The least expensive economy, four continent RTW is about US$2500 from London (low season, US$2800 high season) and US$3800 from the US.
You should read some of the past threads here for information on trip planning. You can download electronic timetables that will help you in planning the trip from
here and
here. The OWE RTW rules are
here.
If you really only need Asia and SWP, something else you should look at is the Cathay Pacific All Asia Pass. You can add on Oz or NZ and once there buy cheap tickets for local travel. Information on it is
here. Note that the AAP does not earn mileage but you can leave from NYC.