Hi,
Add the property to the Container (in this case to the OptionGroup itself), for instance:
1paymentChoices.addContainerProperty("payment_type", String.class, null);
The set the item's property, e.g:
1paymentItem.getItemProperty("payment_type").setValue("billing");
In this case the default Container for OptionGroup (IndexedContainer) creates the Items and adds Properties; this is usually the case because all Items in a Container must have the same Properties.
Hope this helps!
Best Regards,
Marc