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Journal of Applied Mathematics and Stochastic Analysis
Volume 2009 (2009), Article ID 695798, 33 pages
Defaultable Game Options in a Hazard Process Model
1Department of Applied Mathematics, Illinois Institute of Technology, Chicago, IL 60616, USA
2Département de Mathématiques, Université d'Évry Val d'Essonne, 91025 Évry Cedex, France
3Europlace Institute of Finance, Palais Brongniart-28 Place de la Bourse, 75002 Paris, France
4School of Mathematics and Statistics, University of New South Wales, Sydney, NSW 2052, Australia
5Faculty of Mathematics and Information Science, Warsaw University of Technology, 00-661 Warszawa, Poland
Received 22 October 2008; Accepted 4 April 2009
Academic Editor: Salah-Eldin Mohammed
Copyright © 2009 Tomasz R. Bielecki et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.
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 , Ayache et al. , Bielecki et al. [3, 4], Davis and Lischka , Kallsen and Kühn , and Kwok and Lau .
In Bielecki et al. , 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  and Kallsen and Kühn , we showed that the study of an arbitrage price of a defaultable game option can be reduced to the study of the value process of the related Dynkin game under some risk-neutral measure for the primary market model. In this stochastic game, the issuer of a game option plays the role of the minimizer and the holder of the maximizer. In , we dealt with a general market model, which was assumed to be arbitrage-free, but not necessarily complete, so that the uniqueness of a risk-neutral (or martingale) measure was not postulated. In addition, although the default time was introduced, it was left largely unspecified. An explicit specification of the default time will be an important component of the model considered in this work.
As is well known, there are two main approaches to modeling of default risk: the structural approach and the reduced-form approach. In the latter approach, also known as the hazard process approach, the default time is modeled as an exogenous random variable with no reference to any particular economic background. One may object to reduced-form models for their lack of clear reference to economic fundamentals, such as the firm's asset-to-debt ratio. However, the possibility of choosing various parameterizations for the coefficients and calibrating these parameters to any set of CDS spreads and/or implied volatilities makes them very versatile modeling tools, well suited to price and hedge derivatives consistently with plain-vanilla instruments. It should be acknowledged that structural models, with their sound economic background, are better suited for inference of reliable debt information, such as: risk-neutral default probabilities or the present value of the firm's debt, from the equities, which are the most liquid among all financial instruments. The structure of these models, as rich as it may be (and which can include a list of factors such as stock, spreads, default status, and credit events) never rich enough to yield consistent prices for a full set of CDS spreads and/or implied volatilities of related options. As we ultimately aim to specify models for pricing and hedging contracts with optional features (such as convertible bonds), we favor the reduced-form approach in the sequel.
1.1. Outline of the Paper
From the mathematical perspective, the goal of the present paper is twofold. First, we wish to specialize our previous valuation results to the hazard process setup, that is, to a version of the reduced-form approach, which is slightly more general than the intensity-based setup. Hence we postulate that filtration modeling the information flow for the primary market admits the representation , where the filtration is generated by the default indicator process and is some reference filtration. The main tool employed in this section is the effective reduction of the information flow from the full filtration to the reference filtration . The main results in this part are Theorems 3.7 and 3.8, which give convenient pricing formulae with respect to the reference filtration .
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  and Ayache et al. , who worked directly with suitable variational inequalities within the Markovian intensity-based setup.
Our preliminary results for hedging strategies in a hazard process setup, Propositions 4.1 and 4.3, can be informally stated as follows: under the assumption that a related doubly reflected BSDE admits a solution under some risk-neutral measure , for which various sets of sufficient conditions are given in literature, the state-process of the solution is the minimal (pre-default) super-hedging price up to a -sigma (or local) martingale cost process. More specific properties of hedging strategies are subsequently analyzed in Propositions 4.13 and 4.15, in which we resort to suitable Galtchouk-Kunita-Watanabe decompositions of a solution to the related doubly reflected BSDE and discounted prices of primary assets with respect to various risk factors corresponding to systematic, idiosyncratic and event risks. It is noteworthy that these decompositions, though seemingly rather abstract in a general setup considered here, are by no means artificial. On the contrary, they arise naturally in the context of particular Markovian models that are studied in the followup paper by Bielecki et al. [4, 9]. We conclude the paper by briefly commenting on some alternative approaches to hedging of defaultable game options.
1.2. Conventions and Standing Notation
Throughout this paper, we use the concept of the vector stochastic integral, denoted as , as opposed to a more restricted notion of the component-wise stochastic integral, which is defined as the sum of integrals with respect to one-dimensional integrators . For a detailed exposition of the vector stochastic integration, we refer to Shiryaev and Cherny  (see also Chatelain and Stricker  and Jacod ). Given a stochastic basis satisfying the usual conditions, an -valued semimartingale integrator and an -valued (row vector) predictable integrand , the notion of the vector stochastic integral 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 where 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  or Shiryaev and Cherny ).
By default, we denote by the integrals over . Otherwise, we explicitly specify the domain of integration as a subscript of . Note also that, depending on the context, will stand either for a generic stopping time or it will be given as for some specific stopping times and . 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 , 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 , where denotes the statistical probability measure.
Specifically, we consider a primary market composed of the savings account and of risky assets, such that, given a finite horizon date :(i)the discount factor process , that is, the inverse of the savings account, is a -adapted, finite variation, positive, continuous and bounded process,(ii) the risky assets are -semimartingales with càdlàg sample paths.
The primary risky assets, with -valued price process , pay dividends, whose cumulative value process, denoted by , is assumed to be a -adapted, càdlàg and -valued process of finite variation. Given the price process , we define the cumulative price of primary risky assets as In the financial interpretation, the last term in (2.1) represents the current value at time of all dividend payments from the assets over the period , under the assumption that all dividends are immediately reinvested in the savings account. We assume that the primary market model is free of arbitrage opportunities, though presumably incomplete. In view of the First Fundamental Theorem of Asset Pricing (cf. [10, 14]), and accounting in particular for the dividends, this means that there exists a risk-neutral measure , where denotes the set of probability measures for which is a sigma martingale with respect to under (for the definition of a sigma martingale, see [10, Definition 1.9]). The following well-known properties of sigma martingales will be used in the sequel.
Proposition 2.1 (see [10, 15, 16]). (i) The class of sigma martingales is a vector space containing all local martingales. It is stable with respect to (vector) stochastic integration, that is, if is a sigma martingale and is a (predictable) -integrable process then the (vector) stochastic integral 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 ).
We now introduce the concept of a dividend paying game option (see also Kifer ). In broad terms, a dividend paying game option, with the inception date and the maturity date , is a contract with the following cash flows that are paid by the issuer of the contract and received by its holder:(i)a dividend stream with the cumulative dividend at time denoted by , (ii)a terminal put payment made at time if and ; time is called the put time and is chosen by the holder, (iii)a terminal call payment made at time provided that time , known as the call time, is chosen by the issuer and may be subject to the constraint that , where is the lifting time of the call protection, (iv)a terminal payment at maturity made at maturity date provided that .
The (possibly random) time 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 .
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 -valued -stopping time representing the default time of a reference entity, with default indicator process . A defaultable dividend paying game option is a dividend paying game option such that the contract is terminated at , 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 is assumed to include a possible recovery payment made at the default time.
We are interested in the problem of the time evolution of an arbitrage price of the game option. Therefore, we formulate the problem in a dynamic way by pricing the game option at any time . We write to denote the set of all -stopping times with values in and we let stand for the set , where the lifting time of a call protection belongs to .
We are now in the position to state the formal definition of a defaultable game option.
Definition 2.3. A defaultable game option with lifting time of the call protection is a game option with the ex-dividend cumulative discounted cash flows given by the formula, for any and , where and (i)the dividend process equals for some coupon process , which is a -predictable, real-valued, càdlàg process with bounded variation, and some real-valued, -predictable recovery process , (ii)the put payment and the call payment are -adapted, real-valued, càdlàg processes, (iii)the inequality holds for every , (iv)the payment at maturity is a -measurable, real-valued random variable.
The following result easily follows from Definition 2.3.
Lemma 2.4. (i) For any and , the random variable is -measurable.
(ii) For any , the processes and are -adapted.
We further assume that and are bounded from below, so that there exists a constant such that, for every ,
Symmetrically, we should sometimes additionally assume that and are bounded (from below and from above), or that (2.4) is supplemented by the inequality, for every ,
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 , Kifer , and Lepeltier and Maingueneau ). 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 , and extended in  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 “ is an arbitrage price for the game option’’ is in fact to be understood as “ is an arbitrage price for the extended market consisting of the primary market and the game option.’’
Theorem 2.5 (Arbitrage price of a defaultable game option). Assume that a process is a -semimartingale and there exists such that is the value of the Dynkin game related to a game option, meaning that Then is an arbitrage ex-dividend price of the game option, called the -price of the game option. The converse holds true (thus any arbitrage price is a -price for some ) under the following integrability assumption
Note that defaultable American (or European) options can be seen as special cases of defaultable game options.
Definition 2.6. A defaultable American option is a defaultable game option with . A defaultable European option is a defaultable American option such that the process (cf. (2.4) attains its maximum at , that is, for every .
In view of Theorem 2.5, the cash flows of a defaultable European option can be redefined by
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 , and subsequently followed by El Karoui and Quenez , Kifer , Ma and Cvitanić , and Hamadène . 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 .
Recall that (resp., ) 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 such that (i) is a -measurable real-valued random variable representing the initial wealth,(ii) is an -valued, -integrable process representing holdings in primary risky assets.
Remark 2.8. The reason why we do not assume that is trivial (which would, of course, simplify several statements) is that we apply our results in the subsequent work  to situations, where this assumption fails to hold (e.g., when studying convertible bonds with a positive call notice period).
Definition 2.9. The wealth process of a primary trading strategy is given by the formula, for ,
Given the wealth process of a primary strategy , we uniquely specify a -optional process by setting The process represents the number of units held in the savings account at time , when we start from the initial wealth and we use the strategy in the primary risky assets. Recall that we denote .
Definition 2.10. Consider the game option with the ex-dividend cumulative discounted cash flows given by (2.2).
(i)An issuer hedge with cost process is represented by a quadruplet such that(a) is a primary strategy with the wealth process given by (2.9),(b)a cost process is a real-valued, càdlàg -semimartingale with ,(c)a (fixed) call time belongs to ,(d)the following inequality is valid, for every put time , (ii)A holder hedge with cost process is a quadruplet such that(a) is a primary strategy with the wealth process given by (2.9),(b)a cost process is a real-valued, càdlàg -semimartingale with ,(c)a (fixed) put time belongs to ,(d)the following inequality is valid, for every call time ,
Issuer or holder hedges at no cost (i.e., with ) 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) The left-hand side in (2.13) is the value at time of a strategy with a cost process , when the players adopt their respective exercise policies and , whereas the right-hand side represents the payoff to be done by the issuer, including past dividends and the recovery at default.
Remark 2.11. (i) The process is to be interpreted as the (running) financing cost, that is, the amount of cash added to (if ) or withdrawn from (if ) the hedging portfolio in order to get a perfect, but no longer self-financing, hedge. In the special case where is a -martingale under we thus recover the notion of mean self-financing hedge, in the sense of Schweizer .
(ii) Regarding the admissibility of hedging strategies (see, e.g., Delbaen and Schachermayer ), 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 . Likewise, in the case of a bounded payoff (i.e., assuming (2.5), the left-hand side in formula (2.12) is bounded from below for any holder hedge with a cost .
Obviously, the class of all hedges with semimartingale cost processes is too large for any practical purposes. Therefore, we will restrict our attention to hedges with a -sigma martingale cost under a particular risk-neutral measure .
Assumption 2.12. In the sequel, we work under a fixed, but arbitrary, risk-neutral measure .
All the measure-dependent notions like (local) martingale and compensator, implicitly refer to the probability measure . In particular, we define (resp., ) as the set of initial values for which there exists an issuer (resp., holder) hedge of the game option with the initial value (resp., ) and with a -sigma martingale cost under .
The following result gives some preliminary conclusions regarding the initial cost of a hedging strategy for the game option under the present, rather weak, assumptions. In Proposition 4.3, we will see that, under stronger assumptions, the infima are attained and thus we obtain equalities, rather than merely inequalities, in (2.14) and (2.15).
Lemma 2.13. (i) One has (by convention, )
(ii) If inequality (2.5) is valid then
Proof. (i) Assume that for some stopping time the quadruplet is an issuer hedge with a -sigma martingale cost for the game option. It is easily seen from (2.9) and (2.11) that, for any stopping time ,
In particular, by taking , we obtain that, for any ,
The stochastic integral with respect to a -sigma martingale is a -sigma martingale. Hence the stopped process , as well as the process
are -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 (recall that is fixed)
and thus, by the assumed positivity of the process
The required inequality (2.14) is an immediate consequence of the last formula.
(ii) Let be a holder hedge with a -sigma martingale cost for the game option for some stopping time . Then (2.9) and (2.12) imply that, for any , 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 , we obtain so that and this in turn implies (2.15).
3. Valuation in a Hazard Process Setup
In order to get more explicit pricing and hedging results for defaultable game options, we will now study the so-called hazard process setup.
3.1. Standing Assumptions
Given an -valued -stopping time , we assume that , where the filtration is generated by the process and is some reference filtration. As expected, our approach will consist in effectively reducing the information flow from the full filtration to the reference filtration .
Let stand for the process for . The process is a (bounded) -supermartingale, as the optional projection on the filtration of the nonincreasing process (see Jeulin ).
In the sequel, we will work under the following standing assumption.
Assumption 3.1. We assume that the process is (strictly) positive and continuous with finite variation, so that the -hazard process , is well defined and continuous with finite variation.
Remark 3.2. (i) The assumption that is continuous implies that is a totally inaccessible -stopping time (see, e.g., ). Moreover, avoids -stopping times, in the sense that for any -stopping time (see Coculescu and Nikeghbali ).
(ii) If is continuous, the additional assumption that has a finite variation implies in fact that is nonincreasing (see Lemma A.1(i). This lies somewhere between assuming further the (stronger) Hypothesis and assuming further that is an -pseudo-stopping time (see Nikeghbali and Yor ). Recall that the () Hypothesis means that all local -martingales are local -martingales (see, e.g., ), whereas is an -pseudo-stopping time whenever all -local martingales stopped at are -local martingales (see Nikeghbali and Yor  and the appendix).
Definition 3.3. The -stopping time , the -measurable random variable and the -adapted or -predictable process introduced in Lemmas A.2 and A.4 are called the -representatives of and , respectively. In the context of credit risk, where represents the default time of a reference entity, they are also known as the pre-default values of and .
To simplify the presentation, we find it convenient to make additional assumptions. Strictly speaking, these assumptions are superfluous, in the sense that all the results below are true without Assumption 3.4. Indeed, by making use of Lemmas A.2 and A.4 and Definition 3.3, it is always possible to reduce the original problem to the case described in Assumption 3.4. Since this would make the notation heavier, without adding much value, we prefer to work under this standing assumption.
Assumption 3.4. (i) The discount factor process is -adapted.
(ii) The coupon process is -predictable.
(iii) The recovery process is -predictable.
(iv) The payoff processes are -adapted and the random variable is -measurable.
(v) The call protection is an -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 -stopping times can be reduced to the computation of the lower and upper value with respect to -stopping times.
Lemma 3.5. One has that
Proof. For , one has that for some stopping times , where the middle equality follows from Lemma A.4, and the other two from the definition of . Since, clearly, and , we conclude that the lemma is valid.
Under our assumptions, the computation of conditional expectations of cash flows with respect to can be reduced to the computation of conditional expectations of -equivalent cash flows with respect to . Let stand for the credit-risk adjusted discount factor. Note that, similarly to , the process is bounded.
Lemma 3.6. For any stopping times and one has that where is given by, with ,
Note that is an -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 , to the fictitious default-free market, in which cash flows are discounted according to the credit risk adjusted discount factor . Recall that the original cash flows are given as -measurable random variables, whereas the -equivalent cash flows are manifestly -measurable and they depend on the default time only via the hazard process . 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 be the arbitrage ex-dividend -price for a game option. Then one has, for any , where satisfies Hence the Dynkin game with cost criterion on admits the value , which coincides with the pre-default ex-dividend price at time of the game option under the risk-neutral measure .
The following result is the converse of Theorem 3.7. It is an immediate consequence of Lemmas 3.5 and 3.6 and the “if’’ part of Theorem 2.5 (noting also that defined by (3.5) is obviously a -semimartingale if is a -semimartingale).
Theorem 3.8. Let be the value of the Dynkin game with the cost criterion on , for any . Then defined by (3.5) is the value of the Dynkin game with the cost criterion on , for any . If, in addition, is a -semimartingale then is the arbitrage ex-dividend -price for the game option.
3.3. Valuation via Doubly Reflected BSDEs
In this section, we will characterize the arbitrage ex-dividend -price of a game option as a solution to an associated doubly reflected BSDE. To this end, we first recall some auxiliary results concerning the relationship between Dynkin games and doubly reflected BSDEs.
Given an additional -adapted process of finite variation, we consider the following doubly reflected BSDE with the data (see Cvitanić and Karatzas , Hamadène and Hassani , Crépey , Crépey and Matoussi , Bielecki et al. [4, 9]): where the process equals, for ,
Definition 3.9. By a (-) solution to the doubly reflected BSDE (3.7), we mean a triplet such that (i)the state process is a real-valued, -adapted, càdlàg process,(ii) is a real-valued -martingale vanishing at time , (iii) is an -adapted, continuous, finite variation process vanishing at time , (iv)all conditions in (3.7) are satisfied, where in the third line and denote the Jordan components of , and where the convention that is made in the third line.
By the Jordan decomposition, we mean the decomposition , where the nondecreasing continuous processes and vanish at time and define mutually singular measures.
The state process in a solution to (3.7) is clearly an -semimartingale. So there are obvious (though rather artificial) cases in which (3.7) does not admit a solution: it suffices to take and , assumed not to be an -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 and (see, e.g., [31, Remark 4.1]).
Remark 3.10. In applications (see [4, 9, 32, 33]), the input process is typically given in the form of the Lebesgue integral and the component of a solution to (3.7) is usually searched for in the form for some -valued and real-valued square-integrable -martingales and (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 such that on (see, in particular, Crépey and Matoussi , Hamadène and Hassani [31, Theorem 4.1], and previous works in this direction, starting with Cvitanić and Karatzas ). It is thus satisfied when one of the barriers is a quasimartingale and, in particular, when one of the barriers is given as , where is an Itô-Lévy process with square-integrable special semimartingale decomposition components (see ) and is a constant in . This framework covers, for instance, the payoff at call of a convertible bond examined in [3, 4].
Remark 3.11. (i) Since and thus and , are continuous, the minimality conditions (third line) in (3.7) are equivalent to
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 ; see, for example, . In the preprint version  of this work, we defined more general notions of -hedges that were pertaining in the case where there may be jumps in the process . Since in all existing works on doubly reflected BSDEs the process is actually found to be a continuous process (see [4, 30, 31, 33]), we decided to impose here the continuity of in Definition 3.9 and we only consider hedges, as opposed to -hedges. Note, however, that essentially all the results of this paper can be extended to possible jumps in , using the generalized notion of -hedge defined in , and with the minimality conditions stated as (3.9) instead of the third line in condition (3.7) of Definition 3.9.
(ii) Since is a given process, the BSDE (3.7) can be rewritten as where and . This shows that the problem of solving (3.7) can be formally reduced to the case of with suitably modified reflecting barriers and terminal condition . 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 ).
(iii) In the special case where all -martingales are continuous and where the -semimartingale and the barriers and 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 .
(iv) In the context of a Markovian setup, the probabilistic BSDE approach may be complemented by a related analytic variational inequality approach; this issue is dealt with in the followup papers [4, 9]. Note, however, that the variational inequality approach strongly relies on the BSDE approach. Moreover, a simulation method based on the BSDE is the only efficient way of numerically solving the pricing problem whenever the problem dimension (number of model factors) is greater than three or four. Indeed, in that case the computational cost of deterministic numerical schemes based on the variational inequality approach becomes prohibitive.
In order to establish a relationship between a solution to the related doubly reflected BSDE and the arbitrage ex-dividend -price of the defaultable game option, we first recall the general relationship between doubly reflected BSDEs and Dynkin games with purely terminal cost, before applying this result to dividend-paying game options in the fictitious default-free market in Proposition 3.12.
Observe that if solves (3.7) then one has, for any stopping time ,
Proposition 3.12 (Verification principle for a Dynkin game). Let be a solution to (3.7) with . Then is the value of the Dynkin game with cost criterion on , where is the -measurable random variable defined by where . Moreover, for any , the pair of stopping times given by is a saddle-point of this Dynkin game, in the sense that one has, for any ,
Proof. Except for the presence of , the result is standard (see, e.g., Lepeltier and Maingueneau ). Let us first check that the right-hand side inequality in (3.14) is valid for any . Let denote . By the definition of and continuity of , we see that equals 0 on . Since is nondecreasing, (3.11) is applied to yield Taking conditional expectations (recall that is an -martingale), and using also the facts that if if and (recall that , so that and ), we obtain We conclude that for any . 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 , and thus the details are left to the reader.
Let us stress that Assumption 3.13, heroic as it may seem in the general hazard process setup, is in fact a plausible assumption in any reasonable application one may think of (cf. the comments following Definition 3.9).
Lemma 3.14. (i) The process given by (3.20) is -martingale stopped at .
(ii) The process is a -semimartingale.
(iii) The process is a special -semimartingale.
Proof. (i) The triplet satisfies (3.7) with given by in (3.18). Therefore, for every ,
Using Lemma A.5, it is easy to check that one has, for any ,
Since the integral is an -martingale, the process is a -martingale. It is also clear that it is stopped at .
(ii) In view of (3.19), (3.20) and part (i), the process is clearly a -semimartingale.
(iii) By (3.20), one has that where is a -martingale, by (i), and where the second term in the right-hand side is a -adapted and continuous (hence -predictable) processes of finite variation.
Remark 3.15. In view of (3.24) and since is continuous, the process given by (3.20) can equivalently be redefined as the canonical -local martingale component of the discounted cumulative -value process . The processes and are easily seen to coincide on the random interval . Therefore, both and can be interpreted on this interval as the discounted cumulative -value of a defaultable game option.
The following result establishes a useful connection between and the arbitrage ex-dividend -price of the defaultable game option.
Proposition 3.16 (Verification principle for a defaultable game option). The process is the arbitrage ex-dividend -price for the game option. Moreover, for any , the saddle-point for the related Dynkin game (2.6) on is given by
Proof. In view of (3.4), the present assumptions imply that is the value of the Dynkin game (3.6), by Proposition 3.12, with saddle-point . Therefore, by Lemmas 3.5 and 3.6, is the value of the Dynkin game associated with the game option on , with saddle-point . Moreover, is a -semimartingale, by Lemma 3.14(ii). To conclude the proof, it suffices to make use of the last statement in Theorem 3.8.
4. Hedging in a Hazard Process Setup
In the remaining part of this work, we examine in some detail the existence and basic properties of hedging strategies for defaultable game options in a hazard process setup.
4.1. Cost Process of a Hedging Strategy
From now on, we will work under Assumption 3.13. Let thus denote a solution to () and let and be defined by (3.19). In particular, is the arbitrage -price for the game option (by Proposition 3.16) and the left-hand sides in (2.14) and (2.15) are equal to . Finally, recall that the -martingale 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  (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 , whereas conclusions are drawn relative to the filtration .
Proposition 4.1 (Hedging with a local martingale cost). Let be an arbitrary - valued and -integrable process. Then the following statements are valid.
(i) Let the process be given by and Then is an issuer hedge with -sigma (local, in case and are locally bounded) martingale cost.
(ii) Let the process be given by and Then is a holder hedge with a -sigma martingale (local martingale, when and are locally bounded) cost process.
Recall that, according to our convention (see Section 1.2), the -integrability of an -valued stochastic process implies its -predictability. Note also that the equality is valid for any process , since
Proof of Proposition 4.1. The arguments for a holder are essentially symmetrical to those for an issuer; we thus only prove part (i). By Lemma 3.14(i), the process is a -sigma martingale, and a -local martingale if and are locally bounded processes. For the ease of notation, we write . Let denote the wealth process of the primary strategy . By combining (2.9) with (4.1), we obtain and, for every , and thus where the second equality follows from (3.20). Recall that the stopping time is given by (see Proposition 3.16) In order to prove that the quadruplet is an issuer hedge for the game option, it is enough to show that one has for any with (cf. (2.13), From the definition of the minimality conditions in () and the continuity of it follows that and thus on . Since (4.5) thus yields where, by (), one has that In addition, by the definition of , one has that on the event . It is now easy to see that (4.7) is satisfied and thus is indeed an issuer hedge.
Remark 4.2. (i) The situation where can be made equal to zero by the choice of a suitable strategy 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 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 should be minimized in some sense as, for instance, in Schweizer .
(ii) It is possible to introduce the issuer trivial hedge (resp., the holder trivial hedge ) with the -local martingale cost Obviously, this hedge is of no practical interest, since it implicitly assumes that one is not interested in hedging any risks. The trivial hedge or, more precisely, the existence of any hedge is used in the proof of Proposition 4.3, however.
Let us now draw some conclusions from Lemma 2.13 and Proposition 4.1. In the context of specific (Cox-Ross-Rubinstein or Black-Scholes, say) models, analogous results can be found in Kifer . Our main contribution here is an extension of these results to the present setup involving a reduction of filtration, as well as to a fairly general class of semimartingale models. We use here the notation ess min (instead of a more common symbol essinf) in order to emphasize that the respective bounds are in fact attained.
Proposition 4.3. Under the assumptions of Proposition 4.1, the following statements are valid. (i)The equality holds, so that is the minimum of initial wealths of an issuer hedge with a -sigma martingale cost.(ii)One has that . If, in addition, (2.5) holds then and is the minimum of initial wealths of a holder hedge with a -sigma martingale cost.(iii)The above statements are also valid with local martingale instead of sigma martingale therein.
Proof. (i) By applying Proposition 4.1 to the trivial hedge of Remark 4.2(ii), we get, in particular, that , where is also equal to the -value of the related Dynkin game, by Proposition 3.16. Therefore, the infimum is attained and one has equality, rather than inequality, in Lemma 2.13(i).
(ii) In view of (2.5) and Lemma 2.13(ii), the second claim can be proven in the same way as part (i).
(iii) This follows immediately from parts (i) and (ii), since the cost of the trivial hedge is a -local martingale.
Given our definition of hedging with a cost and the definition of , the fact that there exists a hedge with an initial wealth and a -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 .
4.2. Risk Factors of a Defaultable Game Option
Let stand for the compensated default process. Under our standing assumption that the -hazard process of is a continuous and nondecreasing process (cf. Remark 3.2(ii), the process is known to be a -martingale. Recall also that the avoidance property holds, in the sense that for any -stopping time (cf. Remark 3.2(i).
An analysis of hedging strategies in the next section hinges on the following lemma, which yields the risk decomposition of the discounted cumulative value process of a defaultable game option. More formally, the martingale component (cf. Remark 3.15) is represented in terms of the pure jump martingale and a real-valued -martingale , which arise as the second component of a solution to the doubly reflected BSDE (3.7). Intuitively, the process 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 .
Lemma 4.4. The -martingale defined by (3.20) satisfies where the -predictable process equals .
Proof. Let us introduce the Doléans-Dade martingale (see, e.g., ) so that and . Then (cf. (3.19) and (3.20) It may happen that the -semimartingale fails to be also a -semimartingale, so a direct application of the (-)integration by parts formula to is not possible. However, by Lemma A.1(iv), the process stopped at is a -semimartingale. It is also clear that . Hence by applying the integration by parts formula to , we obtain where, in addition, one has that . Using the avoidance property of Remark 3.2(i), formula (3.22), and the assumptions that the coupon process is -predictable and the hazard process is continuous (so that ), we obtain the equality . Using (3.22), we next deduce from (4.13) that Using (2.3) and the equality , we finally arrive at the formula which is the required result.
4.3. Hedging of Risk Factors
In order to study nontrivial cases of hedging strategies for a defaultable game option in the general setup of this paper, we need to impose more assumptions on prices of primary traded assets. Since we are working in a fairly general framework, we will be able to provide only general results concerning hedging strategies. The interested reader is referred to the followup papers [4, 9] for a more detailed analysis of assumptions made in this section and particular examples.
First, we recall that the ex-dividend price of primary risky assets satisfies , for every , where the -valued, -adapted process formally represents the pre-default value of . We thus assume, by convention, that any residual value of the primary asset at is embedded in the recovery part of the dividend process for . We denote by an -valued and -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 stopped at .
Assumption 4.5. The dynamics under of the cumulative price process of primary risky assets are, for every , for some -valued -martingale , where the -valued, -predictable process is given by the equality for every .
By inserting (4.11) and (4.17) into (4.1), we obtain, for every , 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 -martingales and 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 . Therefore, without loss of generality, we may and do assume that is -predictable (see Lemma A.2(ii). This means that the reduction of filtration method can also be applied to hedging of a defaultable game option, and not only to its valuation as was already shown in Section 3.2.
Within the present framework, the event risk factor is common for all traded primary and derivative assets. Therefore, in the next step, we are going to get a closer look on pre-default risks of traded and derivative assets. To this end, we make a further standing assumption, in which the concept of the systematic risk factor (also known as the market risk factor) is introduced.
Assumption 4.7. We are given an -valued -martingale, denoted by , which is aimed to represent the systematic risk factor for the underlying market model. We postulate that the -martingales and of (4.11) and (4.17) satisfy the following decompositions, for every , where (resp., ) is some -adapted, -valued (resp., -valued), -integrable processes and (resp., ) is a real-valued (resp., -valued) -martingale.
It is natural to refer to -martingales and 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 as the event risk factor.
Remark 4.8. A specification of the systematic risk factor 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 and to vanish in (4.19). Once again, for more information on particular models, we refer to [4, 9] (see also Remark 3.10).
Let us denote and let stand for the concatenation of and . 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 and in the dynamics of the cost process of a trading strategy.
Lemma 4.9. For any -valued, -integrable process , the cost process satisfies, for every ,
Example 4.10. To provide some intuition underpinning the present setup, let us first consider a situation where the perfect hedgeability of risks can be achieved, at least in principle. Let us set and we take , so that vanishes. For , we further postulate that and or, equivalently, that for Assume that there exists a -integrable process solving the equation Then it follows from (4.20) (or (4.18) that the cost process vanishes and thus the strategy (resp., ) 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 obtained by solving (4.23) is -predictable (cf. Remark 4.6).
Remark 4.11. In , 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 admits a -integrable solution on . Then the cost process satisfies, for every ,
(ii) Assume that the equation admits a -integrable solution on . Then the dynamics of the cost process are, for every ,
Part (i) in Lemma 4.12 corresponds to the case where the common risks (systematic and event) can be completely eliminated. In contrast, part (ii) refers either to the case of unhedgeable event risk (e.g., when in dynamics (4.17) or to the situation when the issuer (or holder) is not willing to hedge that risk.
As was already mentioned, practically useful decompositions of and will depend on a particular model for the primary market, as well as on the game option under study. In an abstract setup considered here, they may be formally deduced from martingale representation theorems with orthogonal components.
Let thus stand for the class of real-valued -martingales with integrable quadratic variation over or, by a slight abuse of notation, the class of vector-valued processes with mutually strongly orthogonal components in . It is worth recalling here that an -martingale stopped at is also a -local martingale, by virtue of Lemma A.1(iii).
The Galtchouk-Kunita-Watanabe (GKW) decomposition of and with respect to and the filtration (see, e.g., Protter [16, Section IV.3, Corollary 1]) thus yields the decompositions (4.19) of and with and strongly orthogonal to in . Since is meant to represent the systematic risk factor, we may and do assume, without loss of generality, that the idiosyncratic risk factors and are also mutually strongly orthogonal.
The following proposition justifies the informal statement that the strategy (resp., ) hedges the risk factor (resp., ). We use hereafter the standard symbol to denote the square bracket between -semimartingales.
Proposition 4.13 (Orthogonality of risk factors). Assume that the processes and in decompositions (4.19) of and are mutually strongly orthogonal in .
(i) Under assumptions of Lemma 4.12(i), the processes and are orthogonal in , in the sense that is a -sigma martingale (and a -local martingale if is locally bounded).
(ii) Under assumptions of Lemma 4.12(ii), the processes and are orthogonal in , in the sense that is a -sigma martingale (and a -local martingale if and are locally bounded processes).
Proof. We first note that and are -local martingales, by Lemma A.1(iii). Since is strongly orthogonal to in , the process is a -local martingale, as an -local martingale stopped at (cf. Lemma A.1(iii). Furthermore, by Lemma A.6, is a -local martingale. We conclude that is a -local martingale. So are also and since the integral is strongly orthogonal to . Furthermore, by Lemma A.6, is a -local martingale.
Using (4.24), we conclude for part (i) that is a -sigma martingale and thus it follows a -local martingale if is a locally bounded process.
For part (ii), we conclude in view of (4.25) that is a -sigma martingale and thus a -local martingale if , and are locally bounded processes.
4.4. Hedging with Orthogonal Cost
Before concluding this work, let us examine briefly an alternative approach to hedging a defaultable game option, which is formally defined as the problem of finding a strategy that makes the cost process -orthogonal under a given risk-neutral probability measure to a predetermined -valued -local martingale where, without loss of generality, the process is assumed to be stopped at . In reference to Proposition 4.13, by the -orthogonality, we mean here that is a -local martingale under .
Remark 4.14. In the financial interpretation, the process may represent the wealth processes of some preexisting portfolios, rather than risk factors as in Sections 4.2 and 4.3. Admittedly, we consider here a reduced concept of hedging, at least from the theoretical perspective. It is possible to argue, however, that this approach may be of practical relevance, since some kind of a relative hedging (as opposed to replication or superhedging) is a common market practice.
For the purpose of this section, the process 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 or, more generally (cf. Remark 3.15), as the -local martingale component of the discounted cumulative -value process of a game option, provided that is a -special semimartingale.
In the following proposition we denote (whenever well-defined) and
Proposition 4.15 (Hedging with orthogonal cost). Assume that admits the decomposition, for every , with and orthogonal in and an -valued, -integrable process , which is left-invertible on . Let us set, for every , where is the left inverse of the transpose of on . Then the cost process is orthogonal to in .
Proof. By combining (4.1) with (4.27), we obtain Therefore, in order to have the cost orthogonal to in , it suffices to select a strategy for which is -orthogonal to . Relying on the multilinear regression formula, this can be achieved by setting as in (4.28).
The problem of hedging a defaultable game option with respect to 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 in with respect to , provided that the related matrix is left-invertible on