HEC MONTRÉAL. Pricing of synthetic CDO tranches, analysis of base correlations and an introduction to dynamic copulas.

Transcription

1 HEC MONTRÉAL Affiliée à l université de Montréal Pricing of synthetic CDO tranches, analysis of base correlations and an introduction to dynamic copulas Frederic Soustra Sciences de la gestion Mémoire présenté en vue de l obtention du grade de maîtrise ès science (M.Sc.) Septembre 26 Frédéric Soustra, 26

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3 Abstract This thesis studies the pricing of multi-name credit derivatives. The focus is set on the pricing of synthetic Collateralized Debt Obligation (CDO) tranches. A CDO is the name given to the process of securitizing a portfolio of loans or bonds. In our case the components of the portfolio are single name credit derivatives known as Credit Default Swaps (CDS). A CDS is an insurance contract against the default of a particular corporate bond. In the first part we propose our implementation of a pricing model using inverse fourier transforms to recover the distribution of defaults during the life of the portfolio. We also propose a generic monte carlo algorithm for the pricing of the tranches. In a second part we have constructed time series for the Investment Grade CDX index and tranche spreads, we extract base correlation parameters using the previously implemented model and show that there is significant serial dependence. Finally having justified the need for a dynamic model, financially and mathematically, we propose to introduce the framework for dynamic copulas. We present the basic theory of conditional copulas and show how they can be applied to time series context. We look at the gaussian conditional copula and the general Archimedean conditional copula. This thesis provides several innovations to a field that is still considered in its infancy. First we have a full pricing model capable of replicating market quotes. Secondly we provide the reader with a detailed econometric analysis of the base correlation time series of the index returns and the base correlation parameters. This has been very lightly studied in the literature due to the lack of data. Finally we propose a completely new way of looking at copulas and present the basics of dynamic copulas. This framework allows us to model the dynamic dependence of default times. i

4 Résumé Ce mémoire est une étude sur la tarification des dérivés de crédit appelées CDO synthétiques, où CDO veut dire Collateralized Debt Obligation. Un CDO est le nom donné au processus de titrisation de portefeuilles d obligations. Dans le cas de CDO synthétiques les sous-jacents sont des dérivés de crédit. Dans un premier temps nous présentons deux algorithmes de tarification. Le premier est basé sur les transformées de Fourier et permet de retrouver la distribution du nombre de défauts dans un portefeuille de crédit. Le premier modèle utilise une copule à facteur, ce qui permet de réduire la dimensionalité du problème. Le deuxième modèle utilise les copules pour simuler les temps de défaut et tarifer le spread d une tranche du CDO en utilisant la simulation Monte Carlo. Dans un second temps nous étudions les propriétés de la série temporelle de l indice de CDO appellé le CDX, puis nous introduisons le concept de corrélation de base, celle-ci étant l équivalent de la volatilité implicite dans le modèle de Black Scholes, mais appliquée à la tarification de CDO. Nous montrons que le paramètre de corrélation de base est fortement autocorrellé. Ceci est une justification formelle du besoin de modèles à corrélation dynamique. La dernière partie du mémoire s écarte du risque de crédit et propose une introduction à un modèle dynamique basé sur les copules. Nous proposons d étudier la copule associée à une distribution conditionnelle. Nous démontrons que dans le cas de la distribution normale ou les copule archimédiennes, la copule associée à la distribution conditionelle est une copule normale ou archimédienne. Nous illustrons les résultats de ce chapitre en proposant une étude d un portefeuille de neuf CDS, en donnant des résultats de simulations de chaînes de Markov pour chacune des deux modèles de copules dynamiques : la copule normale et la copule de Clayton. ii

5 Acknowledgements I would like to extended my gratitude to Nahed who has been at my side at every moment of despair. Bruno and Nicolas for their wisdom and knowledge. My parents for supporting me during the writing of this thesis. My friends, Nicholas, Alex, Jonathan, Xavier. All errors are that of my own. iii

6 CONTENTS Abstract Résumé Acknowledgements Contents List of Tables List of Figures i ii iii iv vii viii 1 Introduction Credit Risk Credit Derivatives Credit Default Swaps Basket Derivatives Dependent defaults Outline Literature Review Introduction Credit Risk Modelling Structural Models Reduced Form Models Models based on the default Transition Matrix Hybrid models Recovery Rate Portfolio loss distribution and CDO pricing Correlated Defaults in intensity models and structural models Simulation of the joint survival Conditionally independent Defaults Loss distribution estimation techniques Dynamic Credit Risk Dynamic Copulas Other dynamic methods Conditional Copulas Conclusion iv

7 CONTENTS v 3 CDO pricing and Base Correlations analysis Introduction The Data Default Free Yield Curves Standardized CDO tranches Credit Derivative Preliminaries: the CDS Hazard Rates Valuation of a CDS Pricing of Synthetic CDOs Tranche Losses Derivation of the fair spread of a CDO tranche: Pricing synthetic CDOs using Factor Copulas Individual Name default probabilities The Gaussian Copula Homogeneous portfolios of finite size Basic Results from the model Inhomogeneous portfolios of finite size Inputs Monte Carlo pricing of synthetic CDOs Empirical Analysis of the CDX IG Analysis of the CDX Index Level Serial Dependence analysis of base correlations AR and ARMA Models fitting of Base Correlations: Analysis of results Time Series Base correlation Time Series Serial Dependence Tests Introduction of a Dynamic Copula Introduction The conditional Copula Gaussian Conditional Copula Algorithm for generating a Markov chain {U t } T t= with stationary distribution C d,r1 and joint distribution of (U t 1,U t ) C 2d,R Clayton family Algorithm for generating a Markov chain {U t } T t= with stationary distribution C d,θ and joint distribution of (U t 1,U t ) C 2d,θ Archimedean copulas Algorithm for generating a Markov chain {U t } T t= with stationary distribution C d,ϕ and joint distribution of (U t 1,U t ) C 2d,ϕ Special Archimedean families Estimation Application example Gaussian Estimation Clayton Estimation Precision Conclusion Conclusion Comments Innovations Further Work

8 CONTENTS vi A Mathematical appendix 63 A.1 Fourier Transforms A.1.1 Using Discrete Transforms to recover a statistical distribution A.2 Theory of Copulas A.2.1 Elliptical Copulas A.2.2 Archimedean Copulas A.3 Maximum Likelihood estimators A.3.1 Gaussian Case B Tables 72 B.1 CDX Index Returns modelling B.2 CDX Base Correlations: ARMA(p,q) Residuals C Figures 77 C.1 Copulas C.2 CDO Pricing Models C.2.1 Default Surfaces C.2.2 CDX NA IG On-the-run Spreads C.3 Base Correlation Time Series Study C.3.1 Box Ljung Tests Plots Bibliography 87

9 LIST OF TABLES 1.1 S&P One year transition matrix (%), (Source: Roncalli (24)) Nominal Outstanding and semi annual growth of the CDS Market Standard Index Tranches and prices for Investment Grade CDX Tranches Attachment and Detachment points CDS Spreads for AT&T on the 21st of August 26, (Source: Bloomberg Terminal) P-values of Box-Ljung Test on the AR(1) Residuals for the On-the-run base correlations Portfolio of CDS Correlation matrix for the Gaussian copulas Correlation matrix for the dynamic Gaussian copulas Relative error (in %) between the Gaussian copula matrix and the dynamic Gaussian copula matrix Simulation Results Simulation Results for the Clayton copula, θ = B.1 Box Ljung p-values, for several AR fittings of the index return B.2 On the run CDX Index AR Parameters B.3 Box Ljung p-values B.4 P-values of Box-Ljung Test on the ARMA(1,1) Residuals for the first difference of the on-the-run CDX base correlations B.5 P-values of Box-Ljung Test on the ARMA(1,2) Resdiuals for the first difference of the on-the-run CDX base correlations B.6 P-values of Box-Ljung Test on the ARMA(1,3) Resdiuals for the first difference of the on-the-run CDX base correlations B.7 Time series statistics for the example portfolio B.8 Correlation Matrix of the CDS returns vii

10 LIST OF FIGURES 1.1 Nominal amount Outstanding of the CDS market (Source: ISDA) Payoff of a CDS, the seller of protection receives the spread s at each time period until the default or the maturity Typical CDO structure CDX NA IG Index Level Correlation smile of the Investment Grade CDX Tranches Base correlation for the CDX IG Series 5 on January 19th, Default probability as a function of time in years.the underlying CDS has a spread s i = 35bps (Equivalent to high yield bond CDS). And the recovery rate is assumed constant and equal to 4% Default probability as a function of time in years, when the intensity is stepwise constant, or using spline interpolation Loss profile of a Mezzanine tranche Example of the default probability surface using the factor copula model, here we set ρ =.4, and we assume the average CDS spread is 1 bps, the risk free curve is constructed using swap rates CDX NA IG Index Level CDX NA IG Index Returns histogram and estimated density kernels Spline fitted Base correlation curve for the CDX NA IG on August 19, 25, ρ is the correlation between each name and the macroeconomic factor Tranche Spread as a function of Compound Correlation for the CDX Equity Tranche, the horizontal line is the Market Quote Tranche Spread as a function of Compound Correlation for the CDX Senior Tranche (2% - 1%) Tranche Spread as a function of Compound Correlation for the CDX Mezzanine Tranche (3% - 6%), the red line is the Market Quote Base Correlation of the DJ CDX NA IG Series 5, from September 21, 25 to May 17, Log Log plot of the squared error of the MLE estimator Scatter plots of the two two sets of simulations C.1 Gaussian Copula Surface with ρ = C.2 Clayton Copula Surface, with θ = C.3 5 Samples from a bivariate Gaussian Copula C.4 5 Samples from a bivariate student-t Copula with ν = 1 df C.5 5 Samples from a bivariate Clayton Copula C.6 Impact of parameters on the loss distribution C.7 Equity Tranche upfront premium (in % upfront) C.8 Mezzanine Tranche Running Spread in bps viii

11 LIST OF FIGURES ix C.9 Junior Tranche Running Spread in bps C.1 Senior Tranche running spread in basis points C.11 Super Senior running spread in basis points C.12 Base Correlation of the DJ CDX NA IG Series 5, from September 21, 25 to May 17, C.13 Base Correlation of the DJ CDX NA IG Series 5, from September 21, 25 to May 17, C.14 Diagnostic plots of the AR(1) fitting for the Equity tranche base correlation C.15 Diagnostic plots of the ARMA(1,2) fitting for the Equity tranche base correlation

12 CHAPTER ONE Introduction Over the past decade risk management has witnessed a strong period of growth. Since the liquidity of derivatives market has increases as well, the types of risks covered have evolved and secondly the tools used to manage these risks became more reliable and more complex. Events such as Black Monday in 1987 or the LTCM crisis in 1998 show that risk modeling is an essential tool for financial institutions and for firms. It is commonly accepted that there are three types of risks that financial institutions must monitor. We briefly define these three types of risks below: Market Risk: is the risk of potential losses that are due to variations in the market conditions: prices, interest rates, volatility, foreign exchange rates... Operational Risk: We cite here the new definition given by the Bank of International Settlements as part of the the Basel II accords: Operational Risk is the risk of direct or indirect loss resulting from inadequate or failed internal processes, people and systems or from external events Credit Risk: Is the risk of loss due to the change in rating or default of a counterparty, here by default we mean either the default on the payment of a coupon, or the event where a company is unable to repay its outstanding debt. The next section will give a detailed of overview of Credit Risk, and why the world of credit derivatives has been growing at such a pace. 1.1 Credit Risk Credit risk can be broadly defined as the risk that an obligor will not honor his payment obligations. The regulation bodies such as the Bank for International Settlements (BIS) came to an accord of how credit risk is defined (The Basel II accords). Credit risk can be split into three broad categories: downgrade risk, movements in the credit spread and credit events. Most debt securities issued on the market are rated by one of the three rating agencies (Moody s, 1

13 CHAPTER 1. INTRODUCTION 2 Standard & Poors and Fitch Ratings). Usually rating agencies provide transition matrices which describe the probability of a rating change. We present below a simplified transition matrix. The state of default is here an absorbing state, that is once a company has defaulted it will stay in this state. Rating Final Initial Rating AAA AA A BBB BB B CCC D AAA AA A BBB BB B CCC D Table 1.1: S&P One year transition matrix (%), (Source: Roncalli (24)) Table 1.1 above is an example of a Standard and Poors transition matrix for the industry. Rating agencies compile such tables for many different industries. 1.2 Credit Derivatives Credit derivatives can be defined as a bilateral financial contracts that isolates credit risk from financial instruments. In essence credit derivatives are used for hedging corporate debt instruments Billions /21 7/21 1/22 7/22 1/23 7/23 1/24 7/24 1/25 7/25 1/26 Figure 1.1: Nominal amount Outstanding of the CDS market (Source: ISDA) The figure 1.1 and table 1.2 provide a more detailed description of the pace at which the CDS market has grown. Since 21 the annual growth rate has been around 9% per annum.

14 CHAPTER 1. INTRODUCTION 3 Semester ( trillion) s.a. growth % S S S S S S S S S S Table 1.2: Nominal Outstanding and semi annual growth of the CDS Market In the following paragraphs we describe the most popular credit derivatives. We only give a qualitative definition and will focus our attention on Collateralized Debt Obligations Credit Default Swaps The market for derivative securities whose price is contingent on the default risk of an obligor are now part of the mainstream contracts traded over the counter. The most basic credit derivative found in the markets is the credit default swap. Credit Default Swaps (CDS) are the basic material of which the credit derivatives market is made, it is the simplest form of credit risk transfer. We give a brief definition of a CDS below and we will illustrate the payoff function with a graphic. Definition 1.1 (Credit Default Swap). A Credit Default Swap (CDS) is a bilateral agreements between two counterparties (A and B) in which the credit risk of a third party C (usually called the reference entity) is transferred between A and B. During the life of a CDS the buyer of protection, A, periodically pays B a premium until the default of C or until the maturity of the CDS. In case of a well defined credit event, the seller B will pay the protection buyer A. The amount paid by B is commonly known has the loss given default. That is if δ is the recovery rate at the default time τ. The protection seller will pay 1 δ per unit of nominal. The figure 1.2 describes the cashflow structure of a vanilla CDS of nominal N, the payer of protection pays a spread s, usually quarterly. In the graph below, τ is the default time. And δ is the fractional recovery rate. That is the fraction of unit nominal that is recovered in case of a default. At the time of default the seller of protection pays the recovery value to the buyer of protection. The at the money CDS spread or premium is the one in which the initial value of the contract is zero. CDSs are priced using corporate bond data, and depend on the default term structure of a company Basket Derivatives A natural extension of so called single name credit derivatives is to create derivative products whose payoff function depends on the joint behavior of more than one CDS. These types of products are often called correlation products, because their primary use is to hedge against the co-movements of the underlying portfolio. The effect of an event of a portfolio is generally called correlation risk. The n th to

15 CHAPTER 1. INTRODUCTION 4 s(t 1 t ) s(t 2 t 1 ) s(t i t i 1 ) t = t 1 t 2 t i t i+1 τ t N = T (1 δ) Figure 1.2: Payoff of a CDS, the seller of protection receives the spread s at each time period until the default or the maturity. default swap for example is an extension of the basic CDS to a portfolio of reference entities. That is the buyer of protection pays a periodic spread until n components in the portfolio default. In case of a default, the seller of protection pays the fractional recovery value of the defaulted security. The popularity and the complexity of multi-name credit derivatives is increasing at the same pace as the CDS market. Because the industry had now adopted standard models to price single name credit derivatives, the focus is now on multi-name exotic credit derivatives. Since the inception of the Basel II accords, financial institutions must rethink how they manage their balance sheet. Especially because of the McDonough ratio. 1 Credit derivatives are the obvious choice of tool to alleviate capital requirements. Especially CDOs. Collateralized Debt Obligations Collateralized Debt Obligations (CDO) are products which generally belong to the family of securitized products. That is one packages a portfolio of assets into several securities each with a different loss structure. Collateralized Mortgage Obligations or Credit Linked Notes have been around for a while. In a CMO, a mortgage pool is split into several tranches, such as a PO tranche (Principal Only) or an IO (Interest Only) tranche. CLOs are usually used to package letters of credit. In a collateralized debt obligation, a pool of credit risky securities is packaged into several tranches. Each tranche absorbs a certain percentage of the losses incurred during the life of the product. The first-loss-tranche is usually called the equity tranche. Because in a typical CDO deal, the packager is the one who holds the first loss tranche. A tranche is thus defined by its attachment point and detachment point. For example a 2% 5% tranche will absorb losses greater than 2% but lesser than 5%. Suppose an investor has purchased protection on this 2 5 tranche, that is he pays a fixed spread to the seller of protection quarterly. Suppose that there are 1 names in this portfolio. If one name defaults the loss incured represents 1% of the portfolio. Subsequently if two more names default, then the cummulative losses represent 3% of the portfolio. In this case the owner of protection on the 2 5 tranche will be compensated by the seller of protection since the losses have exceeded the attachment point. Of course the worse case scenario is if all the names in the basket default at the same time. In this case each each buyer of protection will 1 The ratio is also called the solvability ratio. As an example the ratio for a bank is: Capital Requirements Market Risk + Credit Risk + Operational Risk 8%

16 CHAPTER 1. INTRODUCTION 5 receive the Loss Given Default that pro-rata of the protection it had previously purchased. Figure 1.3 shows how a typical CDO is structured and broken down into tranches. Figure 1.3: Typical CDO structure

17 CHAPTER 1. INTRODUCTION 6 Cash CDOs In a cash CDO the underlying portfolio is usually composed of corporate bonds. This type of deal is very useful for financial institutions, because it alleviates the capital requirements. 2 Cash CDOs are usually big and complicated deals, this is due to the legal burden of transferring the ownership of the underlying portfolio to an offshore special purpose vehicle. Moreover in a cash CDO deal, all the tranches must be sold to the exception of the equity tranche which is usually held by the arranger of the deal. Synthetic CDOs CDOs are very useful products but cash deals are complicated and the legal burden might sometimes offset the profits from setting up a static CDO. 3 Therefore with the popularity of credit default swaps, one can create a CDO whose underlying portfolio is composed of credit default swaps. This is very practical because it allows the CDO arrangers to forgo the setup of the SPV and the legal burden associated with it. A synthetic CDO is thus considered to be a multi-name credit derivative. The market for synthetic CDOs has been flourishing over the past years especially since the introduction of the CDX and Itraxx indices. Standardized CDOs The popularity of synthetic CDOs in the European and US markets has led to the creation of Single Tranche CDOs (STCDO). These indices are called the CDX in north America and the Itraxx in Europe. They are standardized tranches of a certain predefined pool of CDS that can be actively traded. Table 1.3 shows a market quote on January 26, 26 for the investment grade tranches. The equity tranches are quoted as an upfront payment with a 5 bps running spread. The other tranches are quotes as running spreads. North America Europe Tranche Upfront (%) Running (bp) Tranche Upfront (%) Running (bp) -3% % 29. % 5 3-7% % % % % % % % Table 1.3: Standard Index Tranches and prices for Investment Grade CDOs of CDOs: CDO 2 Sometimes even CDO tranches are packaged into a CDO, we then speak of CDO squared, CDO squared are relatively new, and complex to evaluate, but are gaining in popularity amongst hedge fund, because they are good tools for correlation arbitrage. Custom CDOS: Bespoke tranches The popularity of the CDX indices has pushed the complexity of the CDOs that are traded, custom STCDOs are often refereed to as bespoke CDOs. In order to correctly price these custom tranches one must find a good measure of dependence to be sure that the quoted price is the correct one. 2 As explained earlier 3 In a static deal the package is not touched until the maturity of the CDO, in a market deal a portfolio manager actively trades the underlying portfolio

18 CHAPTER 1. INTRODUCTION Dependent defaults The most complex part of multi-name credit derivatives modeling is the effect of a default or any other credit event of one obligor on the rest of the basket. This is commonly referred in the literature as default dependence. Our aim is to use CDX index data to show that there is serial dependence in the returns of the CDX index and serial dependence in the correlation parameter itself. As a recent example the downgrade of General Motors Corp. and Ford in may 25 has had a very strong effect on basket derivatives where it was included. 8 IG 3 IG 4 IG 5 On the Run 7 6 Idx Level Series 3 Series 4 Series 5 Series 6 2 Sep4 Dec4 Mar5 Jul5 Oct5 Jan6 May6 Aug6 Time Figure 1.4: CDX NA IG Index Level On figure 1.4, we can see the spike which corresponds to the downgrade of the two auto makers. The general consensus in the literature is that the most complex part of credit derivatives valuation is the modeling of default dependence, what happens when one or more names enter a state of default? Implied Correlation In the equity derivatives world, market participants use the traditional Black and Scholes implied volatility to quote options (see Black and Scholes (1973)). In the market for synthetic CDOs participants use a similar approach. The implied correlation or compound correlation is the correlation parameter that replicates the market quote. In essence, one only needs to solve a pricing model for the correlation parameter to obtain compound correlations. The drawback of this correlation measure is that for tranches other than the equity tranche (%,K l %) and the upper tranche (K u %,1%) the solution is not unique. Here K i represents the attachment or detachment point. 4 In fact for the equity tranche, the fair spread is monotonically decreasing with respect to the correlation parameter. Because of this flaw in the compound correlation, the concept of base correlation was introduced in 24 by JP Morgan to standardize the way of quoting the CDX and Itraxx tranches, or any STCDO tranche. Figure 1.5 is a plot of the compound correlation for the CDX IG tranches. Firstly, there is a correlation smile, or skew. The standard Gaussian model yields a flat term structure of correlation. Moreover the close to correlation for the mezzanine tranche is not an error. Because of the distribution of defaults in the underlying basket there is a tranche that has a implied correlation parameter. This concept is usually referred to as being correlation neutral. This would mean that the holder of a tranche whose implied correlation is is not affected by co-movements in the underlying portfolio. 4 The first loss tranche is usually called the equity tranche, that is its attachment point is.

19 CHAPTER 1. INTRODUCTION 8 Base Corr Att. Pt Figure 1.5: Correlation smile of the Investment Grade CDX Tranches Base Correlations Base Correlations are a way to observe the correlation smile effect on a CDO, the base correlation is the parameter which corresponds to the quote of an equity tranche. A bootstrapping method is used to find the base correlation curve. We will present the topic later in the thesis. Base Corr Att. Pt Figure 1.6: Base correlation for the CDX IG Series 5 on January 19th, 26 Figure 1.6 shows the base correlation curve that corresponds to the quotes observed on the indicated day. While our goal in this thesis is not to validate the pricing model for the CDO tranche nor the validity of the base correlation parameter as the true measure of the average dependence amongst the names in the basket, for the purpose of this thesis we will treat base correlation as a quantity that represents the level of the co-movements in the portfolio. Why dynamic dependence? There are many models which can replicate market quotes of the standard multiname credit derivatives. Many authors try to reproduce the correlation smile directly, since the Gaussian copula model produces a flat correlation structure for the tranches. The double t factor copula of Hull and White (24) is able to reproduce the correlation smile observed in the market. Our goal is not to find a model which reproduces the market quotes but rather find a model which takes into account the dynamic form of dependence, that is dependence across time, as well as dependence across the assets in the basket. The CDX tranches are very useful, because they can be used as a benchmark for the prices of synthetic CDOs. The general rule of thumb in the literature is to reproduce the standard tranche spreads to validate a model. Then interpolate of the calibrated model to price non standard tranches.

20 CHAPTER 1. INTRODUCTION Outline In the next chapter we propose a detailed literature review of the pricing of synthetic CDOs. From the modeling of the individual default probabilities, the recovery rate assumptions to the modeling of the aggregate loss distribution. Thus, the next chapter will be ordered as follows: we first briefly describe the principal models for credit risk, we then go over the modeling of the recovery rate, then we present the literature on default dependence and synthetic CDO pricing, the fourth part of the literature review is dedicated to dynamic default dependence modeling. Our goal in this thesis is to present a model for the pricing of CDOs using a dynamic form of default dependence. This approach has two applications which we will briefly present in the third chapter. The third chapter is an implementation of a STCDO pricing model, we will use the implemented model to find a time series of correlation parameters. We hope that our empirical analysis of the later will demonstrate that dynamic dependence is the next step in the modelling of multiname credit derivatives. The fourth chapter is the presentation of our dynamic copula. In this chapter we will depart from the pricing of credit derivatives and rather focus on the establishment of a mathematical framework for the conditional/dynamic copulas and present ideas of how they could be used to price CDOs.

21 CHAPTER TWO Literature Review 2.1 Introduction The pricing of synthetic CDOs requires the knowledge of three categories of data, first the individual name default probability, then the individual recovery rate and finally the portfolio loss distribution, the portfolio loss distribution depends on the first two parameters and requires a choice on the dependence structure of defaults. In the first part of this chapter, we will briefly review the current literature on credit risk modeling. We start with the description of the two main families of models used for the estimation of single name default probabilities: the Structural Models which uses the capital structure of the firm to determine the risk of a default and the reduced form models who model the default by trying to find the distribution of the first time until default. The second section will present a brief description of how the Loss Given Default (or Recovery rate) is modeled in the literature, and what are the usual assumptions that are made in the context of multiname credit derivatives pricing. The third part of this literature review will be a detailed presentation of the different methods used to model the dependence of defaults, and more specifically the portfolio loss distribution over different time horizons. Our focus is on the family of models that use copulas to model the joint behavior of defaults. We justify the need for models that use some form of dynamic dependence structure. 2.2 Credit Risk Modelling Structural Models Structural models rely on the capital structure of the firm to calculate the probability that it will default. They were the first models used to asses the credit risk of corporate bonds. Structural models suppose that a firm can repay its creditors as long as the value of its debt is greater than that of its equity. Therefore equity is seen as an option on the default of the firm. If this option is out of the money, therefore the company is in default. 1

22 CHAPTER 2. LITERATURE REVIEW 11 The Merton Model The Merton model was the first model in the credit risk literature to compute the default probability of a firm using its capital structure. The Merton (1974) model assumes a firm structure in a Modigliani and Miller sense, that is we denote {V t } the firm value at time t. The firm value is then decomposed in two components: V t = S t + D t. Where S t is the value of equity and D t is the value of the liabilities of a firm. In this model, the debt is supposed to be a zero coupon bond B t with nominal N which pays 1 at its maturity T. The Merton (1974) paper uses the same framework than the seminal paper by Black and Scholes (1973). That is, we consider a market with continuous trading that is frictionless. The author also assumes existence of a default free rate r f. Therefore in this model the time horizon the value of the firm follows a geometric Brownian motion, whose underlying wiener process is defined on some filtered probability space (Ω, F,P). The parameters of the process {V t } are constant. Hence, since the market is complete under the risk free measure Q, one can price the corporate bond B t as a claim on the value of the firm. Merton (1974) thus expresses S t and B t as follows: B T = min (N,V T ) = N (N V T ) + (2.1) S T = (V T N) + (2.2) This says that at the maturity of the bond B T the firm defaults if the value of its assets V T is below the face value of the bond: N. The bond is then expressed as the difference between the face value N and a put on the value of the firm with strike N. The equity is considered as a call on the value of the firm with strike price N. It is then apparent that using equations (2.1) and (2.2) one can derive the risk neutral probability of default: Q(V T < N). We refer the reader to the chapter two of Lando (24) for further details. The Merton (1974) model has some drawbacks that do not make it tractable to price a synthetic CDOs, first the default can only occur at the end of the time horizon T, that is obvious since in essence the author proposes to use the Black and Scholes (1973) European option formula to evaluate the corporate liability B t. The implication of this limitation is that short term implied spreads are close to zero. Which is not what is observed in the market. Duffie and Lando (21) explain that this is due to the asymmetry of information between the equity holders and the debt holders is not included in the model. Of course one has to be able to model all of the debt of the firm as a zero coupon bond. Moreover it is unclear which measure of volatility should be chosen. Since we price the value of the firm as an option it is the main factor which will determine the default probability. Generally practitioners use the equity implied volatility and adjust it by a factor to represent the volatility of the debt. Several extensions of the Merton model were proposed Longstaff and Schwartz (1995) and Cox and Black (1976) are two important papers who followed in the steps of Merton (1974). Default Barrier Models Barrier models also make the same assumptions than Merton (1974) about the firm value process, but they introduce a barrier (threshold level). The default occurs when the firm value hits the barrier. Thus the default can occur at any time before the time horizon T. The first paper to describe such a model is Cox and Black (1976), in this model the default time, denoted τ is a stopping stopping time of the

23 CHAPTER 2. LITERATURE REVIEW 12 following form: τ = inf {t > ;V t < H(t)} Where {V t } is the firm value process and H(t) is the new component in the model: the barrier. The barrier process H can be stochastic. This relaxation of the Merton (1974) model allows for the default to occur at any time before the maturity of the credit risky debt. The Black and Cox model has been extended in Longstaff and Schwartz (1995). Longstaff and Schwarz criticize the fact that in Black and Cox the interest rates are held constant and independent of the default time, they propose a model where interest rates are stochastic and correlated to defaults. The firm value follows a geometric brownian motion as in Merton (1974) but the interest rate follows a simple Ornstein Uhlenbeck mean reverting process. The authors find that credit spreads are negatively correlated with interest rates levels. However, the model supposes that the firm value or indirectly the default time is independent of the capital structure of the firm. In the next paragraph we present a recent barrier model which does rely on the capital structure of the firm, but returns to the assumption that interest rates and default time are independent. Finger (22) consider the barrier whose value is based on the debt to equity ratio of the firm. That is the firm value follows a geometric brownian motion with zero drift, the authors note that {V t } is not a real firm value process but rather an index that measures the evolution of the credit quality of the firm. The barrier is LD = LDe λz λ2 /2, where D is the Leverage ratio, and L is log-normal with mean L and standard deviation λ. Then the default is calculated using the initial parameters related to the firm. This model is also known as the CreditGradesçmodel, the main advantage of this model compared to the Merton (1974) model is that in the case the barrier is stochastic, then the instantaneous default probability is non zero, hence the short term credit spread is non zero, this does solve the main problem of the Merton model. Recently Hull et al. (25) use a barrier model similar to the one in Cox and Black (1976) to price multiname credit derivatives, we will give further details about this paper in a later section. Structural models rely on information pertaining to the capital structure of a particular obligor, the other credit risk model family in the literature are the intensity models, these models are fundamentally different from structural model in that they attempt to model the default time of a firm Reduced Form Models Reduced form model propose a completely different view of how default occurs, here the default is considered to be a random event who is governed by a stochastic process commonly called hazard rate or intensity, the relationship between the firm value and the default here is not explicit. Reduced form models are attractive from a mathematical standpoint because they offer closed form formulas in some cases. Also, there is a great affinity between default free term structure modelling and the modeling of credit risky term structures. The simplest case of reduced form model is if the process N t is a homogeneous Poisson process. In this case the default probability can be obtained directly as Pr {τ t} = 1 e λt. This assumption is often made in models who concentrate on the dependence of default. However many researchers consider models where both the jump size and the jump arrival are stochastic, these processes are called Cox Processes. We consider a inhomogeneous Poisson process N with positive intensity h( ) (see Lando

24 CHAPTER 2. LITERATURE REVIEW 13 (1998)). The probability of a jump of size k is then: Pr {N t N s = k} = 1 k! t s { h(u)du exp t s } h(u)du k =,1,... One of the fundamental results of Lando (1998) is that the authors notes that the default time of a Cox Process is: { τ = inf t : t } h(u)du E Where E is a unit exponential variable. In this case, the probability of default by time t is: ( F (t) = Pr {τ t} = 1 exp t ) h(u)du (2.3) and the function S ( ) which is usually referred to as the survival function: ( S (t) = Pr {τ t} = exp t ) h(u)du (2.4) Jarrow and Turnbull (1995) suppose that default free interest rates are independent of credit risk. Therefore their model uses a short rate process to model the risk free curve. They combine this with the reduced form model presented above and provide a methodology to price claims contingent on the default of the firm. Intensity models are very tractable because they provide a similar framework to that of short rate models. That is they can be calibrated to the existing term structure of a company, contrary to the Merton model where one needs to value the assets of the firm and compute a zero coupon bond for the debt of the firm. Intensity models thus provide analytical solutions for the price of defaultable bonds or more complex credit products. We present here a brief description of the classical affine model, a more detailed explanation can be found in Duffie (25). The intensity dynamics is described by the following process. dλ(t) = κ [θ λ(t)] dt + σ λ(t)dw(t) + J(t) (2.5) Here, J(t) denotes the jump process at time t, which is independent of the brownian motion W. J is a classical Poisson process with exponentially distributed jump arrival times of mean µ and jump intensity l. In this case the authors refer to the process λ as an affine process with parameters (κ,θ,σ,µ,l). In this case, the conditional survival probability is given by P t (τ > t + s) = E t [ t+s t ] λ(u)du = e α(s)+β(s)λ(t) In this case the functions α( ) and β( ) can be obtained in closed form. This model provides the default probability of an individual obligor. The parameters can be estimated from market data such as corporate bond prices or CDS spreads. For further reference on intensity models we direct the reader to Bielecki and Rutkowski (22).

25 CHAPTER 2. LITERATURE REVIEW Models based on the default Transition Matrix This type of model is often assimilated with the reduced form models, the default dependence here is modeled using transition matrices (see table 1.1). The individual default probabilities can be modeled using either a structural or a reduced form model. The Jarrow and Turnbull (1995) model was later extended in Jarrow et al. (1997) where the authors suppose that the transition matrix is a generator matrix for a Markov chain. In this paper the bankruptcy process is represented by a discrete Markov chain of the credit ratings. As mentioned earlier, a fundamental difference in this type of model is that under the pricing measure, spot interest rates are considered independent of the bankruptcy process. More recently, Berrada et al. (26) studied the impact of the choice of copula to price standard basket derivatives. The authors use a continuous Markov chain of the rating matrix to model the default dependence Hybrid models There are several cases where researchers used intensity based models where they consider the hazard rate of default as being directly related to the firm value, such a model is found in Madan and Unal (2). The model in Madan and Unal is a two factor hazard rate model where which depends on the capital structure of the firm, this model is able to overcome the problem of short term spreads that is observed in Merton (1974). In the same spirit Duffie and Lando (21) considers a stochastic intensity model where the firm value follows a geometric brownian motion. The resulting spreads of the afore mentioned papers match with the observed data. 2.3 Recovery Rate The recovery rate is the second component needed to price credit derivatives. It represents the percentage value that is recovered in case of a default. We present a brief list of papers that consider a random recovery value in case of a default. According to Moody s investor services, there are three types of recovery assumptions: 1. Recovery of face value This is the main approach in the market, one values the recovery as a fraction of the current face value of the instrument. 2. Recovery of market value This is used to measure the change when the default occurs. To asses this value one needs prices of the security before the default and after the default has occurred. 3. Recovery of treasury In this case, the bond who defaulted is replaced with a treasury bond with same maturity. Andersen and Sidenius (25) considered an extension of the classical Gaussian copula model to include random recovery rate. In Laurent and Gregory (25) and Laurent and Gregory (24) the authors suppose that the recovery rate follows a beta distribution, and show how using discrete fourier transform they can obtain the distribution of default times in the portfolio. The same authors have also extended their model to allow for stochastic correlation. Our focus is on the dependence structure of defaults, we will therefore follow the general assumption which is to use a constant recovery rate. Recovery rates can be estimated from default data obtained from one of the main rating agencies. However most pricing models use a recovery rate δ of 4%.

26 CHAPTER 2. LITERATURE REVIEW Portfolio loss distribution and CDO pricing The third component of multi-name credit derivatives is the specification of the default dependence, this topic is very active in the literature, we will thus restrict ourselves to the papers considered mainstream in the literature. Default dependence studies the effects of the change in credit of one firm in an economy and how the other firms are affected. In the next paragraph, we briefly describe the loss distribution on a portfolio of credit risky securities. The goal here is to compute the loss distribution at time t [,T], T being the maturity of the derivative instrument. We denote the loss by L(t). Each name i = 1,...,n has nominal M i. N i (t) = I(τ i t) is the indicator process of the default time for the ith obligor, and δ i is the fractional recovery rate of name i. n L(t) = (1 δ i ) M i N i (t) (2.6) i=1 The construction of this discrete distribution is the key component of multi name credit derivative pricing or credit portfolio management. Once we obtain the distribution, that is: p(l,t) = Pr {L(t) = l} We can build EL(t) = E [L(t)] the expected loss on the portfolio of n names. In order to value multi-name credit derivatives, (CDOs or nth to default swaps) one has to either estimate the loss distribution over the horizon [,T] or simulate the default times. In the next paragraph we present the main methods used to model the joint dynamics of the default times. We finish this section by presenting some papers who try to model the dynamic dependence structure of the portfolio Correlated Defaults in intensity models and structural models The case of Intensity Models This method that was proposed in Duffie and Gârleanu (21). The authors show how to price cash or synthetic CDO in the diffusion based intensity framework. Recall that in the Duffie affine framework the dynamics of the hazard rate follow equation (2.5). The authors show that if X and Y are two affine processes with parameters (κ,θ X,σ,µ,l X ) and (κ,θ Y,σ,µ,l Y ) then X +Y is a basic affine process with parameters (κ,θ X +θ Y,σ,µ,l X +l Y ). The process X is of the following form: dx(t) = κ(θ X X(t))dt + σ X X(t)dW(t) + J(t) κ is the mean reversion parameter θ X is the long term mean of the process σ X is the volatility W(t) is the Weiner process J(t) is the jump process independent of W