Optimal selling rules for monetary invariant criteria: tracking the maximum of a portfolio with negative drift.

Summary Considering a positive portfolio diffusion $X$ with negative drift, we investigate optimal stopping problems of the form $$ \inf_\theta \Esp{f\left(\frac{X_\theta}{\Sup_{s\in[0,\tau]}{X_s}}\right)}\.,$$ where $f$ is a non-increasing function, $\tau$ is the next random time where the portfolio $X$ crosses zero and $\theta$ is any stopping time smaller than $\tau$. Hereby, our motivation is the obtention of an optimal selling strategy minimizing the relative distance between the liquidation value of the portfolio and its highest possible value before it reaches zero. This paper unifies optimal selling rules observed for quadratic absolute distance criteria with bang-bang type ones. More precisely, we provide a verification result for the general stopping problem of interest and derive the exact solution for two classical criteria $f$ of the literature. For the power utility criterion $f:y \mapsto - {y^\la}$ with $\la>0$, instantaneous selling is always optimal, which is consistent with the observations of \cite{DaiJinZhoZho10} or \cite{ShiXuZho08} for the Black-Scholes model in finite horizon. On the contrary, for a relative quadratic error criterion, $f:y \mapsto {(1-y)^2}$, selling is optimal as soon as the process $X$ crosses a specified function $\varphi$ of its running maximum $X^*$. These results reinforce the idea that optimal stopping problems of similar type lead easily to selling rules of very different nature. Nevertheless, our numerical experiments suggest that the practical optimal selling rule for the relative quadratic error criterion is in fact very close to immediate selling.