Defaultable Game Options in a Hazard Process Model

Abstract

The valuation and hedging of defaultable game options is studied in a hazard process model of credit risk. A convenient pricing formula with respect to a reference filteration is derived. A connection of arbitrage prices with a suitable notion of hedging is obtained. The main result shows that the arbitrage prices are the minimal superhedging prices with sigma martingale cost under a risk neutral measure.

Convertible Bonds in a Defaultable Diffusion Model

“…This was the first motivation for the present study. The second motivation was the fact that all assumptions that we postulated in our previous theoretical works [4,5,6] are satisfied within this set-up; in this sense, the model is consistent with our theory of convertible securities. In particular, we worked in [4,6] under the assumption that the value U cb t of a convertible bond upon a call at time t yields, as a function of time, a well-defined process satisfying some natural conditions.…”

“…This also means that the process γ(t, S t ) is the Fintensity of τ d (see [5,6]). The fact that the default intensity γ may depend on S is crucial, since this dependence actually conveys all the 'equity-to-credit' information in the model.…”

“…(5)) is a game option (see [4,5,6,26,25]) with the ex-dividend cumulative discounted cash flows π(t; τ p , τ c ) given by the formula, for…”

“…In the set-up of this paper, the notions of issuer (super)hedge and holder (super)hedge introduced in [5,6] take the following form. Recall that we denote τ = τ p ∧ τ c .…”

“…Moreover, in [5], we showed that in the hazard process set-up, the theoretical problem of pricing and hedging CS can essentially be reduced to a problem of solving a related doubly reflected Backward Stochastic Differential Equation (BSDE for short). Finally, in [6], we established a formal connection between this BSDE and the corresponding variational inequalities with double obstacle in a generic Markovian intensity model.…”