A Continuous-Exercise Model for American Call Options with Hedging Constraints

In this paper, we derive a parabolic variational inequality with double time-like variables from a continuous exercise model of American call options proposed in Rogers and Schienkman (2007). Using viscosity approach, we prove that the value function is a unique viscosity solution to the variational inequality. For the perpetual problem, we first obtain some conditions which guarantee the problem is well posed.Then, under exponential utility, we explicitly construct the optimal exercise strategy based on the free boundary of the variational inequality. Finally, with the closed-form solutions in two cases that zero interest rate and zero strike price, we prove that the optimal exercise price in one-time exercise model is no smaller than the starting exercise price in the continuous exercise model