I missed this bit of info if it's in there somewhere?
The party is compensated on a contractual basis.
Spoiler:
To start the PCs get room and board, and if they bring Verik back they get 1000 GP, 500 GP if he's dead. For Eel's End she can be talked into giving the PCs as much as 1500 GP, and they can keep whatever they don't use to bribe Devargo. They also get an additional 500 GP for delivering the letters. If they do in the King of Spiders they each get an additional 400 GP in the form of a medal that (if they don't sell it) gives the PCs a +2 modifier on Diplomacy with Korvosans. They get an additional 1000 GP for assisting with the missing Shoanti corpse. After that they aren't so much working for the Korvosan Guard as they are one of the few things left trying to save the city. If you consider things like getting to keep whatever loot you confiscate and having the authority to behave as needed to execute the mission as reward as well, well, tehy get that too!