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

Miryana Grigorova, Marie-Claire Quenez and Agnès Sulem
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Miryana Grigorova: Center for Mathematical Economics, Bielefeld University
Marie-Claire Quenez: Center for Mathematical Economics, Bielefeld University
Agnès Sulem: Center for Mathematical Economics, Bielefeld University

No 607, Center for Mathematical Economics Working Papers from Center for Mathematical Economics, Bielefeld University

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 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 cà dlà g) 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 cà dlà g, 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* ∈ $\mathcal{Q}$ .

Keywords: European options; American options; incomplete markets; non-linear pricing; BSDEs with constraints; constrained re ected BSDEs; Æ’-expectation; control problems with non-linear expectation; optimal stopping with non-linear expectation; non-linear optional decomposition; pricing-hedging duality (search for similar items in EconPapers)
Date: 2019-01-18
New Economics Papers: this item is included in nep-rmg
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https://pub.uni-bielefeld.de/download/2933147/2933148 First Version, 2018 (application/pdf)

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