  When the underlying stock price is a strict local martingale process under an
equivalent local martingale measure, Black-Scholes PDE associated with an
European option may have multiple solutions. In this paper, we study an
approximation for the smallest hedging price of such an European option. Our
results show that a class of rebate barrier options can be used for this
approximation. Among of them, a specific rebate option is also provided with a
continuous rebate function, which corresponds to the unique classical solution
of the associated parabolic PDE. Such a construction makes existing numerical
PDE techniques applicable for its computation. An asymptotic convergence rate
is also studied when the knocked-out barrier moves to infinity under suitable
conditions.
