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Cuchulainn
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Re: Monotone Schemes: what are they and why are they good?

October 8th, 2020, 9:51 am

JohnLeM, 
Here an article on applying exponential fitting to a range of PDEs,

Authors
Le Floc'h
Publication date
2016
Journal
Available at SSRN 2711720
Description
This paper explores nine ways to apply exponential fitting methods to the Black-Scholes PDE. In particular, we will see that many ways lead to undesirable side-effects.
Scholar articles
Pitfalls of Exponential Fitting on the Black-Scholes PDE
L Floc'h - Available at SSRN 2711720, 2016
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The discussion around gamma is not a problem with fitting but using centred differences to compute gamma is weak. Gamma has its own PDE.


Maybe some reading for the weekend, as I suppose Boulevard San Michel is closed(?)
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Cuchulainn
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Re: Monotone Schemes: what are they and why are they good?

October 8th, 2020, 10:05 am

You probably want monotone schemes for the Greeks

https://www.datasim.nl/application/file ... hesis_.pdf
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JohnLeM
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Re: Monotone Schemes: what are they and why are they good?

October 8th, 2020, 10:35 am

You probably want monotone schemes for the Greeks

https://www.datasim.nl/application/file ... hesis_.pdf
@Cuchulainn thank you very much, I will have a read to all this this week end.
 
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Re: Monotone Schemes: what are they and why are they good?

October 8th, 2020, 10:39 am

You probably want monotone schemes for the Greeks

https://www.datasim.nl/application/file ... hesis_.pdf
@Cuchulainn thank you very much, I will have a read to all this this week end.
I spent some time afterwards extending all this stuff. No more divided diferences of V _but_ a FDM scheme that approximates gamma PDE.
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JohnLeM
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Re: Monotone Schemes: what are they and why are they good?

October 10th, 2020, 10:39 am

@Cuchulainn Hello. I will try to read all this, I might not do this this week end.
However here what I think today :
- We can test exponential fitting in a high-dimensional setting : you could toy with any stochastic process, regardless of the number of underlyings (high dimensional), any size of portfolios, computing any kind of risk measures with these schemes.
- This would be a mid-term research project (6 months to one year to produce something serious)
- I don't know what are the impact of working with a fixed grid. First your pricing will not be martingale. Second, we might experience problems in higher dimensions. However I am expecting that exponential fitting should be faster in term of computational weights than our numerical schemes.

Maybe we could enter a discussion over these topics ?
 
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Re: Monotone Schemes: what are they and why are they good?

October 10th, 2020, 12:42 pm

A realistic first case is to produce monotone schemes for 2-factor Heston PDE with non-zero correlation.
This is an open problem AFAIK.

Exponential goes a long way in this regard.

http://wiredspace.wits.ac.za/bitstream/ ... sequence=1

So, extend fitting to non-zero correlation case to produce a monotone scheme.

Here's a hint  or two

viewtopic.php?f=11&t=101316

viewtopic.php?f=4&t=102228&p=854351&hil ... al#p854351
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Re: Monotone Schemes: what are they and why are they good?

October 10th, 2020, 12:57 pm

A realistic first case is to produce monotone schemes for 2-factor Heston PDE with non-zero correlation.
This is an open problem AFAIK.
@Cuchullain, we did 2 factor Heston PDE with non zero correlation with our schemes, and it is published in CRAS - 2017. There is no interest for me to do this experiment twice. But it would be definitively a plan test for your methods.

Anyhow, I did not tell you that I will implement exponential fitting in a general multidimensional setting. I gave you a price: it is the time it would take to develop and test this engine - for me. A student would take a little bit longer. If you have institutions ready to fund this project, I can help.
 
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Re: Monotone Schemes: what are they and why are they good?

October 10th, 2020, 1:20 pm

What a shame! You prefer top-down generalisation to first tackling concrete cases?
Good luck. 
Personally, I think fitting cannot be applied in a  "general multidimensional setting." But I could be wrong.

If you can't get it for Heston )and I don't think you can done) then it is a show-stopper.
I remember the CRAS article.
Last edited by Cuchulainn on October 10th, 2020, 1:28 pm, edited 2 times in total.
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Re: Monotone Schemes: what are they and why are they good?

October 10th, 2020, 1:22 pm

And I don't see how Crank-Nicolson can be made monotone. We are waiting on a proof.
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Re: Monotone Schemes: what are they and why are they good?

October 10th, 2020, 1:37 pm

And I don't see how Crank-Nicolson can be made monotone. We are waiting on a proof.
You will have it when we will publish our paper. To be honest, all is written, but I am not in a hurry to publish our algorithms, and I am taking time watching flies around. But we will, free of research taxes, for the community sake.

But are we talking about my schemes or yours ? Take it as a collaboration proposal to help developing your technology, not mine. Anyhow, I have a deep respect what you achieved with these schemes, the proposal is opened.
Have a nice week end.
 
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Re: Monotone Schemes: what are they and why are they good?

October 10th, 2020, 2:02 pm

I think you are a  bit out of touch how technical took (take?) place on this forum.
The gold standard is how Alan posts his posts.
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Re: Monotone Schemes: what are they and why are they good?

October 12th, 2020, 3:28 pm

The proof of Crank-Nicolson montonicity (for a small enough time-step) is in Pooley et al. "Numerical Convergence Properties of Option Pricing PDEs with Uncertain Volatility". A different more general proof is also in Boley & Crouzeix "Conservation de la positivité lors de la discrétisation des problèmes d’évolution paraboliques".
 
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Re: Monotone Schemes: what are they and why are they good?

October 12th, 2020, 5:34 pm

The proof of Crank-Nicolson montonicity (for a small enough time-step) is in Pooley et al. "Numerical Convergence Properties of Option Pricing PDEs with Uncertain Volatility". A different more general proof is also in Boley & Crouzeix "Conservation de la positivité lors de la discrétisation des problèmes d’évolution paraboliques".
@jhrerekhealy, thank you very much for this reference. The paper of Pooley can be found here.
I am a little bit surprized :
1) I posted a very simple test here with an example of a non monotone CN scheme for the heat equation. I will check this example. *addendum* checked, it seems correct, the computations are very simple.
2) There is a weird result in this paper page 13 : "Lemma 4 (Stability of the CN discretization). If Delta_t < C, then the CN scheme is stable". What is strange is that a CN scheme is unconditionally stable.

I'll have to read this paper more carefully to give a more precise answer. Thanks again.
Last edited by JohnLeM on October 12th, 2020, 10:57 pm, edited 1 time in total.
 
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Re: Monotone Schemes: what are they and why are they good?

October 12th, 2020, 6:27 pm

The proof of Crank-Nicolson montonicity (for a small enough time-step) is in Pooley et al. "Numerical Convergence Properties of Option Pricing PDEs with Uncertain Volatility". A different more general proof is also in Boley & Crouzeix "Conservation de la positivité lors de la discrétisation des problèmes d’évolution paraboliques".
I am starting to have doubt on lemma 3 of Pooley et al. "Numerical Convergence Properties of Option Pricing PDEs with Uncertain Volatility". If one downgrades their equation to a heat equation [$]r=0, \gamma_{ij}=0,\sigma(\Gamma_i^n) S_i = 2, S_i = i \Delta_x [$], I found that 3.44 is now the classical CN scheme for the heat equation

[$]g_i = -U_i^{n+1}+U_i^{n} - \frac{\Delta_t}{\Delta_x^2} (U_{i-1}^{n+1/2} - 2 U_{i}^{n+1/2} - U_{i+1}^{n+1/2})[$], with [$]U_i^{n+1/2} = \frac{U_i^{n+1} + U_i^{n}}{2}[$]

and it does not seem to be monotone : [$]\partial_{U_{i+1}^{n}} g_i = \partial_{U_{i-1}^{n}} g_i = - \frac{\Delta_t}{\Delta_x^2} \le 0 [$]. This small calculus seems to contradict the first computation of the proof in Lemma 3.

@Cuchullain, any view on this topic ?
 
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Re: Monotone Schemes: what are they and why are they good?

October 12th, 2020, 9:10 pm

The proof of Crank-Nicolson montonicity (for a small enough time-step) is in Pooley et al. "Numerical Convergence Properties of Option Pricing PDEs with Uncertain Volatility". A different more general proof is also in Boley & Crouzeix "Conservation de la positivité lors de la discrétisation des problèmes d’évolution paraboliques".
I'm sceptical. B&C is not more general. UVM is nonlinear, unfortunately.To prove your assertion, do it first for a linear PDE.

Coincidentally, I had detailed discussions with Michel Crouzeix in Paris VI on linear parabolic PDE in 1975 when I was an MSc student.I have his thesis in my bookshelf.

Regarding UVM, we did it in 1 factor and two factor (MSc student from the University of Birmingham (UK))  using ADE.

https://wilmott.com/tag/alternating-dir ... it-method/

https://onlinelibrary.wiley.com/doi/abs ... wilm.10014

So, CN is not monotone in my book..
Just prove it for the heat equation and I'll start believing it :-) 
JohnLeM seems to have found  that it doesn't work even  in a simpler case.
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