Hi there,I am playing around with a low-factor BGM-model (i.e. #factors = 1, 2 or 3), calibrated to caps. I am trying to reproduce the cap prices with a Monte Carlo simulation.Thereby I found out that the results are considerably better if the choosen numeraire is the very first one instead of the very last one (which is 30y from today).Well, I expected that both choices of numeraire should lead to comparable results. Does it make sense? Or is it more likely that I made a mistake in my implementation.Maybe anybody knows an answer to this problem.Thanks,BustopherJones

The results should be close, depending on how noisy is your MC engine, how many paths you are using, etc. From what you've said I suspect that you forgot to express the cash flows in the units of the relevant numeraire (even though you may have done the suitable adjustments to the drift terms). As a result, your pricing is accurate for numeraires with very short maturities and off when you pick the last maturity.

how many steps are you doing? 1 or manywhat numeraire are you using after the bond matures?are FRAs priced correctly?

Thanks, Elan.I was running my MC with quite a lot number of paths (~10.000.000). And I also made use of the terms \prod_{k}(1+tau * F_k(T_k)) in order to "roll" the cash flows to the chosen numeraire. So, I think that I expressed the cash flows in units of the relevant numeraire therewith.In order to test the results I chose a covariance matrix with considerably large entries (which are all equal if #factors =1). These large entries produce considerably large drift adjustments (up to sign).*If the chosen numeraire is the very first one there is a large drift adjustment (with positive sign) for the last forward libor simulated. The MC results in this case are really satisfactory, even for the last forward.*If the chosen numeraire is the very last one there is a large drift adjustment (with negative sign) for the first forward libor simulated. The MC results in this case are not so satisfactory, especially for the first forward.I suspected, for reasons of symmetry, that both choices of numeraire should lead to comparable results.I hope I could make this problem clear.Thanks,BustopherJones

Hi MJ,in the terminology of Rebonato I'm performing a "very long jump", i.e. using only one step.FRAs are priced correctly if I make use of the "correct" covariance matrix, which has full rank and requires that#factors = #forwars libors. In this case the drift adjustments are considerably small.If I use a 1-factor model and a covariance matrix where all entries are equal, the drift adjustments areconsiderably large (up to sign) and in this case I can also observe the mismatch desribed below:*If the chosen numeraire is the very first one there is a large drift adjustment (with positive sign) for the last forward libor simulated. The MC results in this case are really satisfactory, even for the last forward.*If the chosen numeraire is the very last one there is a large drift adjustment (with negative sign) for the first forward libor simulated. The MC results in this case are not so satisfactory, especially for the first forward.I am almost sure that my drift calculations are correct (I am using the predictor/corrector scheme) and I also account for the "rolling factors"in order to express the cashflows in the relevant numeraire.Thanks.BustopherJones

How large are your drift adjustments? Under typical market (USD) conditions they don't exceed 40bp. Check whether your implementation of predictor-corrector is not getting in the way (unless you work with lognormal vols, it's not easy to implement). I still suspect that you might have a problem with properly discounting your cashflows under the chosen numeraire.By the way, 10,000,000 MC paths is a lot, you probably don't need that many of them. With well implemented Sobol sequences you need 3,000 - 5,000 paths, with a pseudorandom number generator you should be ok with 20,000 paths.

1) a very jump and first bond as numeraire are an incompatible choice -- you can only use the last or second last bond, your numeraire no longer makes sense after the first year 2) it doesn't make sense to do 1 factor and a very long jump, even if the rates are perfectly correlated, instantaneously, the true covariance matrix to maturiy of the last caplet is full rank. there is some sense in which you are making the later rates very volatile for a year and then stopping them all at the end of the first year, it's not surprising that this gives screwy behaviour

Thank you MJ!1) I don't really understand this point. The formulas for the drift adjustments also allow for the case where the index of the fwd. libor is greater than the index of the chosen numeraire. So, from this point of view I don't understand why the choice of the first bond as numeraire is not allowed for very long jumps.2) is very plausible to me. In which situations is the reduction to fewer factors (say 2 or 3) reasonable? The total covariance of a long jump from T_{i} to T_{i + k} will in general have full rank, so that actually this jump should be driven by as many factors as fwd. libor still alive at T_{i}. So when am I allowed to use, say , 3 factors?Thank you again.BustopherJones

QuoteOriginally posted by: BustopherJonesThank you MJ!1) I don't really understand this point. The formulas for the drift adjustments also allow for the case where the index of the fwd. libor is greater than the index of the chosen numeraire. So, from this point of view I don't understand why the choice of the first bond as numeraire is not allowed for very long jumps.2) is very plausible to me. In which situations is the reduction to fewer factors (say 2 or 3) reasonable? The total covariance of a long jump from T_{i} to T_{i + k} will in general have full rank, so that actually this jump should be driven by as many factors as fwd. libor still alive at T_{i}. So when am I allowed to use, say , 3 factors?Thank you again.BustopherJonesre 1) what does it mean to use a bond as numeraire when it's already reset?re 2) personally i don't recommend doing reduced factor models for very long jumps

Thanks, MJ.re 2) personally i don't recommend doing reduced factor models for very long jumpsOkay. Would you recommend a reduced factor models if we are talking about long jumps (in the sense of Rebonatos book, i.e. the evolution from one price-sensitive event to the next, which can be seperated by as much as several years). Or would you even in this case do without factor reduction?Thanks,BustopherJones