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Hi All,I'm studying the Black Karansinsky model for interest rate.For a personal project i have to calibrate the model from an historical rate series.Could you give me an hint on how I have to proceed in order to find the three parameters of the model (mean reversion level, reversion speed, volatility)?Thank you very much!!!!I'm a bit lost.....

Step 1: Write down the likelihood function. The likelihood of a time series of data {x(i)} is the the product of the probability transition functions p(x(i), x(i-1)). If you're stuck here, you need to figure out the probability transition for your model. Assuming your parametersare constants, I believe this model is simply the OU model, which has a known transition function. p.s. It looks like you will save yourself much grief by considering your 'data' to be x(i) = log r(i)

Also, to build a little intuition and manage your expectations for what you can achieve in a best-case scenario, simulate some rate paths from the model and try your estimator on those. Most likely you will find that you can hope to get a good estimate of the volatility parameter and very noisy estimates of the mean reversion parameters. On real world data you have the additional problem that the parameters are not constant or, more generally, that the model just isn't a good description of interest rate behavior.

Hi,I'm following your suggestion and I'm trying to write the likelihood function for the Black Karasinski model.I started from the discretize version of the SDE:y(t+∆t)=a+(y(t)-a) e^(-k∆t)+√(((1-e^(-2k∆t))σ )/2k)*zwhere a= mean reversion level, k=mean reversion speed , sigma=volatility and z is a standard normal random variable.so y(t+∆t) is normal with mean equal to a+(y(t)-a) e^(-k∆t) and variance ((1-e^(-2k∆t))σ )/2kThen I wrote the likelihood function as a product of the above distribution, Is it right in your opinion to proceed in this way?Thank you very much!!!!Bye

You can check your probability transition function in Karlin & Taylor ("A Second Course), pg 218Take a peek at amazon if you don't own it.