American options in an imperfect complete market with default

Abstract

We study pricing and (super)hedging for American options in an imperfect market model with default, where the imperfections are taken into account via the nonlinearity of the wealth dynamics. The payoff is given by an RCLL adapted process (ξ t ). We define the seller's superhedging price of the American option as the minimum of the initial capitals which allow the seller to build up a superhedging portfolio. We prove that this price coincides with the value function of an optimal stopping problem with nonlinear expectations induced by BSDEs with default jump, which corresponds to the solution of a reflected BSDE with lower barrier. Moreover, we show the existence of a superhedging portfolio strategy. We then consider the buyer's superhedging price, which is defined as the supremum of the initial wealths which allow the buyer to select an exercise time τ and a portfolio strategy ϕ so that he/she is superhedged. Under the additional assumption of left upper semicontinuity along stopping times of (ξ t ), we show the existence of a superhedge (τ, ϕ) for the buyer, as well as a characterization of the buyer's superhedging price via the solution of a nonlinear reflected BSDE with upper barrier.

American options in nonlinear markets

“…The goal of this work is to re-examine and extend the findings from the recent paper by Dumitrescu et al [26] who applied the nonlinear pricing approach developed in El Karoui and Quenez [32]. In contrast to [26] where a particular model with a single jump of the underlying asset was studied, we place ourselves within the setup of a general nonlinear arbitrage-free market with possibly discontinuous asset prices, as introduced in Bielecki et al [9,12] and we examine unilateral acceptable prices for American contracts. We obtain results regarding the pricing, hedging, break-even times and rational exercise times using results on backward stochastic differential equations (BSDEs) from Nie and Rutkowski [62,63], but without explicitly specifying the dynamics of underlying risky assets and funding accounts.…”

“…Notice also in this regard that Dumitrescu et al [25,26] also consider American and game options with default risk but their study focuses on contracts subject to the thirdparty credit risk, as opposed to the counterparty credit risk. In Remark 4.7, we mention that another justification for the study of wealth processes driven by a general RCLL martingale comes from the fact that, typically, several dependent defaults are present in the market (see [11]).…”

European Options in a Nonlinear Incomplete Market Model with Default

“…We recall that in the case of a non-linear complete market model, the seller's (hedging) price of the European option with payoff η and maturity T is given by the non-linear fevaluation (expectation) of η, where f is the non-linear driver of the replicating portfolio (cf. [13] and [17] in the Brownian case, and the recent works [12], [10] and [11] in the default case).…”

Perpetual American Cancellable Standard Options in Models with Last Passage Times

“…The model studied here differs from models studied in existing works such as Szimayer [6], Gapeev and Al Motairi [7], Glover and Hulley [8], Dumitrescu et al [9], and Grigorova et al [10], as neither the immersion hypothesis nor the density hypothesis is satisfied by the random times (or default times) θ and η, and the default intensity process simply does not exists in our setting (see, e.g., Bielecki and Rutkowski [11]). We see clearly in ( 6) and ( 7) that, in the case of zero recovery, this leads to a modified discounting factors, which are no longer functions of the sum of the interest rate and the default intensity rate.…”

“…In addition, the diversion from the immersion hypothesis leads to the appearance of an adjusted dividend rate. Finally, if we were to study the finite horizon problem from a point of view of the backward stochastic differential equations (or BSDEs) as in [9,10], then it could be shown that the dynamics of the no-arbitrage (pre-default) price will no longer satisfy a linear reflected BSDE but rather a linear reflected generalised BSDE where the generalised driver is related to the local time of the underlying asset at h or g.…”