Hi all,I'am calibrating a Vasicek model using an historical series of interest rate. As a result of calibration I have obtained the three parameters of the model: mean reversion speed, mean reversion level and volatility.Then for curiosity I have tried to do the following things: 1. Simulate a path, coherent in term of length with the historical time series, using the parameters find above 2. Insert in my calibration routine the simulate pathI have noticed that, as I expected, the mean reversion level and volatility are basically the same obtained with the calibration on the historical time series; the same does not happen with the mean reversion speed.Is there an explanation for this behavior? For the moment I have only read in literature that is very difficult to find a stable estimate of the mean reversion speedThank you very much for your time!!!Bye

Well, regardless of how you do it, all you get is an estimate: thetahat Then, you have to go on and develop sampling distributions for your estimators.To do that, you just repeat your steps 1. & 2 say 100 times and look atthe histogram of the resulting estimates. Those histograms yield confidenceintervals. If the estimate is 6, the 95% confidence interval for the estimate might be (4,10). For maximum likelihood (ML) estimators, it's indeed true that the sampling distributionfor the mean reversion speed is very wide. It only becomes narrow asymptoticallyas N -> infinity for N equally spaced obs. This is a version of the central limit theorem.Even if you are not using ML, you probably can't do better in this respect.Given dX = c (b - X) dt + a dW, the order from narrowest to widest sampling distribution isabcAs to why that is, first: vol. coefs can be estimated precisely by taking more data with (0,T) fixed.That leaves (b,c)You might work out analytically the ML sampling distributions for (c,a) in dX = -c X dt + a X dW, observed discretely over (0,T) with N obs. What happens as N increases with T held fixed?That will give you a clue about c.Those GBM results are somewhat similar to what happens with (c,a) in dX = c (b - X) dt + a dW

Thank for your answer!!!