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yondaime
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Joined: May 18th, 2020, 8:46 pm

### Interpolation of a Implied Volatility Surface

Hi,

I have an Implied volatility surface. I was using cubic spline interpolation to estimate the volatility at the in between points. However, the volatility data I have is a bit extreme because of rare events like Brexit. This is why the cubic spline isn't working everytime as there are arbitrage opportunities created.

Can you suggest better interpolation techniques such as those probably used by banks to combat such real cases?

Alan
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### Re: Interpolation of a Implied Volatility Surface

There is a nice method by
Note the distinction between interpolating splines and smoothing splines.

Cuchulainn
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### Re: Interpolation of a Implied Volatility Surface

Cubic splines overshoot for sparse data. This is well known.

http://finmod.co.za/Interpolation%20Summary.pdf

I reckon you need a monotonicity-preserving method.

Cuchulainn
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### Re: Interpolation of a Implied Volatility Surface

There is a nice method by
Note the distinction between interpolating splines and smoothing splines.

Alan
Posts: 10462
Joined: December 19th, 2001, 4:01 am
Location: California
Contact:

### Re: Interpolation of a Implied Volatility Surface

There is a nice method by
Note the distinction between interpolating splines and smoothing splines.
Just tried it in Chrome -- loaded fine.
If not working for you, the title is "Arbitrage-Free Smoothing of the Implied Volatility Surface"

Cuchulainn
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### Re: Interpolation of a Implied Volatility Surface

Seems to be working now.Thanks.

fomisha
Posts: 67
Joined: December 30th, 2003, 4:28 pm

### Re: Interpolation of a Implied Volatility Surface

If you need an industrial-strength solution you can check out Vola Dynamics (disclaimer - I am affiliated).

Here is an example of a fully automated fit right before Brexit:

The fits are parametric, arb-free, very robust, and fairly fast.

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

### Re: Interpolation of a Implied Volatility Surface

If you need an industrial-strength solution you can check out Vola Dynamics (disclaimer - I am affiliated).

Here is an example of a fully automated fit right before Brexit:

The fits are parametric, arb-free, very robust, and fairly fast.
Impressive.  I cannot believe any parametric models can achieve such robust results.. liquid or illiquid, ATM or wings..  Are there any public models that can achieve anything similar.. or closer..?

Alan
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### Re: Interpolation of a Implied Volatility Surface

If you need to extrapolate the IV time slices outside the available strikes, I have some nice fits here. See pages  30, 31.

I am using Gaussian mixture model fits, which have some nice properties, esp. for my application (equity risk premium), and lead to interesting extrapolations.

However, calendar arb removal, which is discussed in the paper, is somewhat ad hoc, and would require more work to assure. That was a peripheral issue for me, so haven't pursued it beyond the discussion in the paper. The two solutions discussed above are apparently arb-free.

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

### Re: Interpolation of a Implied Volatility Surface

If you need to extrapolate the IV time slices outside the available strikes, I have some nice fits here. See pages  30, 31.

I am using Gaussian mixture model fits, which have some nice properties, esp. for my application (equity risk premium), and lead to interesting extrapolations.

However, calendar arb removal, which is discussed in the paper, is somewhat ad hoc, and would require more work to assure. That was a peripheral issue for me, so haven't pursued it beyond the discussion in the paper. The two solutions discussed above are apparently arb-free.
Hi Alan, my main objective is to get a smoothed vol smile (I don't worry about the entire vol surface for this particular purpose) that can fits within bid/ask for all the quoted strikes in all cases (ATM/wings, liquid or illiquid). The best result I have so far is Fengler that you recommended above. However there are still many cases that the smoothed vols stay outside of bid/ask no matter how I tune the smoothing factor..

Alan
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### Re: Interpolation of a Implied Volatility Surface

Hi, perhaps you can post a plot that shows a problematic case, showing the IV_bid - IV_ask intervals at each strike. Also, sometimes the cost-of-carry method is a source of problems, so you might explain how you handled that.

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

### Re: Interpolation of a Implied Volatility Surface

Hi, perhaps you can post a plot that shows a problematic case, showing the IV_bid - IV_ask intervals at each strike. Also, sometimes the cost-of-carry method is a source of problems, so you might explain how you handled that.
I wasn't able to find an example quickly.. I observed this many times intraday on some illiquid names. You may be right as some underlyings were often hard-to-borrow which I probably didn't factor-in in the model properly. Sometimes I want to have a simple view on "BS implied vol without cost-of-carry".. [I know this is clearly 'wrong' but this is sometimes easier and more convenient for me to make trading decision.]

Is there any better approach to calibrate the model to find optimal smoothing parameter? The AIC in the paper is based on unconstrainted smoother (i.e, not the optimal choice) and tends to over-smooth so practically not very useful to me. This was explained in the paper but nothing better was recommended. Any thoughts?  Many thanks

Alan
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### Re: Interpolation of a Implied Volatility Surface

I would suggest just using judgment on where the smoothed smiles should lie in relation to the market IV(bid)-IV(ask). In liquid names (AAPL, TSLA) you can likely use Fengler's method to smoothly fit inside every quote. Learn what smoothing parameters accomplish that.

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)

Alan
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### 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.