European options in a non-linear incomplete market model with default

Miryana Grigorova (), Marie-Claire Quenez () and Agnès Sulem ()
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Miryana Grigorova: School of Mathematics [Leeds] - University of Leeds
Marie-Claire Quenez: LPSM (UMR_8001) - Laboratoire de Probabilités, Statistique et Modélisation - SU - Sorbonne Université - CNRS - Centre National de la Recherche Scientifique - UPCité - Université Paris Cité
Agnès Sulem: MATHRISK - Mathematical Risk Handling - UPEM - Université Paris-Est Marne-la-Vallée - ENPC - École nationale des ponts et chaussées - Centre Inria de Paris - Inria - Institut National de Recherche en Informatique et en Automatique

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Abstract: This paper studies the superhedging prices and the associated superhedging strategies for European options in a non-linear incomplete market model 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 processes follow 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 a characterization of the seller's (superhedging) price process as the minimal supersolution of a constrained BSDE with default and a characterization in terms of the minimal weak supersolution of a BSDE with default. By a form of symmetry, we derive corresponding results for the buyer. Our results rely on first establishing a non-linear optional and a non-linear predictable decomposition for processes which are $\mathcal{E}^f$-strong supermartingales under Q, for all Q ∈ Q.

Keywords: European options; Incomplete market; Superhedging; Non-linear pricing; BSDEs with constraints; f-expectation; Control problems with non-linear expectation; Non-linear optional decomposition; Pricing-hedging duality (search for similar items in EconPapers)
Date: 2020-09-02
New Economics Papers: this item is included in nep-rmg
Note: View the original document on HAL open archive server: https://hal.science/hal-02025833v1
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Citations: View citations in EconPapers (2)

Published in SIAM Journal on Financial Mathematics, 2020, 11 (3), pp.849-880. ⟨10.1137/20M1318018⟩

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Persistent link: https://EconPapers.repec.org/RePEc:hal:journl:hal-02025833

DOI: 10.1137/20M1318018

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