Query that references itself - postgresql

Question

I want to create a self referencing and shifting basked of stocks (similar to methodologies used by S&P for the S&P 500). The goal is to create an index that changes structure every month. It guarantees a spot for the first stock by market cap (rank). The 2nd spot goes to the stock that has been in the previous months' lineup and ranks between 2 and 3 this month. If that stock ranks lower than 3, it will get excluded and a new stock slots into its' place. Else, the next closest stock will get chosen.
Give the table below, the index would include the following stocks:
2020-01-01 – AAPL + MSFT
2020-02-01 – AAPL + MSFT
2020-03-01 – APPL + GOOG
In my real data, I obviously have many more stocks and many more months. I am having a very hard time modeling the second case in Postgres since it requires me to create a continuously updated "previous months" table that I need to reference when checking the current month. Any idea how to do this in PostgreSQL? I tried recursive CTEs and those didn't work (due to inner join requirements)
Table with structure below.
date
stock
rank
2020-01-01
AAPL
1
2020-01-01
MSFT
2
2020-01-01
META
3
2020-01-01
GOOG
4
2020-02-01
AAPL
1
2020-02-01
MSFT
3
2020-02-01
META
2
2020-01-01
GOOG
4
2020-03-01
AAPL
1
2020-03-01
MSFT
4
2020-03-01
META
3
2020-01-01
GOOG
2