Perpetual American standard and lookback options with event risk and asymmetric information

We derive closed-form solutions to the perpetual American standard and floating-strike lookback put and call options in an extension of the Black-Merton-Scholes model with event risk and asymmetric information. It is assumed that the contracts are terminated by their writers with linear or fractional recoveries at the last hitting times for the underlying asset price process of its ultimate maximum or minimum over the infinite time interval which are not stopping times with respect to the reference filtration. We show that the optimal exercise times for the holders are the first times at which the asset price reaches some lower or upper stochastic boundaries depending on the current values of its running maximum or minimum. The proof is based on the reduction of the original optimal stopping problems to the associated free-boundary problems and the solution of the latter problems by means of the smooth-fit and normal-reflection conditions. The optimal exercise boundaries are proven to be the maximal or minimal solutions of some first-order nonlinear ordinary differential equations.