Looking for an integrated vBulletin chat plugin

[XYZ500]

I'm looking for a vBulletin chat plugin that acts like any instant messenger. It should be integrated in the site. So whenever users login to the forum, they have the option of logging into messenger as well. Once they are logged in, they can chat with other online members of the forum. All the conversations between members must be saved in a place that admins can view (but obviously not other users). It should also have an option where users cannot create any chat accounts, their forum username and password should be used to login to chat. What I mean is that they should not be allowed to create any usernames on the chat. I don't want this so users cannot impersonate any other members.

My forum is sort of a marketplace where buyers and sellers meet and then exchange their contact info and leave the site. I want to introduce a chat software like that will make the users stay on the forum and for any instant messaging they will have the forum chat. This will retain users on the site for longer periods of time.

Any free/paid userfriendly software would be acceptable to me. It just should not be overpriced.