  In this work, we consider the outer Stefan problem for the short-time
prediction of the spread of a volatile asset traded in a financial market. The
stochastic equation for the evolution of the density of sell and buy orders is
the Heat Equation with a non-smooth noise in the sense of Walsh, posed in a
moving boundary domain with velocity given by the Stefan condition. This
condition determines the dynamics of the spread, and the solid phase
$[s^-(t),s^+(t)]$ defines the bid-ask spread area wherein the transactions
vanish. We introduce a reflection measure and prove existence and uniqueness of
maximal solutions up to stopping times in which the spread $s^+(t)-s^-(t)$
stays a.s. non-negative and bounded. For this, we use a Picard approximation
scheme and some of the estimates of \cite{BH} for the Green's function and the
associated to the reflection measure obstacle problem. Analogous results are
obtained for the equation without reflection corresponding to a signed density.
Additionally, we apply some formal asymptotics when the noise depends only on
time to derive that the spread is given by the integral of the solution of a
linear diffusion stochastic equation.
