Clients like fixed prices because they know what they will spend. Studios dread them because the scope moves. After 60 projects we settled on a process that works for both sides.
Week one: the problem
We interview the people who will use the product, not only the people paying for it. The output is a one-page problem statement both sides sign.
Week two: the smallest useful thing
We map every feature and mark what must exist for the first release to be worth using. The price covers that list. Everything else goes into a backlog with its own estimate.
Since adopting this, 9 out of 10 projects have shipped within 10% of the original quote.