SERVING THE QUANTITATIVE FINANCE COMMUNITY

JohnLeM
Topic Author
Posts: 380
Joined: September 16th, 2008, 7:15 pm

### Re: Are Artificial Intelligence methods (AKA Neural Networks) for PDEs about to rediscover the wheel ?

You seem to be contradicting your LinkedIn post now (or I don't understand something). In it you wrote that "a bounded variation function, a function class for which we know that the convergence rates of a sampling method can not exceed 1 / N, not 1/N^2." The payoff of a call option, (S - K)+, is a bounded variation function, hence the convergence rate should be limited to 1/N.
No : a call option is a function having a gradient of bounded variation. It is one order smoother than a barrier option. Rephrasing : the gradient of a call option is a barrier option.

Cuchulainn
Posts: 62395
Joined: July 16th, 2004, 7:38 am
Location: Amsterdam
Contact:

### Re: Are Artificial Intelligence methods (AKA Neural Networks) for PDEs about to rediscover the wheel ?

A call option does not have a gradient .. it has a derivative which is a Heaviside function????

I see a disconnect between ISM's question and your answer (caveat: I have a bad dose of flu..)

//
Off-topic: I recommend FW in French (published by Gallimard)

Finnegan le Constructeur, stathouder de sa main, maçon des hommes francs, habitait la plus grande rue qu on puisse imarginer en son vieil habit rutilant trop retiré en couleur pour y recevoir des messies, et pendant de nombieuses années cet homme d oiseau, de ciment et d édifices entassa imagifices sur imagifices sur les rives d Hameau Torpeur pour ses foibitants comme dit la chanson.

And the vicus of recirculation from NN to PDE

erre revie, pass'Evant notre Adame, d'erre rive en rêvière, nous recourante via Vico par chaise percée de recirculation vers Howth Castle et Environs

JohnLeM
Topic Author
Posts: 380
Joined: September 16th, 2008, 7:15 pm

### Re: Are Artificial Intelligence methods (AKA Neural Networks) for PDEs about to rediscover the wheel ?

A call option does not have a gradient .. it has a derivative which is a Heaviside function????

I see a disconnect between ISM's question and your answer (caveat: I have a bad dose of flu..)

//
Off-topic: I recommend FW in French (published by Gallimard)

Finnegan le Constructeur, stathouder de sa main, maçon des hommes francs, habitait la plus grande rue qu on puisse imarginer en son vieil habit rutilant trop retiré en couleur pour y recevoir des messies, et pendant de nombieuses années cet homme d oiseau, de ciment et d édifices entassa imagifices sur imagifices sur les rives d Hameau Torpeur pour ses foibitants comme dit la chanson.

And the vicus of recirculation from NN to PDE

erre revie, pass'Evant notre Adame, d'erre rive en rêvière, nous recourante via Vico par chaise percée de recirculation vers Howth Castle et Environs
ok for the flu ...I understand. What is a bounded variation function ? it is basically a function which derivative is a measure (even if it is a little bit more complex). Consider a call payoff (x-K)^+. Take its derivative : {0, x < K, 1, x > K}, that is a heavyside function, also called a barrier. Then take a second derivative : \delta_{K}(x), that is a dirac, being a measure.

Thanks for the traduction, but it was not a French that I can understand. Thus I ordered Finnegans wake ! I enjoyed reading Ulysse when I lived in Trieste, but it is already quite a technical literature for me !

JohnLeM
Topic Author
Posts: 380
Joined: September 16th, 2008, 7:15 pm

### Re: Are Artificial Intelligence methods (AKA Neural Networks) for PDEs about to rediscover the wheel ?

It takes being a physicist to know that translating every problem to PDEs is not necessarily a good approach
I have found some papers, that could interest you, emanating from the Artificial Intelligence community, pointing out the link between PDE and neural network methods. For instance this one seems quite interesting, even if I did not read it fully yet.

In Bishop book chap 5, what is treated are problems that are called in the PDE community optimal control problems. The setting in this book is clearly equivalent to a PDE approach, no doubt about it.

The question I am asking now is : is there exists a single problem, treated by a deep tralala algorithm, that can not be interpreted as a PDE one ? For instance, this paper seems to conclude that both approachs are not equivalent, I will read it carefully.

Cuchulainn
Posts: 62395
Joined: July 16th, 2004, 7:38 am
Location: Amsterdam
Contact:

### Re: Are Artificial Intelligence methods (AKA Neural Networks) for PDEs about to rediscover the wheel ?

What is a bounded variation function ? it is basically a function which derivative is a measure (even if it is a little bit more complex). Consider a call payoff (x-K)^+. Take its derivative : {0, x < K, 1, x > K}, that is a heavyside function, also called a barrier. Then take a second derivative : \delta_{K}(x), that is a dirac, being a measure.

Are you taking to me, a mathematician? A Dirac function is a distribution, not a measure the last time I looked. This is 1st year undergrad stuff.

This statement you wrote looks wrong. What's the story with functions of bounded variation?

a call option is a function having a gradient of bounded variation. It is one order smoother than a barrier option. Rephrasing : the gradient of a call option is a barrier option.

heavyside function, also called a barrier.
No, nay, never.

katastrofa
Posts: 9330
Joined: August 16th, 2007, 5:36 am
Location: Alpha Centauri

### Re: Are Artificial Intelligence methods (AKA Neural Networks) for PDEs about to rediscover the wheel ?

It takes being a physicist to know that translating every problem to PDEs is not necessarily a good approach
I have found some papers, that could interest you, emanating from the Artificial Intelligence community, pointing out the link between PDE and neural network methods. For instance this one seems quite interesting, even if I did not read it fully yet.

In Bishop book chap 5, what is treated are problems that are called in the PDE community optimal control problems. The setting in this book is clearly equivalent to a PDE approach, no doubt about it.

The question I am asking now is : is there exists a single problem, treated by a deep tralala algorithm, that can not be interpreted as a PDE one ? For instance, this paper seems to conclude that both approachs are not equivalent, I will read it carefully.
You can sit and write down all PDEs describing all observable phenomena, square them, move everything to one side and add up. The problem is solving it, vide acoustics.

JohnLeM
Topic Author
Posts: 380
Joined: September 16th, 2008, 7:15 pm

### Re: Are Artificial Intelligence methods (AKA Neural Networks) for PDEs about to rediscover the wheel ?

What is a bounded variation function ? it is basically a function which derivative is a measure (even if it is a little bit more complex). Consider a call payoff (x-K)^+. Take its derivative : {0, x < K, 1, x > K}, that is a heavyside function, also called a barrier. Then take a second derivative : \delta_{K}(x), that is a dirac, being a measure.

Are you taking to me, a mathematician? A Dirac function is a distribution, not a measure the last time I looked. This is 1st year undergrad stuff.
A dirac measure is not a measure ?

Cuchulainn
Posts: 62395
Joined: July 16th, 2004, 7:38 am
Location: Amsterdam
Contact:

### Re: Are Artificial Intelligence methods (AKA Neural Networks) for PDEs about to rediscover the wheel ?

Fair enough. Aka indicator/ Kronecker function.
https://en.wikipedia.org/wiki/Dirac_measure

The question is still: "the derivative of a call option is a barrier option". Is that what you mean?

JohnLeM
Topic Author
Posts: 380
Joined: September 16th, 2008, 7:15 pm

### Re: Are Artificial Intelligence methods (AKA Neural Networks) for PDEs about to rediscover the wheel ?

Fair enough. Aka indicator/ Kronecker function.
https://en.wikipedia.org/wiki/Dirac_measure

The question is still: "the derivative of a call option is a barrier option". Is that what you mean?
(x-K)^+ ' = {0, x< K, 1, x> K}. Can't I call this a barrier option ?

Cuchulainn
Posts: 62395
Joined: July 16th, 2004, 7:38 am
Location: Amsterdam
Contact:

### Re: Are Artificial Intelligence methods (AKA Neural Networks) for PDEs about to rediscover the wheel ?

Fair enough. Aka indicator/ Kronecker function.
https://en.wikipedia.org/wiki/Dirac_measure

The question is still: "the derivative of a call option is a barrier option". Is that what you mean?
(x-K)^+ ' = {0, x< K, 1, x> K}. Can't I call this a barrier option ?
I only see the derivative of a call payoff.

You can call it what you want but it will be your own internalsed definition. Barrier options have the same payoff as yer normal option but with extra boundary checks.

https://en.wikipedia.org/wiki/Barrier_option

http://web.math.ku.dk/~rolf/teaching/th ... rgKani.pdf

And why is this train of thought relevant here? e.g. why do you want to differentiate a payoff.

JohnLeM
Topic Author
Posts: 380
Joined: September 16th, 2008, 7:15 pm

### Re: Are Artificial Intelligence methods (AKA Neural Networks) for PDEs about to rediscover the wheel ?

There's at least one person from the AI community here who's trying to tell you that you're mistaken about the equivalence and explaining clearly why (however difficult it is to pin down what you mean).
I am trying to argue with you, on a mathematical basis, concerning this equivalence. But maybe science is not the good angle here, it seems that AI community is not rational when it comes to this equivalence, I can understand that.

So I would like also to express my feelings: when I read an article entitled "how to solve a PDE problem with a deep learning method", I am feeling:

1) fun, I could entitle the same article "how to solve a deep learning problem with a PDE method"
2) less fun, I am paying taxes to read that.
3) they can call a cat a dog to get grant, but it won't bark.
Last edited by JohnLeM on April 6th, 2019, 12:06 pm, edited 3 times in total.

JohnLeM
Topic Author
Posts: 380
Joined: September 16th, 2008, 7:15 pm

### Re: Are Artificial Intelligence methods (AKA Neural Networks) for PDEs about to rediscover the wheel ?

Fair enough. Aka indicator/ Kronecker function.
https://en.wikipedia.org/wiki/Dirac_measure

The question is still: "the derivative of a call option is a barrier option". Is that what you mean?
(x-K)^+ ' = {0, x< K, 1, x> K}. Can't I call this a barrier option ?
I only see the derivative of a call payoff.

You can call it what you want but it will be your own internalsed definition. Barrier options have the same payoff as yer normal option but with extra boundary checks.

https://en.wikipedia.org/wiki/Barrier_option

http://web.math.ku.dk/~rolf/teaching/th ... rgKani.pdf

And why is this train of thought relevant here? e.g. why do you want to differentiate a payoff.
Fair enough too. You are right, I confused payoff.

katastrofa
Posts: 9330
Joined: August 16th, 2007, 5:36 am
Location: Alpha Centauri

### Re: Are Artificial Intelligence methods (AKA Neural Networks) for PDEs about to rediscover the wheel ?

OMG! The cat looks so cute - I agree with everything you said!
Last edited by katastrofa on April 6th, 2019, 3:52 pm, edited 1 time in total.

JohnLeM
Topic Author
Posts: 380
Joined: September 16th, 2008, 7:15 pm

### Re: Are Artificial Intelligence methods (AKA Neural Networks) for PDEs about to rediscover the wheel ?

>>And why is this train of thought relevant here? e.g. why do you want to differentiate a payoff.

I was trying to explain to @ISayMoo that Calls are somehow one derivative smoothers than Autocalls. Hence a sampling method, meaning a Monte-Carlo like method of kind
$\int_{R^D} P(x) d\mu(x) \sim \frac{1}{N} \sum_{n=1}^N P(x^n)$

will converge with one order more in term of convergence rate : for Autocalls or barrier options, you can't expect more than 1/N, but for Calls, you can find sampling sequences $x^n$ converging at rate 1/N^2

ISayMoo
Posts: 2315
Joined: September 30th, 2015, 8:30 pm

### Re: Are Artificial Intelligence methods (AKA Neural Networks) for PDEs about to rediscover the wheel ?

OK, so just so that I get it straight:
1. call payoff (S - K)+ is not BV and there is NO sampling sequence for it which converges faster than 1/N
2. binary payoff Heaviside(S - K) is BV and there IS a sampling sequence for it which converges faster than 1/N

Correct?