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Bachelier Finance Society One World Seminar
Agnès Sulem (INRIA Paris, Equipe-projet MATHRISK)

Title: Optional pricing in a non-linear incomplete market model with default: the European and American cases

Abstract: We study option pricing in an incomplete market consisting of a risk-free asset and a risky asset driven by a Brownian motion and a compensated default martingale. We consider the case when the portfolio processes follow non-linear dynamics with a non-linear driver f.

We first study the superhedging prices and  associated superhedging strategies for European options. By using a dynamic programming approach, we provide a dual formulation of the seller's (superhedging) price involving a suitable set of equivalent probability measures, which we call f-martingale probability measures. We also establish a characterization of the seller's price as the initial value of the minimal supersolution of a constrained Backward Stochastic Differential Equation with default.

We then study American options with irregular payoff in this market. Both points of view of  the seller and of the buyer are analyzed. We  give a dual representation of the seller's (superhedging) price in terms of the value of a non-linear mixed control/stopping problem,  and  provide two infinitesimal characterizations of the seller's  price process in terms of the minimal supersolution of a constrained reflected BSDE and of an optional reflected BSDE. Under some regularity assumptions on the payoff, we also prove a duality result for the buyer's price in terms of the value of a non-linear control/stopping game problem. (joint work with Miryana Grigorova and  Marie-Claire Quenez)

Jan 14, 2021 07:00 PM in Zurich

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