dc.contributor.author Espinosa, Gilles-Edouard dc.contributor.author Elie, Romuald dc.date.accessioned 2012-07-16T11:35:32Z dc.date.available 2012-07-16T11:35:32Z dc.date.issued 2015 dc.identifier.issn 0960-1627 dc.identifier.uri https://basepub.dauphine.fr/handle/123456789/9751 dc.language.iso en en dc.subject Free boundary PDE dc.subject Mean reverting diffusion dc.subject Verification dc.subject Running maximum dc.subject Optimal prediction dc.subject Optimal stopping dc.subject.ddc 519 en dc.title Optimal selling rules for monetary invariant criteria: tracking the maximum of a portfolio with negative drift, dc.type Article accepté pour publication ou publié dc.contributor.editoruniversityother Departement of Mathematics, ETH Zurich;Suisse dc.contributor.editoruniversityother Centre de Recherche en Économie et Statistique (CREST) http://www.crest.fr/ INSEE – École Nationale de la Statistique et de l'Administration Économique;France dc.description.abstracten 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. dc.relation.isversionofjnlname Mathematical Finance dc.relation.isversionofjnlvol 25 dc.relation.isversionofjnlissue 4 dc.relation.isversionofjnldate 2015 dc.relation.isversionofjnlpages 754-788 dc.relation.isversionofdoi http://dx.doi.org/10.1111/mafi.12036 dc.identifier.urlsite https://hal.archives-ouvertes.fr/hal-00573429 dc.description.sponsorshipprivate oui en dc.relation.isversionofjnlpublisher Blackwell dc.subject.ddclabel Probabilités et mathématiques appliquées en dc.relation.forthcomingprint oui dc.description.ssrncandidate non dc.description.halcandidate oui dc.description.readership recherche dc.description.audience International dc.relation.Isversionofjnlpeerreviewed oui dc.date.updated 2016-11-29T15:45:58Z
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