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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?

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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.

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  • $\begingroup$ Thank you for your answer. What I am in reality asking for is if there exists a method for checking if the simulated forward rates are good estimates or not. You could compare predicted values to market data, but are there any statistical methods that can be used? $\endgroup$ – William Hedén Jul 8 '15 at 11:49
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    $\begingroup$ well the standard thing to do is to price contracts you have closed form formulas for. Eg caplets and digital caplets. The price should be correct within 2 or 3 standard errors. $\endgroup$ – Mark Joshi Jul 8 '15 at 11:52

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