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    The Art of Credit Derivatives: Demystifying the Black Swan
    The Art of Credit Derivatives: Demystifying the Black Swan
    The Art of Credit Derivatives: Demystifying the Black Swan
    Ebook543 pages4 hours

    The Art of Credit Derivatives: Demystifying the Black Swan

    By Joao Garcia and Serge Goossens

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    About this ebook

    Credit derivatives have been instrumental in the recent increase in securitization activity. The complex nature and the size of the market have given rise to very complex counterparty credit risks.  The Lehman failure has shown that these issues can paralyse the financial markets, and the need for detailed understanding has never been greater.

     

    The Art of Credit Derivatives shows practitioners how to put a framework in place which will support the securitization activity.  By showing the models that support this activity and linking them with very practical examples, the authors show why a mind-shift within the quant community is needed - a move from simple modeling to a more hands on mindset where the modeler understands the trading implicitly.

     

    The book has been written in five parts, covering the modeling framework; single name corporate credit derivatives; multi name corporate credit derivatives; asset backed securities and dynamic credit portfolio management.

     

    Coverage includes:

    • groundbreaking solutions to the inherent risks associated with investing in securitization instruments
    • how to use the standardized credit indices as the most appropriate instruments in price discovery processes and why these indices are the essential tools for short term credit portfolio management
    • why the dynamics of systemic correlation and the standardised credit indices are linked with leverage, and consequently the implications for liquidity and solvability of financial institutions
    • how Lévy processes and long term memory processes are related to the understanding of economic activity
    • why regulatory capital should be portfolio dependant and how to use stress tests and scenario analysis to model this
    • how to put structured products in a mark-to market-environment, increasing transparency for accounting and compliance.

     

    This book will be invaluable reading for Credit Analysts, Quantitative Analysts, Credit Portfolio Managers, Academics and anyone interested in these complex yet important markets.

     

    LanguageEnglish
    PublisherWiley
    Release dateFeb 16, 2010
    ISBN9780470684962
    The Art of Credit Derivatives: Demystifying the Black Swan

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      The Art of Credit Derivatives - Joao Garcia

      1

      Introduction

      If you put the federal government in charge of the Sahara Desert, in five years there’d be a shortage of sand.

      Milton Friedman

      The credit derivatives market surged from USD 200 billion in 1997 to an astonishing USD 55 trillion in 2008. The largest growth happened in 2006 and 2007. When associated with the securitization process, the CDS asset class was in the driving seat of the enormous economic and consumption expansion that took place in the world economy in the post-internet bubble years.

      A proper and detailed introduction to credit derivatives can be found in many books already on the market. For an overview of the credit derivatives market, the available instruments, their valuation and trading strategies we refer to the JP Morgan Credit Derivatives Handbook (JP Morgan, 2006) and to the Morgan Stanley Structured Credit Insights books (Morgan Stanley, 2007a; 2007b). For an introduction to stochastic calculus for finance we refer to Shreve (2004a; 2004b), and to Bingham and Kiesel (2004). For an introduction to credit risk modeling we refer to Bluhm et al. (2003). We refer to Schönbucher (2003) and O’Kane (2008) for credit derivatives pricing. A classic work on options, futures and other derivatives is the book by Hull (2003). For an overview of the bond market we refer to Fabozzi (2004).

      This book complements the above references in many respects. First, we focus on the standardized credit indices. Second, we try not to focus only on the instrument and the models but also on the market developments, attempting to adopt a very critical view when using a model. Third, we show models to price instruments, both standardized credit indices and bespoke tranches. Fourth, we show models for portfolio management purposes of bespoke credit portfolios. Fifth, we position the securitization business model as key to the world economy and we describe the processes underlying the activity that need to be well understood. Sixth, we propose a framework to be put in place in financial institutions in order to manage the activity.

      When pricing a single name credit derivative instrument, known as a credit default swap (CDS), one needs to have a default model. There are two widespread approaches in the industry for doing this. The first is based on the equity market, and in the second a default process is postulated. The two approaches are briefly described in Chapter 2. Initially, the market was predominantly a single name protection instrument. However, in the last few years there has been a drive for multiname instruments for portfolio risk management purposes, rating of whole portfolios and for pricing of multiname CDSs, raising the necessity of models for default dependency within a portfolio. The classical solution has been to use the concept of copulas, and this is described in Chapter 3. In both cases we keep the description to the minimum required to understand the remaining chapters.

      We then move forward to Part II of the book where we focus on the pricing of single name credit instruments. In Chapter 4 we show how to price a CDS, the simplest synthetic credit instrument, using the intensity model described in Chapter 2.

      We develop two approaches to calibrate the model to observed market spreads. A first taste of the book can be seen when we go one step further and describe practical reasons why one model is chosen over another. Additionally, we show a table comparing the recovery rates on some defaulted bonds during the credit crunch.

      In Chapter 5 we price a single name credit spread option using trinomial trees typically used for interest rate processes. Although the chapter is based on a published work (see Garcia et al., 2003), and on data from 2001, the study is still very relevant as the option market is still OTC and not yet fully developed. We show again a comparison between model and reality. In addition, we highlight the parallels in terms of modeling purposes between interest rate and the default intensities, and between discount factors and survival probabilities. The subject will become more relevant once the market comes to use the indices for active portfolio management purposes, a key proposal of this book, in which case one will certainly evolve in the direction of term structure of volatilities of the expected loss.

      The collapse of Bear Stearns and the bankruptcy of Lehman Brothers served to highlight the importance of counterparty risk in CDS contracts. In a very short time, protection buyers of CDSs sold by Lehman Brothers realized that their contracts were not as safe as they thought. In order to understand the complex nature of those events, consider a retail bank that sold to its wealthy clients USD 200 million of a capital guaranteed instrument structured by Lehman Brothers (LB). Suppose, for example, that the instrument was a credit constant proportion portfolio insurance (CPPI) issued by LB. The sudden bankruptcy meant that the retail bank got all the exposure to a complex product it might not be able to manage being potentially exposed to any trading loss on the product. The issue of counterparty risk and the so-called credit valuation adjustment (CVA) is addressed in detail in Chapter 6.

      Part III of the book is dedicated to corporate multiname credit derivatives. In Chapter 7 we describe what collateralized debt obligations (CDOs) are, giving a brief overview of the instrument. The chapter addresses very important issues that underlie the current credit crunch. That is, we show the differences between cash and synthetic deals, the cost of regulatory capital showing explicitly how the instrument is suitable for leveraged positions at the cost of systemic risk. Moreover, we point out the issues of concentration, correlation and diversification inherent to the instrument. The chapter is important in order to understand how CDOs can lead to liquidity problems and why the standardized credit indices are needed.

      In Chapter 8 we give a description of the corporate standardized credit indices iTraxx and CDX, focusing on the importance of standardization. In that chapter we give a first intuitive way of pricing the index. The widely-used one factor Gaussian copula algorithm to price tranches of the standardized credit index is described in detail in Chapter 9. We also show how to adapt the model to use Lévy processes. The importance of using Lévy models cannot be emphasized enough. The need for it can be seen in the work of Mandelbrot who was among the first to have studied Lévy processes in finance. We first describe the algorithms used by practitioners. The discussion about self-organized criticality and Mandelbrot is postponed to the final part of the book. The chapter describes the problems of implied compound and base correlation, pointing out the interpolation problems, central to any pricing algorithm for tranches of CDOs.

      A more in-depth study comparing Gaussian copula with Lévy base correlation is presented in Chapter 10. The concept of base correlation solves the problem of pricing bespoke tranches. The problem with the base correlation approach, however, is that it is not an intuitive concept, and neither is it straightforward to guarantee arbitrage free pricing. Those issues can only be guaranteed within the concept of base expected loss described in more detail in Chapter 11. One of the most important applications of the standardized credit indices is the pricing and hedging of bespoke portfolios and for this one needs the concept of correlation mapping. In Chapter 12 we show different methodologies available in the market for choosing the appropriate correlation for pricing purposes of a bespoke tranche. It should be clear that pricing is currently more an art than a science and the user needs to understand the implications prior to choosing one particular algorithm over another.

      In Chapter 13 we show how correlations among tranches are impacted by the assumptions on systemic risk for the underlying collateral. This chapter is very important for risk, regulatory capital and accounting purposes.

      In Chapter 14 we describe cash flow CDOs, presenting a waterfall or indenture in detail. We describe one of the first methodologies to analyze CDOs, the Binomial Expansion Technique (BET), first developed by Moody’s. Although it is current best practice to use Monte Carlo (MC) simulation, we decided to describe the old BET approach in some detail due to its central role in the risk analysis of CDOs that led to the failure of a certain large company during the credit crunch. The curious reader is advised to rush to that chapter.

      Structured credit products such as Constant Proportion Portfolio Insurance (CPPI) and Constant Proportion Debt Obligation (CPDO) are described in Chapter 15. With the credit crunch and the enormous losses suffered by CPDO investors, this instrument became a symbol of a risky product in which models failed. We had foreseen this danger. It could have been detected by comparing the results of simulation driven by Brownian motion with simulations based on jump-driven Lévy processes. This is yet more evidence that pricing means first understanding the nature of the product and only then selecting an appropriate model to catch possible features and hidden risks.

      In Part IV we address CDOs of Asset Backed Securities (ABS). The different protocols used in the market for ABCDSs, that is CDSs of ABSs, are described in Chapter 16. In Chapter 17 we present one credit event model to price CDOs of ABSs, showing the complex problems faced by the industry associated with the input parameters. Given the importance of the asset class, one needs standardized credit indices for pricing and hedging purposes. Some of those indices are described in Chapter 18 and we focus on ABX.HE and TABX.HE, the standardized credit indices for subprime Mortgage Backed Securities (MBSs). In Chapter 19 we show how to adapt the standard market approach for pricing tranches of corporate credit indices to price TABX.HE, the tranches of ABX.HE, both under the Gaussian copula and Lévy models. The deterioration in the subprime MBSs was visible in the TABX.HE tranches. Additionally, we show that, when using the prepayment assumptions taken from the remittance reports there was no value of correlation that would recover observed market prices. An important message of this chapter is that, in order to be able to foster the securitization business model at low cost of capital, key ingredients are the standardized credit indices and transparency in the methodologies for pricing purposes. This also implies the ability to map portions of the bespoke portfolio into the capital structure of standardized credit indices. If the pricing algorithm is one factor then one may use the techniques described in Chapter 12. This implies the assumption by the market of a risk neutral prepayment assumption for pricing purposes. One of the current difficulties in pricing CDOs of ABSs is the input spread parameter from which probabilities of default are implied. Differences in probabilities implied from an ABS bond and ABCDS are due to the cost of funding of the former, the mark to market nature of the latter, and liquidity issues. In Chapter 20 we adapt the techniques widely used for the corporate case to come up with the basis between ABCDSs and the ABS bonds.

      In Part V we point out that a solution for the securitization business model for financial institutions requires understanding the relation between widespread investment in apparently safe AAA securitization instruments and its catastrophic impact on the stability of the whole financial system. To this end, we discuss long-term memory processes and self-organized criticality central to the work of Benoit Mandelbrot and others. An intuitive description of those processes is given in Chapter 21. We also mention the inappropriateness of the Gaussian framework for pricing and portfolio management purposes. We then move to Chapter 22 where we address in detail the credit crunch and its link with securitization. We show via an intuitive example that the process to be followed is the dynamic of systemic correlation that can be monitored via the standardized credit index. It turns out that the dynamics of correlation follow a long-term memory process. We know that the probability of extreme events is much higher than expected under the Gaussian framework. One solution to the stability problem is to significantly increase the cost of capital for securitization instruments.

      This medicine kills the sickness - instability - but also the patient - the securitization activity - and with it a large part of the world economy as we know it. One cannot expect the world to stop thinking in Gaussian terms overnight as all the systems and the mathematical framework in the heads of the practitioners are based on Gaussian distributions. In Chapter 23 we present a solution for the whole puzzle. We show the inadequacy of a regulatory capital framework that is portfolio independent. Moreover, we show the inadequacy of the correlation values that have been used for securitization instruments for both risk management and rating purposes. Next we unveil the implicit assumptions of liquidity adopted by practitioners when rating agency models are used for structuring purposes. This leads us to the necessity of exchange traded standardized credit indices. Continuing along this path, we propose a mark to market approach for securitization instruments within a dynamic credit portfolio management framework as one possible solution for the securitization business model.

      Part I

      Modeling Framework

      2

      Default Models

      Genius, that power that dazzles mortal eyes, is oft but perseverance in disguise.

      Henry Austin

      2.1 INTRODUCTION

      A credit derivative is a derivative whose payoff depends on the credit risk of an underlying reference. In several places in this book we show that the market for credit derivatives has grown enormously in recent years since its formal inception in the mid 1990s. As a consequence of this growth there has been a demand for default models to be used for the evaluation of those instruments. The increased level of sophistication of the credit instruments, as evidenced by the credit crunch of June 2007, brought up the necessity for credit risk systems and models for default probabilities for credit risk purposes. These models do not need to be the same.

      There is already a lot of literature available on this subject and we do not intend to repeat it here. We will give a very brief description of what is behind the modeling approaches in a way that the reader can follow through the remaining chapters. Two traditional references are Schönbucher (2003) and de Servigny and Renault (2004). Additional references will be given at the appropriate place. The chapter is structured as follows. In Section 2.2 we discuss what is called a default. In Section 2.3 we present the two approaches most used to model the default process.

      2.2 DEFAULT

      Generally speaking, an obligor is said to be in default when she cannot honor her legal contractual obligation in a debt instrument. Although intuitively speaking the concept is quite simple, in practice however the default process may be quite complex, and the catch is in the word legal.

      In practice, one says that an obligor is said to be in default when a contractually specified credit event has been triggered. Possible credit events are: bankruptcy, failure to pay, moratorium, debt restructuring, rating downgrade, acceleration of debt payment, or even moves on the credit spread. In order to standardize those contracts and bring liquidity into the market the definitions of what is called a credit event have been documented by the International Swap Derivatives Association (ISDA) and we refer to this organization for legal detail on this topic.

      The importance of those contractual definitions should not be underestimated. Consider, for example, that an insurance portfolio manager sells default insurance on a portfolio of five references, an instrument known as a basket. At the same time, the manager buys individual protection on the entities in the portfolio she feels are most likely to default. Assume, for example, that the entity for which the manager had bought individual protection has a debt restructuring event. Under the single name CDS contract, the seller has the right to call the credit event and, in case of noncash settlement, receive a bond of the buyer. However, it sometimes happens that a debt restructuring may turn out to be a good deal for a company. For the basket contract the one who triggers the default event, however, is the protection buyer and not the seller. In that case it can happen that the manager will have to go in the market and buy underlying name to be able to deliver it to the CDS seller, while still keeping its exposure in the basket open. She will probably have lost money on the deal. The restructuring clause is present in Europe and not in the US and has been the cause of many contentious issues.

      In what follows we do not enter into the legal details of what has triggered a credit default. We define it phenomenologically and assume that the meaning of a credit default is well understood.

      2.3 DEFAULT MODELS

      2.3.1 Overview

      One of the main problems with modeling defaults is that default events are rare and as such not much data is available. Moreover, even if more data becomes available, it will typically represent an historical perspective, more appropriate for a buy and hold strategy. For pricing purposes, however, one is interested in the probabilities of default implied in the prices of instruments available in the market. This means risk neutral measures. From the start, one is left with two possibilities, using information embedded in the prices of either equity or debt instruments. For this reason, for pricing purposes there are basically two widespread approaches to model a default. One approach is called firm or asset value models (AVM) and is based on the original work of Merton (1974) and Black and Scholes (1973). Those are equity market-based models. In the second approach one models the default process explicitly. It is based on the original work of Duffie and Singleton (1999). Those models use debt instruments directly for calibration purposes. In what follows we give a very brief description of the ideas and principles behind those approaches.

      2.3.2 Firm value models

      Firm value models have been around for a long time and the literature is very extensive. They have been very influential in many products available in the market. Both Moody’s KMV and CreditGrade from CreditMetrics are firm value-based models.

      In firm value models, a company is in default when a latent variable, the asset value, breaches some barrier, typically the debt book value. In this approach one needs an assumption for the asset value process and an assumption for the capital structure of a company. Denote by Vt the value of a company, St its equity price and Bt the value of its outstanding debt at time t. Additionally D is the par or notional value of the debt at maturity. The value Vt of the company is given by

      (2.1)

      002

      Under Merton’s assumptions V follows the usual geometric Brownian motion and is given by

      (2.2)

      003

      where µ is the drift, σ is the volatility and W is the driving Brownian motion.

      In the original Merton model the value of the company should not fall below the outstanding debt at maturity. From (2.1) we have that the value of the equity of a company at maturity T is given by

      (2.3)

      004

      which is the payoff of a call option with strike set at D. Analogously for the debt of the company at time T we have:

      (2.4)

      005

      From BS formula one has for St and Dt:

      (2.5)

      006

      and

      (2.6)

      007

      where

      (2.7)

      008

      and r is the risk free interest rate, and N is standard normal cumulative distribution function. Observe that in the Merton model default is associated to the value of the company at the maturity of the debt (T ). Over the years several extensions have been proposed. In one such an extension, proposed by Black and Cox (1976), the default process would be triggered in case the barrier is crossed at any time between t and T .

      Despite its use by some market participants there are very practical problems with this approach. First, the asset value of a company is not an observable, its equity value is. This means that it is common market practice to use the equity process as a proxy for the asset value. Second, the barrier that determines default is not a clear cut value and one needs to have access to the whole capital structure of a company. That is, a real company has several outstanding bonds at different maturities and with different subordination levels, making the model assumptions on the capital structure too simplistic. Third, the model is not easily adaptable for illiquid nonlisted companies for which both equity and debt information is not easily known. Fourth, one cannot use it directly for other asset classes such asset backed securities. Fifth, as we have seen during the credit crunch, many companies have their stock below the supposed book value, and default has not been triggered. That is, the relation between an equity process, the barrier level and default may be a very strong assumption. Sixth, the way to use those models for companies that are typically leveraged, such as financials, is still a matter of discussion.

      An example of a very practical problem using the link between equity and credit for trading purposes is the following. Assume a bank sells an insurance contract on the default of a certain reference name, while deciding to hedge the exposure by buying deeply out of the money equity put options. The rationale behind the strategy is simple. In the case of a default event, share prices fall and the money lost in the insurance side is gained on the put side as the share price will have gone close to zero. As we have seen in the last section, however, the insurance contract gives protection not only to default but also to credit events such as restructuring of debt. Where the company goes through a debt restructuring there are cases in which the credit event is seen as good by the equity holders. In this case the insurance contract may be exercised while the share price will go up. The protection seller will have lost money on both sides of the deal.

      The ultimate problem facing firm value models for pricing purposes is linked to issues of calibration. The link between equity and the default process is in fact a very strong assumption. There is nothing that guarantees that the equity market will move in synchronization with the credit market, such that default probabilities follow the quotes in the credit market.

      A final point on asset value models is as follows. Structured credit products are typically correlation instruments. If one is modeling for pricing purposes the correlation should come from the prices of available liquid market instruments. In Chapter 23 we explain that for doing portfolio analysis one may need to have correlation numbers that do not necessarily need to come from pricing instruments. Given that joint default data is very rare, a framework justifying the use of equity data for the evaluation of correlation is a welcome feature. This justification is addressed via firm value models.

      In this book we do not explore the use of firm value models. The literature is very large and we refer to Chapter 9 of Schönbucher’s book (2003) and to Chapter 3 of O’Kane’s book (2008) and the references therein for additional literature. In the next section we address the ideas behind intensity-based models.

      2.3.3 Intensity models

      In intensity or reduced form models one explicitly proposes a model that is capable of recovering the characteristics of the default process. A first characteristic is that defaults are rare events and the probability of more than one default at the same point in time is assumed to be zero. A second point is that with time going to infinity the probability of default should go to one. A desirable property of the model is a straightforward calibration. Poisson processes are a class of well-known stochastic processes that fit this description. A counting process N(t) for t ≥ 0 is said to be a Poisson process with a rate λ ≥ 0, if it has the following properties:

      1. N(0) = 0;

      2. Stationary and independent increments P(t + Δt, t) = P(Δt);

      3. P [ N (t + Δt ) − N (t ) = n] = n! 1(λΔt ) n exp(−λΔt )

      where, P(t + Δt, t) and P(Δt) are the probabilities of an event occurring between t and t + Δt, and in an interval Δt at any point in time. P [N (t + Δt) − N(t) = n] represents the probability of n events taking place in the interval [t, t + Δt]. These properties have several important consequences (see e.g. Ross, 1996). First, Poisson processes are memoryless as stated by item 2. This means that what happens between t and t + Δt does not depend on what happened prior to time

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