  We discuss global existence and asymptotic behaviour of a price formation
free boundary model introduced by Lasry & Lions in 2007. Our results are based
on a construction which transforms the problem into the heat equation with
specially prepared initial datum. The key point is that the free boundary
present in the original problem becomes the zero level set of this solution.
Using the properties of the heat operator we can show global existence,
regularity and asymptotic results of the free boundary.
