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 [
In Bielecki et al. [
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.
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
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 [
Our preliminary results for hedging strategies in a hazard process setup, Propositions
Throughout this paper, we use the concept of the
By default, we denote by
After recalling some fundamental valuation results from [
We assume throughout that the evolution of the underlying primary market is modeled in terms of stochastic processes defined on a filtered probability space
Specifically, we consider a primary market composed of the savings account and of
the
the
The primary risky assets, with
(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
(ii) Any locally bounded sigma martingale is a local martingale, and any bounded from below sigma martingale is a supermartingale.
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 [
a dividend stream with the
a terminal
a terminal
a terminal
The (possibly random) time
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
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 are now in the position to state the formal definition of a defaultable game option.
A the the the inequality the
The following result easily follows from Definition
(i) For any
(ii) For any
We further assume that
Symmetrically, we should sometimes additionally assume that
We will state the following fundamental pricing result without proof, referring the interested reader to [
Assume that a process
Note that defaultable American (or European) options can be seen as special cases of defaultable game options.
A
In view of Theorem
We adopt the definition of hedging game options stemming from successive developments, starting from the hedging of American options examined by Karatzas [
Recall that
By a (self-financing)
The reason why we do not assume that
The
Given the wealth process
Consider the game option with the ex-dividend cumulative discounted cash flows
An a cost process a (fixed) call time the following inequality is valid, for every put time A a cost process a (fixed) put time the following inequality is valid, for every call time
Issuer or holder hedges
(i) The process
(ii) Regarding the
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
In the sequel, we work under a fixed, but arbitrary, risk-neutral measure
All the measure-dependent notions like
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
(i) One has (by convention,
(ii) If inequality (
(i) Assume that for some stopping time
(ii) Let
In order to get more explicit pricing and hedging results for defaultable game options, we will now study the so-called
Given an
Let
In the sequel, we will work under the following standing assumption.
We assume that the process
(i) The assumption that
(ii) If
Some consequences of Assumption
The
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
(i) The discount factor process
(ii) The coupon process
(iii) The recovery process
(iv) The payoff processes
(v) The call protection
The next lemma shows that the computation of the lower and upper value of the Dynkin games (
One has that
For
Under our assumptions, the computation of conditional expectations of cash flows
For any stopping times
Formula (
Note that
Assuming condition (
The following result is the converse of Theorem
Let
Theorems
In this section, we will characterize the arbitrage ex-dividend
Given an additional
By a ( the all conditions in (
By the
The state process
In applications (see [
Basically, in any model endowed with the martingale representation property, the existence (and uniqueness) of a solution to (
(i) Since
Indeed the related integrands here and in the third line of (
(ii) Since
(iii) In the special case where all
(iv) In the context of a Markovian setup, the probabilistic BSDE approach may be complemented by a related analytic
In order to establish a relationship between a solution to the related doubly reflected BSDE and the arbitrage ex-dividend
Observe that if
Let
Except for the presence of
Let us now apply Proposition
The doubly reflected BSDE (
Let us stress that Assumption
We denote, for
(i) The process
(ii) The process
(iii) The process
(i) The triplet
(ii) In view of (
(iii) By (
In view of (
The following result establishes a useful connection between
The process
In view of (
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.
From now on, we will work under Assumption
Let us stress that some of the key arguments underlying the following result are classical, and they are already contained in Lepeltier and Maingueneau [
Let
(i) Let the process
(ii) Let the process
Recall that, according to our convention (see Section
The arguments for a holder are essentially symmetrical to those for an issuer; we thus only prove part (i). By Lemma
(i) The situation where
(ii) It is possible to introduce the issuer
Let us now draw some conclusions from Lemma
Under the assumptions of Proposition The equality One has that The above statements are also valid with local martingale instead of sigma martingale therein.
(i) By applying Proposition
(ii) In view of (
(iii) This follows immediately from parts (i) and (ii), since the cost
Given our definition of hedging with a cost and the definition of
Let
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
The
Let us introduce the Doléans-Dade martingale (see, e.g., [
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 [
First, we recall that the ex-dividend price
The dynamics under
By inserting (
In what follows, we will only be interested in hedging on the random interval
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
We are given an
It is natural to refer to
A specification of the systematic risk factor
Let us denote
For any
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
In [
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 (
(i) Assume that the equation
(ii) Assume that the equation
Part (i) in Lemma
As was already mentioned, practically useful decompositions of
Let thus
The Galtchouk-Kunita-Watanabe (GKW) decomposition of
The following proposition justifies the informal statement that the strategy
Assume that the processes
(i) Under assumptions of Lemma
(ii) Under assumptions of Lemma
We first note that
Using (
For part (ii), we conclude in view of (
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
In the financial interpretation, the process
For the purpose of this section, the process
In the following proposition we denote (whenever well-defined)
Assume that
By combining (
The problem of hedging a defaultable game option with respect to
The following result examines the special case when
Assume that
Under the present assumptions, we obtain from (
We recognize here a strategy, which is known to arise in the context of the
Recall that an
We work throughout under the standing Assumption
(i)
(ii) The
(iii) Any
(iv) Any
(v) The integral process of a continuous integrand with respect to an
Since
We recall the following well-known results. We refer the interested reader to Bielecki and Rutkowski [
(i) Let
(ii) For any
In the
For any
(i) If
(ii) If
Since
For (ii), let
Let us recall that for any
Assume that
(i) For any
(ii) For any
(iii) For any finite variation
(i) Since
(ii) If suffices to prove the formula for an elementary predictable process of the form
(iii) One has that
In the next result,
For any
Let us write
The research of T. R. Bielecki was supported by NSF Grant 0202851 and Moody’s Corporation grant 5-55411. The research of S. Crépey was supported by Ito33 and the Europlace Institute of Finance. The research of M. Jeanblanc was supported by Ito33, FIRN, and Moody’s Corporation grant 5-55411 and Fédération Bancaire Française. The research of M. Rutkowski was supported by the ARC Discovery Project DP0881460.