Senior Professional Indemnity Underwriter here
Bar a couple of specific professions (lawyers being the main one and even they're slowly changing), PI policies start whenever and generally run for 12 months.l - much like your home or car insurance.
Lawyers used to all renew on 1 October so that was difficult for those that did lawyers and it's still very busy. I've never gone near them. A good chunk have shifted to a 1/4 renewal now that's a busy day too.
The 31 Dec is a very quiet day for renewals. I had a couple of policies that renewed then and even one on Christmas Day however renewals are spread nicely throughout the year.
Any policy that renews anywhere between 15 Dec and 10 Jan can be a pain though as the London Market pretty much shuts up shop so if it's a complex risk that needs marketing it can be an arse.
24 years and counting as a PI underwriter...