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Are there any general arguments to decide whether it is better to use a model with a normal or a lognormal distribution of the short rate? E.g. Hull-White with a normal and Black-Karasinski with a lognormal, is there a way to judge which is better? I am aware of the fact that lognormal models don't describe negative interest rates, but are there any other criteria? By "better" I mean the model's better ability to reproduce actual market data.

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What is your goal ? what market data do you want to study ? Have you tried to plot their distribution ? –  lmorin Jul 16 '13 at 13:21
    
My goal is to use those models in potential future exposure calculations for swaps/swaptions. I have tried plotting distributions, but I didn't come up with any meaningful or clear distinctions. I just want to know if there exists any general knowledge that I can use as guidance beforehand, or if it just boils down to trial and error. –  FQuant Jul 16 '13 at 13:27

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General knowledge:

The reference for short rates models is: Interest Rate Models, by D. Brigo & F. Mercurio, Springer Worth the cost.

You can find a summary of the propeties of the "dr" models p15 & p19: Interest Rate Models: Paradigm shifts in recent years, D. Brigo, Columbia University Seminar

You will see the quote p19: "Pricing models need to be more precise in the distribution properties so lognormal models were usually preferred". Normal Versus Lorgnormal is not the only thing involved in the process of choosing the models.

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Thank you, the slides are particularly helpful. Unfortunately, your link does not work, I found them here: ieor.columbia.edu/pdf-files/Brigo_D.pdf –  FQuant Jul 16 '13 at 15:58

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