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

Grigorova M, Quenez M-C, Sulem A (2018) Center for Mathematical Economics Working Papers; 607.
Bielefeld: Center for Mathematical Economics.

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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 ƒ. 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* ∈ $\mathcal{Q}$ of the ƒ-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 $\mathcal{E}$ƒ -strong supermartingales under *Q*, for all *Q* ∈ $\mathcal{Q}$ .
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Grigorova M, Quenez M-C, Sulem A. Superhedging prices of European and American options in a non-linear incomplete market with default. Center for Mathematical Economics Working Papers. Vol 607. Bielefeld: Center for Mathematical Economics; 2018.
Grigorova, M., Quenez, M. - C., & Sulem, A. (2018). Superhedging prices of European and American options in a non-linear incomplete market with default (Center for Mathematical Economics Working Papers, 607). Bielefeld: Center for Mathematical Economics.
Grigorova, M., Quenez, M. - C., and Sulem, A. (2018). Superhedging prices of European and American options in a non-linear incomplete market with default. Center for Mathematical Economics Working Papers, 607, Bielefeld: Center for Mathematical Economics.
Grigorova, M., Quenez, M.-C., & Sulem, A., 2018. Superhedging prices of European and American options in a non-linear incomplete market with default, Center for Mathematical Economics Working Papers, no.607, Bielefeld: Center for Mathematical Economics.
M. Grigorova, M.-C. Quenez, and A. Sulem, Superhedging prices of European and American options in a non-linear incomplete market with default, Center for Mathematical Economics Working Papers, vol. 607, Bielefeld: Center for Mathematical Economics, 2018.
Grigorova, M., Quenez, M.-C., Sulem, A.: Superhedging prices of European and American options in a non-linear incomplete market with default. Center for Mathematical Economics Working Papers, 607. Center for Mathematical Economics, Bielefeld (2018).
Grigorova, Miryana, Quenez, Marie-Claire, and Sulem, Agnès. Superhedging prices of European and American options in a non-linear incomplete market with default. Bielefeld: Center for Mathematical Economics, 2018. Center for Mathematical Economics Working Papers. 607.
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