- 原书名：Credit Risk: Modeling, Valuation and Hedging (Springer Finance)
- 原出版社： Springer
Part I. Structural Approach
1. Introduction to Credit Risk
1.1 Corporate Bonds
1.1.1 Recovery Rules
1.1.2 Safety Covenants
1.1.3 Credit Spreads
1.1.4 Credit Ratings
1.1.5 Corporate Coupon Bonds
1.1.6 Fixed and Floating Rate Notes
1.1.7 Bank Loans and Sovereign Debt
1.1.8 Cross Default
1.1.9 Default Correlations
1.2 Vulnerable Claims
1.2.1 Vulnerable Claims with Unilateral Default Risk
1.2.2 Vulnerable Claims with Bilateral Default Risk
1.2.3 Defaultable Interest Rate Contracts
1.3 Credit Derivatives
1.3.1 Default Swaps and Options
Although in the first chapter we provide a brief overview of issues related to credit risk, our goal was to introduce the basic concepts and related no-tation, rather than to describe the financial and economical aspects of this important sector of financial market. The interested reader may consult, for instance, Francis et al. (1999) or Nelken (1999) for a much more exhaustive description of the credit derivatives industry.
The main objective of this monograph is to present a comprehensive sur-vey of the past developments in the area of credit risk research, as well as to put forth the most recent advancements in this field. An important aspect of this text is that it attempts to bridge the gap between the mathematical the-ory of credit risk and financial practice, which serves as the motivation for the mathematical modeling studied in this book. Mathematical developments are presented in a thorough manner and cover the structural (value-of-the-firm) and the reduced-form (intensity-based) approaches to credit risk modeling,applied both to single and to multiple defaults. In particular, this book offers a detailed study of various arbitrage-free models of defaultable term struc-tures of interest rates with several rating grades.
This book is divided into three parts. Part I, consisting of Chapters 1-3, is mainly devoted to the classic value-of-the-firm approach to the valuation and hedging of corporate debt. The starting point is the modeling of the dynamics of the total value of the firm's assets (combined value of the firm's debt and equity) and the specification of the capital structure of the assets of the firm. For this reason, the name structural approach is frequently attributed to this approach. For the sake of brevity, we have chosen to follow the latter convention throughout this text.
Modern financial contracts, which are either traded between financial in-stitutions or offered over-the-counter to investors, are typically rather com-plex and they involve risks of several kinds. One of them, commonly referred to as a market risk (such as, for instance, the interest rate risk) is relatively well understood nowadays. Both theoretical and practical methods dealing with this kind of risk are presented in detail, and at various levels of mathe-matical sophistication, in several textbooks and monographs. For this reason,we shall pay relatively little attention to the market risk involved in a given contract, and instead we shall focus on the credit risk component.
As mentioned already, Chapter 1 provides an introduction to the basic concepts that underlie the area of credit risk valuation and management. We introduce the terminology and notation related to defaultable claims, and we give an overview of basic market instruments associated with credit risk.We provide an introductory description of the three types of credit-risk sen-sitive instruments that are subsequently analyzed using mathematical tools presented later in the text. These instruments are: corporate bonds, vul-nerable claims and credit derivatives. So far, most analyses of credit risk have been conducted with direct reference to corporate debt. In this con-text, the contract-selling party is typically referred to as the borrower or the obligor, and the purchasing party is usually termed the creditor or the lender.However, methodologies developed in order to value corporate debt are also applicable to vulnerable claims and credit derivatives.
To value and to hedge credit risk in a consistent way, one needs to develop a quantitative model. Existing academic models of credit risk fall into two broad categories: the structural models and the reduced-form models, also known as the intensity-based models. Our main purpose is to give a thor-ough analysis of both approaches and to provide a sound mathematical basis for credit risk modeling. It is essential to make a clear distinction between stochastic models of credit risk and the less sophisticated models developed by commercial companies for the purpose of measuring and managing the credit risk. The latter approaches are not covered in detail in this text.
The subsequent two chapters are devoted to the so-called structural ap- proach. In Chapter 2, we offer a detailed study of the classic Merton (1974) approach and its variants due to, among others, Geske (1977), Mason and Bhattacharya (i981), Shimko et al. (1993), Zhou (1996), and Buffet (2000). This method is sometimes referred to as the option-theoretic approach, since it was directly inspired by the Black-Scholes-Merton methodology for valua- tion of financial options. Subsequently, in Chapter 3, a detailed study of the Black and Cox (1976) ideas is presented. We also discuss some generaliza- tions of their approach that are due to, among others, Brennan and Schwartz (1977, 1980), Kim et al. (1993a), Nielsen et al. (1993), Longstaff and Schwartz (1995), Briys and de Varenne (1997), and Cathcart and EI-Jahel (1998). Due to the way in which the default time is specified, the models worked out in the references quoted above are referred to as the first-passage-time models.
Within the framework of the structural approach, the default time is defined as the first crossing time of the value process through a default trig-gering barrier. Both the value process and the default triggering barrier are the model's primitives. Consequently, the main issue is the joint modeling of the firm's value and the barrier process that is usually specified in relation to the value of the firm's debt. Since the default time is defined in terms of the model's primitives, it is common to state that it is given endogenously within the model. Another important ingredient in both structural and reduced-form models is the amount of the promised cash flows recovered in case of default, typically specified in terms of the so-called recovery rate at default or,equivalently, in terms of the loss-given-default. Formally, it is thus possible to single out the recovery risk as a specific part of the credit risk; needless to say, the spread, the default and the recovery risks are intertwined both in practice and in most existing models of credit risk. Let us finally men-tion that econometric studies of recovery rates of corporate bond are rather scarce; the interested reader may consult, for instance, the studies by Altman and Kishore (1996) or Carty and Lieberman (1996).
The original Merton model focuses on the case of defaultable debt instru-ments with finite maturity, and it postulates that the default may occur only at the debt's maturity date. By contrast, the first-passage-time technique not only allows valuation of debt instruments with both a finite and an infinite maturity, but, more importantly, it allows for the default to arrive during the entire life-time of the reference debt instrument or entity.
The structural approach is attractive from the economic point of view as it directly links default events to the evolution of the firm's capital structure,and thus it refers to market fundamentals. Another appealing feature of this set-up is that the derivation of hedging strategies for defaultable claims is straightforward. An important aspect of this method is that it allows for a study of the optimal capital structure of the firm. In particular, one can study the most favorable timing for the decision to declare bankruptcy as a dynamic optimization problem. This line of research was originated by Black and Cox (1976), and it was subsequently continued by Leland (1994), An-derson and Sundaresan (1996), Anderson, Sundaresan and Tychon (1996),Leland and Toft (1996), Fall and Sundaresan, (1997), Mella-Barral and Per-raudin (1997), Mella-Barral and Tychon (1999), Ericsson (2000), Anderson,Pan and Sundaresan (2000), Anderson and Sundaresan (2000).
Some authors use this methodology to forecast default events; however,this issue is not discussed in much detail in this text. Let us notice that the structural approach leads to modeling of default times in a way which does not provide any elements of surprise - in the sense that the resulting random times are predictable with respect to the underlying filtrations. This feature is the source of the observed discrepancy between the credit spreads for short maturities predicted by structural models and the market data.
In Part II, we provide a systematic exposition of technical tools that are needed for an alternative approach to credit risk modeling - the reduced-form approach that allows for modeling of unpredictable random times of defaults or other credit events. The main objective of Part II is to work out various mathematical results underlying the reduced-form approach. Much attention is paid to characterization of random times in terms of hazard functions,hazard processes, and martingale hazard processes, as well ss to evaluating relevant (conditional) probabilities and (conditional) expectations in terms of these functions and processes. In this part, the reader will find various pertinent versions of Girsanov's theorem and the martingale representation theorem. Finally, we present a comprehensive study of the problems related to the modeling of several random times within the framework of the intensity-based approach.
The majority of results presented in this part were already known; how-ever, it is not possible to quote all relevant references here. The follow-lng works deserve a special mention: Dellacherie (1970, 1972), Chou and Meyer (1975), Dellacherie and Meyer (1978a, 1978b), Davis (1976), Elliott (1977), Jeulin and Yor (1978), Mazziotto and Szpirglas (1979), Jeulin (1980),Bremaud (1981), Artzner and Delbaen (1995), Duffle et al. (1996), Duffle (1998b), Lando (1998), Knsuoka (1999), Elliott et al. (2000), Belanger et al.(2001), and Israel et al. (2001). Let us emphasize that the exposition in Part II is adapted from papers by Jeanblanc and Rutkowski (2000a, 2000b, 2002).
Part III is dedicated to an investigation of diverse aspects of the reduced-form approach, also commonly referred to as the intensity-based approach.To the best of our knowledge, this approach was initiated by Pye (1974) and Litterman and Iben (1991), and then formalized independently by Lando (1994), Jarrow and Turnbull (1995), and Maclan and Unal (1998). Fur-ther developments of this approach can be found in papers by, among otb-ers, Hull and White (1995), Das and Tufano (1996), Duffle et al. (1996),SchSnbucher (1996), Lando (1997, 1998), Monkkonen (1997), Lotz (1998,1999), and Collin-Dufresne and Solnik (2001).
In many respects, Part III, where we illustrate the developed theory through examples of real-life credit derivatives and we describe market meth-ods related to risk management, is the most practical part of the book. In Chapter 8, we discuss the most fundamental issues regarding the intensity-based valuation and hedging of defaultable claims in case of single reference credit. From the mathematical perspective, the intensity-based modeling of random times hinges on the techniques of modeling random times developed in the reliability theory. The key concept in this methodology is the survival probability of a reference instrument or entity, or, more specifically, the haz- ard rate that represents the intensity of default. In the most simple version of the intensity-based approach, nothing is assumed about the factors generat- ing this hazard rate. More sophisticated versions additionally include factor processes that possibly impact the dynamics of the credit spreads.
Important modeling aspects include: the choice of the underlying proba-ility measure (real-world or risk-neutral - depending on the particular appli-cation), the goal of modeling (risk management or valuation of derivatives),and the source of intensities. In a typical case, the value of the firm is not included in the model; the specification of intensities is based either on the model's calibration to market data or on the estimation based on historical observations. In this sense, the default time is exogenously specified. It is worth noting that in the reduced-form approach the default time is not a predictable stopping time with respect to the underlying information flow.In contrast to the structural approach, the reduced-form methodology thus allows for an element of surprise, which is in this context a practically ap-pealing feature. Also, there is no need to specify the priority structure of the firm's liabilities, ss it is often the case within the structural approach.However, in the so-called hybrid approach, the value of the firm process, or some other processes representing the economic fundamentals, are used to model the hazard rate of default, and thus they are used indirectly to define the default time.
Chapters 9 and 10 deal with the case of several reference credit entities.The main goal is to value basket derivatives and to study default correlations.In case of conditionally independent random times, the closed-form solutions for typical basket derivatives are derived. We also give some formulae related to default correlations and conditional expectations. In a more general situa-tion of mutually dependent intensities of default, we show that the problem of quasi-explicit valuation of defaultable bonds is solvable. This should be con-trssted with the previous results obtained, in particular, by Kusuoka (1999) and Jarrow and Yu (2001), who seemed to suggest that the valuation prob-lem is intractable through the standard approach, without certain additional restrictions.
In view of the important role played in the modeling of credit migrations by the methodologies based on the theory of Markov chains, in Chapter 11 we offer a presentation of the relevant aspects of this theory.
In Chapter 12, we examine various aspects of credit risk models with multiple ratings. Both in case of credit risk management and in case of val-uation of credit derivatives, the possibility of migrations of underlying credit name between different rating grades may need to be accounted for. This reflects the basic feature of the real-life market of credit risk sensitive instru-ments (corporate bonds and loans). In practice, credit ratings are the natural attributes of credit names. Most authors were approaching the issue of mod-eling of the credit migrations from the Markovian perspective. Chapter 12 is mainly devoted to a methodical survey of Markov models developed by,among others, Dss and Tufano (1996), Jarrow et al. (1997), Nakazato (1997),Duffle and Singleton (1998a), Arvanitis et al. (1998), Kijima (1998), Kijima and Komoribayashi (1998), Thomas et al. (1998), Lando (2000a), Wei (2000), and Lando and Skodeberg (2002).