  This work focuses on the indifference pricing of American call option
underlying a non-traded stock, which may be partially hedgeable by another
traded stock. Under the exponential forward measure, the indifference price is
formulated as a stochastic singular control problem. The value function is
characterized as the unique solution of a partial differential equation in a
Sobolev space. Together with some regularities and estimates of the value
function, the existence of the optimal strategy is also obtained. The
applications of the characterization result includes a derivation of a dual
representation and the indifference pricing on employee stock option. As a
byproduct, a generalized Ito's formula is obtained for functions in a Sobolev
space.
