there is 0 month tenor for CDS instruments, it is not clear what does it refer to Can anyone explain what zero month tenor for single-name CDS stands for?
Following the "big bang", the standard CDS maturities are June 20 and December 20. (See: https://www.isda.org/a/vGiDE/amend-single-name-on-the-run-frequency-faq-revised-as-of-12-10.pdf and https://www.isda.org/a/jGiDE/amend-single-name-on-the-run-frequency-faq-revised-as-of-10-5.pdf question 11 for more examples. Single-name contracts used to roll 4 times a year, but now almost all roll only twice a year, like the indices.) (They are sometimes called "IMM dates", which confuses people who think "IMM dates" are 3rd Wednesdays).
For example, if today is June 1, 2020, then a 5-year CDS matures on June 20, 2025.
A 1-year CDS matures on June 20, 2021.
A 6-month CDS matures on December 20, 2020.
Do you see the pattern?
A 0-month CDS matures on June 20, 2020, the next standard maturity date.
If a 3-month and 9-month CDSs are liquid, then they would mature on September 20, 2020 and March 20, 2021 respectively.
Now consider the state of the world 1 month later - on July 1, 2020. The 0-month CDS has matured and does not trade anymore. The other tenors are unchanged.
Now consider the state of the world 6 month later - on December 1, 2020.
All the tenors have shifted by 6 months. A 5-year CDS matures on December 20, 2025, while a 0-month CDS exists again and matures on December 20, 2020.
Observe the 0-month tenor exists for 3 months, then does not exists for 3 months, then exists again (with its date 6 months later) for 3 months...