Nonlinear pricing of European and American options in an imperfect market with default
A talk in the Bielefeld Stochastic Afternoon series by
Marie-Claire Quenez from University of Paris Diderot - Paris 7
| Abstract: | We study pricing and hedging for contingent claims in an imperfect market model with default, where the im- perfections are taken into account via the nonlinearity of the wealth dynamics, expressed in terms of a nonlinear driver $g(y, z, k)$. In this framework, the seller’s (resp. buyer’s) pricing rule for European options corresponds to the nonlinear $g$-expectation $\mathcal{E}^g$ (resp. $\tilde{g}$-expectation $\mathcal{E}^\tilde{g}$)$^1$ , induced by a BSDE with driver $g$ (resp $\tilde{g}$). We also address the case of options which generate intermediate cashflows modeled via an optional finite variational process. We then study the pricing of American options in this framework. The payoff is given by an optional irregular process
($\xi_t$). We define the seller’s price of the American option as the minimum of the initial capitals which allow her/him to build up a superhedging portfolio strategy $\varphi$. We also consider the buyer’s price, defined as the supremum of the initial prices which allow the buyer to select an exercise time $\tau$ and a portfolio strategy $\varphi$ so that she/he is superhedged. We prove that the seller’s (resp. buyer’s) price coincides with the value function of an $\mathcal{E}^g$ -(resp. $\mathcal{E}^\tilde{g}$ -) optimal stopping problem, which corresponds to the solution of a reflected BSDE with obstacle ($\xi_t$) and driver $g$ (resp. $\tilde{g}$). At last, we study the pricing of a game option with irregular payoffs. In this case, the seller’s (resp. buyer’s) price is shown to be equal to the value function of an $\mathcal{E}^g$ -(resp. $\mathcal{E}^\tilde{g}$ -) Dynkin game, which coincides with the solution of a nonlinear doubly reflected BSDEs with driver $g$ (resp. $\tilde{g}$). We also consider the case of ambiguity on the model.
This talk is based on joint works with M. Grigorova, A. Sulem and R. Dumitrescu.
$^1$: where $\tilde{g}(y, z, k) := −g(−y, −z, −k)$
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