Game Options in an Imperfect Market with Default

Abstract

We study pricing and superhedging strategies for game options in an imperfect market with default. We extend the results obtained by Kifer in [23] in the case of a perfect market model to the case of an imperfect market with default, when the imperfections are taken into account via the nonlinearity of the wealth dynamics. We introduce the seller's price of the game option as the infimum of the initial wealths which allow the seller to be superhedged. We prove that this price coincides with the value function of an associated generalized Dynkin game, recently introduced in [14], expressed with a nonlinear expectation induced by a nonlinear BSDE with default jump. We moreover study the existence of superhedging strategies. We then address the case of ambiguity on the model, -for example ambiguity on the default probability -and characterize the robust seller's price of a game option as the value function of a mixed generalized Dynkin game. We study the existence of a cancellation time and a trading strategy which allow the seller to be super-hedged, whatever the model is.

Arbitrage-free pricing of derivatives in nonlinear market models

“…Let us now comment on the existing approaches to the nonlinear valuation of derivatives, as first developed by El and El and later applied by several authors to particular financial models or classes on contracts (see, for instance, Bichuch et al (2018), Brigo and Pallavicini (2014), Crépey (2015a, b), Dumitrescu et al (2017), Mercurio (2013) or Pallavicini et al (2012a, b)). The most common approach to the valuation problem in a nonlinear framework seems to hinge, at least implicitly, on the following steps in which it is usually assumed that the hedger's initial endowment is immaterial and thus it may be set to zero.…”

“…Note, however, that if the wealth process happens to be governed by some simple dynamics with no portfolio constraints or trading adjustments, then there is no need to postulate that this property holds, since it can be deduced from a suitable comparison theorem for ordinary differential equations. For a particular example of a model where Assumption 4 is satisfied, see Lemma 6.2 in the paper by (Dumitrescu et al 2017) who postulate that the wealth process satisfies…”

“…In contrast, only a few papers devoted to nonlinear models of credit risk are available. More recently, Crépey (2015a, b), Dumitrescu et al (2017) and Bichuch et al (2018) used BSDEs with jumps to solve the valuation and hedging problems for derivative contracts exposed to the counterparty credit risk. In Bichuch et al (2018) and Dumitrescu et al (2017), the authors focus on valuation of the full contract, whereas Crépey (2015a, b) examines the problem of the approximate additivity for the credit valuation adjustments.…”

American options in an imperfect complete market with default

“…where f 2 t := f 1 t − A t . Let us apply for fixed ω a comparison result for forward differential equations (see [8] in the Appendix).…”

“…Indeed, no rational agent would pay more than u 0 since there is a cheaper way to achieve at least the same payoff, whatever the exercise time is. Indeed, by investing the amount u 0 + ε and following the strategy ϕ * , whatever the exercise time ν is, he will make the gain V u 0 +ε,ϕ * ν > V u 0 ,ϕ * ν (≥ ξ ν ) a.s. by a strict comparison property for deterministic differential equations (see [8] in the Appendix). Moreover, if the price of the option is equal to u 0 +ε, then, by investing this amount following the strategy ϕ * , whatever the exercise time ν is, the seller will make a gain V u 0 +ε,ϕ * ν −ξ ν > 0 a.s.…”