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

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
Related articles

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(?)
"Compatibility means deliberately repeating other people's mistakes."
David Wheeler

http://www.datasimfinancial.com
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Cuchulainn
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### Re: Monotone Schemes: what are they and why are they good?

You probably want monotone schemes for the Greeks

https://www.datasim.nl/application/file ... hesis_.pdf
"Compatibility means deliberately repeating other people's mistakes."
David Wheeler

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http://www.datasim.nl

JohnLeM
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### Re: Monotone Schemes: what are they and why are they good?

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.

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

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.
"Compatibility means deliberately repeating other people's mistakes."
David Wheeler

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JohnLeM
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### Re: Monotone Schemes: what are they and why are they good?

@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 ?

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

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
"Compatibility means deliberately repeating other people's mistakes."
David Wheeler

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JohnLeM
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### Re: Monotone Schemes: what are they and why are they good?

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.

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

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.
"Compatibility means deliberately repeating other people's mistakes."
David Wheeler

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

And I don't see how Crank-Nicolson can be made monotone. We are waiting on a proof.
"Compatibility means deliberately repeating other people's mistakes."
David Wheeler

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JohnLeM
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### Re: Monotone Schemes: what are they and why are they good?

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.

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

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.
"Compatibility means deliberately repeating other people's mistakes."
David Wheeler

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jherekhealy
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### Re: Monotone Schemes: what are they and why are they good?

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".

JohnLeM
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### Re: Monotone Schemes: what are they and why are they good?

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.

JohnLeM
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### Re: Monotone Schemes: what are they and why are they good?

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 ?

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

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.
"Compatibility means deliberately repeating other people's mistakes."
David Wheeler

http://www.datasimfinancial.com
http://www.datasim.nl