... display "sold out"

A Product will show as "Sold Out" if the following conditions apply

Stock Empty and Clearance Sale is active

When the clearance sale is active, you dont allow any backorders. The flag should be set to true, if you want to sell inventory-managed goods.

No Availability Date and no Presale Startdate set

If there is no availabiltiy and no presale Startdate set, that means the product won't come back. It is not ordered anymore so it is sold out.

Result: