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Joerg
Topic Author
Posts: 2
Joined: October 8th, 2002, 7:34 am

### Simulation of SABR/LMM a la Rebonato

Hi,Has anyone actually implemented the SABR LMM with non-zero correlation for Libor/LiborVol (as suggested in Rebonato's book) ? A MC implementation of Rebonato?s correlation must perform poorly.If beta=1 the SDE is: $dL_i / L_i = m^L_i dt + s_i V_i dW_i$$dV_i / V_i = m^V_i dt + a_i dZ_i$with $E[ dW_i , dZ_i ] = rho_i$the drivers might be factor reduced ($dW_i = sum_k G_{ik} dX_k$ and $dZ_i = sum_k H_{ik} dY_k$ ) , but this is not important.The point is, if rho_i != 0 then a naïve correlation of $dW_i$ and $dZ_i$ (as suggested in Rebonato via a correlation of the driving factors $dX_k$ and $dY_k$ ) will perform poorly. This could be fixed by a scheme similar to Boradie-Kaya scheme which means to cholesky decompose dW :$dW_i = rhoBar_i dU_i + rho_i dZ_i$ with $E[ dU_i , dZ_i ] = 0$ and $rhoBar_i = sqrt( 1 - rho^2_i$ )but then the whole model correlation and model set up changes completely (compared to Rebonato's suggestion). Not a problem, but I'm wondering how people (including Rebonato) got their beautiful simulation results.
Last edited by Joerg on December 17th, 2014, 11:00 pm, edited 1 time in total.

Joerg
Topic Author
Posts: 2
Joined: October 8th, 2002, 7:34 am

### Simulation of SABR/LMM a la Rebonato

No replies can mean:a) Nobody understands what I'm asking.b) Nobody knows the answer or has written a MC simulation for that model.c) Nobody cares about this model.In case it's a) I'll bring it more to the point:Rebonato's correlation specification => very small ( <<< 0.5Y ) time steps required => model is utterly useless in practise (and no point to write a book about it). My question is simply: Does anybody disagrees with this conclusion? The problem has nothing to do with BGM itself. It exists as well in the SABR model. For simplicity take beta=0. dF = s dWds = alpha dZIf you just naively draw correlated random numbers (to simulate correlated dW and dZ) you are forced to use very small time steps, otherwise you will "leak" correlation (and thus invalidate all closed form formulas). The same problem occurs in the SABR BGM "a la Rebonato".There is an easy fix for this, but this changes completely the correlation structure and you end up with quite a different (but usable) model specification. Any comments are welcome.
Last edited by Joerg on December 28th, 2014, 11:00 pm, edited 1 time in total.

Lapsilago
Posts: 259
Joined: October 15th, 2004, 7:36 am
Location: Germany

### Simulation of SABR/LMM a la Rebonato

Hi Joerg,I remember that there is also an issue with the correlation and an advanced scheme (QE) in the Heston model.I have to look this up. Can you send me your email address as private message. Then, we can use emails to discuss and must not spam the forum.Best regards,Jörg

japanstar
Posts: 73
Joined: July 13th, 2009, 7:07 am

### Simulation of SABR/LMM a la Rebonato

There is an upcoming book, published by Palgrave Macmillan, which contains a MC implementation code for SABR LMM (together with other code snippets to be used for calibration etc.). It does not address exactly your problem as a 0 fwd-vol correlation set up has been used throughout the book (for obvious implementation simplifications) but you might want to have a look at it.

Orbit
Posts: 492
Joined: October 14th, 2003, 5:34 pm

### Simulation of SABR/LMM a la Rebonato

QuoteOriginally posted by: japanstarThere is an upcoming book, published by Palgrave Macmillan, which contains a MC implementation code for SABR LMM (together with other code snippets to be used for calibration etc.). It does not address exactly your problem as a 0 fwd-vol correlation set up has been used throughout the book (for obvious implementation simplifications) but you might want to have a look at it.Author?

japanstar
Posts: 73
Joined: July 13th, 2009, 7:07 am

### Simulation of SABR/LMM a la Rebonato

As soon as I get the latest Palgrave catalog I'll tell you the exaxct title, authors and issue date.
Last edited by japanstar on February 23rd, 2015, 11:00 pm, edited 1 time in total.