Test Log-Normality for LIBOR forward rates under the Libor Market Model

Question

As far as I understand, under the Libor Market Model the forward rates are assumed to have a log-normal distribution. Given that I have constructed my LMM model and now have a matrix of:

• k different forward rates, that is, they mature on different dates.
• t time steps
• N different scenarios

How can I make either a chi-squared test or a Kolmogorov-Smirnov test to check for log-normality?

I have already done tests with:

• One forward rate, one time-step and all scenarios at a time

• One forward r ate, all time-steps and one scenario at a time

These two tests reject the hypothesis that they should have a log-normal distribution, and my question is therefore:

In what way are they assumed to be log-normal, that is, with what part of the data do I test?

Worth noticing is that I simulated under the spot measure, does this matter?

Answer 1

well they aren't actually log-normal! if you use the terminal measure and test the last forward rate, it is log-normal.

The essential point is that the drifts that make the ratio of bond prices to nuemraire a martingale are state-dependent. This state dependence destroys log-normality.

You can take the real-world measure rates to be log-normal but the real-world process is not very relevant.