OPTIMAL SELLING RULES FOR MONETARY INVARIANT CRITERIA: TRACKING THE MAXIMUM OF A PORTFOLIO WITH NEGATIVE DRIFT

Considering a positive portfolio diffusion X with negative drift, we investigate optimal stopping problems of the form where f is a nonincreasing function, τ is the next random time where the portfolio X crosses zero and θ is any stopping time smaller than τ. 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 the quadratic absolute distance criteria in this stationary framework with bang–bang type ones observed for monetary invariant criteria but in finite horizon. 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 with , instantaneous selling is always optimal, which is consistent with previous observations for the Black‐Scholes model in finite observation. On the contrary, for a relative quadratic error criterion, , selling is optimal as soon as the process X crosses a specified function φ of its running maximum . 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.