Selling a stock at the ultimate maximum
Selling a stock at the ultimate maximum
Assuming that the stock price Z=(Zt)0≤t≤T follows a geometric Brownian motion with drift μ∈ℝ and volatility σ>0, and letting Mt=max 0≤s≤tZs for t∈[0, T], we consider the optimal prediction problems $$V_{1}=\inf_{0\leq\tau\leq T}\mathsf{E}\biggl(\frac{M_{T}}{Z_{\tau}}\biggr) \quad \mathrm{and} \quad V_{2}=\sup_{0\leq\tau\leq T}\mathsf{E}\biggl(\frac{Z_{\tau}}{M_{T}}\biggr),$$ where the infimum and supremum are taken over all stopping times τ of …