Credit Risk: Modeling, Valuation and Hedging

A default risk is a possibility that a counterparty in a financial contract will not fulfill a contractual commitment to meet her/his obligations stated in the contract. If this actually happens, we say that the party defaults, or that the default event occurs. More generally, by a credit risk we mean the risk associated with any kind of credit-linked events, such as: changes in the credit quality (including downgrades or upgrades in credit ratings), variations of credit spreads, and the default event. The spread risk is thus another components of credit risk. To facilitate the analysis of complex agreements, it is important to make a clear distinction between the reference (credit) risk and the counterparty (credit)risk. The first generic term refers to the situation when both parties of a contract are assumed to be default-free, but due to specific features of the contract the credit risk of some reference entity appears to play an essential role in the contract’s settlement. In other words, the reference risk is that part of the contract’s risk, which is associated with the third party; i.e., with the entity, which is not a party in a given agreement. In the present context, the third party is referred to as the reference entity of a given contract. Credit derivatives are recently developed financial instruments that allow market participants to isolate and trade the reference credit risk. The main goal of a credit derivative is to t ransfer the reference risk, either completely or partially, between the counterparties. In most cases, one of the parties can be seen as a buyer of an insurance against the reference risk. Such a party is called the seller of the reference risk; consequently, the party that bears the reference risk is referred to as its buyer.

The structural approach is primarily directed at pricing the firm’s liabilities. In this methodology, the firm’s liabilities are seen as contingent claims issued against the total value of firm’s assets; for this reason this approach is also referred to as the firm value approach or the option-theoretic approach.

The concepts introduced in the previous chapter will now be extended to a more general set-up, when allowance for a larger flow of information — formally represented by some reference filtration F — is made.

In Sect. 4.5, we have introduced the concept of the martingale hazard function of a random time and we have examined the connection between this concept and the notion of the hazard function. It appeared, that both notions coincide if and only if the cumulative distribution function of τ, and thus also its hazard function, are continuous (see Proposition 4.5.1). In this sense, the martingale hazard function uniquely characterizes the unconditional probability distribution of a continuously distributed random time. On the other hand, we have shown in Sect. 5.1.3 (see Proposition 5.1.3) that if the F-hazard process is continuous, the process H _t − Γ _t⋀τ follows a G-martingale. The main goal of this chapter is to extend the concept to the case of a non-trivial filtration, and to examine whether a continuous F-martingale hazard process uniquely specifies the 1F-conditional survival probabilities of a random time.

In this chapter, we assume throughout that we are given a finite collection τ ₁,...,τ _n of random times, defined on a common probability space (Ω, ς, ℙ) endowed with a filtration F. We define the family of jump processes H ⁱ , i= 1, ..., n by setting \(H_t^i = {11_{\left\{ {{\tau _i}t} \right\}}}\) and we write ℍ ⁱ to denote the filtration generated by the jump process H^t. Let us introduce the enlarged filtration G by setting G=ℍ¹ ν...ν ℍⁿ νF One of our goals is to examine the relationship between the (F,G) -martingale hazard processes of random times τ ₁.... , τ _n , and the (F,G) -martingale hazard process of their minimum, i.e., of the random time τ = min (τ ₁, ...τ _n).

As already mentioned in Sect. 1.4, existing approaches to the modeling of credit (or default) risk may be divided into two broad classes: structural models and reduced-form models. In the former approach, the total value of the firm’s assets is directly used to determine the default event, which occurs when the firm’s value falls through some boundary. It results that here the default time is a predictable stopping time with respect to the reference filtration modeling the information flow available to the traders. This means that the random time of default is announced by an increasing sequence of stopping times. By contrast, in the latter approach, the firm’s value process either is not modeled at all, or it plays only an auxiliary role of a state variable. The default time is modeled as a stopping time that is not predictable; the default event thus arrives as a total surprise. Formally, the random time of default event is given here as a totally inaccessible stopping time, in the terminology of the general theory of stochastic processes (see, e.g., Dellacherie (1972) or Jacod and Shiryaev (1987)). The main tool in this approach is an exogenous specification of the conditional probability of default, given that default has not yet occurred. Since in most cases this is done by means of the hazard rate (or intensity) of default, reduced-form models are also commonly known as hazard rate models or intensity-based models. From a long list of papers devoted to the reduced-form methodology, let us mention a few: Artzner and Delbaen (1995), Jarrow and Turnbull (1995), Duffle et al. (1996), Duffle and Singleton (1997, 1999), Lando (1998), Schlögl (1998), Schönbucher (1998b), Wong (1998), Elliott et al. (2000), and Bélanger et al. (2001).

The next two chapters are devoted to the study of mutually dependent default times within the framework of the intensity-based approach. In case of conditionally independent default times studied in this chapter, we are able to establish closed-form pricing results for the i ^th-to-default contingent claim. In general, the issue becomes much more complicated, and we only provide partial results in the next chapter.

In this chapter, we continue the study of the intensity-based approach to the modeling of dependent defaults. In Sect. 10.1, we shall analyze the ideas presented in a recent paper by Jarrow and Yu (2001). Then, in Sect. 10.2, we will analyze the martingale approach to the valuation of basket credit derivatives. For related results, the interested reader may consult Duffie (1998), Duffle and Singleton (1998b), Hull and White (2000, 2001), as well as to Lando (2000b), who examines the issue of modeling correlated defaults within the framework of a ratings-based model.

Our next goal is to present the most basic notions and results from the theory of discrete- and continuous-time Markov chains. As expected, the emphasis is put here on these properties that are relevant from the viewpoint of credit risk modeling. Throughout this chapter, we fix an underlying probability space (Ω, G, ℚ), as well as a finite set K = {1,..., K}, which plays the role of the state space for all considered Markov chains. Since the state space is finite, it is clear that any function h: K → ℝ is bounded and measurable, provided that we endow the state space with the σ-field of all its subsets.

In Chap. 8, the reduced-form approach was applied to the study of a particular kind of a credit event, namely, the default. In this chapter, we shall consider several possible credit events within the framework of the intensity-based methodology. More specifically, we are going to examine the issue of dynamical modeling of credit migrations of a corporate bond between several possible rating grades (or credit ratings). In other words, we shall focus on the modeling of changes over time in the credit quality of reference names; such changes are henceforth referred to as credit migrations.

In the context of the modeling of the defaultable term structure, the HJM methodology was first examined by Jarrow and Turnbull (1995) and Duffie and Singleton (1999). Their studies were undertaken by Schönbucher (1996, 1998a), who has studied in a systematic way various forms of the no-arbitrage condition between the default-free and defaultable term structures. More recently, some of these results were re-discovered by Maksymiuk and Gątarek (1999) and Pugachevsky (1999), who focused on the arbitrage-free dynamics under the spot martingale measure of the instantaneous forward credit spreads. Subsequently, the HJM methodology was extended by Bielecki and Rutkowski (1999, 2000a, 2000b) and Schönbucher (2000a) to cover the cases of term structure models with multiple ratings for corporate bonds. Eberlein and Õzkan (2001) generalize this approach by considering models driven by Lévy motions (for related results, also see Eberlein and Raible (1999) and Eberlein (2001)). In contrast with models presented in the previous chapter, the credit migration process is not exogenously specified, but it is endogenous in a model. It follows a conditionally Markov process with respect to a reference filtration under the spot (or forward) martingale measure.

In this chapter, we formally introduce several possible interest rate contract structures in the presence of the counterparty risk. Such contracts in turn give rise to several concepts of credit-risk related LIBOR and swap rates, referred to as the defaultable market rates, or more specifically, defaultable LIBOR rates and defaultable swap rates. We examine spot and forward rates associated with single- and multi-period contracts that are subject to either unilateral or bilateral counterparty risk, and we derive several formulae for various kinds of defaultable market rates. The classification and the terminology introduced in what follows is merely tentative, though. It is interesting to note that certain interest rate contracts that are subject to the counterparty credit risk may be equivalently restated as interest rate contracts involving a reference credit risk. However, when we analyze defaultable interest rate swaps, we always assume that the underlying reference floating interest rate is the default-free LIBOR rate.

In Chap. 13, we presented an approach to the modeling of defaultable term structure based on the Heath-Jarrow-Morton modeling methodology. As the underlying building blocks that served to produce a model of default-free and defaultable term structures, we have used there the dynamics of instantaneous, continuously compounded, forward interest rates.