The main reason behind this phenomenon has been the success of sophisticated quantitative methodologies in helping professionals to manage financial risks. The newly developed credit derivatives industry has grown around the need to handle credit risk, which is one of the fundamental factors of financial risk. In recent years, we have witnessed a tremendous acceleration in research efforts aimed at better apprehending, modeling and hedging of this kind of risk.
One of the objectives has been to understand links between credit risk and other major sources of uncertainty, such as the market risk or the liquidity risk. The main objective of this monograph is to present a comprehensive survey ofthe past developments in the area of credit risk research, as well as 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 theory of credit risk and the financial practice, which serves as the motivation for the mathematical modeling studied in the book. Mahtematical 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, the book offers a detailed study of various arbitrage-free models of defaultable term structures with several rating grades.
This book will serve as a valuable reference for financial analysts and traders involved with credit derivatives. Some aspects of the book may also be useful for market practitioners with managing credit-risk sensitives portfolios.
Graduate students and researchers in areas such as finance theory, mathematical finance, financial engineering and probability theory will benefit from the book as well.
On the technical side, readers are assumed to be familiar with graduate level probability theory, theory of stochastic processes, and elements of stochastic analysis and PDEs; some acquaintance with arbitrage pricing theory is also. View 1 excerpt, references background. A new approach to modelling of credit risk, to valuation of defaultable debt, and to pricing of credit derivatives is developed.
Our approach, based on the HJM methodology, uses the available … Expand. This paper provides a Markov model for the term structure of credit risk spreads.
The model is based on Jarrow and Turnbull with the bankruptcy process following a discrete state space Markov … Expand. This paper develops a corporate bond valuation model that takes into account both early default and interest rate risk. It corrects a defect of recent contributions where pricing equations do not … Expand. Modelling default correlation in bond portfolios. The performance of a CBO structure is critically dependent on the correlation of defaults in a medium-sized portfolio of bonds.
We introduce two classes of purely probabilistic models to handle these … Expand. Counterparty Risk and the Pricing of Defaultable Securities. Motivated by recent financial crises in East Asia and the U.
View 2 excerpts, references background. Related Papers. Abstract 1, Citations 47 References Related Papers. By clicking accept or continuing to use the site, you agree to the terms outlined in our Privacy Policy , Terms of Service , and Dataset License.
Exogenous bankruptcy refers to the case when bankruptcy is specified in form of some protective covenants, such as positive net-worth covenant, or when bankruptcy is triggered at an exogenously specified asset value, for instance, the principal value of the debt. We shall discuss some results related to the optimal capital structure in Sect. For a more exhaustive analysis of the strategic debt service, we refer to the original pa- pers by Leland , , Anderson and Sundaresan , , Leland and Toft , Mella-Barral and Perraudin , Ericsson and Reneby , Mella-Barral or Ericsson The mathematical concept of safety covenants associated with a corporate debt was introduced in literature dealing with the structural approach to credit risk in order to specify the default event.
Generally speaking, a safety covenant is modeled as a barrier process also called a threshold process , usually denoted as v in what follows. In most cases, the default event is triggered when the firm value process V falls below the barrier process v either prior, or at the maturity date T. For the purpose of this text, we choose to use the term safety covenants in order to describe any mechanism, which triggers default event before the maturity of the debt. Depending on the situation, a credit spread may be expressed, e.
The generic term term structure of credit spreads will refer to the term structure of such differences. The determination of the credit spread is in fact the ultimate goal of most credit risk models. It is also the topic of several econometric studies. Some authors concentrate on direct modeling of the credit spread, rather than on its derivation from other fundamentals.
Such an approach appears to be very convenient when one deals with these credit derivatives that have the credit spread as the underlying instrument. Distressed securities can thus be defined by directly referring to the high level of credit spreads yielded by some corporate securities, should they not default. A more narrow definition of a distressed security encompasses publicly held and traded debt or equity securities of firms that have defaulted or have filed for protection under the bankruptcy code.
For a detailed analysis of the concept of a distressed security, we refer to Altman Credit ratings are typically identified with elements of a finite set, also referred to as the set of credit classes or credit grades. First, many major financial institutions main- tain their own credit rating systems, based on internally developed method- ologies, and therefore known as the internal ratings.
Finally, the improvement deterioration, resp. For more information on existing rating systems, we refer to Altman , Carty , Crouhy at al. In this text, the generic term credit rating or credit quality is used to describe any classification of corporate debt that can be justified for specific purposes. The former one should be seen as a theoretical construct that is meant to facilitate the analysis of the latter.
The corporate coupon bond with continuous coupon rate is widely used in financial literature, particularly in relation with the structural approach, in order to study quantitative and qualitative behavior of corporate debt. Let us briefly describe a corporate coupon bond with coupon payments occurring at discrete time intervals. It should be stressed here that the coupon payments are only made prior to the default time.
A coupon bond may thus be considered as a portfolio composed of the following securities: — defaultable coupons, which sometimes are equivalent to defaultable zero- coupon bonds with zero recovery,2 — defaultable face value, which can be seen as a defaultable zero-coupon bond with, generally speaking, non-zero recovery. Assume, for instance, that the recovery payment is proportional to the face value and that it is made at maturity T, in case the default event occurs before or at the maturity date.
A corporate coupon bond described by 1. A similar terminological convention applies to defaultable zero-coupon bonds and, indeed, to all kinds of defaultable claims. Consider the two fixed- coupon bonds, a risk-free bond and a defaultable one, with otherwise identical covenants. If both bonds trade at par i. This is indeed observed in the market practice, and the corresponding discrepancy is referred to as the fixed-rate credit spread over Treasury for a given corporate bond.
As already noticed, the credit spread reflects the credit quality of the issuer, as perceived by the market — the financial market requires a higher risk premium for lower quality debt, so that the cost of capital for a debtor of lower credit quality is higher. Let us mention that the terms credit risk and spread risk are used interchangeably by market practitioners. A corporate floating rate note FRNs, for short is another important example of a defaultable debt.
For otherwise comparable notes, the higher level of the credit spread s usually corresponds to the lower credit quality of the issuer. Introduction to Credit Risk Let us examine the credit spread over the risk-free floating rate of an FRN that trades at par.
In view of our current convention concerning the recovery scheme cf. For the sake of computational simplicity, it is frequently assumed in financial modeling that these two random factors are mutually independent. In the market practice, both callable and putable FRNs are common.
The issuer of a callable FRN has the right to redeem the note before its maturity. The holder of a putable FRN has the right to force an early redemption. The changes in credit quality determine whether option exercise is advantageous. This means that typically a floating-rate note has also an embedded credit derivative, specifically, a call or put option on the value of a note. Therefore, a corporate fixed- or floating-rate bond may serve as a natural example of a credit-risk sensitive contract with unilateral default risk.
Syndicated bank loans. Syndicated bank loans SBLs are primarily large, high grade commercial loans. In recent years, a considerable growth has been observed in the secondary trading on the market of syndicated bank loans. This has been paralleled by the emergence of bank loan ratings.
In many respects the syndicated bank loans are similar to corporate bonds, and thus investors are now considering SBLs as substitutes or complements to corpo- rate bonds. We refer to a recent article by Altman and Suggitt , who present a thorough empirical analysis of the default rates on the market of syndicated bank loans. Sovereign debt. As an important sector of the sovereign debt market, let us mention the so-called Brady bonds.
Brady bonds were issued by several less developed countries. They are primarily denominated in U. Typically, they contain various forms of credit guarantees and protections, so that it is rather hard to isolate the country-specific credit spread that is embedded in yields on Brady bonds.
The definition that we want to adopt reflects the fact that the cross-default covenant basically corresponds to provi- sions in loan agreements or bond indentures, which trigger an event of default if the counterparty borrower or issuer defaults on another obligation.
The purpose of this provision is to protect a creditor or counterparty from actions favoring another creditor. For in- stance, abstract corporate bonds with the same maturity date are considered to be different defaultable claims if they have different initial credit ratings, or if their recovery covenants differ. Consider also a period of time contained within the lifetimes of the two claims. Introduction to Credit Risk which takes value 1, if the first claim defaults during the specified period of time, and takes value 0 otherwise; an analogous random variable associated with the second claim is denoted by Y.
By convention, the default correlation between the two defaultable claims is defined as the correlation coefficient be- tween the random variables X and Y.
Default correlations are an important building block of credit risk measurement and management methodologies for credit-risk sensitive portfolios, mentioned in Sect. From the theoret- ical perspective, the issue of modeling correlated defaults was addressed by, among others, Duffie and Singleton , Davis and Lo , , Jarrow et al. We refer also to Sect. The default risk of a counterparty or of both parties is thus an important component of financial risk embedded in a vulnerable claim; it should necessarily be taken it into account in valuation and hedging procedures for vulnerable claims.
On the other hand, the un- derlying reference assets are assumed to be insensitive to credit risk. Credit derivatives are recently developed financial instruments, which allow for a secluded trading in the reference credit risk. In contrast to vulnerable claims, in which the counterparty risk appears as a nuisance or a side effect, credit derivatives are tailored as highly specialized and effective devices to handle or transfer the reference credit risk.
Since credit derivatives are offered over- the-counter, a credit derivative typically represents also a vulnerable claim, though, unless the counterparty risk is negligible. The default risk of the holder of the option is manifestly not relevant. Consider a vulnerable European call option on a default-free U -maturity zero-coupon bond — that is, a vulnerable claim with no reference risk.
The last option may serve as a simple example of a hybrid derivative; its valuation will involve both the reference and the counterparty risks. The prime ex- ample are here swap agreements between two default-prone entities, known as defaultable swaps. Despite the similarity of names, a defaultable swap should not be confused with a default swap, which is in fact a form of insur- ance against the reference risk the latter kind of contracts is explained in Sect.
In contrast to default-free swaps, alternative settlement rules in case of default may largely influence the valuation of defaultable swaps.
Typically, it is assumed that swaps are subordinate to debt in bankruptcy. We shall follow this convention here. Thus, it is natural to assume that if the party that is in default on its original debt is due to make a swap payment, it will default also on the swap contract. If, on the other hand, the party in default is due to receive a swap payment, two alternative settlement rules can be exam- ined: i the swap payment is received, or ii the swap payment is withheld.
If the latter rule is adopted a swap becomes valueless in case of default. In the former case, the swap payment at default has option-like features, and thus the total value of a swap contract depends, in particular, on the value of the embedded option.
We shall now introduce some basic notions related to default-prone interest rate contracts; for more details, see Chap. A more detailed study of selected types of single- and multi-period defaultable interest rate contracts is postponed to Chap.
We shall refer to T as the reset date and to U as the settlement date. The following features of default-free interest rate agreements are worth stressing. First, the actual timing of the payments is not essential. The above equivalences are, of course, valid from the perspective of the in- ception time T only. Second, the covenants of the interest rate agreement described above invoke exchange of principal payments. Essentially, we deal here with a loan in which the debtor the borrower, or the payer in the present context may default on his obligation to repay the debt.
In Chap. The basic type of a spot default-free interest rate swap is the spot fixed- for-floating swap for the accrual period [T, U ], settled in arrears, with the spot default-free LIBOR rate L T being the reference floating rate. It is evident that in the default-free environment the spot swap rate and the spot LIBOR rate coincide. Essentially, in the default-free environment, the loan agreements and the interest rate swaps are equivalent.
As shown in Chap. In other words, in the presence of a counterparty risk, the loan and the swap contract are not equivalent to each other. More specif- ically, the reference security of a credit derivative can be any financial instru- ment that is subject to risk of default or, more generally, to the credit risk.
For example, an actively-traded corporate or sovereign bond, or a portfolio of these bonds, may serve as an underlying asset or index for such a derivative. A credit derivative can also have a loan or a portfolio of loans as the un- derlying reference credit. It is clear that a credit derivative derives its value from the price — and thus from the credit quality — of the underlying default prone credit instrument.
The first agreements for a secluded transfer of credit risk were only signed in the early s. It is thus worthwhile to mention that financial arrange- ments with features similar to credit derivatives — such as the letter of credit or the bond insurance — were largely used by commercial banks much earlier. Under a bond insurance, an issuer pays an insurer to guar- antee the performance on a bond for more details, see, e.
However, in contrast to credit derivatives, these more traditional credit risk protections are not tradeable separately from the underlying obligation. Credit derivatives can be structured in a large variety of ways; they are typically complex agreements, customized to the specific needs of an investor. Due to the rapidly growing demand, in the past few years the credit deriva- tives were the worldwide fastest-growing derivative products.
As estimated by J. Morgan, in April the total nominal value of outstanding credit derivatives has passed the 1, billion U. The common feature of all credit derivatives is the fact that they allow for the transfer of the credit risk from one counterparty to another; they thus constitute a natural and convenient tool to control the credit risk exposure.
The overall risk an investor is concerned with involves two components: market risk and asset- specific credit risk. For an extensive analysis of economical reasons that support the use of these products, as well as expositions of vari- ous aspects of credit derivatives markets, we refer to Duffee and Zhou , Das a, b , Tavakoli , Francis et al.
We shall now focus for a moment on credit derivatives associated with the defaultable term structure. Similarly as in the case of derivative securities associated with the risk-free term structure, we may formally distinguish three main types of agreements: forward contracts, swaps, and options.
In a forward contract, the default risk is normally borne by the long party. If a credit event occurs, the transaction is marked to market and unwound.
In market practice, the most popular credit-sensitive swap contracts are: total rate of return swap, asset swap and default swap explained in some detail in Sect. Credit options are typically embedded in complex credit- sensitive agreements, though the over-the-counter traded credit options, such as: options on an asset swap or credit spread options, are also available.
Credit derivatives can also be classified into three groups. II is adapted from papers by Jeanblanc and Rutkowski a, b,. Credit Risk: Modeling, Valuation and Hedging. Bielecki, Marek Rutkowski. Marek Rutkowski. Structural Models. Mathematical developments are presented in a thorough biellecki and cover the structural value-of-the-firm and the reduced intensity-based approaches to credit risk modeling, applied both to single and to multiple defaults.
An important aspect of this text is that it attempts to bridge the gap between the mathematical theory of credit risk and the financial practice, which serves as the motivation for the mathematical modeling studied in the book. Review quote From the reviews: In particular, the book offers a detailed study of various arbitrage-free models of defaultable term structures with several rating grades. Modeling of Market Rates. Some aspects of the book may also be useful for market practitioners engaged in managing credit-risk sensitive portfolios.
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. Article information Source Ann.
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