LIBOR Market Model - tenors?

In the LIBOR market model, we have a bunch of forward rates $$L_j$$ on $$[T_j, T_{j+1}]$$ for some collection on $$j$$.

My question is, is it the delivery dates or the time to maturities that are fixed? So if I calibrate my model tomorrow, will it have the same underlyings as it did today, or will $$L_j$$ be $$[T_j + \delta t, T_{j+1} + \delta t]$$?

• since it's a quick question i'll just do a comment, time to maturities are fixed same underlying as today Jan 11 '19 at 16:57

In the libor market Model, what are modeled are the forward rates. Hence you have to see the fixing date $$T_j$$ and the maturity date $$T_{j+1}$$ as dates and not durations. Both are fixed dates and are the same at $$t+\delta t$$

Just to be precisely clear, your mathematical formulation will not necessarily capture the nuances of the physical dates that libor is valued between, due to holiday calendars and modification rules.

Take GBP for example. The LIBOR in that currency is subject to a Modified Following rule as well as a Month End Consistency rule.

For example:
Generally 6M Libor starting on any date of the month will roll to the same date of the month 6 months ahead and be modified forwards if it is not a business day. But if this takes it to a new month it is modified backwards. And, in GBP, if it starts on month end it ends on month end:

6M starting Wed Feb 27th 2019 ends on Tues 27th Aug 2019 (no adj. needed)
6M starting Thu Feb 28th 2019 ends on Fri 30th Aug 2019 (month end modified)
6M starting Wed May 29th 2019 ends on Fri 29th Nov 2019 (no adj. needed)
6M starting Thu May 30th 2019 ends on Fri 29th Nov 2019 (modified following)
6M starting Fri May 31st 2019 ends on Fri 29th Nov 2019 (modified following)
6M starting Thu May 16th 2019 ends on Mon 18th Nov 2019 (following)