volatility Surface

Maybe you’re right. It’s been so long since I looked at that method, I may be confused.

jherekhealy
Posts: 15
Joined: December 11th, 2017, 2:25 pm

Re: Interpolation of a Implied Volatility Surface

I don't recall all the details of the Fengler paper, but the core idea is that you trade-off accuracy wrt market quotes vs. smoothness. More importantly, no smoothing at all may not be possible if you want an arbitrage-free interpolation. For example, a cubic smoothing spline with smoothing=0 is just a regular cubic spline, which gives you no control at all over arbitrages. It is the fact that the spline does not interpolate exactly the quotes that gives you the freedom to deal with arbitrages.

Alternative techniques are:
* The good old mixture of lognormal distributions. There are some caveats, but it's reasonably good at fitting a wide variety of quotes. It is described in appendix of Model-free stochastic collocation for an arbitrage-free implied volatility, part I.
* Model-free stochastic collocation for an arbitrage-free implied volatility, part I and II. Part I deals with polynomials (may be enough depending on how crazy are your quotes), part II with B-splines.
* Andreasen-Huge one-step local vol parameterization.
* An arbitrage-free interpolation of class C2 for option prices: similar to Andreasen-Huge technique, taylored to piecewise linear discrete local vol representation.

All are more involved than a cubic spline interpolation, and require some least square minimization.

jkg77
Posts: 6
Joined: September 5th, 2020, 8:46 am

Re: Interpolation of a Implied Volatility Surface

Thanks jherekhealy. I will look at the papers you suggested.