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Edgey
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Swish function call payoff approximation

December 7th, 2020, 4:29 pm

The swish function looks a lot like a call payoff, but is differentiable at x=0 (aiding back propagation in neural network calibration).  

Are there any mathematical finance problems that could benefit from this function's properties?  
 
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Cuchulainn
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Re: Swish function call payoff approximation

December 7th, 2020, 5:44 pm

The sigmoidal functional crops in a number of places

https://onlinelibrary.wiley.com/doi/epd ... wilm.10366

Here (Paul W's brainchild)  we use it to model volatility with memory.

Another use is choosing nice and smooth activation functions (some are more equal than others). and here is one case for Heston. The smoother the a.f. the better the convergence, especially if you want to compute greeks. It could be competitive.

https://www.datasim.nl/application/file ... 423101.pdf

Smooth functions that approximate functions with kinks would be a boon. The whole issue of which a.f. is best seems to be an open issue in ML, which is very much in the 'try it and see what happens' kind of mode in general. 

// I am having trouble relating it to call payoff; is it a smoother around K?
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Edgey
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Re: Swish function call payoff approximation

December 7th, 2020, 11:48 pm

Up close it doesn't look like a good approximation to a call payoff
Image
But zoom out and the smoothness around k (=0) is barely noticeable

https://www.wolframalpha.com/input/?i=x ... 28-x%29%29
 
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Re: Swish function call payoff approximation

December 8th, 2020, 1:07 pm

If the objective is to have a smoothed payoff, don't we need some kind of mollifier?(?)

Image
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Edgey
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Re: Swish function call payoff approximation

December 10th, 2020, 12:51 am

Yes.  The Wikipedia version of the function has a Beta parameter to control the degree of smoothing.  I removed it from the wolfram alpha (you'd think they'd be in beta by now) link to keep the plot as 2D.
 
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Re: Swish function call payoff approximation

December 11th, 2020, 2:27 pm

The swish function looks a lot like a call payoff, but is differentiable at x=0 (aiding back propagation in neural network calibration).  

Are there any mathematical finance problems that could benefit from this function's properties?  
I have to think how to answer this question. The answer almost certainly "no", because I feel We want to smooth the edges, I remember @Alan providing @Erstwhile proposing a smoothing function.

Image

1. activation functions weren't built for this kind of problem.
2. It looks like a call, but that is probably coincidence. Sheet metal design is similar.
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Cuchulainn
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Re: Swish function call payoff approximation

December 11th, 2020, 2:56 pm

Edgey,
Found it. maybe the solution? It is a kind of mollifier.
viewtopic.php?f=4&t=55791
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Edgey
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Re: Swish function call payoff approximation

December 13th, 2020, 9:00 pm

Thanks. The solution in that thread is more generic and useful. It also means that now I don't feel guilty about abandoning this train of thought.
 
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JohnLeM
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Re: Swish function call payoff approximation

February 26th, 2021, 4:40 pm

The swish function looks a lot like a call payoff, but is differentiable at x=0 (aiding back propagation in neural network calibration).  
I understand that this function can help neural network calibration. But if you need to model a true call function, it might indicate that neural networks are inefficient to that purpose. Note that neural networks are suffering from convergence problems for Finance applications.