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ryanmccrickerd / rough_bergomi

A Python implementation of the rough Bergomi model.
MIT License
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Simulate Y #3

Closed anh-tong closed 2 years ago

anh-tong commented 2 years ago

Hi,

Thank you for sharing the code.

I wonder why Y2 has two zeros at the start of the output in this line. I thought it starts with only one zero like Y1.

ryanmccrickerd commented 2 years ago

Hi,

If you can provide the code, or point me to the notebook, which produces what you are referring to, I will take a look at this next week.

Otherwise, if I’m not mistaken, we should expect Y2 to start with one zero like Y1, plus N more zeros where N is the order (kappa) of the hybrid scheme being used. Since the repo uses kappa=1, then we should see 1+1=2 leading zeros in Y2.

Regards,

On Fri, 22 Apr 2022 at 13:02, Anh Tong @.***> wrote:

Hi,

Thank you for sharing the code.

I wonder why Y2 has two zeros at the start of the output in this line. https://github.com/ryanmccrickerd/rough_bergomi/blob/6a04ca5d2d0f6a45df28ae7bad9c882d537ef14e/rbergomi/rbergomi.py#L61 I thought it starts with only one zero like Y1.

— Reply to this email directly, view it on GitHub https://github.com/ryanmccrickerd/rough_bergomi/issues/3, or unsubscribe https://github.com/notifications/unsubscribe-auth/AA257343KJRKXQHV3PLLNE3VGKIO7ANCNFSM5UCDYV5Q . You are receiving this because you are subscribed to this thread.Message ID: @.***>

anh-tong commented 2 years ago

Thank you for replying.

As I understand, Y2 aims to compute the second term in the sum (eq 3.4 in the paper https://arxiv.org/pdf/1507.03004.pdf). If we pick the second point (is not affected by the choice of kappa) on the discretized grid, the second term does not seem to be zero.

image

Here is a script to replicate https://colab.research.google.com/drive/1cwF5u26RB3GApOZg9aX2fg1wEiSQ3qKF?usp=sharing

ryanmccrickerd commented 2 years ago

Thanks for sharing. But Y denotes a truncated Brownian semi-stationary (TBSS) process, so if I'm not mistaken you should be looking at Eq 3.7 in Mikko's paper, not Eq 3.4.

Y2 then corresponds to the second term in Eq 3.7. Setting kappa=1 there, you see that the first two contributed terms (i=0, i=1) are zero, since the applicable sum runs from k=2 to k=0 then k=2 to k=1. If Y was just a BSS, this would not be the case.

This paper is required to understand why the relevant process in the risk-neutral rBergomi model is indeed a TBSS, not a BSS. Hope this all helps

anh-tong commented 2 years ago

Thank you so much for the suggestion and detailed explanation.