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Hedging under gamma constraints by optimal stopping and face-lifting. (English) Zbl 1278.91151
Summary: A super-replication problem with a gamma constraint, introduced in [the authors, SIAM J. Control Optimization 39, No. 1, 73–96 (2000; Zbl 0960.91036)], is studied in the context of the one-dimensional Black-Scholes model. Several representations of the minimal super-hedging cost are obtained using the characterization derived in [P. Cheridito and the authors, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 22, No. 5, 633–666 (2005; Zbl 1078.91010); Ann. Appl. Probab. 15, No. 4, 2472–2495 (2005; Zbl 1099.60027)]. 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.

91G10 Portfolio theory
91G80 Financial applications of other theories
93E20 Optimal stochastic control
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