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Superhedging prices of European and American options in a non-linear incomplete market with default

Abstract : This paper studies the superhedging prices and the associated superhedging strategies for European and American options in a non-linear incomplete market with default. We present the seller's and the buyer's point of view. The underlying market model consists of a risk-free asset and a risky asset driven by a Brownian motion and a compensated default martingale. The portfolio process follows non-linear dynamics with a non-linear driver f. By using a dynamic programming approach, we first provide a dual formulation of the seller's (superhedging) price for the European option as the supremum over a suitable set of equivalent probability measures Q ∈ Q of the f-evaluation/expectation under Q of the payoff. We also provide an infinitesimal characterization of this price as the minimal supersolution of a constrained BSDE with default. By a form of symmetry, we derive corresponding results for the buyer. We also give a dual representation of the seller's (superhedging) price for the American option associated with an irregular payoff (ξ t) (not necessarily càdlàg) in terms of the value of a non-linear mixed control/stopping problem. We also provide an infinitesimal characterization of this price in terms of a constrained reflected BSDE. When ξ is càdlàg, we show a duality result for the buyer's price. These results rely on first establishing a non-linear optional decomposition for processes which are E f-strong supermartingales under Q, for all Q ∈ Q.
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https://hal.inria.fr/hal-02421331
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Submitted on : Friday, December 20, 2019 - 1:44:00 PM
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  • HAL Id : hal-02421331, version 1

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Miryana Grigorova, Marie-Claire Quenez, Agnès Sulem. Superhedging prices of European and American options in a non-linear incomplete market with default. 2018. ⟨hal-02421331⟩

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