Arbitrage pricing of defaultable game options with applications to convertible bonds

Abstract

This paper is the first in a series that we devote to studying the problems of valuation and hedging of defaultable game options in general, and convertible corporate bonds in particular. Here, we present mathematical foundations for our overall study. Specifically, we provide several results characterizing the arbitrage price of a defaultable game option in terms of relevant Dynkin games. In addition, we provide important results regarding price decomposition of defaultable options. These general results are then specified to the case of convertible bonds, yielding in particular a decomposition of convertible bonds in an optional and a bond component.Defaultable game options, Convertible securities, Convertible bonds, Semimartingale market,

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.…”

“…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. In the specific framework of this paper, using uniqueness of arbitrage prices (Propositions 2.1 and 3.1) and a form of continuous aggregation property of the value U cb t of a convertible bond upon a call at time t (Proposition 6.7), we are actually able to prove that this assumption is satisfied, and we also give ways to compute U cb t (Propositions 6.6 and 6.8).…”

“…We now specify to the present model the notion of a convertible security (CS), as formally defined in [4]. Let 0 (resp.…”

“…(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 [4], working in an abstract set-up, we characterized arbitrage prices of generic convertible securities (CS), such as convertible bonds (CB), and we provided a rigorous decomposition of a CB into a straight bond component and a game option component, in order to give a definite meaning to commonly used terms of 'CB spread' and 'CB implied volatility.' 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).…”