# Defaultable Game Options in a Hazard Process Model

Authors: Tomasz R. Bielecki; Stéphane Crépey; Monique Jeanblanc; Marek Rutkowski
Journal: Journal of Applied Mathematics and Stochastic Analysis
 (2009)

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## 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.


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## Body

## 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 ℚ 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 filtration𝔾 modeling the information flow for the primary market admits the representation 𝔾=ℍ∨𝔽, where the filtration ℍ is generated by the default indicator processHt=1{t≥τd} 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, Propositions4.1 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 ℚ, 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,ℚ)-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 thevector stochastic integral,  denoted as ∫HdX, as opposed to a more restricted notion of the component-wise stochastic integral,  which is defined as the sum ∑i=1d∫HidXi 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 ℝd-valued semimartingale integrator X and an ℝ1⊗d-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∫0t 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=τp⋀τc 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.’’


## 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 filtration𝔾 modeling the information flow for the primary market admits the representation 𝔾=ℍ∨𝔽, where the filtration ℍ is generated by the default indicator processHt=1{t≥τd} 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, Propositions4.1 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 ℚ, 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,ℚ)-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 thevector stochastic integral,  denoted as ∫HdX, as opposed to a more restricted notion of the component-wise stochastic integral,  which is defined as the sum ∑i=1d∫HidXi 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 ℝd-valued semimartingale integrator X and an ℝ1⊗d-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∫0t 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=τp⋀τc 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,ℙ), where ℙ denotes the statistical probability measure.Specifically, we consider a primary market composed of the savings account and ofd risky assets, such that, given a finite horizon date  T>0:(i)
thediscount factor  process β, that is, the inverse of the savings account, is a 𝔾-adapted, finite variation, positive, continuous and bounded process,(ii)
therisky assets  are 𝔾-semimartingales with càdlàg  sample paths.The primary risky assets, withℝd-valued price process X, pay dividends, whose cumulative value process, denoted by  𝒟, is assumed to be a 𝔾-adapted, càdlàg  and ℝd-valued process of finite variation. Given the price process X, we define the cumulative priceX̂ of primary risky assets as
(2.1)X̂t=Xt+βt-1∫[0,t]βud𝒟u.
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 measureℚ∈ℳ, where ℳ denotes the set of probability measures ℚ~ℙ for which βX̂  is a  sigma martingale  with respect to 𝔾 under ℚ (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, ifY 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 date0 and the maturity dateT, 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 thecumulative dividend  at time t denoted by Dt,(ii)
a terminalput paymentLt made at time t=τp if τp≤τc and τp<T; time τp is called the put time  and is chosen by the holder,(iii)
a terminalcall paymentUt made at time t=τc provided that τc<τp⋀T; time τc, known as the call time,  is chosen by the issuer and may be subject to the constraint that τc≥τ̅, where τ̅ is the lifting time of the call protection,(iv)
a terminalpayment at maturityξ made at maturity date T provided that T≤τp⋀τc.The (possibly random) timeτ̅ 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,τ̅).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 processHt=1{τd≤t}. 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 timet∈[0,T]. We write 𝒢Tt to denote the set of all 𝔾-stopping times with values in [t,T] and we let 𝒢̅Tt stand for the set {τ∈𝒢Tt;τ⋀τd≥τ̅⋀τd}, where the lifting time of a call protection τ̅ belongs to 𝒢T0.We are now in the position to state the formal definition of a defaultable game option.Definition 2.3.
Adefaultable game option with lifting time of the call protectionτ̅∈𝒢T0 is a game option with the ex-dividend cumulative discounted cash flowsβtπ(t;τp,τc) given by the formula, for any t∈[0,T] and (τp,τc)∈𝒢Tt×𝒢̅Tt,
(2.2)βtπ(t;τp,τc)=∫tτβudDu+1{τ<τd}βτ(1{τ=τp<T}Lτp+1{τ<τp}Uτc+1{τ=T}ξ),
where τ=τp⋀τc and  (i)
thedividend processD=(Dt)t∈[0,T] equals
(2.3)Dt=∫[0,t](1-Hu)dCu+∫[0,t]RudHu=Cτ-1{t≥τ}+Ct1{t<τ}+Rτ1{t≥τ},
for some coupon processC=(Ct)t∈[0,T], which is a 𝔾-predictable, real-valued, càdlàg  process with bounded variation, and some real-valued, 𝔾-predictable recovery processR=(Rt)t∈[0,T],(ii)
theput paymentL=(Lt)t∈[0,T] and the call paymentU=(Ut)t∈[0,T] are 𝔾-adapted, real-valued, càdlàg  processes,(iii)
the inequalityLt≤Ut holds for every t∈[τd⋀τ̅,τd⋀T),(iv)
thepayment at maturityξ is a 𝒢T-measurable, real-valued random variable.The following result easily follows from Definition2.3.Lemma 2.4.
(i) For anyt and (τp,τc)∈𝒢Tt×𝒢̅Tt, the random variable π(t;τp,τc) is 𝒢τ⋀τd-measurable.
(ii) For any(τp,τc)∈𝒢T0×𝒢̅T0, the processes π(0;·,τc) and π(0;τp,·) are 𝔾-adapted.We further assume thatR,L,  and ξ are bounded from below, so that there exists a constant c such that, for every t∈[0,T],
(2.4)βtℒ̂t:=∫[0,t]βudDu+1{t<τd}βt(1{t<T}Lt+1{t=T}ξ)≥-c.Symmetrically, we should sometimes additionally assume thatR,U,  and ξ are bounded (from below and from above), or that  (2.4) is supplemented by the inequality, for every t∈[0,T],
(2.5)βt𝒰̂t:=∫[0,t]βudDu+1{t<τd}βt(1{t<T}Ut+1{t=T}ξ)≤c.


### 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 processΠ is a 𝔾-semimartingale and there exists ℚ∈ℳ such that Π is the value of the Dynkin game related to a game option, meaning that
(2.6)esssupτp∈𝒢Ttessinfτc∈𝒢̅Tt𝔼ℚ(π(t;τp,τc)∣𝒢t)=Πt=essinfτc∈𝒢̅Ttesssupτp∈𝒢Tt𝔼ℚ(π(t;τp,τc)∣𝒢t),t∈[0,T].
Then Π is an arbitrage ex-dividend price of the game option, called the ℚ-price   of the game option. The converse holds true (thus any arbitrage price is a ℚ-price for some ℚ∈ℳ) under the following integrability assumption
(2.7)esssupℚ∈ℳ𝔼ℚ[supt∈[0,T](∫[0,t]βudDu+1{t<τd}βt(1{t<T}Lt+1{t=T}ξ))∣𝒢0]<∞,a.s.Note that defaultable American (or European) options can be seen as special cases of defaultable game options.Definition 2.6.
Adefaultable American option  is a defaultable game option with τ̅=T. A defaultable European option  is a defaultable American option such that the process βℒ̂ (cf.  (2.4) attains its maximum at T, that is, βtℒ̂t≤βTℒ̂T for every t∈[0,T].In view of Theorem2.5, the cash flows ϕ(t) of a defaultable European option can be redefined by
(2.8)βtϕ(t)=∫tTβudDu+1{τd>T}βTξ,t∈[0,T].


### 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 thatX (resp., X̂) 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 ℝ1⊗d-valued, βX̂-integrable process representing holdings in primary risky assets.Remark 2.8.
The reason why we do not assume that𝒢0 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.
Thewealth processV of a primary trading strategy (V0,ζ) is given by the formula, for t∈[0,T],
(2.9)βtVt=β0V0+∫0tζud(βuX̂u).Given the wealth processV of a primary strategy (V0,ζ), we uniquely specify a 𝔾-optional process ζ0 by setting
(2.10)Vt=ζt0βt-1+ζtXt,t∈[0,T].
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=τp⋀τc.Definition 2.10.
Consider the game option with the ex-dividend cumulative discounted cash flowsβπ given by  (2.2).
(i)
Anissuer hedge with cost processρ 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 processρ is a real-valued, càdlàg  𝔾-semimartingale with ρ0=0,(c)
a (fixed) call timeτc belongs to 𝒢̅T0,(d)
the following inequality is valid, for every put timeτp∈𝒢T0,
(2.11)βτVτ+∫0τβudρu≥β0π(0;τp,τc),a.s.(ii)
Aholder hedge with cost processρ 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 processρ is a real-valued, càdlàg  𝔾-semimartingale with ρ0=0,(c)
a (fixed) put timeτp belongs to 𝒢T0,(d)
the following inequality is valid, for every call timeτc∈𝒢̅T0,
(2.12)βτVτ+∫0τβudρu≥-β0π(0;τp,τc),a.s.Issuer or holder hedgesat 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)
(2.13)Vτ+βτ-1∫0τβudρu≥βτ-1∫0τβudDu+1{τ<τd}(1{τ=τp<T}Lτp+1{τ<τp}Uτc+1{τp=τc=T}ξ),a.s.
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 processρ is to be interpreted as the (running) financing cost, that is, the amount of cash added to (if dρt≥0) or withdrawn from (if dρt≤0) 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 ℚ we thus recover the notion of mean self-financing  hedge, in the sense of Schweizer [24].
(ii) Regarding theadmissibility 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 a𝔾-sigma martingale  cost ρ under a particular risk-neutral measure ℚ.Assumption 2.12.
In the sequel, we work under a fixed, but arbitrary, risk-neutral measureℚ∈ℳ.All the measure-dependent notions like(local) martingale  and  compensator,  implicitly refer to the probability measure ℚ. In particular, we define 𝒱0c (resp., 𝒱0p) 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 ℚ.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 Proposition4.3, 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∅=∞)
(2.14)essinfτc∈𝒢̅T0esssupτp∈𝒢T0𝔼ℚ(π(0;τp,τc)∣𝒢0)≤essinfV0∈𝒱0cV0,a.s.
(ii) If inequality  (2.5) is valid then
(2.15)esssupτp∈𝒢T0essinfτc∈𝒢̅T0𝔼ℚ(π(0;τp,τc)∣𝒢0)≥-essinfV0∈𝒱0pV0,a.s.Proof.
(i) Assume that for some stopping timeτ̅c∈𝒢̅T0 the quadruplet (V0,ζ,ρ,τ̅c) 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 τp∈𝒢T0,
(2.16)β0V0=βτp⋀τ̅cVτp⋀τ̅c-∫0τp⋀τ̅cζud(βuX̂u)≥β0π(0;τp,τ̅c)-∫0τp⋀τ̅c(ζud(βuX̂u)+βudρu).
In particular, by taking τp=t, we obtain that, for any t∈[0,T],
(2.17)β0V0=βt⋀τ̅cVt⋀τ̅c-∫0t⋀τ̅cζud(βuX̂u)≥β0π(0;t,τ̅c)-∫0t⋀τ̅c(ζud(βuX̂u)+βudρu).
The stochastic integral ∫0tζud(βuX̂u) with respect to a 𝔾-sigma martingale βX̂ is a 𝔾-sigma martingale. Hence the stopped process ∫0t⋀τ̅cζud(βuX̂u), as well as the process
(2.18)∫0t⋀τ̅c(ζud(βuX̂u)+βudρu)
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 τp∈𝒢T0 (recall that τ̅c is fixed)
(2.19)β0V0≥𝔼ℚ(β0π(0;τp,τ̅c)∣𝒢0),∀τp∈𝒢T0,
and thus, by the assumed positivity of the process β,(2.20)V0≥essinfτc∈𝒢̅T0esssupτp∈𝒢T0𝔼ℚ(π(0;τp,τc)∣𝒢0),a.s.
The required inequality (2.14) is an immediate consequence of the last formula.
(ii) Let(V0,ζ,ρ,τ̅p) be a holder hedge with a 𝔾-sigma martingale cost ρ for the game option for some stopping time τ̅p∈𝒢T0. Then  (2.9) and  (2.12) imply that, for any t∈[τ̅,T],
(2.21)β0V0=βt⋀τ̅pVt⋀τ̅p-∫0t⋀τ̅pζud(βuX̂u)≥-β0π(0;τ̅p,t)-∫0t⋀τ̅p(ζud(βuX̂u)+βudρu).
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 τ̅p∈𝒢T0, we obtain
(2.22)β0V0≥𝔼ℚ(-β0π(0;τ̅p,τc)∣𝒢0),a.s.,∀τc∈𝒢̅T0,
so that
(2.23)V0≥-esssupτp∈𝒢T0essinfτc∈𝒢̅T0𝔼ℚ(π(0;τp,τc)∣𝒢0),a.s.,
and this in turn implies (2.15).


## 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 processΠ is a 𝔾-semimartingale and there exists ℚ∈ℳ such that Π is the value of the Dynkin game related to a game option, meaning that
(2.6)esssupτp∈𝒢Ttessinfτc∈𝒢̅Tt𝔼ℚ(π(t;τp,τc)∣𝒢t)=Πt=essinfτc∈𝒢̅Ttesssupτp∈𝒢Tt𝔼ℚ(π(t;τp,τc)∣𝒢t),t∈[0,T].
Then Π is an arbitrage ex-dividend price of the game option, called the ℚ-price   of the game option. The converse holds true (thus any arbitrage price is a ℚ-price for some ℚ∈ℳ) under the following integrability assumption
(2.7)esssupℚ∈ℳ𝔼ℚ[supt∈[0,T](∫[0,t]βudDu+1{t<τd}βt(1{t<T}Lt+1{t=T}ξ))∣𝒢0]<∞,a.s.Note that defaultable American (or European) options can be seen as special cases of defaultable game options.Definition 2.6.
Adefaultable American option  is a defaultable game option with τ̅=T. A defaultable European option  is a defaultable American option such that the process βℒ̂ (cf.  (2.4) attains its maximum at T, that is, βtℒ̂t≤βTℒ̂T for every t∈[0,T].In view of Theorem2.5, the cash flows ϕ(t) of a defaultable European option can be redefined by
(2.8)βtϕ(t)=∫tTβudDu+1{τd>T}βTξ,t∈[0,T].


## 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 thatX (resp., X̂) 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 ℝ1⊗d-valued, βX̂-integrable process representing holdings in primary risky assets.Remark 2.8.
The reason why we do not assume that𝒢0 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.
Thewealth processV of a primary trading strategy (V0,ζ) is given by the formula, for t∈[0,T],
(2.9)βtVt=β0V0+∫0tζud(βuX̂u).Given the wealth processV of a primary strategy (V0,ζ), we uniquely specify a 𝔾-optional process ζ0 by setting
(2.10)Vt=ζt0βt-1+ζtXt,t∈[0,T].
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=τp⋀τc.Definition 2.10.
Consider the game option with the ex-dividend cumulative discounted cash flowsβπ given by  (2.2).
(i)
Anissuer hedge with cost processρ 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 processρ is a real-valued, càdlàg  𝔾-semimartingale with ρ0=0,(c)
a (fixed) call timeτc belongs to 𝒢̅T0,(d)
the following inequality is valid, for every put timeτp∈𝒢T0,
(2.11)βτVτ+∫0τβudρu≥β0π(0;τp,τc),a.s.(ii)
Aholder hedge with cost processρ 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 processρ is a real-valued, càdlàg  𝔾-semimartingale with ρ0=0,(c)
a (fixed) put timeτp belongs to 𝒢T0,(d)
the following inequality is valid, for every call timeτc∈𝒢̅T0,
(2.12)βτVτ+∫0τβudρu≥-β0π(0;τp,τc),a.s.Issuer or holder hedgesat 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)
(2.13)Vτ+βτ-1∫0τβudρu≥βτ-1∫0τβudDu+1{τ<τd}(1{τ=τp<T}Lτp+1{τ<τp}Uτc+1{τp=τc=T}ξ),a.s.
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 processρ is to be interpreted as the (running) financing cost, that is, the amount of cash added to (if dρt≥0) or withdrawn from (if dρt≤0) 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 ℚ we thus recover the notion of mean self-financing  hedge, in the sense of Schweizer [24].
(ii) Regarding theadmissibility 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 a𝔾-sigma martingale  cost ρ under a particular risk-neutral measure ℚ.Assumption 2.12.
In the sequel, we work under a fixed, but arbitrary, risk-neutral measureℚ∈ℳ.All the measure-dependent notions like(local) martingale  and  compensator,  implicitly refer to the probability measure ℚ. In particular, we define 𝒱0c (resp., 𝒱0p) 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 ℚ.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 Proposition4.3, 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∅=∞)
(2.14)essinfτc∈𝒢̅T0esssupτp∈𝒢T0𝔼ℚ(π(0;τp,τc)∣𝒢0)≤essinfV0∈𝒱0cV0,a.s.
(ii) If inequality  (2.5) is valid then
(2.15)esssupτp∈𝒢T0essinfτc∈𝒢̅T0𝔼ℚ(π(0;τp,τc)∣𝒢0)≥-essinfV0∈𝒱0pV0,a.s.Proof.
(i) Assume that for some stopping timeτ̅c∈𝒢̅T0 the quadruplet (V0,ζ,ρ,τ̅c) 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 τp∈𝒢T0,
(2.16)β0V0=βτp⋀τ̅cVτp⋀τ̅c-∫0τp⋀τ̅cζud(βuX̂u)≥β0π(0;τp,τ̅c)-∫0τp⋀τ̅c(ζud(βuX̂u)+βudρu).
In particular, by taking τp=t, we obtain that, for any t∈[0,T],
(2.17)β0V0=βt⋀τ̅cVt⋀τ̅c-∫0t⋀τ̅cζud(βuX̂u)≥β0π(0;t,τ̅c)-∫0t⋀τ̅c(ζud(βuX̂u)+βudρu).
The stochastic integral ∫0tζud(βuX̂u) with respect to a 𝔾-sigma martingale βX̂ is a 𝔾-sigma martingale. Hence the stopped process ∫0t⋀τ̅cζud(βuX̂u), as well as the process
(2.18)∫0t⋀τ̅c(ζud(βuX̂u)+βudρu)
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 τp∈𝒢T0 (recall that τ̅c is fixed)
(2.19)β0V0≥𝔼ℚ(β0π(0;τp,τ̅c)∣𝒢0),∀τp∈𝒢T0,
and thus, by the assumed positivity of the process β,(2.20)V0≥essinfτc∈𝒢̅T0esssupτp∈𝒢T0𝔼ℚ(π(0;τp,τc)∣𝒢0),a.s.
The required inequality (2.14) is an immediate consequence of the last formula.
(ii) Let(V0,ζ,ρ,τ̅p) be a holder hedge with a 𝔾-sigma martingale cost ρ for the game option for some stopping time τ̅p∈𝒢T0. Then  (2.9) and  (2.12) imply that, for any t∈[τ̅,T],
(2.21)β0V0=βt⋀τ̅pVt⋀τ̅p-∫0t⋀τ̅pζud(βuX̂u)≥-β0π(0;τ̅p,t)-∫0t⋀τ̅p(ζud(βuX̂u)+βudρu).
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 τ̅p∈𝒢T0, we obtain
(2.22)β0V0≥𝔼ℚ(-β0π(0;τ̅p,τc)∣𝒢0),a.s.,∀τc∈𝒢̅T0,
so that
(2.23)V0≥-esssupτp∈𝒢T0essinfτc∈𝒢̅T0𝔼ℚ(π(0;τp,τc)∣𝒢0),a.s.,
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-calledhazard process setup.


### 3.1. Standing Assumptions

Given an[0,+∞]-valued 𝔾-stopping time τd, we assume that 𝔾=ℍ∨𝔽, where the filtration ℍ is generated by the process Ht=1{τd≤t} 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.LetG stand for the process Gt=ℚ(τd>t∣ℱt) for t∈ℝ+. The process G is a (bounded) 𝔽-supermartingale, as the optional projection on the filtration 𝔽 of the nonincreasing process 1-H (see Jeulin [25]).In the sequel, we will work under the following standing assumption.Assumption 3.1.
We assume that the processG is (strictly) positive and continuous with finite variation, so that the 𝔽-hazard processΓt=-ln(Gt),t∈ℝ+, is well defined and continuous with finite variation.Remark 3.2.
(i) The assumption thatG is continuous implies that τd is a totally inaccessible𝔾-stopping time (see, e.g., [26]). Moreover, τdavoids𝔽-stopping times,  in the sense that ℚ(τd=τ)=0 for any 𝔽-stopping time τ (see Coculescu and Nikeghbali [27]).
(ii) IfG 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) (ℋ) Hypothesis and assuming further that τd is an 𝔽-pseudo-stopping time  (see Nikeghbali and Yor [28]). Recall that the (ℋ) 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 atτd are 𝔾-local martingales (see Nikeghbali and Yor [28] and the appendix).Some consequences of Assumption3.1 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.
The𝔽-stopping time τ̃, the ℱt-measurable random variable χ̃ and the 𝔽-adapted or 𝔽-predictable process Ỹ 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 Assumption3.4. 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 processβ is 𝔽-adapted.
(ii) The coupon processC is 𝔽-predictable.
(iii) The recovery processR is 𝔽-predictable.
(iv) The payoff processesL,U are 𝔽-adapted and the random variable ξ is ℱT-measurable.
(v) The call protectionτ̅ 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(3.1)esssupτp∈𝒢Ttessinfτc∈𝒢̅Tt𝔼ℚ(π(t;τp,τc)∣𝒢t)=esssupτp∈ℱTtessinfτc∈ℱ̅Tt𝔼ℚ(π(t;τp,τc)∣𝒢t),essinfτc∈𝒢̅Ttesssupτp∈𝒢Tt𝔼ℚ(π(t;τp,τc)∣𝒢t)=essinfτc∈ℱ̅Ttesssupτp∈ℱTt𝔼ℚ(π(t;τp,τc)∣𝒢t).Proof.
For(τp,τc)∈𝒢Tt×𝒢̅Tt, one has that
(3.2)π(t;τp,τc)=π(t;τp⋀τd,τc⋀τd)=π(t;τ̃p⋀τd,τ̃c⋀τd)=π(t;τ̃p,τ̃c)
for some stopping times (τ̃p,τ̃c)∈ℱTt×ℱ̅Tt, where the middle equality follows from Lemma A.4, and the other two from the definition of π. Since, clearly, ℱTt⊆𝒢Tt and ℱ̅Tt⊆𝒢̅Tt, we conclude that the lemma is valid.Under our assumptions, the computation of conditional expectations of cash flowsπ(t;τp,τc) with respect to 𝒢t can be reduced to the computation of conditional expectations of 𝔽-equivalent cash flows π̃(t;τp,τc) with respect to ℱt. 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 timesτp∈ℱTt and τc∈ℱ̅Tt one has that
(3.3)𝔼ℚ(π(t;τp,τc)∣𝒢t)=1{t<τd}𝔼ℚ(π̃(t;τp,τc)∣ℱt),
where π̃(t;τp,τc) is given by, with τ=τp⋀τc,
(3.4)αtπ̃(t;τp,τc)=∫tταu(dCu+RudΓu)+ατ(1{τ=τp<T}Lτp+1{τ<τp}Uτc+1{τ=T}ξ).Proof.
Formula  (3.3) is an immediate consequence of formula  (2.2) and Lemma A.5.Note thatπ̃(t;τp,τc) is an ℱτ-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 𝒢τ⋀τd-measurable random variables, whereas the 𝔽-equivalent cash flows π̃(t;τp,τc) are manifestly ℱτ-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 ℚ-price for a game option. Then one has, for any t∈[0,T],
(3.5)Πt=1{t<τd}Π̃t,
where Π̃t satisfies
(3.6)esssupτp∈ℱTtessinfτc∈ℱ̅Tt𝔼ℚ(π̃(t;τp,τc)∣ℱt)=Π̃t=essinfτc∈ℱ̅Ttesssupτp∈ℱTt𝔼ℚ(π̃(t;τp,τc)∣ℱt).
Hence the Dynkin game with cost criterion 𝔼ℚ(π̃(t;τp,τc)∣ℱt) on ℱTt×ℱ̅Tt admits the value Π̃t, which coincides with the pre-default ex-dividend price at time t of the game option under the risk-neutral measure ℚ.The following result is the converse of Theorem3.7. 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 Π̃ is a 𝔾-semimartingale).Theorem 3.8.
LetΠ̃t be the value of the Dynkin game with the cost criterion 𝔼ℚ(π̃(t;τp,τc)∣ℱt) on ℱTt×ℱ̅Tt, for any t∈[0,T]. Then Πt defined by  (3.5) is the value of the Dynkin game with the cost criterion 𝔼ℚ(π(t;τp,τc)∣𝒢t) on 𝒢Tt×𝒢̅Tt, for any t∈[0,T]. If, in addition, Π̃ is a 𝔾-semimartingale then Π is the arbitrage ex-dividend ℚ-price for the game option.Theorems3.7 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-dividendℚ-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 additional𝔽-adapted process F of finite variation, we consider the following doubly reflected BSDE with the dataα,F,ξ,L,U,τ̅  (see Cvitanić and Karatzas [30], Hamadène and Hassani [31], Crépey [32], Crépey and Matoussi [33], Bielecki et al. [4, 9]):
(3.7)αtΘt=αTξ+αTFT-αtFt+∫tTαudKu-∫tTαudMu,t∈[0,T],Lt≤Θt≤U̅t,t∈[0,T],∫0T(Θu-Lu)dKu+=∫0T(U̅u-Θu)dKu-=0,
where the process U̅=(U̅t)t∈[0,T] equals, for t∈[0,T],
(3.8)U̅t=1{t<τ̅}∞+1{t≥τ̅}Ut.Definition 3.9.
By a (ℚ-)solution to the doubly reflected BSDE  (3.7), we mean a triplet (Θ,M,K) such that  (i)
thestate processΘ is a real-valued, 𝔽-adapted, càdlàg  process,(ii)
∫0·αdM 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 theJordan 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 processΘ 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 τ̅=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 ℝq-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, 30–33].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 L≤Z≤U 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∨ℓ, where S is an Itô-Lévy process S with square-integrable special semimartingale decomposition components (see [33]) and ℓ is a constant in ℝ∪{-∞}. This framework covers, for instance, the payoff at call of a convertible bond examined in [3, 4].Remark 3.11.
(i) SinceK, and thus K+ and K-, are continuous, the minimality conditions (third line) in  (3.7) are equivalent to
(3.9)∫0T(Θu--Lu-)dKu+=∫0T(U̅u--Θu-)dKu-=0.
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) SinceF is a given process, the BSDE (3.7) can be rewritten as
(3.10)αtΘ̂t=αTξ̂+∫tTαudKu-∫tTαudMu,t∈[0,T],L̂t≤Θ̂t≤Ût,t∈[0,T],∫0T(Θ̂u-L̂u)dKu+=∫0T(Ûu-Θ̂u)dKu-=0,
where Θ̂t=Θt+Ft,ξ̂=ξ+FT,L̂t=Lt+Ft, and Ût=U̅t+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 L̂,Û and terminal condition ξ̂. 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 all𝔽-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 analyticvariational 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-dividendℚ-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 τ∈ℱTt,
(3.11)αtΘt=ατΘτ+ατFτ-αtFt+∫tταudKu-∫tταudMu.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 𝔼ℚ(θ(t;τp,τc)∣ℱt) on ℱTt×ℱ̅Tt, where θ(t;τp,τc) is the ℱτ-measurable random variable defined by
(3.12)αtθ(t;τp,τc)=ατ(1{τ=τp<T}Lτp+1{τ=τc<τp}Uτc+1{τ=T}ξ),
where τ=τp⋀τc. Moreover, for any t∈[0,T], the pair of stopping times (τp*,τc*)∈ℱTt×ℱ̅Tt given by
(3.13)τp*=inf{u∈[t,T];Θu≤Lu}⋀T,τc*=inf{u∈[τ̅∨t,T];Θu≥Uu}⋀T,
is a saddle-point of this Dynkin game, in the sense that one has, for any (τp,τc)∈ℱTt×ℱ̅Tt,
(3.14)𝔼ℚ(θ(t;τp,τc*)∣ℱt)≤Θt≤𝔼ℚ(θ(t;τp*,τc)∣ℱt).Proof.
Except for the presence ofτ̅, 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 τc∈ℱ̅Tt. Let τ*denote τp*⋀τc.  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
(3.15)αtΘt≤ατ*Θτ*-∫tτ*αudMu.
Taking conditional expectations (recall that ∫t·αudMu is an 𝔽-martingale), and using also the facts that Θτp*≤Lτp* if τp*<T,Θτp*=ξ if τp*=T and Θτc≤Uτc (recall that τc∈ℱ̅Tt, so that τc≥τ̅ and U̅τc=Uτc), we obtain
(3.16)αtΘt≤𝔼ℚ(ατ*Θτ*∣ℱt)≤𝔼ℚ(ατ*(1{τ*=τp*<T}Lτp*+1{τ*=τc<τp*}Uτc+1{τ*=T}ξ)∣ℱt).
We conclude that Θt≤𝔼ℚ(θ(t;τp*,τc)∣ℱt) for any τc∈ℱ̅Tt. 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 τ̅, and thus the details are left to the reader.Let us now apply Proposition3.12 to a defaultable game option. To this end, we first rewrite (3.4) as follows
(3.17)αtπ̃(t;τp,τc)=ατF̅τ-αtF̅t+ατ(1{τ=τp<T}Lτp+1{τ<τp}Uτc+1{τ=T}ξ),
where
(3.18)F̅t:=αt-1∫[0,t]αudD̅uwithD̅t:=∫[0,t]dCu+RudΓu.
Let us denote by (ℰ̅) equation  (3.10) with Ft=F̅t, that is,
(ℰ̅)αtΘ̅t=αTξ̂+∫tTαudKu-∫tTαudMu,t∈[0,T],L̂t≤Θ̅t≤Ût,t∈[0,T],∫0T(Θ̅u-L̂u)dKu+=∫0T(Ûu-Θ̅u)dKu-=0,
with ξ̂=ξ+F̅T,L̂t=Lt+F̅t, and Ût=U̅t+F̅t.Assumption 3.13.
The doubly reflected BSDE (ℰ̅) admits a solution (Θ̅,M,K).Let us stress that Assumption3.13, 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, fort∈[0,T],
(3.19)Π̃t=Θ̅t-F̅t,Πt=1{t<τd}Π̃t,Π̂t=Πt+βt-1∫[0,t]βudDu,(3.20)mt=βtΠ̂t+∫0t⋀τdβudKu.
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 processm given by  (3.20) is 𝔾-martingale stopped at τd.
(ii) The processΠ is a 𝔾-semimartingale.
(iii) The processβΠ̂ is a special 𝔾-semimartingale.Proof.
(i) The triplet(Π̃,M,K) satisfies  (3.7) with F given by F̅ in  (3.18). Therefore, for every t∈[0,T],
(3.21)αtΠ̃t=αTξ+∫tTαudD̅u+∫tTαudKu-∫tTαudMu
and thus
(3.22)∫0tαudMu=αtΠ̃t-α0Π̃0+∫0tαudKu+∫0tαudD̅u.
Using Lemma A.5, it is easy to check that one has, for any 0≤t≤u≤T,
(3.23)1{t<τd}eΓt𝔼ℚ(∫tuαvdMv∣ℱt)=𝔼ℚ(mu-mt∣𝒢t).
Since the integral ∫t·αvdMv 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
(3.24)βtΠ̂t=mt-∫0t⋀τdβudKu,
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  canonical𝔾-local martingale component  of the  discounted cumulativeℚ-value processβΠ̂.  The processes m and βΠ̂ are easily seen to coincide on the random interval [0,τc*⋀τp*⋀τd⋀T]. Therefore, both m and βΠ̂ can be interpreted on this interval as the discounted cumulative ℚ-value of a defaultable game option.The following result establishes a useful connection between(Θ̅,M,K) and the arbitrage ex-dividend ℚ-price of the defaultable game option.Proposition 3.16 (Verification principle for a defaultable game option).
The processΠ is the arbitrage ex-dividend ℚ-price for the game option. Moreover, for any t∈[0,T], the saddle-point (τp*,τc*)∈ℱTt×ℱ̅Tt for the related Dynkin game  (2.6) on 𝒢Tt×𝒢̅Tt is given by
(3.25)τp*=inf{u∈[t,T];Π̃u≤Lu}⋀T,τc*=inf{u∈[τ̅∨t,T];Π̃u≥Uu}⋀T.Proof.
In view of  (3.4), the present assumptions imply that Π̃t 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×𝒢̅Tt, 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.


## 3.1. Standing Assumptions

Given an[0,+∞]-valued 𝔾-stopping time τd, we assume that 𝔾=ℍ∨𝔽, where the filtration ℍ is generated by the process Ht=1{τd≤t} 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.LetG stand for the process Gt=ℚ(τd>t∣ℱt) for t∈ℝ+. The process G is a (bounded) 𝔽-supermartingale, as the optional projection on the filtration 𝔽 of the nonincreasing process 1-H (see Jeulin [25]).In the sequel, we will work under the following standing assumption.Assumption 3.1.
We assume that the processG is (strictly) positive and continuous with finite variation, so that the 𝔽-hazard processΓt=-ln(Gt),t∈ℝ+, is well defined and continuous with finite variation.Remark 3.2.
(i) The assumption thatG is continuous implies that τd is a totally inaccessible𝔾-stopping time (see, e.g., [26]). Moreover, τdavoids𝔽-stopping times,  in the sense that ℚ(τd=τ)=0 for any 𝔽-stopping time τ (see Coculescu and Nikeghbali [27]).
(ii) IfG 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) (ℋ) Hypothesis and assuming further that τd is an 𝔽-pseudo-stopping time  (see Nikeghbali and Yor [28]). Recall that the (ℋ) 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 atτd are 𝔾-local martingales (see Nikeghbali and Yor [28] and the appendix).Some consequences of Assumption3.1 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.
The𝔽-stopping time τ̃, the ℱt-measurable random variable χ̃ and the 𝔽-adapted or 𝔽-predictable process Ỹ 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 Assumption3.4. 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 processβ is 𝔽-adapted.
(ii) The coupon processC is 𝔽-predictable.
(iii) The recovery processR is 𝔽-predictable.
(iv) The payoff processesL,U are 𝔽-adapted and the random variable ξ is ℱT-measurable.
(v) The call protectionτ̅ 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(3.1)esssupτp∈𝒢Ttessinfτc∈𝒢̅Tt𝔼ℚ(π(t;τp,τc)∣𝒢t)=esssupτp∈ℱTtessinfτc∈ℱ̅Tt𝔼ℚ(π(t;τp,τc)∣𝒢t),essinfτc∈𝒢̅Ttesssupτp∈𝒢Tt𝔼ℚ(π(t;τp,τc)∣𝒢t)=essinfτc∈ℱ̅Ttesssupτp∈ℱTt𝔼ℚ(π(t;τp,τc)∣𝒢t).Proof.
For(τp,τc)∈𝒢Tt×𝒢̅Tt, one has that
(3.2)π(t;τp,τc)=π(t;τp⋀τd,τc⋀τd)=π(t;τ̃p⋀τd,τ̃c⋀τd)=π(t;τ̃p,τ̃c)
for some stopping times (τ̃p,τ̃c)∈ℱTt×ℱ̅Tt, where the middle equality follows from Lemma A.4, and the other two from the definition of π. Since, clearly, ℱTt⊆𝒢Tt and ℱ̅Tt⊆𝒢̅Tt, we conclude that the lemma is valid.Under our assumptions, the computation of conditional expectations of cash flowsπ(t;τp,τc) with respect to 𝒢t can be reduced to the computation of conditional expectations of 𝔽-equivalent cash flows π̃(t;τp,τc) with respect to ℱt. 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 timesτp∈ℱTt and τc∈ℱ̅Tt one has that
(3.3)𝔼ℚ(π(t;τp,τc)∣𝒢t)=1{t<τd}𝔼ℚ(π̃(t;τp,τc)∣ℱt),
where π̃(t;τp,τc) is given by, with τ=τp⋀τc,
(3.4)αtπ̃(t;τp,τc)=∫tταu(dCu+RudΓu)+ατ(1{τ=τp<T}Lτp+1{τ<τp}Uτc+1{τ=T}ξ).Proof.
Formula  (3.3) is an immediate consequence of formula  (2.2) and Lemma A.5.Note thatπ̃(t;τp,τc) is an ℱτ-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 𝒢τ⋀τd-measurable random variables, whereas the 𝔽-equivalent cash flows π̃(t;τp,τc) are manifestly ℱτ-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 ℚ-price for a game option. Then one has, for any t∈[0,T],
(3.5)Πt=1{t<τd}Π̃t,
where Π̃t satisfies
(3.6)esssupτp∈ℱTtessinfτc∈ℱ̅Tt𝔼ℚ(π̃(t;τp,τc)∣ℱt)=Π̃t=essinfτc∈ℱ̅Ttesssupτp∈ℱTt𝔼ℚ(π̃(t;τp,τc)∣ℱt).
Hence the Dynkin game with cost criterion 𝔼ℚ(π̃(t;τp,τc)∣ℱt) on ℱTt×ℱ̅Tt admits the value Π̃t, which coincides with the pre-default ex-dividend price at time t of the game option under the risk-neutral measure ℚ.The following result is the converse of Theorem3.7. 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 Π̃ is a 𝔾-semimartingale).Theorem 3.8.
LetΠ̃t be the value of the Dynkin game with the cost criterion 𝔼ℚ(π̃(t;τp,τc)∣ℱt) on ℱTt×ℱ̅Tt, for any t∈[0,T]. Then Πt defined by  (3.5) is the value of the Dynkin game with the cost criterion 𝔼ℚ(π(t;τp,τc)∣𝒢t) on 𝒢Tt×𝒢̅Tt, for any t∈[0,T]. If, in addition, Π̃ is a 𝔾-semimartingale then Π is the arbitrage ex-dividend ℚ-price for the game option.Theorems3.7 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-dividendℚ-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 additional𝔽-adapted process F of finite variation, we consider the following doubly reflected BSDE with the dataα,F,ξ,L,U,τ̅  (see Cvitanić and Karatzas [30], Hamadène and Hassani [31], Crépey [32], Crépey and Matoussi [33], Bielecki et al. [4, 9]):
(3.7)αtΘt=αTξ+αTFT-αtFt+∫tTαudKu-∫tTαudMu,t∈[0,T],Lt≤Θt≤U̅t,t∈[0,T],∫0T(Θu-Lu)dKu+=∫0T(U̅u-Θu)dKu-=0,
where the process U̅=(U̅t)t∈[0,T] equals, for t∈[0,T],
(3.8)U̅t=1{t<τ̅}∞+1{t≥τ̅}Ut.Definition 3.9.
By a (ℚ-)solution to the doubly reflected BSDE  (3.7), we mean a triplet (Θ,M,K) such that  (i)
thestate processΘ is a real-valued, 𝔽-adapted, càdlàg  process,(ii)
∫0·αdM 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 theJordan 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 processΘ 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 τ̅=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 ℝq-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, 30–33].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 L≤Z≤U 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∨ℓ, where S is an Itô-Lévy process S with square-integrable special semimartingale decomposition components (see [33]) and ℓ is a constant in ℝ∪{-∞}. This framework covers, for instance, the payoff at call of a convertible bond examined in [3, 4].Remark 3.11.
(i) SinceK, and thus K+ and K-, are continuous, the minimality conditions (third line) in  (3.7) are equivalent to
(3.9)∫0T(Θu--Lu-)dKu+=∫0T(U̅u--Θu-)dKu-=0.
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) SinceF is a given process, the BSDE (3.7) can be rewritten as
(3.10)αtΘ̂t=αTξ̂+∫tTαudKu-∫tTαudMu,t∈[0,T],L̂t≤Θ̂t≤Ût,t∈[0,T],∫0T(Θ̂u-L̂u)dKu+=∫0T(Ûu-Θ̂u)dKu-=0,
where Θ̂t=Θt+Ft,ξ̂=ξ+FT,L̂t=Lt+Ft, and Ût=U̅t+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 L̂,Û and terminal condition ξ̂. 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 all𝔽-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 analyticvariational 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-dividendℚ-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 τ∈ℱTt,
(3.11)αtΘt=ατΘτ+ατFτ-αtFt+∫tταudKu-∫tταudMu.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 𝔼ℚ(θ(t;τp,τc)∣ℱt) on ℱTt×ℱ̅Tt, where θ(t;τp,τc) is the ℱτ-measurable random variable defined by
(3.12)αtθ(t;τp,τc)=ατ(1{τ=τp<T}Lτp+1{τ=τc<τp}Uτc+1{τ=T}ξ),
where τ=τp⋀τc. Moreover, for any t∈[0,T], the pair of stopping times (τp*,τc*)∈ℱTt×ℱ̅Tt given by
(3.13)τp*=inf{u∈[t,T];Θu≤Lu}⋀T,τc*=inf{u∈[τ̅∨t,T];Θu≥Uu}⋀T,
is a saddle-point of this Dynkin game, in the sense that one has, for any (τp,τc)∈ℱTt×ℱ̅Tt,
(3.14)𝔼ℚ(θ(t;τp,τc*)∣ℱt)≤Θt≤𝔼ℚ(θ(t;τp*,τc)∣ℱt).Proof.
Except for the presence ofτ̅, 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 τc∈ℱ̅Tt. Let τ*denote τp*⋀τc.  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
(3.15)αtΘt≤ατ*Θτ*-∫tτ*αudMu.
Taking conditional expectations (recall that ∫t·αudMu is an 𝔽-martingale), and using also the facts that Θτp*≤Lτp* if τp*<T,Θτp*=ξ if τp*=T and Θτc≤Uτc (recall that τc∈ℱ̅Tt, so that τc≥τ̅ and U̅τc=Uτc), we obtain
(3.16)αtΘt≤𝔼ℚ(ατ*Θτ*∣ℱt)≤𝔼ℚ(ατ*(1{τ*=τp*<T}Lτp*+1{τ*=τc<τp*}Uτc+1{τ*=T}ξ)∣ℱt).
We conclude that Θt≤𝔼ℚ(θ(t;τp*,τc)∣ℱt) for any τc∈ℱ̅Tt. 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 τ̅, and thus the details are left to the reader.Let us now apply Proposition3.12 to a defaultable game option. To this end, we first rewrite (3.4) as follows
(3.17)αtπ̃(t;τp,τc)=ατF̅τ-αtF̅t+ατ(1{τ=τp<T}Lτp+1{τ<τp}Uτc+1{τ=T}ξ),
where
(3.18)F̅t:=αt-1∫[0,t]αudD̅uwithD̅t:=∫[0,t]dCu+RudΓu.
Let us denote by (ℰ̅) equation  (3.10) with Ft=F̅t, that is,
(ℰ̅)αtΘ̅t=αTξ̂+∫tTαudKu-∫tTαudMu,t∈[0,T],L̂t≤Θ̅t≤Ût,t∈[0,T],∫0T(Θ̅u-L̂u)dKu+=∫0T(Ûu-Θ̅u)dKu-=0,
with ξ̂=ξ+F̅T,L̂t=Lt+F̅t, and Ût=U̅t+F̅t.Assumption 3.13.
The doubly reflected BSDE (ℰ̅) admits a solution (Θ̅,M,K).Let us stress that Assumption3.13, 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, fort∈[0,T],
(3.19)Π̃t=Θ̅t-F̅t,Πt=1{t<τd}Π̃t,Π̂t=Πt+βt-1∫[0,t]βudDu,(3.20)mt=βtΠ̂t+∫0t⋀τdβudKu.
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 processm given by  (3.20) is 𝔾-martingale stopped at τd.
(ii) The processΠ is a 𝔾-semimartingale.
(iii) The processβΠ̂ is a special 𝔾-semimartingale.Proof.
(i) The triplet(Π̃,M,K) satisfies  (3.7) with F given by F̅ in  (3.18). Therefore, for every t∈[0,T],
(3.21)αtΠ̃t=αTξ+∫tTαudD̅u+∫tTαudKu-∫tTαudMu
and thus
(3.22)∫0tαudMu=αtΠ̃t-α0Π̃0+∫0tαudKu+∫0tαudD̅u.
Using Lemma A.5, it is easy to check that one has, for any 0≤t≤u≤T,
(3.23)1{t<τd}eΓt𝔼ℚ(∫tuαvdMv∣ℱt)=𝔼ℚ(mu-mt∣𝒢t).
Since the integral ∫t·αvdMv 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
(3.24)βtΠ̂t=mt-∫0t⋀τdβudKu,
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  canonical𝔾-local martingale component  of the  discounted cumulativeℚ-value processβΠ̂.  The processes m and βΠ̂ are easily seen to coincide on the random interval [0,τc*⋀τp*⋀τd⋀T]. Therefore, both m and βΠ̂ can be interpreted on this interval as the discounted cumulative ℚ-value of a defaultable game option.The following result establishes a useful connection between(Θ̅,M,K) and the arbitrage ex-dividend ℚ-price of the defaultable game option.Proposition 3.16 (Verification principle for a defaultable game option).
The processΠ is the arbitrage ex-dividend ℚ-price for the game option. Moreover, for any t∈[0,T], the saddle-point (τp*,τc*)∈ℱTt×ℱ̅Tt for the related Dynkin game  (2.6) on 𝒢Tt×𝒢̅Tt is given by
(3.25)τp*=inf{u∈[t,T];Π̃u≤Lu}⋀T,τc*=inf{u∈[τ̅∨t,T];Π̃u≥Uu}⋀T.Proof.
In view of  (3.4), the present assumptions imply that Π̃t 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×𝒢̅Tt, 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 Assumption3.13. Let thus (Θ̅,M,K) denote a solution to (ℰ̅) and let Π̃ and Π be defined by  (3.19). In particular, Π is the arbitrage ℚ-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).
Letζ be an arbitrary ℝ1⊗d- valued and βX̂-integrable process. Then the following statements are valid.
(i) Let the processρ(ζ) be given by ρ0(ζ)=0 and
(4.1)βtdρt(ζ)=dmt-ζtd(βtX̂t).
Then (Π0,ζ,ρ(ζ),τc*) is an issuer hedge with 𝔾-sigma (local, in case βX̂ and ζ are locally bounded) martingale cost.
(ii) Let the processρ̅(ζ) be given by ρ̅0(ζ)=0 and
(4.2)βtdρ̅t(ζ)=-dmt-ζtd(βtX̂t).
Then (-Π0,ζ,ρ̅(ζ),τp*) is a holder hedge with a 𝔾-sigma martingale (local martingale, when βX̂ and ζ are locally bounded) cost process.Recall that, according to our convention (see Section1.2), the βX̂-integrability of an ℝ1⊗d-valued stochastic process ζ implies its 𝔾-predictability. Note also that the equality ρ̅(-ζ)=-ρ(ζ) is valid for any process ζ, since
(4.3)-βtdρt(ζ)=-dmt+ζtd(βtX̂t)=-dmt-(-ζt)d(βtX̂t).Proof of Proposition4.1.
The arguments for a holder are essentially symmetrical to those for an issuer; we thus only prove part (i). By Lemma3.14(i), the process ρ(ζ) is a 𝔾-sigma martingale, and a 𝔾-local martingale if βX̂ 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],
(4.4)d(βtVt)=ζtd(βtX̂t)=dmt-βtdρt
and thus
(4.5)βtVt+∫0tβudρu=mt+β0(Π0-Π̂0)=βtΠ̂t+∫0t⋀τdβudKu+β0(Π0-Π̂0),
where the second equality follows from (3.20). Recall that the stopping time τc*∈ℱ̅T0 is given by (see Proposition 3.16)
(4.6)τc*=inf{t∈[τ̅,T];Π̃t≥Ut}⋀T.
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 τp∈ℱT0, with τ=τp⋀τc* (cf.  (2.13),
(4.7)βτVτ+∫0τβu(dρu-dDu)≥1{τ<τd}βτ(1{τ=τp<T}Lτp+1{τ<τp}Uτc*+1{τp=τc*=T}ξ).
From the definition of τc*, the minimality conditions in (ℰ̅) and the continuity of K- it follows that K-=0 and thus K≥0 on [0,τc*]. Since τ≤τc*,  (4.5) thus yields
(4.8)βτVτ+∫0τβu(dρu-dDu)=βτΠτ+∫[0,τ⋀τd]βudKu≥βτΠτ=1{τ<τd}βτΠ̃τ,
where, by (ℰ̅), one has that
(4.9)Π̃τ≥1{τ<T}Lτ+1{τ=T}ξ.
In addition, by the definition of τc*, one has that Π̃τc*≥Uτc* 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 whereρ 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 ρ≠0 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 issuertrivial hedge(Π0,0,ρ0,τc*) (resp., the holder trivial hedge(-Π0,0,-ρ0,τp*)) with the 𝔾-local martingale cost
(4.10)ρt0=∫0tβu-1dmu,t∈[0,T].
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 Lemma2.13 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 Proposition4.1, the following statements are valid. (i)
The equalityΠ0=essmin𝒱0c holds, so that Π0 is the minimum of initial wealths of an issuer hedge with a 𝔾-sigma martingale cost.(ii)
One has that-Π0∈𝒱0p. If, in addition,  (2.5) holds then Π0=-essmin𝒱0p 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 Proposition4.1 to the trivial hedge of Remark 4.2(ii), we get, in particular, that Π0∈𝒱0c, where Π0 is also equal to the ℚ-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 costρ0 of the trivial hedge is a 𝔾-local martingale.Given our definition of hedging with a cost and the definition ofΠ0, 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

LetNd=H-Γ·⋀τd 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 ℚ(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 componentm (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.
The𝔾-martingale m defined by  (3.20) satisfies
(4.11)dmt=1{t≤τd}βt(dMt+YtdNtd),
where the 𝔽-predictable process Y equals Yt=Rt-Π̃t-.Proof.
Let us introduce the Doléans-Dade martingale (see, e.g., [29])
(4.12)ℰt=1{t<τd}eΓt=1-∫0tℰu-dNud,
so that αtℰt=βt1{t<τd} and αtℰt-=βt1{t≤τd}. Then (cf.  (3.19) and (3.20)
(4.13)dmt=d(βtΠ̂t)+1{t≤τd}βtdKt=d(ℰtαtΠ̃t)+1{t≤τd}βtdKt+βtdDt.
It may happen that the 𝔽-semimartingale αΠ̃ fails to be also a 𝔾-semimartingale, so a direct application of the (𝔾-)integration by parts formula to ℰαΠ̃ is not possible. However, by Lemma A.1(iv), the process αΠ̃stopped atτd is a 𝔾-semimartingale. It is also clear that ℰαΠ̃=ℰα·⋀τdΠ̃·⋀τd. Hence by applying the integration by parts formula to ℰα·⋀τdΠ̃·⋀τd, we obtain
(4.14)d(ℰtαt⋀τdΠ̃t⋀τd)=ℰt-(d(αt⋀τdΠ̃t⋀τd)-αtΠ̃t-dNtd)+d[ℰ,α·⋀τdΠ̃·⋀τd]t,
where, in addition, one has that [ℰ,α·⋀τdΠ̃·⋀τd]t=-eΓτdατdΔΠ̃τdHt. 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 ΔΠ̃τd=0. Using  (3.22), we next deduce from (4.13) that
(4.15)dmt=ℰt-(d(αt⋀τdΠ̃t⋀τd)-αtΠ̃t-dNtd)+1{t≤τd}βtdKt+βtdDt=1{t≤τd}βt(-dKt-dCt-RtdΓt+dMt-Π̃t-dNtd)+1{t≤τd}βtdKt+βtdDt=1{t≤τd}βt(-dCt-RtdΓt+dMt-Π̃t-dNtd)+βtdDt.
Using  (2.3) and the equality ΔCτd=0, we finally arrive at the formula
(4.16)dmt=1{t≤τd}βt(dMt+(Rt-Π̃t-)dNtd),
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 priceX of primary risky assets satisfies Xt=(1-Ht)X̃t, for every t∈[0,T], where the ℝd-valued, 𝔽-adapted process X̃ 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 ℛ an ℝd-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 X̂ stopped at τd⋀T.Assumption 4.5.
The dynamics underℚ of the cumulative price process X̂ of primary risky assets are, for every t∈[0,T⋀τd],
(4.17)d(βtX̂t)=βt(dM̂t+ŶtdNtd)
for some ℝd-valued 𝔽-martingale M̂, where the ℝd-valued, 𝔽-predictable process Ŷ is given by the equality  Ŷt=ℛt-X̃t- for every t∈[0,T].By inserting  (4.11) and  (4.17) into (4.1), we obtain, for every t∈[0,T⋀τd],
(4.18)dρt(ζ)=dMt-ζtdM̂t+(Yt-ζtŶt)dNtd.
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 M̂ 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,τd⋀T]. 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 thesystematic risk factor  (also known as the market risk factor) is introduced.Assumption 4.7.
We are given anℝq-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 M̂ of (4.11) and (4.17) satisfy the following decompositions, for every t∈[0,T],
(4.19)dMt=ZtdNt+dnt,dM̂t=ẐtdNt+dn̂t,
where Z (resp.,  Ẑ) is some 𝔽-adapted, ℝ1⊗q-valued (resp., ℝd⊗q-valued), N-integrable processes and n (resp., n̂) is a real-valued (resp., ℝd-valued) 𝔽-martingale.It is natural to refer to𝔽-martingales n and n̂ 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 factorN 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 Ẑ 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 denote𝒩=[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 n̂ in the dynamics of the cost process of a trading strategy.Lemma 4.9.
For anyℝ1⊗d-valued, βX̂-integrable process ζ, the cost process satisfies, for every t∈[0,T⋀τd],
(4.20)dρt(ζ)=([Zt,Yt]-ζt[Ẑt,Ŷt])d𝒩t+dnt-ζtdn̂t.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 setq=1 and we take dMt=ZtdNt, so that n vanishes. For d=2, we further postulate that n̂=0 and
(4.21)dM̂t=d[M̂t1M̂t2]=[Ẑt1Ẑt2]dMt
or, equivalently, that for i=1,2(4.22)d(βtX̂ti)=βt(ẐtidMt+ŶtidNtd).
Assume that there exists a βX̂-integrable process ζ solving the equation
(4.23)ζt[Ẑt,Ŷt]=[ζt1,ζt2][Ẑt1Ŷt1Ẑt2Ŷt2]=[Zt,Yt].
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[Ẑ,Ŷ] admits a βX̂-integrable solution ζ̂ on [0,T⋀τd]. Then the cost process ρ̂=ρ(ζ̂) satisfies, for every t∈[0,T⋀τd],
(4.24)dρ̂t=dnt-ζ̂tdn̂t.
(ii) Assume that the equationZ=ζẐ admits a βX̂-integrable solution ζ̌ on [0,T⋀τd]. Then the dynamics of the cost process ρ̌=ρ(ζ̌) are, for every t∈[0,T⋀τd],
(4.25)dρť=(Yt-ζ̌tŶt)dNtd+dnt-ζ̌tdn̂t.Part (i) in Lemma4.12 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 Ŷ=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 ofM and M̂ 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 thusℋ2 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 ℋ2. 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 ofM and M̂ 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 M̂ with n and n̂ strongly orthogonal to N in ℋ2. 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 n̂ are also mutually strongly orthogonal.The following proposition justifies the informal statement that the strategyζ̂ (resp., ζ̌) 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 processesN,n, and n̂ in decompositions  (4.19) of M and M̂ are mutually strongly orthogonal in ℋ2.
(i) Under assumptions of Lemma4.12(i), the processes ρ̂ and 𝒩·⋀τd are  orthogonal   in 𝔾, in the sense that [ρ̂,𝒩·⋀τd] is a 𝔾-sigma martingale (and a 𝔾-local martingale if ζ̂ is locally bounded).
(ii) Under assumptions of Lemma4.12(ii), the processes ρ̌ and N·⋀τd are  orthogonal   in 𝔾, in the sense that [ρ̌,N·⋀τd] is a 𝔾-sigma martingale (and a 𝔾-local martingale if ζ̌,R, and ℛ are locally bounded processes).Proof.
We first note thatn·⋀τd and N·⋀τd are 𝔾-local martingales, by Lemma A.1(iii). Since n is strongly orthogonal to N in ℋ2, the process [n·⋀τd,N·⋀τd] is a 𝔾-local martingale, as an 𝔽-local martingale stopped at τd (cf. Lemma A.1(iii). Furthermore, by Lemma A.6, [n·⋀τd,Nd] is a 𝔾-local martingale. We conclude that [n·⋀τd,𝒩·⋀τd] is a 𝔾-local martingale. So are also [n̂,N·⋀τd] and [n̂,𝒩·⋀τd], since the integral ∫0·βtdn̂t is strongly orthogonal to 𝒩·⋀τd. Furthermore, by Lemma A.6, [N·⋀τd,Nd] is a 𝔾-local martingale.
Using (4.24), we conclude for part (i) that [ρ̂,𝒩·⋀τd] is a 𝔾-sigma martingale and thus it follows a 𝔾-local martingale if ζ̂ is a locally bounded process.
For part (ii), we conclude in view of  (4.25) that [ρ̌,N·⋀τd] is a 𝔾-sigma martingale and thus a 𝔾-local martingale if ζ̌, R and ℛ 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 strategyζ that makes the cost process 𝔾-orthogonal under a given risk-neutral probability measure ℚ to a predetermined ℝq-valued 𝔾-local martingale N̅ where, without loss of generality, the process N̅ is assumed to be stopped at τd. In reference to Proposition 4.13, by the 𝔾-orthogonality, we mean here that [ρ,N̅] is a 𝔾-local martingale under ℚ.Remark 4.14.
In the financial interpretation, the processN̅ 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 processm 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 ℚ-value process βΠ̂ of a game option, provided that βΠ̂ is a 𝔾-special semimartingale.In the following proposition we denote (whenever well-defined)(4.26)ℂovt(dXt,dYt)=limh→0h-1ℂov(Xt+h-Xt,Yt+h-Yt∣𝒢t)
and 𝕍art(dXt)=ℂovt(dXt,dXt).Proposition 4.15 (Hedging with orthogonal cost).
Assume thatX̂ admits the decomposition, for every t∈[0,T⋀τd],
(4.27)d(βtX̂t)=βtZ̅tdN̅t+βtdn̅t
with n̅ and N̅ orthogonal in 𝔾 and an ℝd⊗q-valued, N̅-integrable process Z̅, which is left-invertible on [0,T⋀τd]. Let us set, for every t∈[0,T⋀τd],
(4.28)ζt=ℂovt(dmt,βtdN̅t)𝕍art(βtdN̅t)-1Λt,
where Λ is the left inverse of the transpose of Z̅ on [0,T⋀τd]. Then the cost process ρ(ζ) is orthogonal to N̅ in 𝔾.Proof.
By combining (4.1) with (4.27), we obtain
(4.29)βtdρt(ζ)=dmt-ζtβtZ̅tdN̅t-βtζtdn̅t.
Therefore, in order to have the cost ρ orthogonal to N̅ in 𝔾, it suffices to select a strategy ζ for which m-∫0·βtζtZ̅tdN̅t is 𝔾-orthogonal to N̅. Relying on the multilinear regression formula, this can be achieved by setting ζ as in  (4.28).The problem of hedging a defaultable game option with respect toN̅ can thus be solved, at least formally, provided that one can find a decomposition  (4.27) with the required properties. Such a decomposition can be obtained as the GKW decomposition of βX̂ in 𝔾 with respect to N̅, provided that the related matrix Z̅ is left-invertible on [0,T⋀τd]. It is then natural to conjecture that the strategy given by  (4.28) for N̅=𝒩·⋀τd (resp., N̅=N·⋀τd) will coincide with the strategy ζ̂ (resp., ζ̌) of Proposition 4.13.The following result examines the special case whenn̅=0 in (4.27). As can be seen from formula (4.31) below, this corresponds to the assumption that the process N̅ represents the wealth processes of some portfolios of primary traded assets.Corollary 4.16 (Min-variance hedging).
Assume thatn̅=0 in (4.27) and Z̅ is left-invertible on [0,T⋀τd]. Then the strategy ζ of Proposition 4.15 becomes
(4.30)ζt=ℂovt(dmt,d(βtX̂t))𝕍art(d(βtX̂t))-1.Proof.
Under the present assumptions, we obtain from (4.27)
(4.31)βtdN̅t=Λtd(βtX̂t),t∈[0,T⋀τd].
Consequently, formula (4.28) reduces to
(4.32)ζt=ℂovt(dmt,Λtd(βtX̂t))𝕍art(Λtd(βtX̂t))-1Λt,
and this in turn yields (4.31).We recognize here a strategy, which is known to arise in the context of themin-variance hedging  in incomplete markets. In the present setup, it was not derived by suitable optimization arguments, however, but obtained by simply postulating that the cost process should be orthogonal to prices of primary assets under the preselected risk-neutral  probability measure ℚ. It should be noted that in the context of a game (or even American) option, the min-variance hedging approach should also incorporate optimization with respect to exercise times. This would lead to the optimization problem of the form (from the issuer's perspective, cf.  (2.11)
(4.33)essinfτc∈𝒢̅T0,ζesssupτp∈𝒢T0𝔼ℙ((βτVτ-β0π(0;τp,τc))2∣𝒢0)
for a given level V0 of the initial wealth, where τ=τc⋀τp⋀τd and where the expectation is taken under the statistical  probability measure ℙ. In a discrete-time setup, this problem was recently proposed and solved by Dolinsky and Kifer [36], who have also observed that the corresponding problem in a continuous-time framework is very difficult to deal with. Whether such an approach in continuous time is amenable to mathematical and practical solution is indeed far from trivial, so it is left as an open challenging problem.


## 4.1. Cost Process of a Hedging Strategy

From now on, we will work under Assumption3.13. Let thus (Θ̅,M,K) denote a solution to (ℰ̅) and let Π̃ and Π be defined by  (3.19). In particular, Π is the arbitrage ℚ-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).
Letζ be an arbitrary ℝ1⊗d- valued and βX̂-integrable process. Then the following statements are valid.
(i) Let the processρ(ζ) be given by ρ0(ζ)=0 and
(4.1)βtdρt(ζ)=dmt-ζtd(βtX̂t).
Then (Π0,ζ,ρ(ζ),τc*) is an issuer hedge with 𝔾-sigma (local, in case βX̂ and ζ are locally bounded) martingale cost.
(ii) Let the processρ̅(ζ) be given by ρ̅0(ζ)=0 and
(4.2)βtdρ̅t(ζ)=-dmt-ζtd(βtX̂t).
Then (-Π0,ζ,ρ̅(ζ),τp*) is a holder hedge with a 𝔾-sigma martingale (local martingale, when βX̂ and ζ are locally bounded) cost process.Recall that, according to our convention (see Section1.2), the βX̂-integrability of an ℝ1⊗d-valued stochastic process ζ implies its 𝔾-predictability. Note also that the equality ρ̅(-ζ)=-ρ(ζ) is valid for any process ζ, since
(4.3)-βtdρt(ζ)=-dmt+ζtd(βtX̂t)=-dmt-(-ζt)d(βtX̂t).Proof of Proposition4.1.
The arguments for a holder are essentially symmetrical to those for an issuer; we thus only prove part (i). By Lemma3.14(i), the process ρ(ζ) is a 𝔾-sigma martingale, and a 𝔾-local martingale if βX̂ 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],
(4.4)d(βtVt)=ζtd(βtX̂t)=dmt-βtdρt
and thus
(4.5)βtVt+∫0tβudρu=mt+β0(Π0-Π̂0)=βtΠ̂t+∫0t⋀τdβudKu+β0(Π0-Π̂0),
where the second equality follows from (3.20). Recall that the stopping time τc*∈ℱ̅T0 is given by (see Proposition 3.16)
(4.6)τc*=inf{t∈[τ̅,T];Π̃t≥Ut}⋀T.
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 τp∈ℱT0, with τ=τp⋀τc* (cf.  (2.13),
(4.7)βτVτ+∫0τβu(dρu-dDu)≥1{τ<τd}βτ(1{τ=τp<T}Lτp+1{τ<τp}Uτc*+1{τp=τc*=T}ξ).
From the definition of τc*, the minimality conditions in (ℰ̅) and the continuity of K- it follows that K-=0 and thus K≥0 on [0,τc*]. Since τ≤τc*,  (4.5) thus yields
(4.8)βτVτ+∫0τβu(dρu-dDu)=βτΠτ+∫[0,τ⋀τd]βudKu≥βτΠτ=1{τ<τd}βτΠ̃τ,
where, by (ℰ̅), one has that
(4.9)Π̃τ≥1{τ<T}Lτ+1{τ=T}ξ.
In addition, by the definition of τc*, one has that Π̃τc*≥Uτc* 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 whereρ 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 ρ≠0 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 issuertrivial hedge(Π0,0,ρ0,τc*) (resp., the holder trivial hedge(-Π0,0,-ρ0,τp*)) with the 𝔾-local martingale cost
(4.10)ρt0=∫0tβu-1dmu,t∈[0,T].
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 Lemma2.13 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 Proposition4.1, the following statements are valid. (i)
The equalityΠ0=essmin𝒱0c holds, so that Π0 is the minimum of initial wealths of an issuer hedge with a 𝔾-sigma martingale cost.(ii)
One has that-Π0∈𝒱0p. If, in addition,  (2.5) holds then Π0=-essmin𝒱0p 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 Proposition4.1 to the trivial hedge of Remark 4.2(ii), we get, in particular, that Π0∈𝒱0c, where Π0 is also equal to the ℚ-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 costρ0 of the trivial hedge is a 𝔾-local martingale.Given our definition of hedging with a cost and the definition ofΠ0, 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

LetNd=H-Γ·⋀τd 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 ℚ(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 componentm (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.
The𝔾-martingale m defined by  (3.20) satisfies
(4.11)dmt=1{t≤τd}βt(dMt+YtdNtd),
where the 𝔽-predictable process Y equals Yt=Rt-Π̃t-.Proof.
Let us introduce the Doléans-Dade martingale (see, e.g., [29])
(4.12)ℰt=1{t<τd}eΓt=1-∫0tℰu-dNud,
so that αtℰt=βt1{t<τd} and αtℰt-=βt1{t≤τd}. Then (cf.  (3.19) and (3.20)
(4.13)dmt=d(βtΠ̂t)+1{t≤τd}βtdKt=d(ℰtαtΠ̃t)+1{t≤τd}βtdKt+βtdDt.
It may happen that the 𝔽-semimartingale αΠ̃ fails to be also a 𝔾-semimartingale, so a direct application of the (𝔾-)integration by parts formula to ℰαΠ̃ is not possible. However, by Lemma A.1(iv), the process αΠ̃stopped atτd is a 𝔾-semimartingale. It is also clear that ℰαΠ̃=ℰα·⋀τdΠ̃·⋀τd. Hence by applying the integration by parts formula to ℰα·⋀τdΠ̃·⋀τd, we obtain
(4.14)d(ℰtαt⋀τdΠ̃t⋀τd)=ℰt-(d(αt⋀τdΠ̃t⋀τd)-αtΠ̃t-dNtd)+d[ℰ,α·⋀τdΠ̃·⋀τd]t,
where, in addition, one has that [ℰ,α·⋀τdΠ̃·⋀τd]t=-eΓτdατdΔΠ̃τdHt. 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 ΔΠ̃τd=0. Using  (3.22), we next deduce from (4.13) that
(4.15)dmt=ℰt-(d(αt⋀τdΠ̃t⋀τd)-αtΠ̃t-dNtd)+1{t≤τd}βtdKt+βtdDt=1{t≤τd}βt(-dKt-dCt-RtdΓt+dMt-Π̃t-dNtd)+1{t≤τd}βtdKt+βtdDt=1{t≤τd}βt(-dCt-RtdΓt+dMt-Π̃t-dNtd)+βtdDt.
Using  (2.3) and the equality ΔCτd=0, we finally arrive at the formula
(4.16)dmt=1{t≤τd}βt(dMt+(Rt-Π̃t-)dNtd),
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 priceX of primary risky assets satisfies Xt=(1-Ht)X̃t, for every t∈[0,T], where the ℝd-valued, 𝔽-adapted process X̃ 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 ℛ an ℝd-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 X̂ stopped at τd⋀T.Assumption 4.5.
The dynamics underℚ of the cumulative price process X̂ of primary risky assets are, for every t∈[0,T⋀τd],
(4.17)d(βtX̂t)=βt(dM̂t+ŶtdNtd)
for some ℝd-valued 𝔽-martingale M̂, where the ℝd-valued, 𝔽-predictable process Ŷ is given by the equality  Ŷt=ℛt-X̃t- for every t∈[0,T].By inserting  (4.11) and  (4.17) into (4.1), we obtain, for every t∈[0,T⋀τd],
(4.18)dρt(ζ)=dMt-ζtdM̂t+(Yt-ζtŶt)dNtd.
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 M̂ 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,τd⋀T]. 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 thesystematic risk factor  (also known as the market risk factor) is introduced.Assumption 4.7.
We are given anℝq-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 M̂ of (4.11) and (4.17) satisfy the following decompositions, for every t∈[0,T],
(4.19)dMt=ZtdNt+dnt,dM̂t=ẐtdNt+dn̂t,
where Z (resp.,  Ẑ) is some 𝔽-adapted, ℝ1⊗q-valued (resp., ℝd⊗q-valued), N-integrable processes and n (resp., n̂) is a real-valued (resp., ℝd-valued) 𝔽-martingale.It is natural to refer to𝔽-martingales n and n̂ 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 factorN 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 Ẑ 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 denote𝒩=[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 n̂ in the dynamics of the cost process of a trading strategy.Lemma 4.9.
For anyℝ1⊗d-valued, βX̂-integrable process ζ, the cost process satisfies, for every t∈[0,T⋀τd],
(4.20)dρt(ζ)=([Zt,Yt]-ζt[Ẑt,Ŷt])d𝒩t+dnt-ζtdn̂t.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 setq=1 and we take dMt=ZtdNt, so that n vanishes. For d=2, we further postulate that n̂=0 and
(4.21)dM̂t=d[M̂t1M̂t2]=[Ẑt1Ẑt2]dMt
or, equivalently, that for i=1,2(4.22)d(βtX̂ti)=βt(ẐtidMt+ŶtidNtd).
Assume that there exists a βX̂-integrable process ζ solving the equation
(4.23)ζt[Ẑt,Ŷt]=[ζt1,ζt2][Ẑt1Ŷt1Ẑt2Ŷt2]=[Zt,Yt].
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[Ẑ,Ŷ] admits a βX̂-integrable solution ζ̂ on [0,T⋀τd]. Then the cost process ρ̂=ρ(ζ̂) satisfies, for every t∈[0,T⋀τd],
(4.24)dρ̂t=dnt-ζ̂tdn̂t.
(ii) Assume that the equationZ=ζẐ admits a βX̂-integrable solution ζ̌ on [0,T⋀τd]. Then the dynamics of the cost process ρ̌=ρ(ζ̌) are, for every t∈[0,T⋀τd],
(4.25)dρť=(Yt-ζ̌tŶt)dNtd+dnt-ζ̌tdn̂t.Part (i) in Lemma4.12 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 Ŷ=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 ofM and M̂ 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 thusℋ2 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 ℋ2. 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 ofM and M̂ 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 M̂ with n and n̂ strongly orthogonal to N in ℋ2. 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 n̂ are also mutually strongly orthogonal.The following proposition justifies the informal statement that the strategyζ̂ (resp., ζ̌) 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 processesN,n, and n̂ in decompositions  (4.19) of M and M̂ are mutually strongly orthogonal in ℋ2.
(i) Under assumptions of Lemma4.12(i), the processes ρ̂ and 𝒩·⋀τd are  orthogonal   in 𝔾, in the sense that [ρ̂,𝒩·⋀τd] is a 𝔾-sigma martingale (and a 𝔾-local martingale if ζ̂ is locally bounded).
(ii) Under assumptions of Lemma4.12(ii), the processes ρ̌ and N·⋀τd are  orthogonal   in 𝔾, in the sense that [ρ̌,N·⋀τd] is a 𝔾-sigma martingale (and a 𝔾-local martingale if ζ̌,R, and ℛ are locally bounded processes).Proof.
We first note thatn·⋀τd and N·⋀τd are 𝔾-local martingales, by Lemma A.1(iii). Since n is strongly orthogonal to N in ℋ2, the process [n·⋀τd,N·⋀τd] is a 𝔾-local martingale, as an 𝔽-local martingale stopped at τd (cf. Lemma A.1(iii). Furthermore, by Lemma A.6, [n·⋀τd,Nd] is a 𝔾-local martingale. We conclude that [n·⋀τd,𝒩·⋀τd] is a 𝔾-local martingale. So are also [n̂,N·⋀τd] and [n̂,𝒩·⋀τd], since the integral ∫0·βtdn̂t is strongly orthogonal to 𝒩·⋀τd. Furthermore, by Lemma A.6, [N·⋀τd,Nd] is a 𝔾-local martingale.
Using (4.24), we conclude for part (i) that [ρ̂,𝒩·⋀τd] is a 𝔾-sigma martingale and thus it follows a 𝔾-local martingale if ζ̂ is a locally bounded process.
For part (ii), we conclude in view of  (4.25) that [ρ̌,N·⋀τd] is a 𝔾-sigma martingale and thus a 𝔾-local martingale if ζ̌, R and ℛ 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 strategyζ that makes the cost process 𝔾-orthogonal under a given risk-neutral probability measure ℚ to a predetermined ℝq-valued 𝔾-local martingale N̅ where, without loss of generality, the process N̅ is assumed to be stopped at τd. In reference to Proposition 4.13, by the 𝔾-orthogonality, we mean here that [ρ,N̅] is a 𝔾-local martingale under ℚ.Remark 4.14.
In the financial interpretation, the processN̅ 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 processm 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 ℚ-value process βΠ̂ of a game option, provided that βΠ̂ is a 𝔾-special semimartingale.In the following proposition we denote (whenever well-defined)(4.26)ℂovt(dXt,dYt)=limh→0h-1ℂov(Xt+h-Xt,Yt+h-Yt∣𝒢t)
and 𝕍art(dXt)=ℂovt(dXt,dXt).Proposition 4.15 (Hedging with orthogonal cost).
Assume thatX̂ admits the decomposition, for every t∈[0,T⋀τd],
(4.27)d(βtX̂t)=βtZ̅tdN̅t+βtdn̅t
with n̅ and N̅ orthogonal in 𝔾 and an ℝd⊗q-valued, N̅-integrable process Z̅, which is left-invertible on [0,T⋀τd]. Let us set, for every t∈[0,T⋀τd],
(4.28)ζt=ℂovt(dmt,βtdN̅t)𝕍art(βtdN̅t)-1Λt,
where Λ is the left inverse of the transpose of Z̅ on [0,T⋀τd]. Then the cost process ρ(ζ) is orthogonal to N̅ in 𝔾.Proof.
By combining (4.1) with (4.27), we obtain
(4.29)βtdρt(ζ)=dmt-ζtβtZ̅tdN̅t-βtζtdn̅t.
Therefore, in order to have the cost ρ orthogonal to N̅ in 𝔾, it suffices to select a strategy ζ for which m-∫0·βtζtZ̅tdN̅t is 𝔾-orthogonal to N̅. Relying on the multilinear regression formula, this can be achieved by setting ζ as in  (4.28).The problem of hedging a defaultable game option with respect toN̅ can thus be solved, at least formally, provided that one can find a decomposition  (4.27) with the required properties. Such a decomposition can be obtained as the GKW decomposition of βX̂ in 𝔾 with respect to N̅, provided that the related matrix Z̅ is left-invertible on [0,T⋀τd]. It is then natural to conjecture that the strategy given by  (4.28) for N̅=𝒩·⋀τd (resp., N̅=N·⋀τd) will coincide with the strategy ζ̂ (resp., ζ̌) of Proposition 4.13.The following result examines the special case whenn̅=0 in (4.27). As can be seen from formula (4.31) below, this corresponds to the assumption that the process N̅ represents the wealth processes of some portfolios of primary traded assets.Corollary 4.16 (Min-variance hedging).
Assume thatn̅=0 in (4.27) and Z̅ is left-invertible on [0,T⋀τd]. Then the strategy ζ of Proposition 4.15 becomes
(4.30)ζt=ℂovt(dmt,d(βtX̂t))𝕍art(d(βtX̂t))-1.Proof.
Under the present assumptions, we obtain from (4.27)
(4.31)βtdN̅t=Λtd(βtX̂t),t∈[0,T⋀τd].
Consequently, formula (4.28) reduces to
(4.32)ζt=ℂovt(dmt,Λtd(βtX̂t))𝕍art(Λtd(βtX̂t))-1Λt,
and this in turn yields (4.31).We recognize here a strategy, which is known to arise in the context of themin-variance hedging  in incomplete markets. In the present setup, it was not derived by suitable optimization arguments, however, but obtained by simply postulating that the cost process should be orthogonal to prices of primary assets under the preselected risk-neutral  probability measure ℚ. It should be noted that in the context of a game (or even American) option, the min-variance hedging approach should also incorporate optimization with respect to exercise times. This would lead to the optimization problem of the form (from the issuer's perspective, cf.  (2.11)
(4.33)essinfτc∈𝒢̅T0,ζesssupτp∈𝒢T0𝔼ℙ((βτVτ-β0π(0;τp,τc))2∣𝒢0)
for a given level V0 of the initial wealth, where τ=τc⋀τp⋀τd and where the expectation is taken under the statistical  probability measure ℙ. In a discrete-time setup, this problem was recently proposed and solved by Dolinsky and Kifer [36], who have also observed that the corresponding problem in a continuous-time framework is very difficult to deal with. Whether such an approach in continuous time is amenable to mathematical and practical solution is indeed far from trivial, so it is left as an open challenging problem.



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*Source: 695798-2009-07-21.xml*