Selling a stock at the ultimate maximum

Assuming that the stock price Z=(Z_t)_{0\leq t\leq T} follows a geometric Brownian motion with drift \mu\in\mathbb{R} and volatility \sigma>0, and letting M_t=\max_{0\leq s\leq t}Z_s for t\in[0,T], we consider the optimal prediction problems \[V_1=\inf_{0\leq\tau\leq T}\mathsf{E}\biggl(\frac{M_T}{Z_{\tau}}\biggr)\quadand\quad V_2=\sup_{0\leq\tau\leq T}\mathsf{E}\biggl(\frac{Z_{\tau}}{M_T}\biggr),\] where the infimum and supremum are taken over all stopping times \tau of Z.... Show more