SERVING THE QUANTITATIVE FINANCE COMMUNITY

• 1
• 2

trc
Posts: 95
Joined: April 4th, 2002, 2:28 pm

### Implementation of BGM/Libor market model

TREES:In the driftless model it is extremely easy to implement a recombining tree for the model.To do this we have to drop the number of factors.Suppose we decide that 4 factors are good enough.This means that four of the state variables determine all the others as deterministic functions of these four and it is easy to make this explicit.Set Y_j=log(U_j). Then the Y_j follow a dynamicsdY_j(t)=-0.5*sigma^2_j(t)dt + sigma_j(t)u_jdW(t)with deterministic drift. So we need to kow only the driftless "volatility parts" V_j(t)V_j(0)=Y_j(0), dV_j(t)=sigma_j(t)u_jdW(t).At any time t the vector V(t) is multinormal with covariance matrix C of rank four, that is,C=RR' where R is of dimension k by 4.Since the vector V(t) is multinormal we have V(t)=RZ, where Z is a 4 dimensional standard normal vector.If you want to determine the value of the variables V_j(t) from any four of them simply use these four to solve for Zand use the values of Z to compute all the others. Of course we could formulate this in terms of multiplication with a matrix inverse.We then add the deterministic drift to get Y_j(t) and from this U_j(t)=exp(Y_j(t)).Consequently we only have to evolve 4 variables, say V_{n-4},...,V_{n-1} and these form a Gaussian process.To construct a tree for that is even easier than constructing a tree for a log-Gaussian dynamicsdS(t)=S(t)sigma(t)dW(t).If we use a simple up-down model each of the four state variables has t+1 possible states at time step t.This gives us (t+1)^4 possible nodes at time t -- still manageable.Four factors cover the four largest eigenvalues of the covariance matrix and from what I have seen in mycovariance matrices this should get about 97% of the variability.I believe the availability of trees will tilt the scale in favor of the driftless Libor market model.I'll try to implement it before I release the C++ code.
Last edited by trc on June 4th, 2003, 10:00 pm, edited 1 time in total.

trc
Posts: 95
Joined: April 4th, 2002, 2:28 pm

### Implementation of BGM/Libor market model

Folks,I have just uploaded my free book "Monte Carlo Simuylation with Java and C++" on my websitehttp://martingale.berlios.de/Martingale.html (follow the links to Download, the files are MCBook.pdf.gz, MCBook.ps.gz).I have written up a possible approach to tree modelling of the Libor market model.Please let me know if you find any mistakes or feel that some ideas there are not sound for one reason oranother. PS.: I have finally put links to the mailing lists in place.Please subscribe.
Last edited by trc on June 13th, 2003, 10:00 pm, edited 1 time in total.

trc
Posts: 95
Joined: April 4th, 2002, 2:28 pm

### Implementation of BGM/Libor market model

The Berlios servers are terminally ill.I am attaching an updated version of the FastLibor paper containing what I hope tobe a viable method of constructing a tree for the LMM.
Last edited by trc on July 5th, 2003, 10:00 pm, edited 1 time in total.

trc
Posts: 95
Joined: April 4th, 2002, 2:28 pm

### Implementation of BGM/Libor market model

The method which evolves n state variables from only two of them using the correlation matricesas described in the old attachment seems to be unsound. Consequently I am unable to implement a tree for the Libor market model with time varying volatilities.This is a big disappointment. The good news is that the case of constant volatilities is easy. I have this working for two and three factor models.I have removed the old version of the paper and attached the updated one.
Attachments
FastLibor-7-5-03.zip
Last edited by trc on July 5th, 2003, 10:00 pm, edited 1 time in total.

trc
Posts: 95
Joined: April 4th, 2002, 2:28 pm

### Implementation of BGM/Libor market model

Ladies and Gentlemen,Please obtain the first release of my C++ libor package (libor-0.4) from http://martingale.berlios.de.The problems with the predictor-corrector algorithm disappeared after some seeminglyunrelated code rewrites. I no longer have any reason to believe that this algorithm is not sound.
Last edited by trc on September 23rd, 2003, 10:00 pm, edited 1 time in total.

AlanB
Posts: 662
Joined: July 14th, 2002, 3:00 am

### Implementation of BGM/Libor market model

tried to get to the website. access was denied.

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

### Implementation of BGM/Libor market model

Hi trc,I am very interesting in the Libor project and want to more learn on it.Especially, I am interested in your calibration procedure. On what source is it based?Is there a VC++ project available?Best wishes, Lapsilago

Wilmott.com has been "Serving the Quantitative Finance Community" since 2001. Continued...

 JOBS BOARD

Looking for a quant job, risk, algo trading,...? Browse jobs here...

GZIP: On