European Options in a Nonlinear Incomplete Market Model with Default

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 fevaluation/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 E f-strong supermartingales under Q, for all Q ∈ Q.

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.…”