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HEDGING UNDER GAMMA CONSTRAINTS BY OPTIMAL STOPPING AND FACE‐LIFTING
Author(s) -
Soner H. Mete,
Touzi Nizar
Publication year - 2007
Publication title -
mathematical finance
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.98
H-Index - 81
eISSN - 1467-9965
pISSN - 0960-1627
DOI - 10.1111/j.1467-9965.2007.00294.x
Subject(s) - constraint (computer aided design) , stochastic game , upper and lower bounds , context (archaeology) , optimal stopping , mathematical optimization , simple (philosophy) , mathematical economics , mathematics , function (biology) , face (sociological concept) , computer science , mathematical analysis , paleontology , philosophy , social science , geometry , epistemology , evolutionary biology , sociology , biology
A super‐replication problem with a gamma constraint, introduced in Soner and Touzi, is studied in the context of the one‐dimensional Black and Scholes model. Several representations of the minimal super‐hedging cost are obtained using the characterization derived in Cheridito, Soner, and Touzi. It is shown that the upper bound constraint on the gamma implies that the optimal strategy consists in hedging a conveniently face‐lifted payoff function. Further an unusual connection between an optimal stopping problem and the lower bound is proved. A formal description of the optimal hedging strategy as a succession of periods of classical Black–Scholes hedging strategy and simple buy‐and‐hold strategy is also provided.