[snowwrestler]

There's a big misunderstanding here, which is that fixed price implies a waterfall methodology. This is not true. All a fixed price does is cap the financial scope of the project.

But the tradeoff is that you need to vary the technical scope. For this reason, agile methodology is actually the best way to do a fixed-price contract. Every sprint is an opportunity to reconsider the technical scope to stay within schedule and budget.

This does reduce risk for the client. Most companies have much more serious limitations on budget and schedule, than they do on their imagination for what an ideal site should do.

And even if the client does decide they want to spend more money to get more functionality, they can just execute a change request or amendment--a process that is much more controlled and transparent (from the client perspective) than just working a few more hours.