Applying a standard Euler sheme on the logarithm of the asset (or interest rate) we want to evolve in a Monte Carlo simulation procedure, what happend to the bias and to the estimate standard error if we take extra time step in the discretization?I know what is the standard error of the estimates, as it can be very easily derived starting from the n replications of the price we are simualting. What about the bias? Can someone try to exlain me what is it?Thank you in advance.Regards

Can you be more precise?Do some searching on Wilmott (Numerical methods, Technical) Kloeden's book discusses this as well.

To be more precise, I have implemented in matlab the Libor market model fro simulating forward Libors with standard tenor (6 months) resetting on fixed dates in the future.To perform the simualtion I used an Euler sheme applyed to the log of the forward rates... what I have to choose is the number of replications (I used a MC based on 100,000 replications) and the time step. As far as teh time step I compared replications with a delta(t) of 0.5 (6 motnhs) 0.25 /3 months) and 0.125 (1 month and a half)... as taking extar steps one should observe a smaller discretization error in the estimate, but what about its bias?I hope not to have ill-posed my question.Thanks for your time!

I am not familiar with the details, but didn't Mark Joshi (on Wilmott 'mj') do some work on LMM,using Predictor Corrector method. In general, Euler is very erratic and unpredictable. The bias is bad.HTH

Hi, Mark Joshi implemented the LIBOR market model in Quantlib. Quantib is in C++.F.

I have implemented a very basic Libor Market Model in Matlab (as unfortunately I don't know very well C++) using an Euler scheme discretization. Results are not bad, both tests on ZCB and caplet prices have matched almodt accuartely market quotes.As far as I have understood, the choice of a time-discretization always poses a trade-off: using a scheme with a higher number of time steps requires more computing time per path but decrease the discretization bias resulting from time-discretization of the continuous-time dynamics of the underlying model. The number of paths, on the other side, affects the standard error of any estimates we compute, but has no effect on the discretization bias, which is determined by our choice of time-step in the scheme. Thus we face a trade-off between reducing bias and reducing variance. I know Euler scheme is just the simplest method for approximating simulation of stochastic differential equations, anyway I think that also other schemes (of higher orders) suffer from this trade-off. Thus I think that my dicretization will be enough taking into account my objective. Regards

Indeed there are two sources of error, and they arise from different parameters. You generally need a lot of MC paths to see discretization error clearly since standard error tends to swamp it, but that depends on the step size, of course. In my experience, using predictor-corrector results in much less discretization bias at a cost of each step taking about 1.5 times as long. It sufficiently effective that I just use it and never worry about the discretization bias. The additional coding effort is tiny too. I have a couple of papers on this topic on www.markjoshi.com and one www.quarchome.org