Valuation of exotic options under shortselling constraints

Options with discontinuous payoffs are generally traded above their theoretical Black-Scholes prices because of the hedging difficulties created by their large delta and gamma values. A theoretical method for pricing these options is to constrain the hedging portfolio and incorporate this constraint into the pricing by computing the smallest initial capital which permits super-replication of the option. We develop this idea for exotic options, in which case the pricing problem becomes one of stochastic control. Our motivating example is a call which knocks out in the money, and explicit formulas for this and other instruments are provided.