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Alan
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### Re: Smoothing splines (clamped spline)

The data sample is not draws from a pdf, but option prices.

However you approach the problem, you have to tackle the issue of extrapolation (of the risk-neutral pdf) to non-marketed strikes.

Good point. Does this mean that kernel functions are not 'suitable' for data with no discernible underlying density? I need to think a bit more about this. It might be the wrong solution for the wrong problem.

This article does discuss a related problem it seems and authors are using slice kernel for interpolation and even a reference to extrapolation.

https://mathfinance.com/wp-content/uplo ... elling.pdf

Re your question, with options there is a density via the Breeden-Litzenberger relation -- which can be applied in various forms once you've got a fitting algorithm. As to whether kernel functions are suitable, I noticed the authors said: "Normally the slice kernel produces reasonable output smiles based on a maximum of seven delta-volatility points".  The "maximum of seven" gives me pause for my data -- but I haven't tried their method so don't know why they say that. What I am doing is somewhat related though.

jherekhealy
Posts: 9
Joined: December 11th, 2017, 2:25 pm

### Re: Smoothing splines (clamped spline)

The use of kernels to model the RND (risk neutral density) is well known in the litterature, especially, a mixture of lognormals leads to a simple linear combination of Black-Scholes formulas. Wystup kernel method is a bit stranger, as I think it is applied to vols in delta directly. The implied density is not explicitely computed and in fact there is no reason why it would be well defined (positive, etc.).

The paper Model-free stochastic collocation for an arbitrage-free implied volatility: Part I gives a summary of various ways to imply the risk neutral density from option prices.

Alan
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Joined: December 19th, 2001, 4:01 am
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### Re: Smoothing splines (clamped spline)

Thanks for the link. Yes, what I ended up doing is mixtures of normals.

Cuchulainn
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### Re: Smoothing splines (clamped spline)

The term "Slice Kernel" *Definition 2.1 is confusing at best and mathematically irresponsible. There is no reason to fabricate a new name since it is the well-known Gaussian radial basis function (RBF).

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jherekhealy
Posts: 9
Joined: December 11th, 2017, 2:25 pm

### Re: Smoothing splines (clamped spline)

You are a bit harsh on Wystup. There is the notion of smoothing kernel in maths: https://en.wikipedia.org/wiki/Kernel_smoother
Although I agree that his method is really a Gaussian RBF interpolation, as he chooses the weights to pass exactly through the points.

Cuchulainn
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### Re: Smoothing splines (clamped spline)

It has nothing to do with harshness. It has to do with unambiguous definitions and sources (BTW that article does not mention an author, so I did not assume it was Uwe W, but you seem to be claiming that).

Anyhoo, call it a RBF if that is what it is! Why introduce redundant notation etc. That's my "gripe". A referee in a journal would ask for a reference.

//
"kernel smoother" has a noun and a verb but "slice kernel" has an adjective and a noun. Are these the same thing?
(BTW I spent some time on the net surfing for "slice kernel" ..) Once you know it's a RBF then the statement "Normally the slice kernel produces reasonable output smiles" becomes a bit more clear!
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