his simple interest rate model in Q measure isr is r(t) for abbreviationdr = mu r dt + sigma r dWLet's assume that mu and sigma is constant through time.If I was given a historical time series of 1 day rates (approximately short rates), In a GBM, the log-returns are normally distributed so the ML estimators are:mu = gamma + (sigma^2)/2sigma^2 = 1/T * Sum(log(rt)-log(r_t-1)-gamma)^2 (variance )gamma = 1/T Sum(log(rt)-log(r_t-1)) (mean of time series)Are the parameters estimated for the Q (risk neutral) measure or P measure?I'm slightly confused because a lot of posts say historical time series will only give us P measure.However, if one assumes that parameters are stable through time, can we use these parameters for the Q measure?

Hi,The rates time serie is the real world (P measure) time serie and has to be used for real world applications: forecasting, stress or back-testing...The observable options prices, that depend on an implicit Q world time series (people who trade derivatives have their own vision for the rates future evolution that differs from the real world) will not be coherent with your P measure time series, unless the market price of interest rate risk is zero. Now when you move to probability Q (called the risk-neutral and uses the cash as numeraire). You will have the following link, Note that short rate is not a traded asset so it is not easy to get mu^Q:- mu^P = mu^Q - lambda*sigma : drift changes, where lambda is the risk premium)- sigma will be similar for both measures (this holds for the cash numeraire but false for other numeraires !)Now the question about how to estimate lambda ?? is an open question !! and I did not see a convenient practical methods. This maybe why people stipulate their models in Q and use calibration to implicit parameters rather than historical estimation that is hard to link, does not give stable parameters and maybe does not recover quoted prices.

What exactly do you want to estimate in term of model, product?

QuoteOriginally posted by: ZhuLiAnWhat exactly do you want to estimate in term of model, product?I thinks he want to estimate r0, mu, sigma the pricing diffusion parameters (under Q) using the time serie of short rate in the real world P.