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

by Romuald ELIE & Gilles-Edouard ESPINOSA

Considering a positive portfolio diffusion X with negative drift, we investigate optimal stopping problems of the form (…) This paper unifies optimal selling rules observed by [5] for quadratic absolute distance criteria with bang-bang type ones observed in [1, 4, 9]. 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. (…). 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.

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