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JohnLeM
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Posts: 515
Joined: September 16th, 2008, 7:15 pm

The Bachelier problem

March 29th, 2021, 9:15 am

Following @bearish advise, I open a dedicated thread to discuss a numerical study of the Bachelier problem, that can be found here at SSRN.
Indeed, I did not complete still the conclusion section of this paper, this thread could help here. My first idea was to include a basic argumentation to sustain the numerical analysis, like
1) Why are the first two methods non convergent ?
2) Why the two other methods are convergent ?

The paper abstract is: We present a general approach to compute conditional expectations, materialized in this report by a python function [$]p(z | x) = \Pi^k(x,z)[$], where [$]x,z[$] are any iid samples of a martingale process at two different times, and [$]k[$] is a kernel. This function is embedded in a framework, CodPy, and we benchmark it against two other methods. The first one is a Neural Network (NN) based method, using an open-source implementation. The second one is a kernel-based methods, that is quite similar to the neural network one. We will show numerically that these two last methods provides non convergent methods, since the function [$]\Pi^k[$] provides a convergent method at statistical rate  [$]\frac{1}{\sqrt{N}} \%[$], where [$]N := card(x)[$] is the number of samples used. We illustrate also how to boost convergence rates using sharp discrepancy sequences.
 
User avatar
JohnLeM
Topic Author
Posts: 515
Joined: September 16th, 2008, 7:15 pm

Re: The Bachelier problem

April 4th, 2021, 8:53 am

Here is the good link to this technical analysis, the previous link seems erroneous.