Increasing computational speed in pricing single tranche CDOs

Contents

List of abbreviations

List of tables

List of figures

List of symbols

1 Introduction

2 Credit Derivatives
2.1 Credit default swap
2.2 Credit-linked note
2.3 Total return swap
2.4 Collateralized debt obligation
2.4.1 Definition
2.4.2 Classification

3 The Merton model for quantifying default probabilities of a single obligor

4 The Copula function as a tool to determine joint default probabilities
4.1 Gaussian copula
4.2 Single factor Gaussian copula

5 Pricing credit derivatives
5.1 Hazard rate
5.2 Risk neutral pricing
5.3 Quasi analytical determination of joint default distribution
5.4 Nth to default swap
5.5 CDO tranches

6 Risk Management of a STCDO
6.1 Risk exposure with respect to spread changes in single name CDS (delta)
6.2 Risk exposure with respect to changes in correlation
6.3 Risk exposure with respect to changes in recovery rates

7 Conclusion

Appendix: Recursion algorithm introduced by Hull & White [28]

References

Insurance

List of abbreviations

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List of tables

Table 2.1: Example: details of individual tranches in a CDO

Table 2.2: Comparison of tranching for typical cash and synthetic CDOs

Table 5.1: Default probabilities for constant hazard rates

Table 5.2: nth to default swap spreads for varying hazard rates

Table 5.3: nth to default swap spreads for varying correlation levels

Table 5.4: CDO spreads for varying correlation levels

Table 6.1: Delta positions for the equity tranche wrt changes in risk variables

Table 6.2: Delta positions for the mezzanine tranche wrt changes in risk variables

Table 6.3: Delta positions for the senior and super senior tranche wrt changes in risk variables

Table 6.4: Risk position in a long STCDO with respect to changes in recovery rates

List of figures

Figure 1.1: Global CDO Issuance, Source: Deutsche Bank Securitization Research

Figure 2.1: Structure of a CDS

Figure 2.2: Structure of a CLN

Figure 2.3: Structure of a TRS

Figure 2.4: Typical structure of a CDO

Figure 2.5: Example: payoff profiles for a protection buyer in a CDO

Figure 2.6: Classification of CDOs and their historic market volume, figures from Lucas [35]

Figure 4.1: Gaussian copula vs. independence copula with normal marginals

Figure 4.2: Cond. default probability on default probability, Bluhm, Overbeck & Wagner [11]

Figure 5.1: Cash flow structure of a CDS

Figure 5.2: Example: Binomial tree for one credit

Figure 5.3: Example: Binomial tree for two credits

Figure 5.4 :Expected loss distribution for different correlation levels

Figure 6.1: Delta surfaces for the CDO example

Figure 6.2: Correlation exposure of CDO tranches

Figure 6.3: Double meaning of quoted market spreads for implied correlation

Figure 6.4: Risk exposure with respect to changing recovery rates

List of symbols

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1 Introduction

In recent years enormous write offs in bank’s credit portfolios stimulated the demand for products that allow for an active trading of credit risk within the field of capital management. Securitization is a tool to reduce credit risk embedded on balance sheets. Thereby various assets are pooled in a portfolio that serves as collateral for issued notes. These asset backed securities (ABS) were initially aimed to securitize mainly mortgage and consumer loans of financial institutions in the early 1980s (Tavakoli [50]).

A collateralized debt obligation (CDO) is a type of ABS that was first set up to securitize junk bonds (below investment grade investments) in the late 1980s (Moore[41]). The growing demand for securitizing credit risk during the 1990s led to a tremendous rise in CDO issuance which was further stimulated by the introduction of synthetic CDOs whose portfolios consists of credit derivatives such as credit default swaps (CDS). In 2003 the CDO issuance volume was USD 94 billion, a rise of 27% compared to 2002.

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Figure 1.1: Global CDO Issuance, Source: Deutsche Bank Securitization Research

Major risks involved in the evaluation of CDOs are liquidity risk, credit risk and the risk of the joint defaults in the credit portfolio (European Central Bank [17] and Basel Committee on Banking Supervision [9]). In addition, a CDOs bears pricing risk related to the applied model. Pricing a CDO requires complex valuation models. Estimating joint default probabilities in the underlying asset portfolio is a part of the pricing process. Common practice to derive a value for a CDO is to apply a Monte Carlo Simulation whose accuracy is linked to the number of simulations and can take too much time for a trader in times of strong market movements. For this purpose, this paper explains the risk neutral pricing of a CDO based on a quasi-analytical method introduced by Andersen, Sidenius & Basu [6]. Thereby a single factor Gaussian copula is applied to determine joint default probabilities.

To approach the topic of this paper, chapter 2 gives an introduction into the basics of credit derivatives and CDOs in particular. Following this, we discuss the Merton model for the calculation of default probabilities of a single obligor in chapter 3. The concept of copulas and particularly a single factor Gaussian copula for the determination of joint default probabilities is discussed in chapter 4. Chapter 5 contains theoretical background for the valuation of credit derivatives and presents a recursion algorithm that reduces computational time for the calculation of joint default probabilities. Moreover, we develop the pricing formula for nth to default swaps as well as for CDO tranches and implement the recursion algorithm together with the pricing formula in order to present sample results for both credit derivatives. In chapter 6 we investigate the risk exposure in CDO tranches with respect to changes in single name credit spreads, correlation and recovery rates before the results are summarized in Chapter 7.

2 Credit Derivatives

Credit derivatives are financial instruments that allow transferring pure credit risk in a bilateral contract between two parties.

The market for credit derivatives grew rapidly over the past decade. As of June 2004 notional amounts outstanding of credit derivatives equalled USD 4.5 trillion, compared to USD 0.7 trillion in June 2001 (Bank for International Settlement[8]). The reason for the rise in volume is that credit derivatives are an efficient tool for risk allocation and capital management (Heidorn [27]). Generally, credit derivatives enhance the stability of financial markets (Effenberger [15]). However, as the market is still in its growth period, there are certain risks involved, e.g. liquidity and operational risk that have to be addressed by the market participants (Basel Committee on Banking Supervision [9]). In order to understand how these products work, major credit derivatives are explained in this chapter.^[1]

2.1 Credit default swap

Traditional credit products such as loans and bonds require an initial investment. With a CDS it is possible to create credit risk positions without funding. By the end of 2001 CDS had with 44% the largest share in the credit derivative market (Effenberger[15]).

The buyer of a CDS (protection buyer) acquires protection for a time period up to time illustration not visible in this excerptagainst any losses resulting from a credit event in the reference credit. This requires him or her to pay a fixed periodic premium to the seller of the CDS (protection seller) at the end of each period illustration not visible in this excerptas long as no default occurred in the period illustration not visible in this excerpt. If there is a default at time illustration not visible in this excerptthe protection seller receives an accrued spread at the time of default. The protection buyer receives a settlement payment from the protection seller in case a credit event occurs before the CDS matures. The settlement procedure can involve (i) a cash payment by the CDS buyer for the recovery of the reference credit (cash settlement) or (ii) the delivery of the defaulted asset (physical settlement) in return for the notional amount of the reference credit (Schonbucher [47]). Common market practice is a cash settlement between the CDS contract parties (Heidorn [27]), i.e. the protection buyer receives the notional of the credit exposure. Commonly, the credit event is the default of the reference credit but can be also defined differently, e.g. as a change in credit quality. For the definition of the relevant credit events it is often referred to the ISDA regulations as it provides internationally accepted standard definitions (ISDA [30]). A CDS is unfunded as the protection seller creates credit synthetically without an initial investment. Figure 2.1 illustrates the structure of a CDS.

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Figure 2.1: Structure of a CDS

For example, consider an investor who enters into a CDS with a bank on a EUR500million bond of DaimlerChrysler for five years. Thereby, the investor commits himself to pay a notional of EUR500million in case DaimlerChrysler defaults within five years. Therefore he or she receives periodic payments of 1.2% on the notional at the end of each year as long as no default occurs whereas the bank is hedged in terms of credit risk.

The fair value of a CDS can be calculated by a no arbitrage relationship. When a CDS is set up, the spread is determined so that it is worth zero. A more detailed view on pricing credit derivatives is given in section 5.

When referring not to only one reference entity but to a portfolio of underlying reference credits, the product is called a credit basket. One type of a credit basket is the nth to default swap where the premium payments from the protection buyer stop with the occurrence of the nth default in the credit portfolio (Schonbucher [47]). The default payment of the protection seller is then determined by the loss incurred by defaulted obligor, usually the notional amount of the credit minus the recovery amount. With credit derivatives that have multiple obligors it is crucial to have a model for the joint defaults in the credit portfolio (see section 4).

Although the protection buyer transfers the credit risk of the reference credit or reference basket, he or she is still exposed to counterparty risk. Claiming collateral or choosing a highly rated counterparty reduces this risk. Other products such as credit-linked notes (CLN) have features that minimize counterparty risk.

2.2 Credit-linked note

A credit-linked note (CLN) is a structured product, thereby composed of a CDS and a low risk bond. As the CLN involves an initial investment it is also called a funded CDS. The investor in a CLN is exposed to credit risk in the CDS as well as in the low risk bond. Thereby, the low risk bond serves as collateral in case of credit event occurs. The residual, after paying the CDS counterparty for any incurred losses, is transferred back to the CLN investor at maturity. Hence, the counterparty risk reduces to the risk in the decline in value for the low risk bond. For paying a periodic amount, consisting of the coupon payment from the low risk bond and the spread received from the CDS, the issuer of a CLN benefits from being protected against any credit risk. Figure 2.2 illustrates the structure of a CLN.

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Figure 2.2: Structure of a CLN

2.3 Total return swap

The investor in a total return swap (TRS) is, contrary to a CDS and a CLN, exposed to market and credit risk. In this, the seller of a TRS (TR payer) provides the investor (TR receiver) with fixed periodic payments whereas he or she is ensured against any decline in value in the reference credit. Moreover, the TR payer waives any upside from the reference credit, as he or she has to transfer any positive price changes to the TR receiver. Therefore the TR payer receives variable interest payments (EURIBOR or LIBOR plus spread) and is rewarded for any negative price changes of the reference credit. The TRS ceases to exist in case of default. In this case the TR receiver has to provide the TR payer for the losses incurred from the reference credit.

As a TRS is not recorded on the balance sheet, it is an attractive investment for investors that seek for returns from synthetic credit investments but do not want to increase their balance sheet in terms of total assets. The structure of a TRS is shown in Figure 2.3.

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Figure 2.3: Structure of a TRS

2.4 Collateralized debt obligation

2.4.1 Definition

A CDO is a type of securitization in which notes of different rating categories are issued on a portfolio of defaultable products such as e.g. bonds, loans or CDS. When the originator of a CDO transaction prefers to take credit risky positions out of his balance sheet the assets are transferred via a true sale to a special purpose vehicle (SPV) that is financed by the issuance of notes backed by the asset portfolio.^[2] The SPV is thereby ring fenced from the originator, i.e. the assets in a CDO do not serve as collateral in case the originator defaults. A typical CDO structure is illustrated in Figure 2.4.

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Figure 2 . 4 : Typical structure of a CDO

For an originator not willing to reduce the amount of total assets in his or her balance sheet it is also possible to securitize the credit portfolio via credit derivatives. Applying this structure would not necessarily require to set up a SPV.

Similar to a CDS, the investor (protection seller) in a CDO receives a periodic payment, usually EURIBOR or LIBOR plus spread, as long as there is no default in the underlying asset portfolio. Thereby, the SPV (protection buyer) pays a premium to receive protection in case of default. Whereas an investor in a CDS must provide the seller with the face value of the reference entity in case of any default event the investor in a CDO only faces losses when the portfolio losses exceed the subordination level of the note. As shown in Figure 2.4, the SPV issues notes with different seniority. This procedure of creating different asset classes is also called tranching. Thereby, the size and the subordination of each tranche is determined by rating agencies (Adelson & Whetten [1]). Typically, a CDO consists of three rated tranches and the most risky equity tranche which is not rated (Lucas [35]). The tranches with higher seniority compared to the equity tranche are usually called subordinate, mezzanine and senior tranche (Adelson & Whetten [1]). The equity tranche, which is also known as first loss piece, is affected by the first losses occurring in the asset portfolio. After losses reach the upper bound illustration not visible in this excerpt(detachment point) of a tranche, the note is completely wiped off and further losses are absorbed by the more senior tranches. The attachment point illustration not visible in this excerptdefines the level that portfolio losses need to exceed in order to affect the tranche. When we denote illustration not visible in this excerptas the cumulative portfolio loss, the payoff illustration not visible in this excerptfor the protection buyer can be written formally as^[3]

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When illustration not visible in this excerptaffects the protection seller but is below the illustration not visible in this excerptin the first place, any increase of illustration not visible in this excerptup to illustration not visible in this excerptbefore maturity leads to an additional default payment by the protection seller. To illustrate a CDO structure, consider a CDO with a credit portfolio of EUR°1°billion and four tranches with the following characteristics:

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Table 2.1: Example: details of individual tranches in a CDO

The investor in the equity tranche receives a periodic payment of EURIBOR plus 14.73% on the notional. If losses occur in the underlying asset portfolio the spread is paid to the reduced notional, e.g. when losses are equal to EUR 10 million the spread of 14.73% (assuming that the spread stays at this level) is paid on a notional of EUR20million instead of EUR30million. The subordinated tranche absorbs losses from the point they exceed EUR30million and ceases to exist when they reach EUR60million. As the tranche is not affected as long as portfolio losses remain below EUR30million the risk of investing in this tranche is lower. This circumstance explains the lower spread payment compared to the first loss piece. Same mechanics apply to the mezzanine and senior tranche. While assuming a recovery rate of 0%, the relevant payoff profile for the protection buyer is illustrated for each tranche in Figure 2.5.

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Figure 2.5: Example: payoff profiles for a protection buyer in a CDO

Thereby the payoff to the protection buyer can be interpreted as a call spread in which the attachment point marks the strike level and the detachment point the maximum payout. Likewise, the protection seller is short the call spread and receives a spread payment for bearing the risk of being hit by losses in the credit portfolio.

Any cash flows to meet interest and repayment claims of the CDO investors are distributed among the tranches in accordance to their subordination level, i.e. the tranche with the highest seniority receives the relevant payments first. If there is any cash flow left the next lower rated tranche is served. This procedure is also known as a waterfall principle (Basel Committee on Banking Supervision [9]).

2.4.2 Classification

Firstly, CDOs can be differentiated according to the assets included in the underlying credit portfolio. If it consists solely out of loans or bonds the CDO is called collateralized loan obligation (CLO) and collateralized bond obligation (CBO), respectively. Financial institutions introduced CLOs in the early 1980s to securitize mortgage and consumer loans. CBOs were set up in the late 1980s to sell off junk bonds from balance sheets. The underlying portfolio can further contain ABS, mortgages and credit derivatives such as CDO themselves.

Further, CDOs can be categorized regarding their motivation, their function and their type of assets included in the underlying credit portfolio (Adelson & Whetten[1], Lucas [35] and Tavakoli [50]). The following chart illustrates the structures that are commonly traded in the market.

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Figure 2.6: Classification of CDOs and their historic market volume, figures from Lucas [35]

Theoretically, it is possible to structure eight different types of CDOs. However, balance sheet transactions are structured usually as a cash flow structure as it allows for more leverage. Additionally, market value structures set up for arbitrage purposes do usually not contain synthetic positions (Lucas [35]).

Motivation

The motivation for setting up a CDO is twofold. On the one hand the originator is able to generate liquidity by selling off credit-linked assets in order to lower regulatory capital. A CDO set up for this purpose is called a balance sheet CDO. In this transaction the originator of a transaction sells the credit linked assets to a SPV that is financed by the issuance of notes backed by these assets.

In arbitrage CDOs, the motivation behind is not making risk free profits from market imperfections, as the name would suggest. Here, the underlying credit portfolio is expected to earn a higher yield compared to the interest paid on the CDO tranches. In order to do so, active management of the portfolio is often allowed.

The motivation for buying these products is that tranches of CDOs can be customized in accordance to the risk and return profile of investors. For an investor especially the rating, expected losses as well as the underlying asset pool are decisive for the performance of a CDO investment (Heidorn & Koenig [26]).

Function

In a cash flow CDO, income streams from the underlying credit portfolio such as amortization and interest payments received from loans or bonds as well as premium payments received from credit derivatives are pooled in the SPV and paid out to the CDO investors in accordance to their subordination level. Investors only face losses if there is a default in the asset portfolio. Crucial input factors for setting up a cash flow CDO are the default probability, default correlation and default severity.

In a market value CDO, the underlying credit portfolio consists of tradable assets so that a mark to market can be calculated at all times. Hence, the price of a market value CDO is more volatile. Further, the credit products included in the CDO portfolio are also dependent on market risk, e.g. interest rate movements, and changes in the credit spread of underlying obligors. Interest payments on the CDO tranches are made out of gains from actively managing the CDO portfolio. Usually, restrictions are set in a market value CDO to avoid that payments to the CDO investors cannot be made. These are for instance over-collateralization measures where the market value of the CDO is compared to its notional. If the value drops under a predefined threshold, low rated assets are exchanged for low risk assets to prevent from a further decline in value.

Asset type

CDOs can further be classified according to the asset types included in the underlying credit portfolio. Therefore, a cash CDO is backed by securities that are traded in the cash market such as bonds. In a fully synthetic CDO, the credit portfolio of the SPV is created via selling protection on single names. For a financial institution trying to lower regulatory capital without reducing total assets there is no need for setting up a SPV. CDO tranches can be created by financial institutions themselves and directly marketed to investors. However, a SPV is preferable to exclude the credit risk of the originator.

Synthetic CDOs are superior to cash CDOs with respect to a credit investment as they provide a pure exposure to credit risk whereas the value of cash products also depends on other risk factors such as liquidity risk. Another characteristic of synthetic CDOs is the creation of a super senior tranche that usually makes up to 90% of a transaction. This tranche has an even higher seniority than the AAA rated senior tranche what is reflected in a single digit spread payment. Despite having a greater seniority there is no clear approach by rating agencies how to set an attachment point to have such a tranche (Tavakoli [50]). To prevent investors from providing protection in case of a default that occurs at a random point in time, a synthetic CDO can require an initial investment of the notional which is invested in low risk assets. These serve as collateral in case of a default in the CDO portfolio so that the structure is comparable to that of a CLN. A typical tranche structure for synthetic and cash CDOs is shown in Table 2.2.

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Table 2.2: Comparison of tranching for typical cash and synthetic CDOs

Further to the explained categorization, CDOs can have static or managed underlying credit portfolios. In its origins CDOs contained static portfolios, i.e. the underlying credit portfolio did not change. The advantage of a static portfolio is obviously, having lower administrative costs over the lifetime. At the same time, investors can suffer from deteriorating credit products in the CDO portfolio. As a result managed CDOs emerged where asset managers actively trade the CDO portfolio to create additional value. Whereas 8% out of all synthetic CDOs had managed portfolios in 2002, the ratio increased to 22% in 2003 (Moore [41]).

Single tranche CDOs

Unlike conventional CDOs where all tranches are sold to investors, only one tranche of the credit portfolio is carved out in a single tranche CDO (STCDO). Hence, the originator is left with the remaining capital structure that has to be hedged dynamically (Adelson & Whetten [2]). A more comprehensive view on risk management of a STCDO is given in section 6. A reason to set up this type of product is that it can be originated quickly as a rating agency has to evaluate only one tranche and only one investor needs to be found. The advantage of a STCDO is that it can be tailored to the investor’s risk preference. The investor can determine the subordination level and size by negotiating attachment and detachment points. The difference between the attachment and the detachment point defines the notional amount of the tranche.

For example, assume an investor buys a 8%-15% tranche of a EUR 1 billion CDO portfolio with a maturity of five years. Therefore, he or she receives annual payments of 2.5% on the notional at the end of each year. The tranche faces losses when portfolio losses exceed 8% and must be completely written off when they reach 15%. Thereby, the notional of the tranche is EUR70million.

In this structure, it is possible to choose between static and managed portfolios. In an unfunded scheme the STCDO is equivalent to a CDS where no initial investment is required. As STCDOs are usually structured on a bilateral basis between originator and investors, it is difficult to track volumes (Adelson & Whetten [2], Basel Committee on Banking Supervision [9], and Moore [41]).

3 The Merton model for quantifying default probabilities of a single obligor

In order to price a credit derivative a model to estimate the probability of default has to be set up. Therefore, the model developed by Merton [40] is introduced to estimate default probabilities of a single obligor. In this model, a default is triggered when the asset value illustration not visible in this excerptof an obligor is lower than the book value of debt illustration not visible in this excerptat maturity illustration not visible in this excerpt. Hence, one can observe at timeillustration not visible in this excerptwhether an obligor is in default or not. The default probability illustration not visible in this excerptcan be then written as:

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Thereby, we denote illustration not visible in this excerptas the risk free interest rate, illustration not visible in this excerptas the asset volatility and illustration not visible in this excerptas a Wiener process (also known as a Brownian Motion). The Wiener process is a random process that is equal to illustration not visible in this excerptwith illustration not visible in this excerptbeing a random component that is distributed standard normal (Crosbie & Bohn [13]). In case the asset volatility is equal to zero the process of illustration not visible in this excerptwould be determined solely by the risk free interest rate. Hence, the asset volatility is the component that influences the degree of uncertainty in the process. As illustration not visible in this excerptcannot be observed in the market the asset volatility can be derived from the equity volatility (Crosbie & Bohn [13]). The result of applying Ito’s lemma to equation (3.2) is

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Setting the asset value of the obligor at time illustration not visible in this excerptto illustration not visible in this excerptand as of time 0 to illustration not visible in this excerptequation(3.3) can be written as:

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Given the asset value illustration not visible in this excerptand debt value illustration not visible in this excerptat time 0, it is possible to calculate the probability of default. When illustration not visible in this excerptis replaced in equation (3.1) by its process, the probability of default can then be written as:

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In the following the formula is rearranged to isolate the random component illustration not visible in this excerptof the asset return:

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As the Black & Scholes formula assumes a standard normal distribution of illustration not visible in this excerpt, the probability of default can be written in terms of the normal density function.

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The upper bound of the density function is equal to the illustration not visible in this excerptof the Black & Scholes formula. Thereby, illustration not visible in this excerptequals the term illustration not visible in this excerptand is equivalent to a barrier for the random component of the asset return illustration not visible in this excerpt. A default is triggered when illustration not visible in this excerptfalls below the barrier illustration not visible in this excerpt. By extracting the default probability illustration not visible in this excerptout of the market from e.g. credit spreads one can solve backwards for the implied default boundary illustration not visible in this excerpt.

By applying option pricing theory one has to be clear about the assumptions made in this approach. One major assumption is that the calculation is based on the no arbitrage relationship and hence, the price of an option on the asset value can be replicated by trading the asset value in the market. This assumption is obviously not met so that the results of this approach must be taken as an approximation.

4 The Copula function as a tool to determine joint default probabilities

In order to generalize the idea of extracting single default probabilities, we present a method to determine joint default probabilities in this section. Specifically, we determine the joint default probability for a credit portfolio by a single factor copula. Firstly, the concept of copulas is explained in general. Then the Gaussian copula as one type of a copula is illustrated before a further variation, the single factor model, is applied.

For equity markets, correlation is an adequate tool to measure the linear relationship of two or more entities (Embrechts, McNeil & Straumann [16] and McGinty, Beinstein, Ahuwalia & Watts [39]). Thereby equity returns are assumed to be normally distributed and thus provide symmetry around their mean. In order to determine the joint probability of defaults, copulas have become a popular tool in the credit market and were first applied by Li [32] and [33].

The risk neutral cumulative default probabilities up to time illustration not visible in this excerptillustration not visible in this excerptfor one credit product (marginal default probability) can be extracted e.g. out of quoted credit spreads (see section 5). For the evaluation of a credit portfolio it is essential to model the dependency between defaults of the underlying credits. Accordingly, the joint distribution of the default times illustration not visible in this excerptthat are random variables has to be determined for the relevant credit portfolio with illustration not visible in this excerptcredits.

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In the next step the joint distribution has to be linked to the marginal default probabilities illustration not visible in this excerptwhat is done by the copula function (Nelsen [42]). The copula expresses the joint distribution of the marginals in a functional form. Thereby the marginal dependence of the illustration not visible in this excerptis separated from its joint distribution.

Given that the cumulative default probability illustration not visible in this excerpthas a uniform distribution in the interval illustration not visible in this excerptthe joint default distribution can be written in terms of a copula.

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Thereby, illustration not visible in this excerptis the copula function and can also be written as

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To set up a model for the joint distribution, each default time illustration not visible in this excerptis defined as illustration not visible in this excerptwith illustration not visible in this excerptbeing the inverse function of illustration not visible in this excerptfor the uniform random variables illustration not visible in this excerpt.

The underlying theorem of copulas was introduced by Sklar [48] pointing out that any multivariate distribution illustration not visible in this excerpt with marginals illustration not visible in this excerptcan be expressed by a unique copula function as long as its marginal distribution is continuous. Thereby the copula function contains uniforms illustration not visible in this excerptsuch that illustration not visible in this excerpt.

[...]

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^[1] For a comprehensive view on credit derivatives refer to O’Kane [43] or Schonbucher [47]

^[2] For a discussion on legal issues involved in setting up a CDO see Zahn & Lemke [51]

^[3] We refer again to this equation in section 5.5