When LIBOR Forward Rates Are Lognormal Under the Market Model

Summary

The document clarifies a distributional assumption in the Libor Market Model and explains why simulated forward rates may fail lognormality tests. Although the model is often described as using lognormal forward rates, the answer emphasizes that this property depends on the chosen pricing measure and on which forward rate is examined.

Under the terminal measure, the final forward rate is lognormal. For other rates, state-dependent drifts required to make bond-price ratios relative to the numeraire martingales can break lognormality. The question reports rejection of lognormality tests on sampled rates, but the response does not present test procedures or empirical results beyond that observation. It also says real-world rates may be modeled as lognormal, while cautioning that the real-world process is not the central object for the pricing model. The distinction limits any blanket interpretation of the LMM assumption.

Key ideas

• Lognormality in the Libor Market Model depends on the forward rate and the pricing measure.
• The final forward rate is lognormal under the terminal measure.
• State-dependent drift terms can prevent other forward rates from having lognormal distributions.
• A blanket lognormality test across rates, times, or scenarios may not match the model’s assumptions.
• The real-world rate process is distinct from the measure used for pricing.

Tags

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


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












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 by Mark Joshi (score 1, accepted)

https://quant.stackexchange.com/a/18770

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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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.