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.

1. Introduction

The goal of this work is to analyze valuation and hedging of defaultable contracts with game option features within a hazard process model of credit risk. Our motivation for considering American or game clauses together with defaultable features of an option is not that much a quest for generality, but rather the fact that the combination of early exercise features and defaultability is an intrinsic feature of some actively traded assets. It suffices to mention here the important class of convertible bonds, which were studied by, among others, Andersen and Buffum [1], Ayache et al. [2], Bielecki et al. [3, 4], Davis and Lischka [5], Kallsen and Kühn [6], and Kwok and Lau [7].

In Bielecki et al. [3], we formally defined a defaultable game option, that is, a financial contract that can be seen as an intermediate case between a general mathematical concept of a game option and much more specific convertible bond with credit risk. We concentrated there on developing a fairly general framework for valuing such contracts. In particular, building on results of Kifer [8] and Kallsen and Kühn [6], we showed that the study of an arbitrage price of a defaultable game option can be reduced to the study of the value process of the related Dynkin game under some risk-neutral measure for the primary market model. In this stochastic game, the issuer of a game option plays the role of the minimizer and the holder of the maximizer. In [3], we dealt with a general market model, which was assumed to be arbitrage-free, but not necessarily complete, so that the uniqueness of a risk-neutral (or martingale) measure was not postulated. In addition, although the default time was introduced, it was left largely unspecified. An explicit specification of the default time will be an important component of the model considered in this work.

As is well known, there are two main approaches to modeling of default risk: the structural approach and the reduced-form approach. In the latter approach, also known as the hazard process approach, the default time is modeled as an exogenous random variable with no reference to any particular economic background. One may object to reduced-form models for their lack of clear reference to economic fundamentals, such as the firm's asset-to-debt ratio. However, the possibility of choosing various parameterizations for the coefficients and calibrating these parameters to any set of CDS spreads and/or implied volatilities makes them very versatile modeling tools, well suited to price and hedge derivatives consistently with plain-vanilla instruments. It should be acknowledged that structural models, with their sound economic background, are better suited for inference of reliable debt information, such as: risk-neutral default probabilities or the present value of the firm's debt, from the equities, which are the most liquid among all financial instruments. The structure of these models, as rich as it may be (and which can include a list of factors such as stock, spreads, default status, and credit events) never rich enough to yield consistent prices for a full set of CDS spreads and/or implied volatilities of related options. As we ultimately aim to specify models for pricing and hedging contracts with optional features (such as convertible bonds), we favor the reduced-form approach in the sequel.

1.1. Outline of the Paper

From the mathematical perspective, the goal of the present paper is twofold. First, we wish to specialize our previous valuation results to the hazard process setup, that is, to a version of the reduced-form approach, which is slightly more general than the intensity-based setup. Hence we postulate that filtration 𝔾 modeling the information flow for the primary market admits the representation 𝔾=𝔽, where the filtration is generated by the default indicator process 𝐻𝑡=1{𝑡𝜏𝑑} and 𝔽 is some reference filtration. The main tool employed in this section is the effective reduction of the information flow from the full filtration 𝔾 to the reference filtration 𝔽. The main results in this part are Theorems 3.7 and 3.8, which give convenient pricing formulae with respect to the reference filtration 𝔽.

The second goal is to study the issue of hedging of a defaultable game option in the hazard process setup. Some previous attempts to analyze hedging strategies for defaultable convertible bonds were done by Andersen and Buffum [1] and Ayache et al. [2], who worked directly with suitable variational inequalities within the Markovian intensity-based setup.

Our preliminary results for hedging strategies in a hazard process setup, Propositions 4.1 and 4.3, can be informally stated as follows: under the assumption that a related doubly reflected BSDE admits a solution (Θ,𝑀,𝐾) under some risk-neutral measure , for which various sets of sufficient conditions are given in literature, the state-process Θ of the solution is the minimal (pre-default) super-hedging price up to a (𝔾,)-sigma (or local) martingale cost process. More specific properties of hedging strategies are subsequently analyzed in Propositions 4.13 and 4.15, in which we resort to suitable Galtchouk-Kunita-Watanabe decompositions of a solution to the related doubly reflected BSDE and discounted prices of primary assets with respect to various risk factors corresponding to systematic, idiosyncratic and event risks. It is noteworthy that these decompositions, though seemingly rather abstract in a general setup considered here, are by no means artificial. On the contrary, they arise naturally in the context of particular Markovian models that are studied in the followup paper by Bielecki et al. [4, 9]. We conclude the paper by briefly commenting on some alternative approaches to hedging of defaultable game options.

1.2. Conventions and Standing Notation

Throughout this paper, we use the concept of the vector stochastic integral, denoted as 𝐻𝑑𝑋, as opposed to a more restricted notion of the component-wise stochastic integral, which is defined as the sum 𝑑𝑖=1𝐻𝑖𝑑𝑋𝑖 of integrals with respect to one-dimensional integrators 𝑋𝑖. For a detailed exposition of the vector stochastic integration, we refer to Shiryaev and Cherny [10] (see also Chatelain and Stricker [11] and Jacod [12]). Given a stochastic basis satisfying the usual conditions, an 𝑑-valued semimartingale integrator 𝑋 and an 1𝑑-valued (row vector) predictable integrand 𝐻, the notion of the vector stochastic integral 𝐻𝑑𝑋 allows one to take into account possible “interferences’’ of local martingale and finite variation components of a (scalar) integrator process, or of different components of a multidimensional integrator process. Well-defined vector stochastic integrals include, in particular, all integrals with a predictable and locally bounded integrand (e.g., any integrand of the form 𝐻=𝑌 where 𝑌 is an adapted càdlàg process, see He et al. [13, Theorem  7.7]). The usual properties of stochastic integral, such as: linearity, associativity, invariance with respect to equivalent changes of measures and with respect to inclusive changes of filtrations, are known to hold for the vector stochastic integral. Moreover, unlike other kinds of stochastic integrals, vector stochastic integrals form a closed space in a suitable topology. This feature makes them well adapted to many problems arising in the mathematical finance, such as Fundamental Theorems of Asset Pricing (see, e.g., Delbaen and Schachermayer [14] or Shiryaev and Cherny [10]).

By default, we denote by 𝑡0 the integrals over (0,𝑡]. Otherwise, we explicitly specify the domain of integration as a subscript of . Note also that, depending on the context, 𝜏 will stand either for a generic stopping time or it will be given as 𝜏=𝜏𝑝𝜏𝑐 for some specific stopping times 𝜏𝑐 and 𝜏𝑝. Finally, we consider the right-continuous and completed versions of all filtrations, so that they satisfy the so-called “usual conditions.’’

2. Semimartingale Setup

After recalling some fundamental valuation results from [3], we will examine basic features of hedging strategies for defaultable game options that are valid in a general semimartingale setup. The important special case of a hazard process framework is studied in the next section.

We assume throughout that the evolution of the underlying primary market is modeled in terms of stochastic processes defined on a filtered probability space (Ω,𝔾,), where denotes the statistical probability measure.

Specifically, we consider a primary market composed of the savings account and of 𝑑 risky assets, such that, given a finite horizon date 𝑇>0:

(i)the discount factor process 𝛽, that is, the inverse of the savings account, is a 𝔾-adapted, finite variation, positive, continuous and bounded process,(ii) the risky assets are 𝔾-semimartingales with càdlàg sample paths.

The primary risky assets, with 𝑑-valued price process 𝑋, pay dividends, whose cumulative value process, denoted by 𝒟, is assumed to be a 𝔾-adapted, càdlàg and 𝑑-valued process of finite variation. Given the price process 𝑋, we define the cumulative price 𝑋 of primary risky assets as 𝑋𝑡=𝑋𝑡+𝛽𝑡1[0,𝑡]𝛽𝑢𝑑𝒟𝑢.(2.1) In the financial interpretation, the last term in (2.1) represents the current value at time 𝑡 of all dividend payments from the assets over the period [0,𝑡], under the assumption that all dividends are immediately reinvested in the savings account. We assume that the primary market model is free of arbitrage opportunities, though presumably incomplete. In view of the First Fundamental Theorem of Asset Pricing (cf. [10, 14]), and accounting in particular for the dividends, this means that there exists a risk-neutral measure , where denotes the set of probability measures for which 𝛽𝑋 is a sigma martingale with respect to 𝔾 under (for the definition of a sigma martingale, see [10, Definition  1.9]). The following well-known properties of sigma martingales will be used in the sequel.

Proposition 2.1 (see [10, 15, 16]). (i) The class of sigma martingales is a vector space containing all local martingales. It is stable with respect to (vector) stochastic integration, that is, if 𝑌 is a sigma martingale and 𝐻 is a (predictable) 𝑌-integrable process then the (vector) stochastic integral 𝐻𝑑𝑌 is a sigma martingale.
(ii) Any locally bounded sigma martingale is a local martingale, and any bounded from below sigma martingale is a supermartingale.

Remark 2.2. In the same vein, we recall that stochastic integration of predictable, locally bounded integrands preserves local martingales (see, e.g., Protter [16]).

We now introduce the concept of a dividend paying game option (see also Kifer [8]). In broad terms, a dividend paying game option, with the inception date 0 and the maturity date 𝑇, is a contract with the following cash flows that are paid by the issuer of the contract and received by its holder:

(i)a dividend stream with the cumulative dividend at time 𝑡 denoted by 𝐷𝑡, (ii)a terminal put payment 𝐿𝑡 made at time 𝑡=𝜏𝑝 if 𝜏𝑝𝜏𝑐 and 𝜏𝑝<𝑇; time 𝜏𝑝 is called the put time and is chosen by the holder, (iii)a terminal call payment 𝑈𝑡 made at time 𝑡=𝜏𝑐 provided that 𝜏𝑐<𝜏𝑝𝑇; time 𝜏𝑐, known as the call time, is chosen by the issuer and may be subject to the constraint that 𝜏𝑐𝜏, where 𝜏 is the lifting time of the call protection, (iv)a terminal payment at maturity 𝜉 made at maturity date 𝑇 provided that 𝑇𝜏𝑝𝜏𝑐.

The (possibly random) time 𝜏 in (iii) is used to model the restriction that the issuer of a game option may be prevented from making a call on some random time interval [0,𝜏).

Of course, there is also the initial cash flow, namely, the purchasing price of the contract, which is paid at the initiation time by the holder and received by the issuer.

Let us now be given an [0,+]-valued 𝔾-stopping time 𝜏𝑑 representing the default time of a reference entity, with default indicator process 𝐻𝑡=1{𝜏𝑑𝑡}. A defaultable dividend paying game option is a dividend paying game option such that the contract is terminated at 𝜏𝑑, if it has not been put or called and has not matured before. In particular, there are no more cash flows related to this contract after the default time. In this setting, the dividend stream 𝐷 is assumed to include a possible recovery payment made at the default time.

We are interested in the problem of the time evolution of an arbitrage price of the game option. Therefore, we formulate the problem in a dynamic way by pricing the game option at any time 𝑡[0,𝑇]. We write 𝒢𝑡𝑇 to denote the set of all 𝔾-stopping times with values in [𝑡,𝑇] and we let 𝒢𝑡𝑇 stand for the set {𝜏𝒢𝑡𝑇𝜏;𝜏𝑑𝜏𝜏𝑑}, where the lifting time of a call protection 𝜏 belongs to 𝒢0𝑇.

We are now in the position to state the formal definition of a defaultable game option.

Definition 2.3. A defaultable game option with lifting time of the call protection 𝜏𝒢0𝑇 is a game option with the ex-dividend cumulative discounted cash flows 𝛽𝑡𝜋(𝑡;𝜏𝑝,𝜏𝑐) given by the formula, for any 𝑡[0,𝑇] and (𝜏𝑝,𝜏𝑐)𝒢𝑡𝑇×𝒢𝑡𝑇, 𝛽𝑡𝜋𝑡;𝜏𝑝,𝜏𝑐=𝜏𝑡𝛽𝑢𝑑𝐷𝑢+1{𝜏<𝜏𝑑}𝛽𝜏1{𝜏=𝜏𝑝<𝑇}𝐿𝜏𝑝+1{𝜏<𝜏𝑝}𝑈𝜏𝑐+1{𝜏=𝑇}𝜉,(2.2) where 𝜏=𝜏𝑝𝜏𝑐 and (i)the dividend process 𝐷=(𝐷𝑡)𝑡[0,𝑇] equals 𝐷𝑡=[]0,𝑡1𝐻𝑢𝑑𝐶𝑢+[]0,𝑡𝑅𝑢𝑑𝐻𝑢=𝐶𝜏1{𝑡𝜏}+𝐶𝑡1{𝑡<𝜏}+𝑅𝜏1{𝑡𝜏},(2.3) for some coupon process 𝐶=(𝐶𝑡)𝑡[0,𝑇], which is a 𝔾-predictable, real-valued, càdlàg process with bounded variation, and some real-valued, 𝔾-predictable recovery process 𝑅=(𝑅𝑡)𝑡[0,𝑇], (ii)the put payment 𝐿=(𝐿𝑡)𝑡[0,𝑇] and the call payment 𝑈=(𝑈𝑡)𝑡[0,𝑇] are 𝔾-adapted, real-valued, càdlàg processes, (iii)the inequality 𝐿𝑡𝑈𝑡 holds for every 𝑡[𝜏𝑑𝜏,𝜏𝑑𝑇), (iv)the payment at maturity 𝜉 is a 𝒢𝑇-measurable, real-valued random variable.

The following result easily follows from Definition 2.3.

Lemma 2.4. (i) For any 𝑡 and (𝜏𝑝,𝜏𝑐)𝒢𝑡𝑇×𝒢𝑡𝑇, the random variable 𝜋(𝑡;𝜏𝑝,𝜏𝑐) is 𝒢𝜏𝜏𝑑-measurable.
(ii) For any (𝜏𝑝,𝜏𝑐)𝒢0𝑇×𝒢0𝑇, the processes 𝜋(0;,𝜏𝑐) and 𝜋(0;𝜏𝑝,) are 𝔾-adapted.

We further assume that 𝑅,𝐿, and 𝜉 are bounded from below, so that there exists a constant 𝑐 such that, for every 𝑡[0,𝑇], 𝛽𝑡𝑡=[0,𝑡]𝛽𝑢𝑑𝐷𝑢+1{𝑡<𝜏𝑑}𝛽𝑡1{𝑡<𝑇}𝐿𝑡+1{𝑡=𝑇}𝜉𝑐.(2.4)

Symmetrically, we should sometimes additionally assume that 𝑅,𝑈, and 𝜉 are bounded (from below and from above), or that (2.4) is supplemented by the inequality, for every 𝑡[0,𝑇], 𝛽𝑡𝒰𝑡=[0,𝑡]𝛽𝑢𝑑𝐷𝑢+1{𝑡<𝜏𝑑}𝛽𝑡1{𝑡<𝑇}𝑈𝑡+1{𝑡=𝑇}𝜉𝑐.(2.5)

2.1. Valuation of a Defaultable Game Option

We will state the following fundamental pricing result without proof, referring the interested reader to [3, Proposition  3.1 and Theorem  4.1] for more details. The goal is to characterize the set of arbitrage ex-dividend prices of a game option in terms of values of related Dynkin games (for the general theory of Dynkin games, see, e.g., Dynkin [17], Kifer [18], and Lepeltier and Maingueneau [19]). The notion of an arbitrage price of a game option referred to in Theorem 2.5 is the dynamic notion of arbitrage price for game options, as defined in Kallsen and Kühn [6], and extended in [3] to the case of dividend-paying primary assets and dividend-paying game options by resorting to the transformation of prices into cumulative prices. Note that in the sequel, the statement “(Π𝑡)𝑡[0,𝑇] is an arbitrage price for the game option’’ is in fact to be understood as “(𝑋𝑡,Π𝑡)𝑡[0,𝑇]is an arbitrage price for the extended market consisting of the primary market and the game option.’’

Theorem 2.5 (Arbitrage price of a defaultable game option). Assume that a process Π is a 𝔾-semimartingale and there exists such that Π is the value of the Dynkin game related to a game option, meaning that esssup𝜏𝑝𝒢𝑡𝑇essinf𝜏𝑐𝒢𝑡𝑇𝔼𝜋𝑡;𝜏𝑝,𝜏𝑐𝒢𝑡=Π𝑡=essinf𝜏𝑐𝒢𝑡𝑇esssup𝜏𝑝𝒢𝑡𝑇𝔼𝜋𝑡;𝜏𝑝,𝜏𝑐𝒢𝑡[].,𝑡0,𝑇(2.6) Then Π is an arbitrage ex-dividend price of the game option, called the -price of the game option. The converse holds true (thus any arbitrage price is a -price for some ) under the following integrability assumption esssup𝔼sup𝑡[0,𝑇][0,𝑡]𝛽𝑢𝑑𝐷𝑢+1{𝑡<𝜏𝑑}𝛽𝑡1{𝑡<𝑇}𝐿𝑡+1{𝑡=𝑇}𝜉𝒢0<,a.s.(2.7)

Note that defaultable American (or European) options can be seen as special cases of defaultable game options.

Definition 2.6. A defaultable American option is a defaultable game option with 𝜏=𝑇. A defaultable European option is a defaultable American option such that the process 𝛽 (cf. (2.4) attains its maximum at 𝑇, that is, 𝛽𝑡𝑡𝛽𝑇𝑇 for every 𝑡[0,𝑇].

In view of Theorem 2.5, the cash flows 𝜙(𝑡) of a defaultable European option can be redefined by 𝛽𝑡𝜙(𝑡)=𝑇𝑡𝛽𝑢𝑑𝐷𝑢+1{𝜏𝑑>𝑇}𝛽𝑇[]𝜉,𝑡0,𝑇.(2.8)

2.2. Hedging of a Defaultable Game Option

We adopt the definition of hedging game options stemming from successive developments, starting from the hedging of American options examined by Karatzas [20], and subsequently followed by El Karoui and Quenez [21], Kifer [8], Ma and Cvitanić [22], and Hamadène [23]. One of our goals is to show that this definition is consistent with the concept of arbitrage valuation of a defaultable game option in the sense of Kallsen and Kühn [6].

Recall that 𝑋 (resp., 𝑋) is the price process (resp., cumulative price process) of primary traded assets, as given by (2.1). The following definitions are standard, accounting for the dividends on the primary market.

Definition 2.7. By a (self-financing) primary trading strategy we mean any pair (𝑉0,𝜁) such that (i)𝑉0 is a 𝒢0-measurable real-valued random variable representing the initial wealth,(ii)𝜁 is an 1𝑑-valued, 𝛽𝑋-integrable process representing holdings in primary risky assets.

Remark 2.8. The reason why we do not assume that 𝒢0 is trivial (which would, of course, simplify several statements) is that we apply our results in the subsequent work [4] to situations, where this assumption fails to hold (e.g., when studying convertible bonds with a positive call notice period).

Definition 2.9. The wealth process 𝑉 of a primary trading strategy (𝑉0,𝜁) is given by the formula, for 𝑡[0,𝑇], 𝛽𝑡𝑉𝑡=𝛽0𝑉0+𝑡0𝜁𝑢𝑑𝛽𝑢𝑋𝑢.(2.9)

Given the wealth process 𝑉 of a primary strategy (𝑉0,𝜁), we uniquely specify a 𝔾-optional process 𝜁0 by setting 𝑉𝑡=𝜁0𝑡𝛽𝑡1+𝜁𝑡𝑋𝑡[],𝑡0,𝑇.(2.10) The process 𝜁0 represents the number of units held in the savings account at time 𝑡, when we start from the initial wealth 𝑉0 and we use the strategy 𝜁 in the primary risky assets. Recall that we denote 𝜏=𝜏𝑝𝜏𝑐.

Definition 2.10. Consider the game option with the ex-dividend cumulative discounted cash flows 𝛽𝜋 given by (2.2).
(i)An issuer hedge with cost process 𝜌 is represented by a quadruplet (𝑉0,𝜁,𝜌,𝜏𝑐) such that(a)(𝑉0,𝜁) is a primary strategy with the wealth process 𝑉 given by (2.9),(b)a cost process 𝜌 is a real-valued, càdlàg 𝔾-semimartingale with 𝜌0=0,(c)a (fixed) call time 𝜏𝑐 belongs to 𝒢0𝑇,(d)the following inequality is valid, for every put time 𝜏𝑝𝒢0𝑇, 𝛽𝜏𝑉𝜏+𝜏0𝛽𝑢𝑑𝜌𝑢𝛽0𝜋0;𝜏𝑝,𝜏𝑐,a.s.(2.11)(ii)A holder hedge with cost process 𝜌 is a quadruplet (𝑉0,𝜁,𝜌,𝜏𝑝) such that(a)(𝑉0,𝜁) is a primary strategy with the wealth process 𝑉 given by (2.9),(b)a cost process 𝜌 is a real-valued, càdlàg 𝔾-semimartingale with 𝜌0=0,(c)a (fixed) put time 𝜏𝑝 belongs to 𝒢0𝑇,(d)the following inequality is valid, for every call time 𝜏𝑐𝒢0𝑇, 𝛽𝜏𝑉𝜏+𝜏0𝛽𝑢𝑑𝜌𝑢𝛽0𝜋0;𝜏𝑝,𝜏𝑐,a.s.(2.12)

Issuer or holder hedges at no cost (i.e., with 𝜌=0) are thus in effect issuer or holder superhedges. A more explicit form of condition (2.11) reads (for (2.12), we need to insert the minus sign in the right-hand side of (2.13) 𝑉𝜏+𝛽𝜏1𝜏0𝛽𝑢𝑑𝜌𝑢𝛽𝜏1𝜏0𝛽𝑢𝑑𝐷𝑢+1𝜏<𝜏𝑑1𝜏=𝜏𝑝<𝑇𝐿𝜏𝑝+1𝜏<𝜏𝑝𝑈𝜏𝑐+1𝜏𝑝=𝜏𝑐=𝑇𝜉,a.s.(2.13) The left-hand side in (2.13) is the value at time 𝜏 of a strategy with a cost process 𝜌, when the players adopt their respective exercise policies 𝜏𝑝 and 𝜏𝑐, whereas the right-hand side represents the payoff to be done by the issuer, including past dividends and the recovery at default.

Remark 2.11. (i) The process 𝜌 is to be interpreted as the (running) financing cost, that is, the amount of cash added to (if 𝑑𝜌𝑡0) or withdrawn from (if 𝑑𝜌𝑡0) the hedging portfolio in order to get a perfect, but no longer self-financing, hedge. In the special case where 𝜌 is a 𝔾-martingale under we thus recover the notion of mean self-financing hedge, in the sense of Schweizer [24].
(ii) Regarding the admissibility of hedging strategies (see, e.g., Delbaen and Schachermayer [14]), note that the left-hand side in formula (2.11) (discounted wealth process inclusive of financing costs) is bounded from below for any issuer hedge with a cost (𝑉0,𝜁,𝜌,𝜏𝑐). Likewise, in the case of a bounded payoff 𝜋 (i.e., assuming (2.5), the left-hand side in formula (2.12) is bounded from below for any holder hedge with a cost (𝑉0,𝜁,𝜌,𝜏𝑝).

Obviously, the class of all hedges with semimartingale cost processes is too large for any practical purposes. Therefore, we will restrict our attention to hedges with a 𝔾-sigma martingale cost 𝜌 under a particular risk-neutral measure .

Assumption 2.12. In the sequel, we work under a fixed, but arbitrary, risk-neutral measure .

All the measure-dependent notions like (local) martingale and compensator, implicitly refer to the probability measure . In particular, we define 𝒱𝑐0 (resp., 𝒱𝑝0) as the set of initial values 𝑉0 for which there exists an issuer (resp., holder) hedge of the game option with the initial value 𝑉0 (resp., 𝑉0) and with a 𝔾-sigma martingale cost under .

The following result gives some preliminary conclusions regarding the initial cost of a hedging strategy for the game option under the present, rather weak, assumptions. In Proposition 4.3, we will see that, under stronger assumptions, the infima are attained and thus we obtain equalities, rather than merely inequalities, in (2.14) and (2.15).

Lemma 2.13. (i) One has (by convention, essinf=) essinf𝜏𝑐𝒢0𝑇esssup𝜏𝑝𝒢0𝑇𝔼𝜋0;𝜏𝑝,𝜏𝑐𝒢0essinf𝑉0𝒱𝑐0𝑉0,a.s.(2.14)
(ii) If inequality (2.5) is valid then esssup𝜏𝑝𝒢0𝑇essinf𝜏𝑐𝒢0𝑇𝔼𝜋0;𝜏𝑝,𝜏𝑐𝒢0essinf𝑉0𝒱𝑝0𝑉0,a.s.(2.15)

Proof. (i) Assume that for some stopping time 𝜏𝑐𝒢0𝑇 the quadruplet (𝑉0,𝜁,𝜌,𝜏𝑐) is an issuer hedge with a 𝔾-sigma martingale cost 𝜌 for the game option. It is easily seen from (2.9) and (2.11) that, for any stopping time 𝜏𝑝𝒢0𝑇, 𝛽0𝑉0=𝛽𝜏𝑝𝜏𝑐𝑉𝜏𝑝𝜏𝑐𝜏𝑝𝜏𝑐0𝜁𝑢𝑑𝛽𝑢𝑋𝑢𝛽0𝜋0;𝜏𝑝,𝜏𝑐𝜏𝑝𝜏𝑐0𝜁𝑢𝑑𝛽𝑢𝑋𝑢+𝛽𝑢𝑑𝜌𝑢.(2.16) In particular, by taking 𝜏𝑝=𝑡, we obtain that, for any 𝑡[0,𝑇], 𝛽0𝑉0=𝛽𝑡𝜏𝑐𝑉𝑡𝜏𝑐𝑡𝜏𝑐0𝜁𝑢𝑑𝛽𝑢𝑋𝑢𝛽0𝜋0;𝑡,𝜏𝑐𝑡𝜏𝑐0𝜁𝑢𝑑𝛽𝑢𝑋𝑢+𝛽𝑢𝑑𝜌𝑢.(2.17) The stochastic integral 𝑡0𝜁𝑢𝑑(𝛽𝑢𝑋𝑢) with respect to a 𝔾-sigma martingale 𝛽𝑋 is a 𝔾-sigma martingale. Hence the stopped process 𝑡𝜏𝑐0𝜁𝑢𝑑(𝛽𝑢𝑋𝑢), as well as the process 𝑡𝜏𝑐0𝜁𝑢𝑑𝛽𝑢𝑋𝑢+𝛽𝑢𝑑𝜌𝑢(2.18) are 𝔾-sigma martingales. The latter process is bounded from below (this follows from (2.2)–(2.4) and (2.17), so that it is a bounded from below local martingale [15, page 216] and thus also a supermartingale. By taking conditional expectations in (2.16), we thus obtain for any stopping time 𝜏𝑝𝒢0𝑇 (recall that 𝜏𝑐 is fixed) 𝛽0𝑉0𝔼𝛽0𝜋0;𝜏𝑝,𝜏𝑐𝒢0,𝜏𝑝𝒢0𝑇,(2.19) and thus, by the assumed positivity of the process 𝛽,𝑉0essinf𝜏𝑐𝒢0𝑇esssup𝜏𝑝𝒢0𝑇𝔼𝜋0;𝜏𝑝,𝜏𝑐𝒢0,a.s.(2.20) The required inequality (2.14) is an immediate consequence of the last formula.
(ii) Let (𝑉0,𝜁,𝜌,𝜏𝑝) be a holder hedge with a 𝔾-sigma martingale cost 𝜌 for the game option for some stopping time 𝜏𝑝𝒢0𝑇. Then (2.9) and (2.12) imply that, for any 𝑡[𝜏,𝑇], 𝛽0𝑉0=𝛽𝑡𝜏𝑝𝑉𝑡𝜏𝑝𝑡𝜏𝑝0𝜁𝑢𝑑𝛽𝑢𝑋𝑢𝛽0𝜋0;𝜏𝑝,𝑡𝑡𝜏𝑝0𝜁𝑢𝑑𝛽𝑢𝑋𝑢+𝛽𝑢𝑑𝜌𝑢.(2.21) Under condition (2.5), the stochastic integral in the last formula is bounded from below and thus we conclude, by the same arguments as in part (i) that it is a supermartingale. Consequently, for a fixed stopping time 𝜏𝑝𝒢0𝑇, we obtain 𝛽0𝑉0𝔼𝛽0𝜋0;𝜏𝑝,𝜏𝑐𝒢0,a.s.,𝜏𝑐𝒢0𝑇,(2.22) so that 𝑉0esssup𝜏𝑝𝒢0𝑇essinf𝜏𝑐𝒢0𝑇𝔼𝜋0;𝜏𝑝,𝜏𝑐𝒢0,a.s.,(2.23) and this in turn implies (2.15).

3. Valuation in a Hazard Process Setup

In order to get more explicit pricing and hedging results for defaultable game options, we will now study the so-called hazard process setup.

3.1. Standing Assumptions

Given an [0,+]-valued 𝔾-stopping time 𝜏𝑑, we assume that 𝔾=𝔽, where the filtration is generated by the process 𝐻𝑡=1{𝜏𝑑𝑡} and 𝔽 is some reference filtration. As expected, our approach will consist in effectively reducing the information flow from the full filtration 𝔾 to the reference filtration 𝔽.

Let 𝐺 stand for the process 𝐺𝑡=(𝜏𝑑>𝑡𝑡) for 𝑡+. The process 𝐺 is a (bounded) 𝔽-supermartingale, as the optional projection on the filtration 𝔽 of the nonincreasing process 1𝐻 (see Jeulin [25]).

In the sequel, we will work under the following standing assumption.

Assumption 3.1. We assume that the process 𝐺 is (strictly) positive and continuous with finite variation, so that the 𝔽-hazard process Γ𝑡=ln(𝐺𝑡),𝑡+, is well defined and continuous with finite variation.

Remark 3.2. (i) The assumption that 𝐺 is continuous implies that 𝜏𝑑 is a totally inaccessible 𝔾-stopping time (see, e.g., [26]). Moreover, 𝜏𝑑 avoids 𝔽-stopping times, in the sense that (𝜏𝑑=𝜏)=0 for any 𝔽-stopping time 𝜏 (see Coculescu and Nikeghbali [27]).
(ii) If 𝐺 is continuous, the additional assumption that 𝐺 has a finite variation implies in fact that 𝐺 is nonincreasing (see Lemma A.1(i). This lies somewhere between assuming further the (stronger) () Hypothesis and assuming further that 𝜏𝑑 is an 𝔽-pseudo-stopping time (see Nikeghbali and Yor [28]). Recall that the () Hypothesis means that all local 𝔽-martingales are local 𝔾-martingales (see, e.g., [29]), whereas 𝜏𝑑 is an 𝔽-pseudo-stopping time whenever all 𝔽-local martingales stopped at 𝜏𝑑 are 𝔾-local martingales (see Nikeghbali and Yor [28] and the appendix).

Some consequences of Assumption 3.1 useful for this work are summarized in Lemma A.1. The next definition refers to some auxiliary results, which are relegated to the appendix.

Definition 3.3. The 𝔽-stopping time ̃𝜏, the 𝑡-measurable random variable 𝜒 and the 𝔽-adapted or 𝔽-predictable process 𝑌 introduced in Lemmas A.2 and A.4 are called the 𝔽-representatives of 𝜏,𝜒 and 𝑌, respectively. In the context of credit risk, where 𝜏𝑑 represents the default time of a reference entity, they are also known as the pre-default values of 𝜏,𝜒 and 𝑌.

To simplify the presentation, we find it convenient to make additional assumptions. Strictly speaking, these assumptions are superfluous, in the sense that all the results below are true without Assumption 3.4. Indeed, by making use of Lemmas A.2 and A.4 and Definition 3.3, it is always possible to reduce the original problem to the case described in Assumption 3.4. Since this would make the notation heavier, without adding much value, we prefer to work under this standing assumption.

Assumption 3.4. (i) The discount factor process 𝛽 is 𝔽-adapted.
(ii) The coupon process 𝐶 is 𝔽-predictable.
(iii) The recovery process 𝑅 is 𝔽-predictable.
(iv) The payoff processes 𝐿,𝑈 are 𝔽-adapted and the random variable 𝜉 is 𝑇-measurable.
(v) The call protection 𝜏 is an 𝔽-stopping time.

3.2. Reduction of a Filtration

The next lemma shows that the computation of the lower and upper value of the Dynkin games (2.6) with respect to 𝔾-stopping times can be reduced to the computation of the lower and upper value with respect to 𝔽-stopping times.

Lemma 3.5. One has that esssup𝜏𝑝𝒢𝑡𝑇essinf𝜏𝑐𝒢𝑡𝑇𝔼𝜋𝑡;𝜏𝑝,𝜏𝑐𝒢𝑡=esssup𝜏𝑝𝑡𝑇essinf𝜏𝑐𝑡𝑇𝔼𝜋𝑡;𝜏𝑝,𝜏𝑐𝒢𝑡,essinf𝜏𝑐𝒢𝑡𝑇esssup𝜏𝑝𝒢𝑡𝑇𝔼𝜋𝑡;𝜏𝑝,𝜏𝑐𝒢𝑡=essinf𝜏𝑐𝑡𝑇esssup𝜏𝑝𝑡𝑇𝔼𝜋𝑡;𝜏𝑝,𝜏𝑐𝒢𝑡.(3.1)

Proof. For (𝜏𝑝,𝜏𝑐)𝒢𝑡𝑇×𝒢𝑡𝑇, one has that 𝜋𝑡;𝜏𝑝,𝜏𝑐=𝜋𝑡;𝜏𝑝𝜏𝑑,𝜏𝑐𝜏𝑑=𝜋𝑡;̃𝜏𝑝𝜏𝑑,̃𝜏𝑐𝜏𝑑=𝜋𝑡;̃𝜏𝑝,̃𝜏𝑐(3.2) for some stopping times (̃𝜏𝑝,̃𝜏𝑐)𝑡𝑇×𝑡𝑇, where the middle equality follows from Lemma A.4, and the other two from the definition of 𝜋. Since, clearly, 𝑡𝑇𝒢𝑡𝑇 and 𝑡𝑇𝒢𝑡𝑇, we conclude that the lemma is valid.

Under our assumptions, the computation of conditional expectations of cash flows 𝜋(𝑡;𝜏𝑝,𝜏𝑐) with respect to 𝒢𝑡 can be reduced to the computation of conditional expectations of 𝔽-equivalent cash flows 𝜋(𝑡;𝜏𝑝,𝜏𝑐) with respect to 𝑡. Let 𝛼𝑡=𝛽𝑡exp(Γ𝑡) stand for the credit-risk adjusted discount factor. Note that, similarly to 𝛽, the process 𝛼 is bounded.

Lemma 3.6. For any stopping times 𝜏𝑝𝑡𝑇 and 𝜏𝑐𝑡𝑇 one has that 𝔼𝜋𝑡;𝜏𝑝,𝜏𝑐𝒢𝑡=1{𝑡<𝜏𝑑}𝔼𝜋𝑡;𝜏𝑝,𝜏𝑐𝑡,(3.3) where 𝜋(𝑡;𝜏𝑝,𝜏𝑐) is given by, with 𝜏=𝜏𝑝𝜏𝑐, 𝛼𝑡𝜋𝑡;𝜏𝑝,𝜏𝑐=𝜏𝑡𝛼𝑢𝑑𝐶𝑢+𝑅𝑢𝑑Γ𝑢+𝛼𝜏1{𝜏=𝜏𝑝<𝑇}𝐿𝜏𝑝+1{𝜏<𝜏𝑝}𝑈𝜏𝑐+1{𝜏=𝑇}𝜉.(3.4)

Proof. Formula (3.3) is an immediate consequence of formula (2.2) and Lemma A.5.

Note that 𝜋(𝑡;𝜏𝑝,𝜏𝑐) is an 𝜏-measurable random variable. A comparison of formulae (2.2) and (3.4) shows that we have effectively moved our considerations from the original market subject to the default risk, in which cash flows are discounted according to the discount factor 𝛽, to the fictitious default-free market, in which cash flows are discounted according to the credit risk adjusted discount factor 𝛼. Recall that the original cash flows 𝜋(𝑡;𝜏𝑝,𝜏𝑐) are given as 𝒢𝜏𝜏𝑑-measurable random variables, whereas the 𝔽-equivalent cash flows 𝜋(𝑡;𝜏𝑝,𝜏𝑐) are manifestly 𝜏-measurable and they depend on the default time 𝜏𝑑 only via the hazard process Γ. For the purpose of computation of the ex-dividend price of a defaultable game option these two market models are in fact equivalent. This follows from the next result, which is obtained by combining Theorem 2.5 with Lemmas 3.5 and 3.6.

Theorem 3.7 (Pre-default price of a defaultable game option). Assuming condition (2.7), let Π be the arbitrage ex-dividend -price for a game option. Then one has, for any 𝑡[0,𝑇], Π𝑡=1{𝑡<𝜏𝑑}Π𝑡,(3.5) where Π𝑡 satisfies esssup𝜏𝑝𝑡𝑇essinf𝜏𝑐𝑡𝑇𝔼𝜋𝑡;𝜏𝑝,𝜏𝑐𝑡=Π𝑡=essinf𝜏𝑐𝑡𝑇esssup𝜏𝑝𝑡𝑇𝔼𝜋𝑡;𝜏𝑝,𝜏𝑐𝑡.(3.6) Hence the Dynkin game with cost criterion 𝔼(𝜋(𝑡;𝜏𝑝,𝜏𝑐)𝑡) on 𝑡𝑇×𝑡𝑇 admits the value Π𝑡, which coincides with the pre-default ex-dividend price at time 𝑡 of the game option under the risk-neutral measure .

The following result is the converse of Theorem 3.7. It is an immediate consequence of Lemmas 3.5 and 3.6 and the “if’’ part of Theorem 2.5 (noting also that Π defined by (3.5) is obviously a 𝔾-semimartingale if Π is a 𝔾-semimartingale).

Theorem 3.8. Let Π𝑡 be the value of the Dynkin game with the cost criterion 𝔼(𝜋(𝑡;𝜏𝑝,𝜏𝑐)𝑡) on 𝑡𝑇×𝑡𝑇, for any 𝑡[0,𝑇]. Then Π𝑡 defined by (3.5) is the value of the Dynkin game with the cost criterion 𝔼(𝜋(𝑡;𝜏𝑝,𝜏𝑐)𝒢𝑡) on 𝒢𝑡𝑇×𝒢𝑡𝑇, for any 𝑡[0,𝑇]. If, in addition, Π is a 𝔾-semimartingale then Π is the arbitrage ex-dividend -price for the game option.

Theorems 3.7 and 3.8 thus allow us to reduce the study of a game option to the study of Dynkin games (3.6) with respect to the reference filtration 𝔽.

3.3. Valuation via Doubly Reflected BSDEs

In this section, we will characterize the arbitrage ex-dividend -price of a game option as a solution to an associated doubly reflected BSDE. To this end, we first recall some auxiliary results concerning the relationship between Dynkin games and doubly reflected BSDEs.

Given an additional 𝔽-adapted process 𝐹 of finite variation, we consider the following doubly reflected BSDE with the data 𝛼,𝐹,𝜉,𝐿,𝑈,𝜏 (see Cvitanić and Karatzas [30], Hamadène and Hassani [31], Crépey [32], Crépey and Matoussi [33], Bielecki et al. [4, 9]): 𝛼𝑡Θ𝑡=𝛼𝑇𝜉+𝛼𝑇𝐹𝑇𝛼𝑡𝐹𝑡+𝑇𝑡𝛼𝑢𝑑𝐾𝑢𝑇𝑡𝛼𝑢𝑑𝑀𝑢[],𝐿,𝑡0,𝑇𝑡Θ𝑡𝑈𝑡[],,𝑡0,𝑇𝑇0Θ𝑢𝐿𝑢𝑑𝐾+𝑢=𝑇0𝑈𝑢Θ𝑢𝑑𝐾𝑢=0,(3.7) where the process 𝑈=(𝑈𝑡)𝑡[0,𝑇] equals, for 𝑡[0,𝑇], 𝑈𝑡=1{𝑡<𝜏}+1{𝑡𝜏}𝑈𝑡.(3.8)

Definition 3.9. By a (-) solution to the doubly reflected BSDE (3.7), we mean a triplet (Θ,𝑀,𝐾) such that (i)the state process Θ is a real-valued, 𝔽-adapted, càdlàg process,(ii)0𝛼𝑑𝑀 is a real-valued 𝔽-martingale vanishing at time 0, (iii)𝐾 is an 𝔽-adapted, continuous, finite variation process vanishing at time 0, (iv)all conditions in (3.7) are satisfied, where in the third line 𝐾+ and 𝐾 denote the Jordan components of 𝐾, and where the convention that 0×±=0 is made in the third line.

By the Jordan decomposition, we mean the decomposition 𝐾=𝐾+𝐾, where the nondecreasing continuous processes 𝐾+ and 𝐾 vanish at time 0 and define mutually singular measures.

The state process Θ in a solution to (3.7) is clearly an 𝔽-semimartingale. So there are obvious (though rather artificial) cases in which (3.7) does not admit a solution: it suffices to take 𝜏=0 and 𝐿=𝑈, assumed not to be an 𝔽-semimartingale. It is also clear that a solution would not necessarily be unique if we did not impose the condition of a mutual singularity of the nonnegative measures defined by 𝐾+ and 𝐾 (see, e.g., [31, Remark  4.1]).

Remark 3.10. In applications (see [4, 9, 32, 33]), the input process 𝐹 is typically given in the form of the Lebesgue integral 𝛼𝐹=𝛼𝑓𝑑𝑢 and the component 𝑀 of a solution to (3.7) is usually searched for in the form 𝑀=𝑍𝑑𝑁+𝑛 for some 𝑞-valued and real-valued square-integrable 𝔽-martingales 𝑁 and 𝑛 (see also Assumption 4.7 in Section 4.3). For more explicit (in particular, Markovian) specifications of the present setup and sufficient conditions for the existence and uniqueness of a solution to (3.7), the interested reader is referred to, for example, [4, 3033].

Basically, in any model endowed with the martingale representation property, the existence (and uniqueness) of a solution to (3.7) (supplemented by suitable integrability conditions on the data and the solution) is equivalent to the so-called Mokobodski condition, namely, the existence of a quasimartingale 𝑍 such that 𝐿𝑍𝑈 on [0,𝑇] (see, in particular, Crépey and Matoussi [33], Hamadène and Hassani [31, Theorem  4.1], and previous works in this direction, starting with Cvitanić and Karatzas [30]). It is thus satisfied when one of the barriers is a quasimartingale and, in particular, when one of the barriers is given as 𝑆, where 𝑆 is an Itô-Lévy process 𝑆 with square-integrable special semimartingale decomposition components (see [33]) and is a constant in {}. This framework covers, for instance, the payoff at call of a convertible bond examined in [3, 4].

Remark 3.11. (i) Since 𝐾, and thus 𝐾+ and 𝐾, are continuous, the minimality conditions (third line) in (3.7) are equivalent to 𝑇0Θ𝑢𝐿𝑢𝑑𝐾+𝑢=𝑇0𝑈𝑢Θ𝑢𝑑𝐾𝑢=0.(3.9)
Indeed the related integrands here and in the third line of (3.7) differ on an at most countable set whereas the integrators define atomless measures on [0,𝑇]; see, for example, [33]. In the preprint version [34] of this work, we defined more general notions of 𝜀-hedges that were pertaining in the case where there may be jumps in the process 𝐾. Since in all existing works on doubly reflected BSDEs the process 𝐾 is actually found to be a continuous process (see [4, 30, 31, 33]), we decided to impose here the continuity of 𝐾 in Definition 3.9 and we only consider hedges, as opposed to 𝜀-hedges. Note, however, that essentially all the results of this paper can be extended to possible jumps in 𝐾, using the generalized notion of 𝜀-hedge defined in [34], and with the minimality conditions stated as (3.9) instead of the third line in condition (3.7) of Definition 3.9.
(ii) Since 𝐹 is a given process, the BSDE (3.7) can be rewritten as 𝛼𝑡Θ𝑡=𝛼𝑇̂𝜉+𝑇𝑡𝛼𝑢𝑑𝐾𝑢𝑇𝑡𝛼𝑢𝑑𝑀𝑢[],𝐿,𝑡0,𝑇𝑡Θ𝑡𝑈𝑡[],,𝑡0,𝑇𝑇0Θ𝑢𝐿𝑢𝑑𝐾+𝑢=𝑇0𝑈𝑢Θ𝑢𝑑𝐾𝑢=0,(3.10) where Θ𝑡=Θ𝑡+𝐹𝑡,̂𝜉=𝜉+𝐹𝑇,𝐿𝑡=𝐿𝑡+𝐹𝑡, and 𝑈𝑡=𝑈𝑡+𝐹𝑡. This shows that the problem of solving (3.7) can be formally reduced to the case of 𝐹=0 with suitably modified reflecting barriers 𝑈𝐿, and terminal condition ̂𝜉. However, the freedom to choose the driver of a related BSDE associated with a game option is important from the point of view of applications (this is apparent in the followup papers [4, 9]; see also [34]).
(iii) In the special case where all 𝔽-martingales are continuous and where the 𝔽-semimartingale 𝐹 and the barriers 𝐿 and 𝑈 are continuous (see [4, 30, 35]), it is natural to look for a continuous solution of (3.7), that is, a solution of (3.7) given by a triplet of continuous processes (Θ,𝑀,𝐾).
(iv) In the context of a Markovian setup, the probabilistic BSDE approach may be complemented by a related analytic variational inequality approach; this issue is dealt with in the followup papers [4, 9]. Note, however, that the variational inequality approach strongly relies on the BSDE approach. Moreover, a simulation method based on the BSDE is the only efficient way of numerically solving the pricing problem whenever the problem dimension (number of model factors) is greater than three or four. Indeed, in that case the computational cost of deterministic numerical schemes based on the variational inequality approach becomes prohibitive.

In order to establish a relationship between a solution to the related doubly reflected BSDE and the arbitrage ex-dividend -price of the defaultable game option, we first recall the general relationship between doubly reflected BSDEs and Dynkin games with purely terminal cost, before applying this result to dividend-paying game options in the fictitious default-free market in Proposition 3.12.

Observe that if (Θ,𝑀,𝐾) solves (3.7) then one has, for any stopping time 𝜏𝑡𝑇, 𝛼𝑡Θ𝑡=𝛼𝜏Θ𝜏+𝛼𝜏𝐹𝜏𝛼𝑡𝐹𝑡+𝜏𝑡𝛼𝑢𝑑𝐾𝑢𝜏𝑡𝛼𝑢𝑑𝑀𝑢.(3.11)

Proposition 3.12 (Verification principle for a Dynkin game). Let (Θ,𝑀,𝐾) be a solution to (3.7) with 𝐹=0. Then Θ𝑡 is the value of the Dynkin game with cost criterion 𝔼(𝜃(𝑡;𝜏𝑝,𝜏𝑐)𝑡) on 𝑡𝑇×𝑡𝑇, where 𝜃(𝑡;𝜏𝑝,𝜏𝑐) is the 𝜏-measurable random variable defined by 𝛼𝑡𝜃𝑡;𝜏𝑝,𝜏𝑐=𝛼𝜏1{𝜏=𝜏𝑝<𝑇}𝐿𝜏𝑝+1{𝜏=𝜏𝑐<𝜏𝑝}𝑈𝜏𝑐+1{𝜏=𝑇}𝜉,(3.12) where 𝜏=𝜏𝑝𝜏𝑐. Moreover, for any 𝑡[0,𝑇], the pair of stopping times (𝜏𝑝,𝜏𝑐)𝑡𝑇×𝑡𝑇 given by 𝜏𝑝[]=inf𝑢𝑡,𝑇;Θ𝑢𝐿𝑢𝑇,𝜏𝑐=inf𝑢𝜏𝑡,𝑇;Θ𝑢𝑈𝑢𝑇,(3.13) is a saddle-point of this Dynkin game, in the sense that one has, for any (𝜏𝑝,𝜏𝑐)𝑡𝑇×𝑡𝑇, 𝔼𝜃𝑡;𝜏𝑝,𝜏𝑐𝑡Θ𝑡𝔼𝜃𝑡;𝜏𝑝,𝜏𝑐𝑡.(3.14)

Proof. Except for the presence of 𝜏, the result is standard (see, e.g., Lepeltier and Maingueneau [19]). Let us first check that the right-hand side inequality in (3.14) is valid for any 𝜏𝑐𝑡𝑇. Let 𝜏denote 𝜏𝑝𝜏𝑐. By the definition of 𝜏𝑝 and continuity of 𝐾+, we see that 𝐾+ equals 0 on [𝑡,𝜏]. Since 𝐾 is nondecreasing, (3.11) is applied to yield 𝛼𝑡Θ𝑡𝛼𝜏Θ𝜏𝜏𝑡𝛼𝑢𝑑𝑀𝑢.(3.15) Taking conditional expectations (recall that 𝑡𝛼𝑢𝑑𝑀𝑢 is an 𝔽-martingale), and using also the facts that Θ𝜏𝑝𝐿𝜏𝑝 if 𝜏𝑝<𝑇,Θ𝜏𝑝=𝜉 if 𝜏𝑝=𝑇 and Θ𝜏𝑐𝑈𝜏𝑐 (recall that 𝜏𝑐𝑡𝑇, so that 𝜏𝑐𝜏 and 𝑈𝜏𝑐=𝑈𝜏𝑐), we obtain 𝛼𝑡Θ𝑡𝔼𝛼𝜏Θ𝜏𝑡𝔼𝛼𝜏1𝜏=𝜏𝑝<𝑇𝐿𝜏𝑝+1𝜏=𝜏𝑐<𝜏𝑝𝑈𝜏𝑐+1𝜏=𝑇𝜉𝑡.(3.16) We conclude that Θ𝑡𝔼(𝜃(𝑡;𝜏𝑝,𝜏𝑐)𝑡) for any 𝜏𝑐𝑡𝑇. This completes the proof of the right-hand side inequality in (3.14). The left-hand side inequality can be shown similarly. It is in fact standard, since it does not involve 𝜏, and thus the details are left to the reader.

Let us now apply Proposition 3.12 to a defaultable game option. To this end, we first rewrite (3.4) as follows 𝛼𝑡𝜋𝑡;𝜏𝑝,𝜏𝑐=𝛼𝜏𝐹𝜏𝛼𝑡𝐹𝑡+𝛼𝜏1{𝜏=𝜏𝑝<𝑇}𝐿𝜏𝑝+1{𝜏<𝜏𝑝}𝑈𝜏𝑐+1{𝜏=𝑇}𝜉,(3.17) where 𝐹𝑡=𝛼𝑡1[0,𝑡]𝛼𝑢𝑑𝐷𝑢with𝐷𝑡=[0,𝑡]𝑑𝐶𝑢+𝑅𝑢𝑑Γ𝑢.(3.18) Let us denote by () equation (3.10) with 𝐹𝑡=𝐹𝑡, that is, 𝛼𝑡Θ𝑡=𝛼𝑇̂𝜉+𝑇𝑡𝛼𝑢𝑑𝐾𝑢𝑇𝑡𝛼𝑢𝑑𝑀𝑢[],𝐿,𝑡0,𝑇𝑡Θ𝑡𝑈𝑡[],,𝑡0,𝑇𝑇0Θ𝑢𝐿𝑢𝑑𝐾+𝑢=𝑇0𝑈𝑢Θ𝑢𝑑𝐾𝑢(=0,) with ̂𝜉=𝜉+𝐹𝑇,𝐿𝑡=𝐿𝑡+𝐹𝑡, and 𝑈𝑡=𝑈𝑡+𝐹𝑡.

Assumption 3.13. The doubly reflected BSDE () admits a solution (Θ,𝑀,𝐾).

Let us stress that Assumption 3.13, heroic as it may seem in the general hazard process setup, is in fact a plausible assumption in any reasonable application one may think of (cf. the comments following Definition 3.9).

We denote, for 𝑡[0,𝑇], Π𝑡=Θ𝑡𝐹𝑡,Π𝑡=1{𝑡<𝜏𝑑}Π𝑡,Π𝑡=Π𝑡+𝛽𝑡1[0,𝑡]𝛽𝑢𝑑𝐷𝑢,𝑚(3.19)𝑡=𝛽𝑡Π𝑡+𝑡𝜏𝑑0𝛽𝑢𝑑𝐾𝑢.(3.20) The following lemma is crucial in what follows (Lemma 3.14(i) is actually the key of the proof of Proposition 4.1 below).

Lemma 3.14. (i) The process 𝑚 given by (3.20) is 𝔾-martingale stopped at 𝜏𝑑.
(ii) The process Π is a 𝔾-semimartingale.
(iii) The process 𝛽Π is a special 𝔾-semimartingale.

Proof. (i) The triplet (Π,𝑀,𝐾) satisfies (3.7) with 𝐹 given by 𝐹 in (3.18). Therefore, for every 𝑡[0,𝑇], 𝛼𝑡Π𝑡=𝛼𝑇𝜉+𝑇𝑡𝛼𝑢𝑑𝐷𝑢+𝑇𝑡𝛼𝑢𝑑𝐾𝑢𝑇𝑡𝛼𝑢𝑑𝑀𝑢(3.21) and thus 𝑡0𝛼𝑢𝑑𝑀𝑢=𝛼𝑡Π𝑡𝛼0Π0+𝑡0𝛼𝑢𝑑𝐾𝑢+𝑡0𝛼𝑢𝑑𝐷𝑢.(3.22) Using Lemma A.5, it is easy to check that one has, for any 0𝑡𝑢𝑇, 1{𝑡<𝜏𝑑}𝑒Γ𝑡𝔼𝑢𝑡𝛼𝑣𝑑𝑀𝑣𝑡=𝔼𝑚𝑢𝑚𝑡𝒢𝑡.(3.23) Since the integral 𝑡𝛼𝑣𝑑𝑀𝑣 is an 𝔽-martingale, the process 𝑚 is a 𝔾-martingale. It is also clear that it is stopped at 𝜏𝑑.
(ii) In view of (3.19), (3.20) and part (i), the process Π is clearly a 𝔾-semimartingale.
(iii) By (3.20), one has that 𝛽𝑡Π𝑡=𝑚𝑡𝑡𝜏𝑑0𝛽𝑢𝑑𝐾𝑢,(3.24) where 𝑚 is a 𝔾-martingale, by (i), and where the second term in the right-hand side is a 𝔾-adapted and continuous (hence 𝔾-predictable) processes of finite variation.

Remark 3.15. In view of (3.24) and since 𝐾 is continuous, the process 𝑚 given by (3.20) can equivalently be redefined as the canonical 𝔾-local martingale component of the discounted cumulative -value process 𝛽Π. The processes 𝑚 and 𝛽Π are easily seen to coincide on the random interval [0,𝜏𝑐𝜏𝑝𝜏𝑑𝑇]. Therefore, both 𝑚 and 𝛽Π can be interpreted on this interval as the discounted cumulative -value of a defaultable game option.

The following result establishes a useful connection between (Θ,𝑀,𝐾) and the arbitrage ex-dividend -price of the defaultable game option.

Proposition 3.16 (Verification principle for a defaultable game option). The process Π is the arbitrage ex-dividend -price for the game option. Moreover, for any 𝑡[0,𝑇], the saddle-point (𝜏𝑝,𝜏𝑐)𝑡𝑇×𝑡𝑇 for the related Dynkin game (2.6) on 𝒢𝑡𝑇×𝒢𝑡𝑇 is given by 𝜏𝑝[];Π=inf𝑢𝑡,𝑇𝑢𝐿𝑢𝑇,𝜏𝑐=inf𝑢;Π𝜏𝑡,𝑇𝑢𝑈𝑢𝑇.(3.25)

Proof. In view of (3.4), the present assumptions imply that Π𝑡 is the value of the Dynkin game (3.6), by Proposition 3.12, with saddle-point (𝜏𝑝,𝜏𝑐). Therefore, by Lemmas 3.5 and 3.6, Π𝑡 is the value of the Dynkin game associated with the game option on 𝒢𝑡𝑇×𝒢𝑡𝑇, with saddle-point (𝜏𝑝,𝜏𝑐). Moreover, Π is a 𝔾-semimartingale, by Lemma 3.14(ii). To conclude the proof, it suffices to make use of the last statement in Theorem 3.8.

4. Hedging in a Hazard Process Setup

In the remaining part of this work, we examine in some detail the existence and basic properties of hedging strategies for defaultable game options in a hazard process setup.

4.1. Cost Process of a Hedging Strategy

From now on, we will work under Assumption 3.13. Let thus (Θ,𝑀,𝐾) denote a solution to () and let Π and Π be defined by (3.19). In particular, Π is the arbitrage -price for the game option (by Proposition 3.16) and the left-hand sides in (2.14) and (2.15) are equal to Π0. Finally, recall that the 𝔾-martingale 𝑚 is defined by (3.20).

Let us stress that some of the key arguments underlying the following result are classical, and they are already contained in Lepeltier and Maingueneau [19] (see, in particular, Theorem  11 therein). Proposition 4.1 can thus be seen as a natural extension of their results to the defaultable case, in which two filtrations are involved. It is notable that our assumptions are made relative to the filtration 𝔽, whereas conclusions are drawn relative to the filtration 𝔾.

Proposition 4.1 (Hedging with a local martingale cost). Let 𝜁 be an arbitrary 1𝑑- valued and 𝛽𝑋-integrable process. Then the following statements are valid.
(i) Let the process 𝜌(𝜁) be given by 𝜌0(𝜁)=0 and 𝛽𝑡𝑑𝜌𝑡(𝜁)=𝑑𝑚𝑡𝜁𝑡𝑑𝛽𝑡𝑋𝑡.(4.1) Then (Π0,𝜁,𝜌(𝜁),𝜏𝑐) is an issuer hedge with 𝔾-sigma (local, in case 𝛽𝑋 and 𝜁 are locally bounded) martingale cost.
(ii) Let the process 𝜌(𝜁) be given by 𝜌0(𝜁)=0 and 𝛽𝑡𝑑𝜌𝑡(𝜁)=𝑑𝑚𝑡𝜁𝑡𝑑𝛽𝑡𝑋𝑡.(4.2) Then (Π0,𝜁,𝜌(𝜁),𝜏𝑝) is a holder hedge with a 𝔾-sigma martingale (local martingale, when 𝛽𝑋 and 𝜁 are locally bounded) cost process.

Recall that, according to our convention (see Section 1.2), the 𝛽𝑋-integrability of an 1𝑑-valued stochastic process 𝜁 implies its 𝔾-predictability. Note also that the equality 𝜌(𝜁)=𝜌(𝜁) is valid for any process 𝜁, since 𝛽𝑡𝑑𝜌𝑡(𝜁)=𝑑𝑚𝑡+𝜁𝑡𝑑𝛽𝑡𝑋𝑡=𝑑𝑚𝑡𝜁𝑡𝑑𝛽𝑡𝑋𝑡.(4.3)

Proof of Proposition 4.1. The arguments for a holder are essentially symmetrical to those for an issuer; we thus only prove part (i). By Lemma 3.14(i), the process 𝜌(𝜁) is a 𝔾-sigma martingale, and a 𝔾-local martingale if 𝛽𝑋 and 𝜁 are locally bounded processes. For the ease of notation, we write 𝜌=𝜌(𝜁). Let 𝑉 denote the wealth process of the primary strategy (Π0,𝜁). By combining (2.9) with (4.1), we obtain 𝑉0=Π0 and, for every 𝑡[0,𝑇], 𝑑𝛽𝑡𝑉𝑡=𝜁𝑡𝑑𝛽𝑡𝑋𝑡=𝑑𝑚𝑡𝛽𝑡𝑑𝜌𝑡(4.4) and thus 𝛽𝑡𝑉𝑡+𝑡0𝛽𝑢𝑑𝜌𝑢=𝑚𝑡+𝛽0Π0Π0=𝛽𝑡Π𝑡+𝑡𝜏𝑑0𝛽𝑢𝑑𝐾𝑢+𝛽0Π0Π0,(4.5) where the second equality follows from (3.20). Recall that the stopping time 𝜏𝑐0𝑇 is given by (see Proposition 3.16) 𝜏𝑐=inf𝑡;Π𝜏,𝑇𝑡𝑈𝑡𝑇.(4.6) In order to prove that the quadruplet (Π0,𝜁,𝜌,𝜏𝑐) is an issuer hedge for the game option, it is enough to show that one has for any 𝜏𝑝0𝑇, with 𝜏=𝜏𝑝𝜏𝑐 (cf. (2.13), 𝛽𝜏𝑉𝜏+𝜏0𝛽𝑢𝑑𝜌𝑢𝑑𝐷𝑢1𝜏<𝜏𝑑𝛽𝜏1𝜏=𝜏𝑝<𝑇𝐿𝜏𝑝+1𝜏<𝜏𝑝𝑈𝜏𝑐+1𝜏𝑝=𝜏𝑐=𝑇𝜉.(4.7) From the definition of 𝜏𝑐, the minimality conditions in () and the continuity of 𝐾 it follows that 𝐾=0 and thus 𝐾0 on [0,𝜏𝑐]. Since 𝜏𝜏𝑐, (4.5) thus yields 𝛽𝜏𝑉𝜏+𝜏0𝛽𝑢𝑑𝜌𝑢𝑑𝐷𝑢=𝛽𝜏Π𝜏+[0,𝜏𝜏𝑑]𝛽𝑢𝑑𝐾𝑢𝛽𝜏Π𝜏=1{𝜏<𝜏𝑑}𝛽𝜏Π𝜏,(4.8) where, by (), one has that Π𝜏1{𝜏<𝑇}𝐿𝜏+1{𝜏=𝑇}𝜉.(4.9) In addition, by the definition of 𝜏𝑐, one has that Π𝜏𝑐𝑈𝜏𝑐 on the event {𝜏𝑐<𝑇}. It is now easy to see that (4.7) is satisfied and thus (𝑉0,𝜁,𝜌,𝜏𝑐) is indeed an issuer hedge.

Remark 4.2. (i) The situation where 𝜌 can be made equal to zero by the choice of a suitable strategy 𝜁 in Proposition 4.1 corresponds to a particular form of hedgeability of a game option in which an issuer and a holder are able to hedge all risks embedded in a defaultable game option. The case where 𝜌0 corresponds either to nonhedgeability of a game option or to the situation in which an issuer (or a holder) is able to hedge, but she prefers not to hedge all risks associated with a game option, for instance, she may be willing to take some directional bets regarding specific risks. For this reason, we decided not to postulate a priori that 𝜌 should be minimized in some sense as, for instance, in Schweizer [24].
(ii) It is possible to introduce the issuer trivial hedge (Π0,0,𝜌0,𝜏𝑐) (resp., the holder trivial hedge (Π0,0,𝜌0,𝜏𝑝)) with the 𝔾-local martingale cost 𝜌0𝑡=𝑡0𝛽𝑢1𝑑𝑚𝑢[],𝑡0,𝑇.(4.10) Obviously, this hedge is of no practical interest, since it implicitly assumes that one is not interested in hedging any risks. The trivial hedge or, more precisely, the existence of any hedge is used in the proof of Proposition 4.3, however.

Let us now draw some conclusions from Lemma 2.13 and Proposition 4.1. In the context of specific (Cox-Ross-Rubinstein or Black-Scholes, say) models, analogous results can be found in Kifer [8]. Our main contribution here is an extension of these results to the present setup involving a reduction of filtration, as well as to a fairly general class of semimartingale models. We use here the notation ess min (instead of a more common symbol essinf) in order to emphasize that the respective bounds are in fact attained.

Proposition 4.3. Under the assumptions of Proposition 4.1, the following statements are valid. (i)The equality Π0=essmin𝒱𝑐0 holds, so that Π0 is the minimum of initial wealths of an issuer hedge with a 𝔾-sigma martingale cost.(ii)One has that Π0𝒱𝑝0. If, in addition, (2.5) holds then Π0=essmin𝒱𝑝0 and Π0 is the minimum of initial wealths of a holder hedge with a 𝔾-sigma martingale cost.(iii)The above statements are also valid with local martingale instead of sigma martingale therein.

Proof. (i) By applying Proposition 4.1 to the trivial hedge of Remark 4.2(ii), we get, in particular, that Π0𝒱𝑐0, where Π0 is also equal to the -value of the related Dynkin game, by Proposition 3.16. Therefore, the infimum is attained and one has equality, rather than inequality, in Lemma 2.13(i).
(ii) In view of (2.5) and Lemma 2.13(ii), the second claim can be proven in the same way as part (i).
(iii) This follows immediately from parts (i) and (ii), since the cost 𝜌0 of the trivial hedge is a 𝔾-local martingale.

Given our definition of hedging with a cost and the definition of Π0, the fact that there exists a hedge with an initial wealth Π0 and a 𝔾-sigma martingale cost (or a local martingale cost, in suitable cases) is by no means surprising. The minimality statement establishes a connection between arbitrage prices and hedging in a general incomplete market. Let us conclude this section by mentioning that one could state analogous definitions and results regarding hedging strategies for a defaultable game option starting at any date 𝑡[0,𝑇].

4.2. Risk Factors of a Defaultable Game Option

Let 𝑁𝑑=𝐻Γ𝜏𝑑 stand for the compensated default process. Under our standing assumption that the 𝔽-hazard process Γ of 𝜏𝑑 is a continuous and nondecreasing process (cf. Remark 3.2(ii), the process 𝑁𝑑 is known to be a 𝔾-martingale. Recall also that the avoidance property holds, in the sense that (𝜏𝑑=𝜏)=0 for any 𝔽-stopping time 𝜏 (cf. Remark 3.2(i).

An analysis of hedging strategies in the next section hinges on the following lemma, which yields the risk decomposition of the discounted cumulative value process of a defaultable game option. More formally, the martingale component 𝑚 (cf. Remark 3.15) is represented in terms of the pure jump martingale 𝑁𝑑 and a real-valued 𝔽-martingale 𝑀, which arise as the second component of a solution to the doubly reflected BSDE (3.7). Intuitively, the process 𝑀 models the pre-default risk associated with a defaultable game option, as opposed to the event risk, which is due to an unexpected occurrence of the default event, and which is modeled through the jump martingale 𝑁𝑑.

Lemma 4.4. The 𝔾-martingale 𝑚 defined by (3.20) satisfies 𝑑𝑚𝑡=1𝑡𝜏𝑑𝛽𝑡𝑑𝑀𝑡+𝑌𝑡𝑑𝑁𝑑𝑡,(4.11) where the 𝔽-predictable process 𝑌 equals 𝑌𝑡=𝑅𝑡Π𝑡.

Proof. Let us introduce the Doléans-Dade martingale (see, e.g., [29]) 𝑡=1{𝑡<𝜏𝑑}𝑒Γ𝑡=1𝑡0𝑢𝑑𝑁𝑑𝑢,(4.12) so that 𝛼𝑡𝑡=𝛽𝑡1{𝑡<𝜏𝑑} and 𝛼𝑡𝑡=𝛽𝑡1{𝑡𝜏𝑑}. Then (cf. (3.19) and (3.20) 𝑑𝑚𝑡𝛽=𝑑𝑡Π𝑡+1{𝑡𝜏𝑑}𝛽𝑡𝑑𝐾𝑡=𝑑𝑡𝛼𝑡Π𝑡+1{𝑡𝜏𝑑}𝛽𝑡𝑑𝐾𝑡+𝛽𝑡𝑑𝐷𝑡.(4.13) It may happen that the 𝔽-semimartingale 𝛼Π fails to be also a 𝔾-semimartingale, so a direct application of the (𝔾-)integration by parts formula to Π𝛼 is not possible. However, by Lemma A.1(iv), the process 𝛼Πstopped at 𝜏𝑑 is a 𝔾-semimartingale. It is also clear that 𝛼Π=𝛼𝜏𝑑Π𝜏𝑑. Hence by applying the integration by parts formula to 𝛼𝜏𝑑Π𝜏𝑑, we obtain 𝑑𝑡𝛼𝑡𝜏𝑑Π𝑡𝜏𝑑=𝑡𝑑𝛼𝑡𝜏𝑑Π𝑡𝜏𝑑𝛼𝑡Π𝑡𝑑𝑁𝑑𝑡+𝑑,𝛼𝜏𝑑Π𝜏𝑑𝑡,(4.14) where, in addition, one has that [,𝛼𝜏𝑑Π𝜏𝑑]𝑡=𝑒Γ𝜏𝑑𝛼𝜏𝑑ΔΠ𝜏𝑑𝐻𝑡. Using the avoidance property of Remark 3.2(i), formula (3.22), and the assumptions that the coupon process 𝐶 is 𝔽-predictable and the hazard process Γ is continuous (so that Δ𝐶𝜏𝑑=ΔΓ𝜏𝑑=0), we obtain the equality ΔΠ𝜏𝑑=0. Using (3.22), we next deduce from (4.13) that 𝑑𝑚𝑡=𝑡𝑑𝛼𝑡𝜏𝑑Π𝑡𝜏𝑑𝛼𝑡Π𝑡𝑑𝑁𝑑𝑡+1{𝑡𝜏𝑑}𝛽𝑡𝑑𝐾𝑡+𝛽𝑡𝑑𝐷𝑡=1{𝑡𝜏𝑑}𝛽𝑡𝑑𝐾𝑡𝑑𝐶𝑡𝑅𝑡𝑑Γ𝑡+𝑑𝑀𝑡Π𝑡𝑑𝑁𝑑𝑡+1{𝑡𝜏𝑑}𝛽𝑡𝑑𝐾𝑡+𝛽𝑡𝑑𝐷𝑡=1𝑡𝜏𝑑𝛽𝑡𝑑𝐶𝑡𝑅𝑡𝑑Γ𝑡+𝑑𝑀𝑡Π𝑡𝑑𝑁𝑑𝑡+𝛽𝑡𝑑𝐷𝑡.(4.15) Using (2.3) and the equality Δ𝐶𝜏𝑑=0, we finally arrive at the formula 𝑑𝑚𝑡=1{𝑡𝜏𝑑}𝛽𝑡𝑑𝑀𝑡+𝑅𝑡Π𝑡𝑑𝑁𝑑𝑡,(4.16) which is the required result.

4.3. Hedging of Risk Factors

In order to study nontrivial cases of hedging strategies for a defaultable game option in the general setup of this paper, we need to impose more assumptions on prices of primary traded assets. Since we are working in a fairly general framework, we will be able to provide only general results concerning hedging strategies. The interested reader is referred to the followup papers [4, 9] for a more detailed analysis of assumptions made in this section and particular examples.

First, we recall that the ex-dividend price 𝑋 of primary risky assets satisfies 𝑋𝑡=(1𝐻𝑡)𝑋𝑡, for every 𝑡[0,𝑇], where the 𝑑-valued, 𝔽-adapted process 𝑋 formally represents the pre-default value of 𝑋. We thus assume, by convention, that any residual value of the primary asset at 𝜏𝑑 is embedded in the recovery part of the dividend process for 𝑋. We denote by an 𝑑-valued and 𝔽-predictable process, which is aimed to represent the recovery processes of primary risky assets. Inspired by decomposition (4.11) of Lemma 4.4, we make also the following natural postulate regarding the behavior of the cumulative price process 𝑋 stopped at 𝜏𝑑𝑇.

Assumption 4.5. The dynamics under of the cumulative price process 𝑋 of primary risky assets are, for every 𝜏𝑡[0,𝑇𝑑], 𝑑𝛽𝑡𝑋𝑡=𝛽𝑡𝑑𝑀𝑡+𝑌𝑡𝑑𝑁𝑑𝑡(4.17) for some 𝑑-valued 𝔽-martingale 𝑀, where the 𝑑-valued, 𝔽-predictable process 𝑌 is given by the equality 𝑌𝑡=𝑡𝑋𝑡 for every 𝑡[0,𝑇].

By inserting (4.11) and (4.17) into (4.1), we obtain, for every 𝜏𝑡[0,𝑇𝑑], 𝑑𝜌𝑡(𝜁)=𝑑𝑀𝑡𝜁𝑡𝑑𝑀𝑡+𝑌𝑡𝜁𝑡𝑌𝑡𝑑𝑁𝑑𝑡.(4.18) At this stage, we were only able to separate the two principal components of the cost process that correspond to pre-default and default event risks, respectively, where the pre-default risk is now modeled by the 𝔽-martingales 𝑀 and 𝑀 associated with a game option and primary traded assets, respectively.

Remark 4.6. In what follows, we will only be interested in hedging on the random interval [0,𝜏𝑑𝑇]. Therefore, without loss of generality, we may and do assume that 𝜁 is 𝔽-predictable (see Lemma A.2(ii). This means that the reduction of filtration method can also be applied to hedging of a defaultable game option, and not only to its valuation as was already shown in Section 3.2.

Within the present framework, the event risk factor is common for all traded primary and derivative assets. Therefore, in the next step, we are going to get a closer look on pre-default risks of traded and derivative assets. To this end, we make a further standing assumption, in which the concept of the systematic risk factor (also known as the market risk factor) is introduced.

Assumption 4.7. We are given an 𝑞-valued 𝔽-martingale, denoted by 𝑁, which is aimed to represent the systematic risk factor for the underlying market model. We postulate that the 𝔽-martingales 𝑀 and 𝑀 of (4.11) and (4.17) satisfy the following decompositions, for every 𝑡[0,𝑇], 𝑑𝑀𝑡=𝑍𝑡𝑑𝑁𝑡+𝑑𝑛𝑡𝑀,𝑑𝑡=𝑍𝑡𝑑𝑁𝑡+𝑑̂𝑛𝑡,(4.19) where 𝑍 (resp., 𝑍) is some 𝔽-adapted, 1𝑞-valued (resp., 𝑑𝑞-valued), 𝑁-integrable processes and 𝑛 (resp., ̂𝑛) is a real-valued (resp., 𝑑-valued) 𝔽-martingale.

It is natural to refer to 𝔽-martingales 𝑛 and ̂𝑛 appearing in Assumption 4.7 as idiosyncratic risk factors associated with a defaultable game option and primary traded assets, respectively. In this context, we find it convenient to refer to 𝑁𝑑 as the event risk factor.

Remark 4.8. A specification of the systematic risk factor 𝑁 depends on a particular market model and on a problem at hand, so that it is not possible to make it more explicit in the abstract setup considered here. As it will become apparent in the sequel, the idiosyncratic risk factors are expected to be in some sense orthogonal to the systematic risk factor. For this reason, one cannot simply make 𝑍 and 𝑍 to vanish in (4.19). Once again, for more information on particular models, we refer to [4, 9] (see also Remark 3.10).

Let us denote 𝒩=[𝑁𝑁𝑑] and let [𝑍,𝑌] stand for the concatenation of 𝑍 and 𝑌. The next lemma is an immediate consequence of (4.18) and (4.19). The idea behind formula (4.20) is the separation of risk factors 𝑁,𝑁𝑑,𝑛, and ̂𝑛 in the dynamics of the cost process of a trading strategy.

Lemma 4.9. For any 1𝑑-valued, 𝛽𝑋-integrable process 𝜁, the cost process satisfies, for every 𝜏𝑡[0,𝑇𝑑], 𝑑𝜌𝑡𝑍(𝜁)=𝑡,𝑌𝑡𝜁𝑡𝑍𝑡,𝑌𝑡𝑑𝒩𝑡+𝑑𝑛𝑡𝜁𝑡𝑑̂𝑛𝑡.(4.20)

Example 4.10. To provide some intuition underpinning the present setup, let us first consider a situation where the perfect hedgeability of risks can be achieved, at least in principle. Let us set 𝑞=1 and we take 𝑑𝑀𝑡=𝑍𝑡𝑑𝑁𝑡, so that 𝑛 vanishes. For 𝑑=2, we further postulate that ̂𝑛=0 and 𝑑𝑀𝑡𝑀=𝑑1𝑡𝑀2𝑡=𝑍1𝑡𝑍2𝑡𝑑𝑀𝑡(4.21) or, equivalently, that for 𝑖=1,2𝑑𝛽𝑡𝑋𝑖𝑡=𝛽𝑡𝑍𝑖𝑡𝑑𝑀𝑡+𝑌𝑖𝑡𝑑𝑁𝑑𝑡.(4.22) Assume that there exists a 𝛽𝑋-integrable process 𝜁 solving the equation 𝜁𝑡𝑍𝑡,𝑌𝑡=𝜁1𝑡,𝜁2𝑡𝑍1𝑡𝑌1𝑡𝑍2𝑡𝑌2𝑡=𝑍𝑡,𝑌𝑡.(4.23) Then it follows from (4.20) (or (4.18) that the cost process 𝜌(𝜁) vanishes and thus the strategy 𝜁 (resp., 𝜁) is an issuer's (resp., holder's) superhedge for a defaultable game option, in the sense of Definition 2.10. Note that the first (resp., the second) equation in formula (4.23) is used to eliminate the pre-default risk (resp., the event risk). As was expected, the strategy 𝜁 obtained by solving (4.23) is 𝔽-predictable (cf. Remark 4.6).

Remark 4.11. In [4], we further specify the setup of Example 4.10, by examining the exact replication of a convertible bond with the equity and the credit default swap on the underlying credit name in an equity-to-credit intensity-based model, in which the systematic risk factor is modeled by the Brownian motion driving the equity value and all processes appearing in (4.23) can be computed explicitly.

In the foregoing result, we examine two typical situations regarding the partial hedgeability of risk factors when superhedging is either not possible or not desirable. The case considered in part (i) refers to elimination of event and systematic risks. In contrast, part (ii) deals with hedging of the systematic risk only. Of course, it is also possible to hedge the event risk only, but we do not formulate here the corresponding result. Since the proof of the lemma follows easily from (4.20), it is omitted.

Lemma 4.12 (Hedging of risk factors). (i) Assume that the equation [𝑍,𝑌]=𝜁[𝑍,𝑌] admits a 𝛽𝑋-integrable solution ̂𝜁 on 𝜏[0,𝑇𝑑]. Then the cost process ̂̂𝜌=𝜌(𝜁) satisfies, for every 𝜏𝑡[0,𝑇𝑑], 𝑑̂𝜌𝑡=𝑑𝑛𝑡̂𝜁𝑡𝑑̂𝑛𝑡.(4.24)
(ii) Assume that the equation 𝑍𝑍=𝜁 admits a 𝛽𝑋-integrable solution ̌𝜁 on 𝜏[0,𝑇𝑑]. Then the dynamics of the cost process ̌̌𝜌=𝜌(𝜁) are, for every 𝜏𝑡[0,𝑇𝑑], 𝑑𝜌𝑡=𝑌𝑡̌𝜁𝑡𝑌𝑡𝑑𝑁𝑑𝑡+𝑑𝑛𝑡̌𝜁𝑡𝑑̂𝑛𝑡.(4.25)

Part (i) in Lemma 4.12 corresponds to the case where the common risks (systematic and event) can be completely eliminated. In contrast, part (ii) refers either to the case of unhedgeable event risk (e.g., when 𝑌=0 in dynamics (4.17) or to the situation when the issuer (or holder) is not willing to hedge that risk.

As was already mentioned, practically useful decompositions of 𝑀 and 𝑀 will depend on a particular model for the primary market, as well as on the game option under study. In an abstract setup considered here, they may be formally deduced from martingale representation theorems with orthogonal components.

Let thus 2 stand for the class of real-valued 𝔽-martingales with integrable quadratic variation over [0,𝑇] or, by a slight abuse of notation, the class of vector-valued processes with mutually strongly orthogonal components in 2. It is worth recalling here that an 𝔽-martingale stopped at 𝜏𝑑 is also a 𝔾-local martingale, by virtue of Lemma A.1(iii).

The Galtchouk-Kunita-Watanabe (GKW) decomposition of 𝑀 and 𝑀 with respect to 𝑁 and the filtration 𝔽 (see, e.g., Protter [16, Section IV.3, Corollary  1]) thus yields the decompositions (4.19) of 𝑀 and 𝑀 with 𝑛 and ̂𝑛 strongly orthogonal to 𝑁 in 2. Since 𝑁 is meant to represent the systematic risk factor, we may and do assume, without loss of generality, that the idiosyncratic risk factors 𝑛 and ̂𝑛 are also mutually strongly orthogonal.

The following proposition justifies the informal statement that the strategy ̂𝜁 (resp., ̌𝜁) hedges the risk factor 𝒩 (resp., 𝑁). We use hereafter the standard symbol [,] to denote the square bracket between 𝔾-semimartingales.

Proposition 4.13 (Orthogonality of risk factors). Assume that the processes 𝑁,𝑛, and ̂𝑛 in decompositions (4.19) of 𝑀 and 𝑀 are mutually strongly orthogonal in 2.
(i) Under assumptions of Lemma 4.12(i), the processes ̂𝜌 and 𝒩𝜏𝑑 are orthogonal in 𝔾, in the sense that [̂𝜌,𝒩𝜏𝑑] is a 𝔾-sigma martingale (and a 𝔾-local martingale if ̂𝜁 is locally bounded).
(ii) Under assumptions of Lemma 4.12(ii), the processes ̌𝜌 and 𝑁𝜏𝑑 are orthogonal in 𝔾, in the sense that [̌𝜌,𝑁𝜏𝑑] is a 𝔾-sigma martingale (and a 𝔾-local martingale if ̌𝜁,𝑅, and are locally bounded processes).

Proof. We first note that 𝑛𝜏𝑑 and 𝑁𝜏𝑑 are 𝔾-local martingales, by Lemma A.1(iii). Since 𝑛 is strongly orthogonal to 𝑁 in 2, the process [𝑛𝜏𝑑,𝑁𝜏𝑑] is a 𝔾-local martingale, as an 𝔽-local martingale stopped at 𝜏𝑑 (cf. Lemma A.1(iii). Furthermore, by Lemma A.6, [𝑛𝜏𝑑,𝑁𝑑] is a 𝔾-local martingale. We conclude that [𝑛𝜏𝑑,𝒩𝜏𝑑] is a 𝔾-local martingale. So are also [̂𝑛,𝑁𝜏𝑑] and [̂𝑛,𝒩𝜏𝑑], since the integral 0𝛽𝑡𝑑̂𝑛𝑡 is strongly orthogonal to 𝒩𝜏𝑑. Furthermore, by Lemma A.6, [𝑁𝜏𝑑,𝑁𝑑] is a 𝔾-local martingale.
Using (4.24), we conclude for part (i) that [̂𝜌,𝒩𝜏𝑑] is a 𝔾-sigma martingale and thus it follows a 𝔾-local martingale if ̂𝜁 is a locally bounded process.
For part (ii), we conclude in view of (4.25) that [̌𝜌,𝑁𝜏𝑑] is a 𝔾-sigma martingale and thus a 𝔾-local martingale if ̌𝜁, 𝑅 and are locally bounded processes.

4.4. Hedging with Orthogonal Cost

Before concluding this work, let us examine briefly an alternative approach to hedging a defaultable game option, which is formally defined as the problem of finding a strategy 𝜁 that makes the cost process 𝔾-orthogonal under a given risk-neutral probability measure to a predetermined 𝑞-valued 𝔾-local martingale 𝑁 where, without loss of generality, the process 𝑁 is assumed to be stopped at 𝜏𝑑. In reference to Proposition 4.13, by the 𝔾-orthogonality, we mean here that [𝜌,𝑁] is a 𝔾-local martingale under .

Remark 4.14. In the financial interpretation, the process 𝑁 may represent the wealth processes of some preexisting portfolios, rather than risk factors as in Sections 4.2 and 4.3. Admittedly, we consider here a reduced concept of hedging, at least from the theoretical perspective. It is possible to argue, however, that this approach may be of practical relevance, since some kind of a relative hedging (as opposed to replication or superhedging) is a common market practice.

For the purpose of this section, the process 𝑚 arising in (4.1) may be defined either by (3.20), in reference to a solution of a related doubly reflected BSDE with respect to the filtration 𝔽 or, more generally (cf. Remark 3.15), as the 𝔾-local martingale component of the discounted cumulative -value process 𝛽Π of a game option, provided that 𝛽Π is a 𝔾-special semimartingale.

In the following proposition we denote (whenever well-defined) ov𝑡𝑑𝑋𝑡,𝑑𝑌𝑡=lim01𝑋ov𝑡+𝑋𝑡,𝑌𝑡+𝑌𝑡𝒢𝑡(4.26) and 𝕍ar𝑡(𝑑𝑋𝑡)=ov𝑡(𝑑𝑋𝑡,𝑑𝑋𝑡).

Proposition 4.15 (Hedging with orthogonal cost). Assume that 𝑋 admits the decomposition, for every 𝜏𝑡[0,𝑇𝑑], 𝑑𝛽𝑡𝑋𝑡=𝛽𝑡𝑍𝑡𝑑𝑁𝑡+𝛽𝑡𝑑𝑛𝑡(4.27) with 𝑛 and 𝑁 orthogonal in 𝔾 and an 𝑑𝑞-valued, 𝑁-integrable process 𝑍, which is left-invertible on 𝜏[0,𝑇𝑑]. Let us set, for every 𝜏𝑡[0,𝑇𝑑], 𝜁𝑡=ov𝑡𝑑𝑚𝑡,𝛽𝑡𝑑𝑁𝑡𝕍ar𝑡𝛽𝑡𝑑𝑁𝑡1Λ𝑡,(4.28) where Λ is the left inverse of the transpose of 𝑍 on 𝜏[0,𝑇𝑑]. Then the cost process 𝜌(𝜁) is orthogonal to 𝑁 in 𝔾.

Proof. By combining (4.1) with (4.27), we obtain 𝛽𝑡𝑑𝜌𝑡(𝜁)=𝑑𝑚𝑡𝜁𝑡𝛽𝑡𝑍𝑡𝑑𝑁𝑡𝛽𝑡𝜁𝑡𝑑𝑛𝑡.(4.29) Therefore, in order to have the cost 𝜌 orthogonal to 𝑁 in 𝔾, it suffices to select a strategy 𝜁 for which 𝑚0𝛽𝑡𝜁𝑡𝑍𝑡𝑑𝑁𝑡 is 𝔾-orthogonal to 𝑁. Relying on the multilinear regression formula, this can be achieved by setting