Vorwort Heft 2-09

Hans-Christoph Grunau

Fur¨ die gegenw ärtige Wirtschaftskrise werden h äufig auch Kreditderivate – oder genauer, der allzu sorglose und unkontrollierte Umgang mit diesen – verantwortlich gemacht. Der Ubersichtsartikel¨ von Kay Giesecke An over- ” view of credit derivatives“ greift diese Problematik auf. Er gibt zun ächst einen Uberblick¨ uber¨ die Rolle und Funktionsweise von Kreditderivaten und fuhrt¨ dann in die Mathematik zur Bewertung von Ausfallrisiken und zur Preisermittlung dieser Finanzmarktprodukte ein. Am 13. August 2008 verstarb das DMV-Ehrenmitglied Henri Cartan, der insbesondere die komplexe Analysis und algebraische Geometrie des 20. Jahrhunderts maßgeblich mitgestaltet hat. Klaus Hulek und Thomas Peter- nell gehen in ihrem Nachruf auf einige der wichtigsten Ergebnisse Cartans ein und stellen dabei insbesondere seine freundschaftlichen Beziehungen zur Munsteraner¨ Schule um Heinrich Behnke in den Mittelpunkt. Nach all dem Elend und Ungl uck,¨ das Nazi-Deutschland uber¨ die Welt und auch uber¨ die Familie Cartans gebracht hat, war diese Freundschaft eine wunderbare Ge- legenheit f ur¨ die Mathematiker im Nachkriegsdeutschland, ihre Isolation zu uberwinden¨ und wieder Anschluss an das internationale wissenschaftliche Le- ben zu gewinnen. Die Buchbesprechungen nehmen in diesem Heft einen relativ breiten Raum ein. Schließlich m öchte ich Sie noch auf die erweiterte Pr äsenz des Jahresbe- richtes im Internet – zun ächst im Probebetrieb – aufmerksam machen: http://www.dmv.mathematik.de/index.php?Itemid=641. Dort sollen die Informationen uber¨ k unftig¨ erscheinende Artikel bereits vor deren Druck verf ugbar¨ sein. Soweit sich das realisieren l ässt, sollen ausf uhr-¨ liche Zusammenfassungen Lust auf das Lesen und Interesse an den vollst än- digen Artikeln wecken.

1 bersichtsartikel Historische Beitrge Berichte aus der Forschung Buchbesprechungen

An Overview of Credit Derivatives

Kay Giesecke*

Abstract

- Mathematics Subject Classification: 60-01, 60-08, 60G55, 60H99, 91-01 - Keywords and Phrases: Credit derivative, credit swap, collateralized debt obligation, point process, measure change, transform

Credit risk is the distribution of financial loss due to a broken financial agreement, for example failure to pay interest or principal on a loan or bond. It pervades virtually all financial transactions, and therefore plays a significant role in financial markets. A credit derivative is a security that allows investors to transfer credit risk to other investors who are willing to take it. By facilitating the allocation of risk, these instruments have an important economic function. Yet they have hit the headlines recently. This paper gives an overview of credit derivatives. It discusses the mechanics of standard contracts, describes their application, and outlines the mathematical challenges associated with their analysis.

*I am grateful to Xiaowei Ding, Baeho Kim and Jack Kim for research assistance, Eymen Errais for providing market data, Alexander Schied for comments, and particularly Stefan Weber for sti- mulating discussions, comments and suggestions. Im also grateful for the suggestions of an anony- mous referee.

Eingegangen: 18.09.2008

Stanford University, Department of Management Science & Engineering, JAHRESBERICHT Stanford, CA 94305-4026, USA, Phone (650) 7239265, Fax (650) 7231614, DER DMV [email protected], www.stanford.edu/~giesecke Vieweg+Teubner 2009

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1 Introduction

The financial crisis in the United States has repercussions on a global scale. Credit derivatives, especially collateralized debt obligations, have been pictured as one of the main culprits. This article provides an overview of credit derivatives. It discusses the mechanics of standard contracts, describes their application, and highlights the mathematical challenges associated with their analysis. A credit derivative is a financial instrument whose cash flows are linked to the financial losses due to default in a pool of reference credit securities such as loans, mortgages, bonds issued by corporations or governments, or even other credit derivatives. The term “default” refers to an event that adversely affects the position of an investor in the reference securities. Examples include bankruptcy, failure to pay interest or principal according to schedule, debt moratorium, and restructuring of an issuer. Credit derivatives facilitate the trading of credit risk, and therefore the allocation of risk among market participants. They resemble bilateral insurance contracts, with one party buying protection against default losses, and the other party selling that protection. This structure enables investors to take different sides and implement various investment and hedging strategies. For example, a fixed income investor may buy protection to hedge the default risk associated with a corporate bond position. An insurance company or hedge fund may act as the counterparty to this deal, and promise to pay potential default losses. The seller of protection speculates on the survival of the bond issuer, and gains investment exposure without having to commit the capital required to actually buy the bonds in the cash markets. The growth in the volume of credit derivative transactions has exceeded expectations year by year after they were introduced in the early nineties. The trading of credit derivatives reached a peak in January 2008, when industry sources estimated the total notional of credit derivatives outstanding at 62 trillion dollars. During 2008, the near failure of investment bank Bear Stearns, the collapse of investment bank Lehman Brothers, insurance firm American International Group, and other market participants exposed the latent threats that credit derivatives can pose for the global financial system. The contract netting and unwinding motivated by these events and the development of a worldwide financial and economic crisis has pushed the outstanding contract notional to under 20 trillion dollars at the end of November 2008. Governments and regulatory authorities call for more transparency and basic regulation of the credit derivatives market. While market liquidity has dried up, the economics behind a credit derivative contract remain sound. For example, banks making loans will continue to have a basic need for hedging loan exposures. Therefore, we expect the credit derivatives market to survive in a leaner, more transparent form that is subject to basic regulation. In this market, the risk management of exposures will likely have a more prominent role in the trading process than it currently has. The risk ratings of complex structured deals issued by rating agencies such as Moody’s, Standard & Poor’s or Fitch will likely be re-structured and improved. This development generates important research themes for economists, financial engineers, statisticians and mathematicians.

58 JB 111. Band (2009), Heft 2 Kay Giesecke: An Overview of Credit Derivatives

The primary objective of this article is to give a concise overview of the current credit derivative landscape. Section 2 illustrates some of the main features, application cases and potential problems of credit derivatives by providing an informal discussion of credit swaps, which reference a single credit obligation. Section 3 discusses a probabilistic model of default timing, on which our formal analysis in the subsequent sections is based. Section 4 analyzes corporate bonds, and then treats credit swaps in more detail. Section 5 discusses multi-name credit derivatives, which are referenced on a pool of issuers whose defaults are correlated.

2 Credit default swaps

To provide some perspective, this introductory section starts with an informal discussion of credit default swaps. Many more complex credit derivative instruments share the basic features of credit swaps. A credit swap is a financial agreement that is individually negotiated between two investors. As indicated in Figure 1, it resembles an insurance contract. The protection seller compensates the protection buyer for the financial loss, if the issuer of a specified reference security defaults on his obligation before the maturity of the swap. The reference security can be a loan, a bond issued by a corporation or a sovereign nation, or a more complex instrument. The protection buyer obtains insurance against default of the reference security, and must pay a premium for this insurance. The premium is called the credit swap spread, is stated as a fraction of the swap notional per annum, and is typically paid quarterly until default or maturity, whichever is earlier. Here is a basic example. A protection buyer purchases 5 year protection on an issuer with notional $10 million at an annual swap spread of 300 basis points (one basis point corresponds to 0.01% of the notional). Suppose the reference issuer defaults 4 months after inception, and that the reference obligation has a recovery rate of 45%. This means that 55 cents on every dollar lent to the reference issuer are lost at default, an estimate typically obtained through a poll of market participants. Thus, 3 months after inception, the protection buyer makes the first spread payment, roughly equal to $10 million 0:03 0:25 ¼ $75; 000.1 At default, the protection seller compensates the buyer for the loss by paying $10 million ð100% 45%Þ¼$5:5 million, assuming the contract is settled in cash.2 At the same time, the protection buyer pays to the seller the premium accrued since the last payment date, roughly equal to $10 million 0:03 1=12 ¼ $25; 000. The payments are netted. With these cash flows the swap expires; there are no further obligations in the contract.

1 In practice, the payment depends on the ratio of the actual number of days in the given quarter to the total number of days in the year, which is fixed at 360. We have approximated that ratio by 0.25. 2 An alternative settlement convention is physical delivery. Here, the buyer delivers to the seller an asset of this choice from a specified pool of reference securities, in exchange for the notional. The terms of the swap may also stipulate a fixed cash payment at default.

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Figure 1. Mechanics of a credit default swap (CDS).

The swap investors have opposing positions. The protection buyer has sold the credit risk associated with the reference obligation. If the buyer actually owns the reference security, then he has hedged the position against default. For a bank hedging its loans, this can lead to economic and regulatory capital relief, i.e. a decline in the capital required to support the bank’s lending activities. If the buyer does not own the reference security, then he has entered into a speculative short position that seeks to benefit from a deterioration of the issuer’s creditworthiness. If the target scenario unfolds, the buyer simply sells the contract to another party, for a profit because default protection commands a higher premium after a deterioration of the issuer’s credit quality. The possibility of buying default insurance on a bond without actually owning the bond distin- guishes the credit swap from a classical insurance contract. The protection seller has bought the credit risk of the reference security. Relative to buying the reference security directly, the swap position has the advantage of not requir- ing a capital outlay at inception. To make this clear, think of an investor who wishes to collect compensation for assuming the credit risk of a given corporation. The investor could purchase a coupon bond issued by the corporation by paying the face value (or current price) of the bond, and then collecting the coupons paid by the issuer. The investor could alternatively assume exposure to the issuer’s credit risk by selling protection in a credit swap referenced on the issuer’s bond. This would involve only a commitment to compensate the swap protection buyer for the potential losses due to default of the issuer before the swap maturity, but no initial cash flow. In return for this commitment, the seller collects the swap spread from the swap protection buyer. Since the credit swap is unfunded, it does not appear as a liability on the balance sheet of the swap protection seller. This “off-balance sheet” nature of the contract is a main feature of many credit derivatives. While attractive to many investors, this feature can also be liability for the financial system. This is because an investor can act as a protection seller in a large number of contracts without disclosing these deals, and accumu- late a tremendous exposure to defaults that is practically “invisible” to regulators and

60 JB 111. Band (2009), Heft 2 Kay Giesecke: An Overview of Credit Derivatives other market participants. Although a protection seller’s economic position is effec- tively that of an insurance firm, the seller is not subject to the insurance industry’s regulation. Insurance regulation imposes capital requirements compatible with the insurance risks. An insurer’s capital limits the policies a firm can write, and therefore the total risk that can be taken. Without transparency and regulatory capital constraints, the total exposure of a protection seller is not limited, and few defaults can exhaust the capital of an over-exposed investor. When an investor is wiped out, the remaining swap contracts become null, leaving the corresponding protection buyers exposed to further defaults. A chain reaction might ensue, jeopardizing the stability of the financial system. Recently, such a meltdown became a real possibility with the near-default of Bear Stearns, an investment bank that had written protection in a large number of deals. A disaster could eventually be avoided since JP Morgan, another bank, bought Bear Stearns and entered into its contracts, backed by guarantees from the US government. The government chose not to provide similar support to the investment bank Lehman Brothers. Lehman had sold protection on a large number of firms and was itself a reference entity to countless other contracts, and its default brought the system to a near breakdown. The collapse of American International Group, an insurance firm that had written protection on a large scale as part of its investment strategy, was only averted by massive government intervention. These events call for meaningful regulation of the credit swap and general credit derivatives market, an effort that governments, central banks and regulatory authorities actively discuss. The basic swap pricing problem is to determine the fair swap spread at inception. The swap spread must equate the present value at inception of the premium payments and the present value of the payments at default. After inception, the swap must be marked to the market, i.e. its current market value must be determined. For the protection seller, the mark-to-market value is given by the difference of the present values of premium and default payments evaluated at the prevailing market conditions.3 If, for example, the credit risk of the reference issuer has deteriorated after inception, then the fixed swap spread does not provide sufficient compensation for the default risk any- more. The protection seller could obtain a higher spread at the current conditions. Therefore, the seller’s mark-to-market value may be negative in this scenario. Of course, the loss of the seller is the gain of the protection buyer, whose mark-to-market value is the negative of the seller’s mark-to-market value. If the buyer has implemented a short credit position as described above, then in this scenario the buyer could sell his position, with a profit equal to the current mark-to-maket value, less the value of the premia al- ready paid. In a liquid swap market, the swap rates provide a sensitive measure of an issuer’s creditworthiness, from which we can extract issuer default probabilities for multiple horizons. These market-implied probabilities reflect investors’ forward-looking expectations about a firm’s credit quality. They can be used to value more complex, less liquid credit derivative instruments, and to design credit investment strategies. To facilitate

3 Marking a position to the market may be difficult when the market is illiquid and there are few buyers or sellers of default risk.

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these applications, we require a probabilistic model of default timing, to which we turn next.

3 Default point process

The uncertainty in the economy is modeled by a complete probability space ð; F; PÞ, where is a set that represents the possible states of the world, F is a sigma-field on , and P is a probability measure on F. The flow of information accessible to investors is modeled by a filtration F, i.e. an increasing family ðFtÞt0 of sub-sigma-fields of F. In- tuitively, Ft represents the events observable at t. We assume that F is right-continuous and contains all P-null sets in F; see Dellacherie & Meyer (1982). The default time of a firm is modeled by a stopping time S. This is a non-negative random variable with the property that fS > tg2Ft. We consider a sequence of stopping times T k with T 0 ¼ 0 that strictly increases to 1, almost surely. The Tk represent the ordered default times in a pool of firms. They generate a counting process N given by X

Nt ¼ 1fTktg: ð1Þ k1

Here and below, 1B is the indicator function of an event B 2F. That is, 1Bð!Þ is equal to one if ! 2 B and zero otherwise. A credit derivative is a security with cash flows that depend on the value of the process N at a set of times. For example, the contract may stipulate the payment of NT at a maturity date T. In this case, the buyer of the derivative is protected against the defaults in the pool, assuming the loss at a default is less than unity. To determine the fair value of a credit derivative, we require the distribution of N. Below, we develop a formula for the Laplace transform of N. The distribution of N is obtained by transform inversion, and can be used to calculate the value of a derivative with payoff gðNT Þ at maturity T, where g is an integrable function on f0; 1; 2; ...g specified by the derivative terms. We illustrate this in Sections 4 and 5. Section 3.3 comments on alternative computational methods for the valuation of credit derivatives whose cash flows take a complicated form. The development of the transform formula requires some prerequisits from the theory of stochastic processes, which is treated in Dellacherie & Meyer (1982) , Karatzas & Shreve (1988), Protter (2004) and others. Readers that are not interested in the technical details but rather in the structure of standard credit derivatives can skip this section and proceed directly to Sections 4 and 5.

3.1 Transform analysis

Since the default process N has increasing sample paths, the Doob-Meyer decomposi- tion theorem guarantees the existence of an increasing, predictable process A starting at

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0 such that M ¼ N A is a local martingale. The process A counteracts the increasing tendency of the default process; it is called the compensator to N. The compensator depends on the underlying filtration F and the reference measure P; it is transformed with a change of F or P. The analytic properties of A correspond to probabilistic properties of the default stopping times. The sample paths of A are continuous if and only if the Tk are totally inaccessible. Totally inaccessible stopping times formalize the intuitive concept of random times that come unannounced and take investors by surprise. There is empirical evidence in support of this property, see Sarig & Warga (1989), for example. Assume the default times are totally inaccessible and that expðAT Þ is integrable for a fixed horizon T > 0. We develop a formula for the conditional Laplace transform

uðN NtÞ ’ðu; t; TÞ¼Eðe T jFtÞ; u 0. (2) The corresponding conditional distribution can be recovered from the relation

1 k PðNT Nt ¼ k jFtÞ¼lim @v ’ð log v; t; TÞ; k ¼ 0; 1; 2; ... ð3Þ v#0 k! Alternatively, we can extend (2) to the complexR plane and invert the Fourier transform 1 so obtained: PðNT Nt ¼ k jFtÞ¼ð2Þ ’ðiv; t; TÞ expðikvÞdv. This formulation facilitates the application of the Fast Fourier Transform, which may have computational advantages over formula (3), depending on the particular setting. For u 0 define the process ZðuÞ by

ZtðuÞ¼expð ðuÞAt uNtÞ; ðuÞ¼1 expðuÞ: ð4Þ Since A is continuous, an application of Stieltjes integration by parts on the product ZtðuÞ¼expð ðuÞAtÞ expðuNtÞ leads to the alternative expression Z t ZtðuÞ¼1 ðuÞ Zs ðuÞdMs; 0 showing that ZðuÞ is the stochastic exponential of the scaled local martingale ðuÞM. Hence, ZðuÞ is a local martingale itself. The integrability condition on the compensator guarantees that the stopped process ZT ðuÞ is a uniformly integrable martingale. The family of martingales ZðuÞ indexed by u induces a family of equivalent prob- u u u ability measures P on FT by P ðBÞ¼EðZT ðuÞ1BÞ for B 2FT . Each measure P corresponds to a conditional Laplace transform of the compensator,

u u vðA AtÞ L ðv; t; TÞ¼E ðe T jFtÞ; u; v 0: ð5Þ The P-Laplace transform (2) can be expressed in terms of the Pu-Laplace transform (5),

ðuÞðA AtÞ ’ðu; t; TÞ¼Eðe T ZT ðuÞ=ZtðuÞjFtÞ u ðuÞðA AtÞ ¼ E ðe T jFtÞ ð6Þ ¼Luð ðuÞ; t; TÞ:

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Formula (6) states that the Laplace transform of a counting process with totally inaccessible arrivals is given by the Laplace transform of the compensator evaluated at the Laplace exponent of the standard Poisson process. The Laplace transform of A is taken under an equivalent measure Pu. To calculate that transform, consider the specification of A in terms of a non-negative intensity process such that Z t At ¼ s ds ð7Þ 0 almost surely. The process is the density of the random measure associated with A relative to Lebesgue measure. It can be interpreted as the conditional event arrival rate, in the sense that EðNtþ Nt jFtÞt for “small” > 0, see Bre´maud (1980). With the representation (7), the Laplace transform (5) takes the form R T u u v sds L ðv; t; TÞ¼E ðe t jFtÞ: ð8Þ The expectation on the right hand side of equation (8) is a familiar expression in finance; see Duffie (1966), for example. Consider a government bond that pays 1 at time T. The price of this bond at t T is analogous to formula (8) once we regard the process v as the short-term rate of interest. The explicit calculation of the bond price is well understood for a large class of interest rate processes, see Duffie, Pan & Singleton (2000) and Leippold & Wu (2002). This observation motivates us to adopt a parametric model formulation from this bond pricing literature for the purpose of specifying the intensity . However, before we can apply the results from the bond pricing literature, we need to understand how the dynamics of the intensity are adjusted when the measure is changed to Pu. This is necessary since the dynamics of are typically specified under P, while the Laplace transform (8) is taken under Pu. To discuss the adjustment of the dynamics of , let V be a local martingale relative to the reference measure P. Girsa- nov’s theorem as stated in Dellacherie & Meyer (1982) implies that the process V þ ðuÞhV; Nið9Þ is a Pu-local martingale. Here, hV; Ni is the P-conditional covariation, i.e. the compensator of the quadratic variation ½V; N relative to P. Consider a local martingale V that does not have jumps in common with N. Here ½V; N¼0, and therefore hV; Ni¼0. Thus, V remains a local martingale under Pu. Now consider the local martingale V ¼ N A, whose jumps coincide with those of N. The quadratic variation ½V; N¼ ½N; N¼N, whose compensator is hV; Ni¼A. Thus, from (9), V þ ðuÞhV; Ni¼V þ ðuÞA ¼ N euA is a local martingale under Pu, meaning N has Pu-intensity eu. The measure change calls for a deterministic scaling of the intensity that depends on the variable u. In Section 3.2 below, we show how these observations can be applied to calculate the Pu-dynamics of for a classical specification of under P adopted from the bond pricing literature. The measure change underlying the transform formula (6) is not necessary when N is a doubly-stochastic Poisson process. The process N is called a doubly-stochastic Pois-

64 JB 111. Band (2009), Heft 2 Kay Giesecke: An Overview of Credit Derivatives son process if the compensator A is adapted to a right-continuous and complete sub-filtration G ¼ðGtÞt0 of F, and the conditional distribution of NT Nt given Ft _GT is the Poisson distribution with parameter AT At, see Bre´maud (1980). More precisely, for all t < T and u 0 we have

uðN NtÞ ðuÞðA AtÞ Eðe T jFt _GT Þ¼e T : ð10Þ

Taking conditional expectations relative to ðP; FtÞ on both sides of equation (10), we get a formula for the Laplace transform of a doubly-stochastic Poisson process:

ðuÞðA AtÞ ’ðu; t; TÞ¼Eðe T jFtÞ ð11Þ ¼L0ð ðuÞ; t; TÞ where L0ðv; t; TÞ is the Laplace transform of the compensator under P ¼ P0. Compare formula (11) with the general transform formula (6). In the general, non doubly-stochastic case, the measure change leads to a transform formula that has the same structure as the doubly-stochastic formula (11). In this sense, the measure change preserves the doubly-stochastic setting. The special feature of this doubly-stochastic setting is that the arrivals of N are not allowed to influence the dynamics of A. This means that a doubly-stochastic Poisson process cannot be self-exciting: an arrival of the process cannot increase the likelihood of further arrivals. This is because the doubly-stochastic compensator is adapted to G, and the event fNt ¼ kg can never be contained in Gt. Since the change of measure to Pu “absorbs” any feedback from N to the dynamics of A,we call Pu the correlation-neutral probability measure. The measure change argument underlying the transform formula (6) is analogous to a similar argument proposed by Carr & Wu (2004) to calculate the Fourier transform of a time-changed Lvy process. Carr & Wu (2004) suggest that this transform is given by the Laplace transform of the time change evaluated at the characteristic exponent of the Lvy process. The Laplace transform is calculated under a complex measure defined by the time-changed Wald martingale associated with the Lvy process. This argument leads to a formula for the Fourier transform of a counting process N if N is realized as a time-changed Poisson process. In this case the compensator A takes the role of the time change and Z coincides with the time-changed Wald martingale.

3.2 Affine point processes

The transform formula (6) is of practical use whenever the Pu-Laplace transform (5) of the point process compensator A can be explicitly calculated. In this section we discuss an example. We consider a default point process N whose intensity is driven by an affine jump diffusion in the sense of Duffie et al. (2000). The resulting transform formula is a special case of the transform formula for general affine jump diffusions developed in Duffie et al. (2000) and Duffie, Filipovic & Schachermayer (2003). Our measure change argument provides an alternative route to the formula for a finite-activity affine jump process.

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Suppose the observation filtration F is generated by a Markov process X that is a strong solution to the stochastic differential equation

dXt ¼ ðXtÞ dt þ ðXtÞ dWt þ dJt; X0 2 R: ð12Þ Here W is a standard Brownian motion, ðXÞ isP the drift process, ðXÞ is the volatility Nt k process, 0 is a sensitivity parameter and Jt ¼ k¼0 J is a point process whose jump times are those of the default counting process N. The random jump sizes Jk are drawn, independently of one another, from a distribution on Rþ that has no mass at zero. We interpret X as a risk factor process that describes the state of the economy. De- faults influence the economic state in that X jumps at each event. The size of the impact is random, and the sensitivity is controlled by . Suppose the compensator A of N has intensity ¼ ðXÞ for some non-negative function on R. Then, there is feedback from N to its intensity via X. The default point process N is self-exciting, in the sense that an event tends to increase the likelihood of further events. By formula (6), the calculation of the Laplace transform of N reduces to the calculation of the Laplace transform (5) of A under the correlation-neutral measure. To obtain an explicit expression for the transform (5), we impose an additional structure on X and . We assume that X and the functions , 2 and satisfy 2 ðxÞ¼K0 þ K1x;ðxÞ ¼ H0 þ H1x; ðxÞ¼0 þ 1x; R T for constant coefficients such that ðXÞ is non-negative and expð 0 ðXsÞdsÞ is integrable for the horizon T > 0. Under these assumptions, the state process X is an affine jump diffusion in the sense of Duffie et al. (2000), and the counting process N is called an affine point process. Special cases of N include the Poisson process (H0 ¼ H1 ¼ ¼ 0), the birth process (K0 ¼ K1 ¼ H0 ¼ H1 ¼ 0), and the Hawkes process (H0 ¼ H1 ¼ 0). Proposition 1 in Duffie et al. (2000) states technical conditions such that R T RðXsÞdsþzXT aðtÞþbðtÞXt Eðe t jFtÞ¼e ð13Þ where RðxÞ¼0 þ 1x for constants 0 and 1, and the coefficient functions bðtÞ¼ bðz; t; TÞ and aðtÞ¼aðz; t; TÞ satisfy the ordinary differential equations 1 @ bðtÞ¼ K bðtÞ H bðtÞ2 ððbðtÞÞ 1Þð14Þ t 1 1 2 1 1 1 @ aðtÞ¼ K bðtÞ H bðtÞ2 ððbðtÞÞ 1Þð15Þ t 0 0 2 0 0 with boundary conditions bðTÞ¼z and aðTÞ¼0 and jump transform Z ðcÞ¼ eczdðzÞ; c 2 R; ð16Þ Rþ whenever this integral is well-defined. For 0 ¼ 1 ¼ 0, formula (13) yields the transform of the compound birth process X ¼ J (set K0 ¼ K1 ¼ H0 ¼ H1 ¼ 0 in (14)–(15)). To get the transform for more general point processes whose intensity is driven by X, we need to construct an enlarged state vector ðX; NÞ by appending the counting process

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N to the original state X, see Errais, Giesecke & Goldberg (2006). Proposition 1 in Duffie et al. (2000) is then applied to ðX; NÞ. Rather than following this route, we show how to use our transform formula (6) in the affine setting. Formula (6) requires the Laplace transform (5) relative to the correlation-neutral measure Pu. We can apply Proposition 1 of Duffie et al. (2000) to obtain a formula for u this Laplace transform. To this end, note that the P -dynamics of theP state X are de- u Nt k scribed by equation (12), where W is a P -Brownian motion, Jt ¼ k¼0 J is a point process whose jump times arrive with Pu-intensity u u u e ðXÞ¼e 0 þ e 1X; and where the Jk are drawn, independently of one another, from the distribution relative to Pu. Here, we use the fact that W has no jumps in common with N and therefore remains a Brownian motion under Pu, that the distribution of the Jk is not affected by the measure change, and that the compensator of N is scaled by eu when the measure is changed. Now we apply Proposition 1 in Duffie et al. (2000), taking Pu as reference measure, to get R T u u v ðXsÞds ðtÞþðtÞXt L ðv; t; TÞ¼E ðe t jFtÞ¼e ð17Þ where the coefficient functions ðtÞ¼ðu; v; t; TÞ and ðtÞ¼ðu; v; t; TÞ satisfy the ordinary differential equations 1 @ ðtÞ¼v K ðtÞ H tÞ2 eu ðððtÞÞ 1Þð18Þ t 1 1 2 1 1 1 @ ðtÞ¼v K ðtÞ H ðtÞ2 eu ðððtÞÞ 1Þð19Þ t 0 0 2 0 0 with boundary conditions ðTÞ¼ðTÞ¼0 and jump transform (16). Thus, the conditional Laplace transform (2) of an affine point process N is given in terms of the solu- tions to the equations (18)–(19) as u ’ðu; t; TÞ¼Lð ðuÞ; t; TÞ¼expððu; ðuÞ; t; TÞþðu; ðuÞ; t; TÞXtÞ: ð20Þ Compare equations (17)–(19) with equations (13)–(15). The measure change influ- ences only the jump terms in the ODEs (18)–(19). It has no effect when ¼ 0 in the risk factor dynamics (12). In this case, N is doubly-stochastic. There is no feedback from N to its intensity, and the Pu-Laplace transform (8) agrees with its counterpart under P.

3.3 Credit derivatives valuation

There are two alternative formulations of the credit derivatives valuation problem: (1) Model the default point process N under an equivalent martingale measure, relative to which the discounted price process of any traded security is a martingale. Apply the transform formula (6) with the martingale measure compensator of N.

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(2) Model N under the actual probability measure that represents the empirical likelihood of default events. Specify a martingale measure equivalent to the physical measure, and calculate the martingale measure compensator from the physical measure compensator by Girsanov’s theorem. Apply (6). Below we follow the first approach, and specify the “risk-neutral” behavior of N under P, which we take to be an equivalent martingale measure relative to a constant risk- free interest rate r. To exemplify this approach, fix a parametric model for the P-compensator A, for instance the affine specification analyzed in Section 3.2 above. Consider a credit derivative with payoff gðNT Þ at maturity T, where g is an integrable function on f0; 1; 2; ...g specified by the terms of the derivative contract. By the martingale property of discounted prices under P, an arbitrage-free value of the derivative at time t T is given by the conditional expectation X rðTtÞ rðTtÞ e EðgðNT Þj FtÞ¼e gðNt þ kÞPðNT Nt ¼ k jFtÞð21Þ k0 where the conditional probabilities PðNT Nt ¼ k jFtÞ are obtained from the Laplace transform (2) of N via the inversion formula (3). If default arrivals are totally inaccessible and expðAT Þ is integrable, then the Laplace transform (2) is given by the Laplace transform of A with respect to the correlation-neutral measure, see formula (6). In practice, the parameters of the model A are chosen in a calibration procedure. Here (21) is applied to a set of marketed credit derivatives, i.e., different payoff functions g available for trading. The parameters are chosen so as to minimize the discre- pancy between the prices of the marketed derivatives and the prices generated by the model for these securities. In practice, the calibrated model is often used to assign prices to non-traded “exotic” credit derivatives. This approach, however, obscures the fact that the credit derivatives market is typically incomplete. That is, not every credit derivative can be perfectly hedged by dynamic trading in the marketed securities. A derivative carries intrinsic risk that cannot be hedged away. In this situation, the practice of pricing exotic derivatives via (21) does not systematically account for the costs of hedging or the residual risks. The pricing rule (21) assigns to the exotic derivative a price that pretends that the security can be perfectly hedged. The proper quantification and valuation of the residual risk motivates the design of incomplete market models. We refer to Fllmer & Schied (2004) for a comprehensive treatment of this topic and Staum (2008) for a survey paper. The second approach outlined above is more complete than the first approach, and would be used for empirical time-series applications that require the distribution of N under the actual measure and the specification and estimation of the risk premia parameters that determine the change of measure from actual to risk-neutral probabilities, as in Berndt, Douglas, Duffie, Ferguson & Schranz (2005), Eckner (2007b), and Aziz- pour & Giesecke (2008a).

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3.4 Simulation

The Laplace transform approach to the valuation of credit derivatives is applicable if the derivative payoff takes a relatively simple form. It covers many standard contracts. Monte Carlo simulation of the default times T k can alternatively be used to estimate the price of a credit derivative. It is the method of choice when the derivative payoff takes a complicated form. For example, the payoff at time t T may be path-depen- dent, and take the form f ðt; ðNsÞ0st; ...Þ for some function f . A powerful simulation approach is related to the time change theorem of Meyer (1971), which implies that if the compensator A is continuous and increases to 1 almost surely, then N can be transformed into a standard Poisson process by a change of time that is given by A. For the simulation of N, we appeal to the converse of this result. The arrivals of N can be generated by re-scaling standard Poisson arrivals with A. This approach is discussed in Daley & Vere-Jones (2003); see also Duffie & Singleton (1998). It requires the simulation of the continuous-time process A. If this process is approximated on a discrete time grid, then the simulation results may suffer from discretization bias. Alternative exact simulation methods are developed by Giesecke & Kim (2007) and Giesecke, Kakavand & Mousavi (2008). These exact methods avoid the discretization of A and lead to unbiased simulation results.

4 Single-name credit derivatives

This section discusses the valuation of credit derivatives referenced on a given firm. The issuer’s default time is taken to be the first jump time T 1 of the counting process N. The corresponding default process is N1 ¼ minðN; 1Þ, the process N stopped at its first jump. The financial loss at default is modeled by an FT1 -measurable random variable ‘1, which is independent of T 1 and has expectation ‘ ¼ Eð‘1Þ. At any time t < T1 ^ T ¼ minðT; T1Þ, the firm’s risk-neutral conditional survival probability satisfies almost surely 1 u PðT > T jFtÞ¼PðNT ¼ 0 jFtÞ¼lim ’ðu; t; TÞ¼lim L ð ðuÞ; t; TÞð22Þ u"1 u"1 where the last equality is valid if default arrivals are totally inaccessible and expðAT Þ is integrable, see Section 3.1 above. If, for example, N is an affine point process as in Sec- tion 3.2, then Luð ðuÞ; t; TÞ is given by formula (20), and with the corresponding coefficient equations (18) and (19), for t < T1 ^ T we then get the formula 1 PðT > T jFtÞ¼expðaðt; TÞþbðt; TÞXtÞð23Þ where the coefficient functions bðtÞ¼bðt; TÞ and aðtÞ¼aðt; TÞ satisfy the ODEs 1 @ bðtÞ¼ K bðtÞ H bðtÞ2 ð24Þ t 1 1 2 1 1 @ aðtÞ¼ K bðtÞ H bðtÞ2 ð25Þ t 0 0 2 0

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with boundary conditions bðTÞ¼aðTÞ¼0. Equations (24) and (25) indicate that the feedback from N to its intensity, which comes from the jump term J in the state equation (12), plays no role for the survival probability (23). Intuitively, the conditional distribution of T 1 is governed by the dynamics of the intensity before T 1. The behavior of the intensity at the event time is not relevant for this distribution.

4.1 Corporate zero-coupon bond

Corporate bonds are issued by firms to raise funds. While they are not credit derivatives in a strict sense, we use them as basic building blocks for credit derivatives. A zero coupon bond with unit face value and maturity T pays

- The face value 1 at T < T 1, - The recovery ð1 ‘1Þ of face value at T1 T. Since P is an equivalent martingale measure, the value of the survival cash flow at a time t prior to default is given by the discounted P-expected payoff 1 1 EðFðt; TÞð1 NT ÞjFtÞ¼Fðt; TÞPðT > T jFtÞ; ð26Þ where Fðt; TÞ¼expðrðT tÞÞ is the price at time t of a unit face value, T-maturity zero coupon government bond without default risk. The corporate bond is worth less than the government bond to account for the issuer’s risk of default. If N is doubly-stochasticR with intensity , then the value of the survival cash flow T ðrþsÞds (26) takes the form Eðe t jFtÞ, which can be interpreted as the value of the cer- tain cash flow 1 when the discount rate is r þ rather than the risk-free rate r. Thus, the intensity can be interpreted as a short-term credit spread that compensates the bond investor for assuming the issuer’s risk of default over an infinitesimal period. The value of the recovery cash flow at a time t prior to default is 1 1 1 EðFðt; T Þð1 ‘ ÞNT jFtÞ¼ð1 ‘ÞRtðTÞð27Þ 1 where RtðTÞ is the pre-default value of a unit recovery payment at T T. By Stieltjes integration by parts and Fubini’s theorem, Z T 1 RtðTÞ¼E Fðt; sÞdNs Ft t Z 28 T ð Þ 1 1 ¼ Fðt; TÞPðT T jFtÞþr Fðt; sÞPðT s jFtÞds: t

The pre-default value of the bond BtðTÞ is given by the value of the survival cash flow (26) plus the value of the recovery cash flow (27). If the survival probability (22) is analytically tractable then so is the corporate bond pricing problem. The zero coupon credit spread CS is the difference between the (promised) yield on a corporate zero bond and the yield on a government zero bond with matching face value and maturity. With Fðt; TÞ the price of the government bond,

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1 BtðTÞ CS ðTÞ¼ log ð29Þ t T t Fðt; TÞ at any time t < T prior to default. Thus, the value CStðTÞ is the excess yield at time t over the risk-free yield r demanded by the bond investor for bearing the default risk over the remaining term T t. Figure plots the term structure of credit spreads T ! CS0ðTÞ when N is an affine point process driven by a state variable that follows a mean-reverting Feller diffusion process; see Feller (1951) and Section 3.2 above.

4.2 Corporate coupon bond

Most corporate bonds are issued with a coupon that stipulates a stream of interest payments. A corporate coupon bond with unit face value, annualized coupon rate c, coupon dates ðtmÞ and maturity T pays 1 - The coupon cCm at each tm < T where Cm is the day count fraction for period m, - The face value 1 at T < T 1, - The recovery ð1 ‘1Þ of face value at T1 T, T1t - The accrued coupon m1 cC at T 1 if t < T 1 t , where t t . m m m1 m m ¼ m m1

The coupon stream is a portfolio of zero-recovery zero bonds with maturities tm and face values cC . Therefore, at any time t before default, the coupon stream has value X m 1 c Fðt; tmÞCmPðT > tm jFtÞ: tmt The principal payment is a zero-recovery zero bond with maturity T and face value 1 1 and therefore its value is Fðt; TÞPðT > T jFtÞ at any time t before default. The pre-default value of the recovery is ð1 ‘ÞRtðTÞ. Suppose the issuer defaults between two coupon dates. The accrued coupon covers the interest that has accumulated since the last coupon date. Typically, the accrued coupon is not paid separately, but may be sub- sumed into the recovery payment at default. Although artificial, here we treat it separately since it will play a role for credit swaps. The pre-default value of the accrued coupon is X Z C tm c m E Fðt; sÞðs t dN1 F : m1Þ s t tmt m t_tm1 The value of the corporate coupon bond is given by the sum of the values of the four parts; at any time t before default we get 1 Fðt; TÞPðT > T jFtÞþcVtðTÞþð1 ‘ÞRtðTÞ where VtðTÞ is the risky DV01, the value at any time t before default of a unit stream at 1 coupon times ðtmÞ until T ^ T plus any accruals, given by

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kappa=1.00, theta=1.00, sigma=0.30, recovery=0.00 kappa=1.00, theta=1.00, sigma=0.30, recovery=0.40 5 3 lamda0=0.10 lamda0=0.10 lamda0=0.50 lamda0=0.50 2.5 4 lamda0=1.00 lamda0=1.00 lamda0=1.50 lamda0=1.50 lamda0=5.00 2 lamda0=5.00 3 1.5 2 1

1 0.5

0 0 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10

Figure 2. Term structure of zero coupon credit spreads (29) at t ¼ 0 when N is an affine point process with intensityffiffiffiffiffi ¼ X, where X follows a mean-reverting Feller diffusion process: dXt ¼ p 2 ðc XtÞdt þ XtdWt: That is, K0 ¼ c; K1 ¼; H0 ¼ 0; H1 ¼ and ¼ 0 in the risk factor dynamics (12). The parameter controls the speed of mean reversion, c prescribes the mean reversion level, and governs the diffusive volatility of X. The condition 2c 2 guarantees positivity 1 of X. The conditional survival probability PðT > T jFtÞ is given by formula (23). The associated ODEs (24) and (25) can be solved analytically to give a closed formula for the bond price and credit spread. The risk-free rate r ¼ 5%;¼ c ¼ 1 and ¼ 0:3. Left panel: Zero recovery, i.e. ‘ ¼ 1: Right panel: Expected recovery 0.4, i.e. ‘ ¼ 0:6:

X Cm 1 VtðTÞ¼ Fðt; tmÞm Fðt; t _ tm1Þðt _ tm1 tm1ÞPðT t _ tm1 jFtÞ m Ztmt tm 1 Fðt; sÞð1 rðs tm 1ÞÞPðT s jFtÞds : (30) t_tm1 . Now we are ready to analyze a credit swap.

4.3 Credit swap

A credit swap with unit notional, annualized swap spread S, premium payment dates ðtmÞ and maturity T is a bilateral contract in which

- The protection seller pays the default loss ‘1 at T1 T (default leg), 1 - The protection buyer pays the swap spread SCm at each tm < T plus any accruals (premium leg). The value of the premium leg at any time t before default is

PtðSÞ¼SVtðTÞð31Þ

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where VtðTÞ is the risky DV01 given in formula (30). The value of the default leg at any time t before default is

Dt ¼ ‘RtðTÞð32Þ 1 where RtðTÞ is the value of a unit recovery payment at T T given in formula (28). The fair spread S equates the values at inception of the default and premium legs. Since there is no cash flow at inception, the fair swap spread at inception date t is the solution S ¼ StðTÞ to the equation Dt ¼ PtðSÞ. Thus, StðTÞ¼‘RtðTÞ=VtðTÞ at any time t prior to default. Note that, to derive this formula, we have only assumed that the firm’s default time and recovery rate are independent, that the risk-free rate of interest is constant, and that the default risk of the protection seller is negligible.4 Consider an investor who buys protection at t ¼ 0 for the period ½0; T for a swap spread of S0ðTÞ. The mark-to-market value of the investor’s position at time t 0, when the market spread is StðTÞ, is given by VtðTÞðStðTÞS0ðTÞÞ ¼ Dt PtðS0ðTÞÞ at any time t prior to default. The mark-to-market value of the protection seller’s position is the negative of the buyer’s value. Note that the mark-to-market value can be negative. This occurs when the credit quality of the reference name has improved since inception, and default protection is cheaper at current conditions. Suppose we observe a firm’s swap spreads in the credit swap market. We can extract the firm’s (risk-neutral) survival probabilities from these spread quotes. That is, we fix a parametric family of intensity models, and choose model parameters so as to match the observed spreads as closely as possible. The calibrated model implies survival probabilities for all maturities. This procedure is illustrated in Figure 3 (next site), where we extract survival probabilities for the investment bank Lehman Brothers and the telecom- munications firm Comcast, based on market spreads for several maturities on Septem- ber 2, 2008. Lehman’s spread term structure is decreasing, indicating the market’s awareness of Lehman’s difficulties at that time. Lehman defaulted on September 15, 2008. The market implied survival probabilities have multiple applications. For example, they can be used to design credit trading strategies. One strategy is to exploit different valuations of a firm’s creditworthiness that may exist in different markets. For instance, the swap market implied survival probabilities can be contrasted with survival probabilities extracted from the firm’s equity prices, in order to identify those firms for which the gap is relatively large. The investor can bet on the gap to narrow over time. Another important application of the implied survival probabilities is the valuation of other credit-sensitive securities that are issued by or referenced on the firm in question. One such instrument is considered in the next section.

4 Counterparty risk, the risk that the protection seller fails with the reference entity, has become a serious issue in the current market environment. Intuitively, it lowers the fair swap spread StðTÞ calculated under the assumption that counterparty risk is neligible.

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100 700 Lehman Brothers Lehman Brothers Comcast Comcast 95 600 90

500 85

400 80

300 75

CDS Spread in Basis Points 200 Survival Probability in Percent 70

100 65

0 60 0 2 4 6 8 10 0 2 4 6 8 10 Maturity Maturity Figure 3. Extracting the survival probabilities of Lehman Brothers and Comcast from market credit swap spreads. Left panel: Mid-market spread quotes on September 2, 2008, for each of several maturities. Source: Barclays Capital. To fit these quotes perfectly, we assume that the process N has piece-wise constant intensity ðtÞ, whose intervals of constancy are given by the inter-maturity times of the given spread quotes. With a deterministic intensity,R N is a non-homogeneous Poisson 1 T process and the survival probability PðT > T jFtÞ¼expð t ðsÞdsÞ: We sequentially determine the values of the intensity on its intervals of constancy from spread quotes of the corresponding maturities, starting with the shortest maturity, by finding the intensity value that equates a quote to the model spread for a fixed expected loss ‘ ¼ 0:6, the standard industry value. This in- verse problem is usually well-posed. If it is not, then that indicates arbitrage opportunities. Right panel: Survival probabilities PðT 1 > TÞ implied by the calibrated model.

4.4 Forward credit swap

In a regular (spot) credit swap the parties agree on a price for protection that starts im- mediately. In a forward credit swap, the parties agree on a price for protection that starts at a specified time in the future. More precisely, in a forward credit swap with unit notional, annualized forward spread S, premium payment dates ðtmÞ, forward start date U and maturity date T > U,

- The protection seller pays the default loss ‘1 at T1 2½U; T (default leg), 1 - The protection buyer pays the forward swap spread SCm at each tm 2ðU; T Þ plus any accruals (premium leg). Note that cash flows occur only between times U and T. Of course, if U is the time at which the spread is negotiated, then the forward swap is a spot swap. Also note that the contract is canceled (“knocked out”) in case the reference issuer defaults before U. At any time t before default, the premium leg has value

PtðSÞ¼SðVtðTÞVtðUÞÞ The default leg covers the loss over ½U; T and is equal to the difference between the default leg of a spot swap maturing at T and the default leg of a spot swap maturing at U.

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Therefore, its pre-default value must be equal to the difference of the pre-default values of the corresponding premium legs:

Dt ¼ StðTÞVtðTÞStðUÞVtðUÞ

The fair forward spread at time t is the solution S ¼ StðU; TÞ to the equation Dt ¼ PtðSÞ so at any time t prior to default

StðTÞVtðTÞStðUÞVtðUÞ StðU; TÞ¼ VtðTÞVtðUÞ:

Using this formula, we can estimate the forward spread StðU; TÞ from the market spot swap spreads StðUÞ and StðTÞ, and the corresponding risky DV01s VtðUÞ and VtðTÞ, which are calculated via formula (30) based on the market implied survival probabilities.

4.5 Credit swaption

A forward swap obligates the parties to enter into the swap at the forward start date. An option on a credit swap, or swaption with maturity T imparts the right but not the obligation to enter into a swap at a future expiration date U < T at a strike spread S.A payer swaption gives the right to buy protection at U for a spread S. As a forward swap, it can include a knock-out provision, which would stipulate that the contract is canceled in case the reference issuer defaults between the trade date and option’s expiry date U. While a payer swaption allows the investor to monetize widening spreads (deteriorating credit quality), a receiver swaption imparts the right to sell protection at a spread S, and thus allows the investor to bet on tightening spreads.

5 Multi-name credit derivatives

Corporate defaults are correlated because firms are exposed to common or correlated economic risk factors, see Duffie, Saita & Wang (2006). The movements of the risk factors generate correlated changes in firms’ conditional default rates. This relationship is illustrated in Figure 4, which shows the number of defaults of Moody’s rated firms in any given year between January 1970 and September 2008. Some of the risk factors may not be observable, and the uncertainty associated with unobservable factors generates an additional channel of default correlation, see Duffie, Eckner, Horel & Saita (2008). Furthermore, a default may have a contagious impact on the surviving firms that is channeled through the complex web of contractual relationships in the economy, see Azizpour & Giesecke (2008b). The idea is that the failure of a firm tends to weaken the others. The ongoing crisis in credit markets provides evidence of the existence of contagion phenomena. Multi-name or portfolio credit derivatives are referenced on a pool of 5 to several hundred firms. They are often structured as swaps, very much like a credit swap refer-

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Figure 4. Annual defaults of Moody's rated firms between January 1970 and September 8, 2008. Source: Moody's Default Risk Service. The peak in 1970 represents a cluster of 24 railway defaults triggered by the collapse of Penn Central Railway on June 21, 1970. The fallout of the 1987 crash is indicated by the peak in the early nineties. The burst of the internet bubble caused many defaults during 2001-2002. From a trough in 2007, default rates started to increase significantly in 2008. enced on a single obligation, or single-name swap. The instruments are designed to facilitate the transfer of the correlated default risk in the reference portfolio. From a modeling perspective, the challenge is to capture the sources of default clustering mentioned above while maintaining the computational tractability of the valuation relations, which require the distributions of the point processes describing portfolio defaults and losses. There are two different modeling philosophies. In a bottom-up approach, the default processes of the portfolio constituents are the modeling primitives, see Duffie & Garlea- nu (2001), Eckner (2007a), Mortensen (2005) and others. That is, for each constituent i we specify a counting process, NðiÞ say, most conveniently through an intensity ðiÞ. The dependence between the constituent intensity processes reflects the default correlation in the portfolio. The first jump of NðiÞ models the default time of firm i, as in Sec- tion 4 above. Single-firm default probabilities are given by a formula analogous to (22) in terms of ðiÞ. Unfortunately, unless the dependence between the intensities ðiÞ has a specialP structure, the distribution of the portfolio default counting process N ¼ iðNðiÞ^1Þ is much harder to calculate. For example, it is often difficult to calculate this distribution in the presence of contagion effects; see however Collin-Dufresne, Goldstein & Hugonnier (2004) in this regard. In an alternative top-down approach, we specify the portfolio default counting process N directly in terms of an intensity , without reference to the portfolio constituents, see Arnsdorf & Halperin (2007), Cont & Minca (2008), Errais et al. (2006), and others. The default correlation among the portfolio constituents is reflected in the dynamics of . For example, we could specify N as an affine point process; see Section 3.2 above.

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The jumps in the intensity would reflect the contagious impact of an event, while the diffusive movements of the inter-arrival intensity would reflect firms’ exposure to a risk factor. The inversion formula (3) would give the distribution of the portfolio default process. A caveat of this approach is that it requires additional steps to obtain the survival probabilities of the portfolio constituents. Giesecke & Goldberg (2005) propose random thinning to disintegrate into the constituent intensities ðiÞ, and develop a corresponding formula for the constituent survival probabilities. Below we take a top-down approach, with N representing the portfolio default process, to analyze index and tranche swaps. These swaps are referenced on a portfolio of single-name credit swaps rather than bonds or loans. The collection of tranche swaps referenced on a given portfolio is called a collateralized debt obligation, or CDO.

5.1 Index swaps

An index swap is the most liquid portfolio derivative. It is based on a portfolio of n single-name credit swaps. The constituent swaps have common notional that we normalize to 1, common maturity date T and common premium payment dates ðtmÞ. A default in the reference obligation of a constituent swap translates into a loss for the constituentP N k swap protection seller. This loss is recorded by the portfolio loss process L ¼ k¼0 ‘ , where ‘k 2½0; 1 is the normalized random loss at the kth default. In an index swap with swap spread S, the cash flows are as follows:

- The protection seller covers portfolio losses as they occur, i.e. the increments of the portfolio loss process L (default leg),

- The protection buyer pays SCmðn Ntm Þ at each date tm (premium leg). The protection seller assumes exposure to the correlated default risk associated with the reference portfolio. There are multiple families of standard reference portfolios, called indices. The CDX family covers North American issuers. There are various CDX sub-indices, including the Investment Grade (125 constituents) and High Yield (100 constituents) indices. The iTraxx index family represents European and Asian portfolios. The standardization of reference portfolios helps to increase liquidity in the index swap market, which benefits all market participants. The value Dt at time t T of the index swap default leg is given by the discounted cumulative losses. Note that, unlike a single-name swap, an index swap is not termi- nated at a default event. By integration by parts and Fubini, we get Z T Dt ¼ E Fðt; sÞdLs jFt t Z 33 T ð Þ ¼ Fðt; TÞEðLT jFtÞLt þ r Fðt; sÞEðLs jFtÞ ds: t

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Figure 5. Term structure of index swap spreads at t ¼ 0 when N is a self-exciting affine point process with intensity ¼ X, where X follows the process dXt ¼ ðc XtÞdt þ dJt, see Errais et al. (2006). That is, K0 ¼ c; K1 ¼, and H0 ¼ H1 ¼ 0 in the risk factor dynamics (12), and N is a Hawkes process. The intensity jumps in response to a default, with random jump sizes that are drawn independently of one another from a uniform distribution on {0.4, 0.6, 0.8, 1}. The Laplace transform ’ðu; t; TÞ of N is given by formula (20). The corresponding satisfy the ODEs (19) and (18) are solved numerically. The mean EðNT jFtÞ¼Nt @u’ðu; t; TÞju¼0. The mean loss k EðLT jFtÞ¼‘EðNT jFtÞ, assuming that the loss variables ‘ are drawn independently of one another and independently of N and J from a common distribution with mean ‘. We take that distribution to be the uniform distribution on {0.4, 0.6, 0.8, 1}. The number of names n ¼ 125. The risk- free rate r ¼ 5%. Unless stated otherwise, we set X0 ¼ 0:5; c ¼ ¼ ¼ 1: Left panel: Index spread term structure for each of several values of , which governs the exponential decay of the intensity after an event. The higher , the faster the intensity reverts back to the base intensity c after a default. That is, the quicker the negative impact of a default on the economy fades away, and the lower the risk of a default cluster. Thus, index spreads are decreasing in : Right panel: Index spread term structure for each of several values of , which governs the sensitivity of the intensity to a default. The higher , the larger the mean impact of a default on the economic state, and the higher the probability of a default cluster, which generates large losses. Thus, index spreads are increasing in .

The value at time t T of the premium leg is given by X PtðSÞ¼S Fðt; tmÞCmðn EðNtm jFtÞÞ: ð34Þ tmt

The fair index swap spread at time t is the solution S ¼ StðTÞ to the equation Dt ¼ PtðSÞ. Formulae (33) and (34) indicate that the index spread depends only on expected defaults and losses for horizons between t and T. Figure 5 plots the term structure T ! S0ðTÞ of index swap spreads when the default process N is a self-exciting affine point process.

5.2 Tranche swaps

The index protection seller is exposed to the correlated default risk associated with the entire portfolio. Investors seeking narrower risk profiles can trade a tranche swap refer-

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Figure 6. Paths of the portfolio loss process L and the ordered default times (T k), and payments of the protection seller in a senior tranche swap. The protection seller is exposed to the correlated default risk in the reference portfolio. The risk profile is determined by the location of the attachment points K and K. The riskiest tranche is the equity tranche, for which K ¼ 0 and K is typically 3 or 10%. The least risky tranche is the super senior tranche, for which K ¼ 1 and K is typically 30 or 35%. This tranche bears losses only if all the other tranches are wiped out. The collection of all tranches is called a collateralized debt obligation, or CDO. enced on the portfolio. A tranche swap is specified by a lower attachment point K 2½0; 1 and an upper attachment point K 2ðK; 1. The product of the difference K ¼ K K and the portfolio notional n is the tranche notional. In a tranche swap with upfront rate G and swap spread S, the cash flows are as follows:

- The protection seller covers tranche losses as they occur, i.e. the increments of the þ þ tranche loss Ut ¼ðLt KnÞ ðLt KnÞ (default leg), - The protection buyer pays GKn at inception and SCmðKn Utm Þ at each date tm (premium leg, assuming K < 1). Figure 6 shows a sample path of the loss process and the corresponding payments of the protection seller. Unlike an index protection seller, a tranche protection seller is exposed only to a slice of the loss that is specified by the lower and upper attachment points. The higher the lower attachment point, the lower is the risk of payouts for the protection seller, and the lower is the fee that the protection seller can require. This is because the losses must first eat through the subordinated tranches, which provide a safety cushion. The equity tranche, whose lower attachment point is zero, has no safety cushion: the equity protection seller must bear any losses until the equity notional is ex- hausted. It is the riskiest tranche, and commands the highest premium. In a customized or bespoke deal, the attachment points are set so that an investor’s target risk profile is met.

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Figure 7: Term structure of mezzanine tranche spreads S0ð3%; 7%; 0; TÞ at t ¼ 0 when N is a self- exciting affine point process with intensity ¼ X, where X follows the process dXt ¼ ðc XtÞdt þ dJt, see Errais et al. (2006). That is, K0 ¼ c; K1 ¼ and H0 ¼ H1 ¼ 0 in the risk factor dynamics (12), and N is a Hawkes process, as in Figure 5. The distribution of NT Nt is ob- uðL LtÞ tainedP via formula (3), and used to calculate the Laplace transform Eðe T jFtÞ¼ k k k0ð f ðuÞÞ PðNT Nt ¼ k jFtÞ of the loss, assuming that the loss variables ‘ are drawn independently of one another and independently of N and J from a common distribution with Laplace transform f(u). We take that distribution to be the uniform distribution on {0.4, 0.6, 0.8, 1}. The Laplace transform of the loss is inverted to obtain theR loss distribution, which is used to calculate þ n the (undiscounted) call value EððLs cÞ jFtÞ¼ c ðx cÞdPðLs x jFtÞ for 0 c n: The number of names n ¼ 125. The risk-free rate r ¼ 5%. Unless stated otherwise, we set X0 ¼ 0:5; c ¼ ¼ ¼ 1: Left panel: Mezzanine spread term structure for each of several values of , which governs the exponential decay of the intensity after an event. The higher , the faster the intensity reverts back to the base intensity c after a default. Right panel: Mezzanine spread term structure for each of several values of , which governs the propensity of defaults to cluster. The more senior the tranche (i.e., the higher the lower attachment point K, the more sensitive is the tranche spread to . This is because the mass in the tail of the loss distribution is increasing in , and the more senior the tranche the more sensitive it is to scenarios with large losses.

The value at time t of the default leg is Z T DtðK; KÞ¼Fðt; TÞEðUT jFtÞUt þ r Fðt; sÞEðUs jFtÞ ds: ð35Þ t This formula is analogous to formula (33) for the value of an index swap default leg. The latter can be viewed as the default leg of a tranche swap for which K ¼ 0 and K ¼ 1. The value of the premium leg is given by X PtðK; K; G; SÞ¼GKn þ S Fðt; tmÞCmðKn EðUtm jFtÞÞ: ð36Þ tmt For a fixed upfront payment rate G, the fair tranche spread S is the solution S ¼ StðK; K; G; TÞ to the equation DtðK; KÞ¼PtðK; K; G; SÞ. Similarly, for a fixed tranche spread S, the fair tranche upfront rate G is the solution G ¼ GtðK; K; S; TÞ to the equation DtðK; KÞ¼PtðK; K; G; SÞ. The fair spread and upfront rate depend only on the value of call spreads þ þ Fðt; sÞEðUs jFtÞ¼Fðt; sÞðEððLs KnÞ jFtÞEððLs KnÞ jFtÞÞ

80 JB 111. Band (2009), Heft 2 Kay Giesecke: An Overview of Credit Derivatives

Contract MaBid MaAsk Model 0–10 70.50% 70.75% 71.48% 10–15 34.25% 34.50% 32.74% 15–25 316.00 319.00 311.43 25–35 79.00 81.00 77.34 Index 262.85 263.10 262.97 AAPE 2.24%

Figure 8. Index and tranche market calibration at t ¼ 0 when N is a self-exciting affine point process with intensity ¼ X, where X follows the process dXt ¼ ðc XtÞdt þ dJt, as analyzed in Figures 5 and 7. The loss variables ‘k are drawn independently of one another and independently of N and J from a common uniform distribution over {a, b} with 0 < a < b < 1. Consistent with market practice, we set the expected loss at default ‘ ¼ Eð‘kÞ¼0:6. The parameterP vector to be calibrated is ¼ðX0; c;;;aÞ. We solve the optimization problem min2 iðMaMidðiÞ Modelði;ÞÞ2=ðMaAskðiÞMaBidðiÞÞ2; where ¼½0; 540:2; 0:6 and the sum ranges over the data points. The market mid quote MaMid is the arithmetic average of the observed MaAsk and MaBid quotes. The risk-free rate r ¼ 5%. More details are given in Giesecke & Kim (2007). Left panel: Market bid and ask quotes for the 5 year maturity index and tranches on the CDX High Yield portfolio of n ¼ 100 names, observed on 5/11/2007, together with model calibrated values at the optimal parameter vector ¼ð0:75; 1:60; 2:58; 2:94; 0:24Þ. The quotes are in basis points ex- cept for the [0, 10] and [10, 15] percent tranches, which are quoted in terms of a percentage upfront fee. We report the average absolute percentage fitting error AAPE. Right panel: Smoothed portfolio loss distribution PðLT 2 dxÞ for T between 2 and 10 years implied by the calibrated model.

on the portfolio loss Ls with strikes Kn and Kn and maturities s between t and T. Figure 7 shows the term structure T ! S0ð3%; 7%; 0; TÞ of mezzanine tranche spreads when the default process N is a self-exciting affine point process, as in Figure 5. In analogy to the single-name swap case, we can calibrate a parametric model of the risk-neutral intensity of the portfolio default process N from market spreads of index and tranche swaps with different maturities and attachment points, all referenced on the same portfolio. The calibration procedure is illustrated in Figure 8 for the self-exciting affine point process considered in Figures 5 and 7, for which is an affine function of an affine jump diffusion. The calibrated intensity model induces the market-implied risk-neutral distributions of the portfolio default and loss processes. It is used to estimate the arbitrage-free value of a non-traded portfolio derivative, for example a tranche referenced on the given portfolio but with non-standard attachment points. There is often model risk associated with this procedure, in that there may be many distinct parametric intensity models that fit the given market data but imply different prices for a given non-traded portfolio derivative.

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5.3 Other instruments

There is a range of other portfolio credit derivatives that parallel the instruments in the single-name space. In particular, index and tranche swaps are also traded on a forward basis. The forward spread curve can be constructed from the spot spread curve using the arbitrage argument given in Section 4.4 for the single-name case. There are also op- tions on index and tranche swaps, which are analyzed in Ding, Giesecke & Tomecek (2006). A collateralized debt obligation can be structured in several ways. Above we have analyzed a synthetic CDO, which is backed by credit swaps. In a cash CDO, the pool of reference securities contains straight bonds, loans or other credit obligations. While the rationale of the cash CDO is similar to that of the synthetic CDO, the cash structure often has additional features that complicate the analysis. For example, cash instruments such as bonds or loans generate interest payments that increase collateral. Mortgages can be retired before their maturity, generating additional pre-payment risk. In order to account for such features, the transform analytic methods described above must be abandoned for simulation methods as in Duffie & Garleanu (2001). Here, simulation is used to generate cash flow scenarios according to an underlying stochastic model. Simu- lation schemes for the affine point processes of Section 3.2 are described in Giesecke & Kim (2007) and Giesecke et al. (2008).

References

Arnsdorf, Matthias & Igor Halperin (2007), BSLP: markovian bivariate spread-loss model for portfolio credit derivatives. Journal of Computational Finance, forthcoming. Azizpour, Shahriar & Kay Giesecke (2008a), Premia for correlated default risk. Working Paper, Stanford University. Azizpour, Shahriar & Kay Giesecke (2008b), Self-exciting corporate defaults: Contagion vs. frailty. Working Paper, Stanford University. Berndt, Antje, Rohan Douglas, Darrell Duffie, Mark Ferguson & David Schranz (2005), Measur- ing default risk premia from default swap rates and EDF’s. Working Paper, Stanford Univer- sity. Brmaud, Pierre (1980), Point Processes and Queues - Martingale Dynamics, Springer-Verlag, New York. Carr, Peter & Liuren Wu (2004), Time-changed Lvy processes and option pricing’, Journal of Fi- nancial Economics 71, 113–141. Collin-Dufresne, Pierre, Robert Goldstein & Julien Hugonnier (2004), A general formula for the valuation of defaultable securities’, Econometrica 72, 1377–1407. Cont, Rama & Andreea Minca (2008), Extracting portfolio default rates from CDO spreads. Working Paper, Columbia University. Daley, Daryl & David Vere-Jones (2003), An Introduction to the Theory of Point Processes, Sprin- ger-Verlag, New York. Dellacherie, Claude & Paul-Andre Meyer (1982), Probabilities and Potential, North Holland, Ams- terdam. Ding, Xiaowei, Kay Giesecke & Pascal Tomecek (2006), Time-changed birth processes and multi- name credit derivatives. Operations Research, forthcoming.

82 JB 111. Band (2009), Heft 2 Kay Giesecke: An Overview of Credit Derivatives

Duffie, Darrell (1996), Dynamic Asset Pricing Theory, second edn, Princeton University Press, Princeton, New Jersey. Duffie, Darrell, Andreas Eckner, Guillaume Horel & Leandro Saita (2008), Frailty correlated default. Journal of Finance, forthcoming. Duffie, Darrell, Damir Filipovic & Walter Schachermayer (2003), Affine processes and applications in finance’, Annals of Applied Probability 13, 984–1053. Duffie, Darrell, Jun Pan & Kenneth Singleton (2000), Transform analysis and asset pricing for affine jump-diusions’, Econometrica 68, 1343–1376. Duffie, Darrell & Kenneth J. Singleton (1998), Simulating correlated defaults. Working Paper, GSB, Stanford University. Duffie, Darrell, Leandro Saita & Ke Wang (2006), Multi-period corporate default prediction with stochastic covariates’, Journal of Financial Economics 83(3), 635–665. Duffie, Darrell & Nicolae Garleanu (2001), Risk and valuation of collateralized debt obligations’, Financial Analysts Journal 57(1), 41–59. Eckner, Andreas (2007a), Computational techniques for basic affine models of portfolio credit risk. Working Paper, Stanford University. Eckner, Andreas (2007b), Risk premia in structured credit derivatives. Working Paper, Stanford University. Errais, Eymen, Kay Giesecke & Lisa Goldberg (2006), Affine point processes and portfolio credit risk. Working Paper, Stanford University. Feller, William (1951), Two singular diusion problems’, Annals of Mathematics 54, 173–182. Fllmer, Hans & Alexander Schied (2004), Stochastic Finance: An Introduction in Discrete Time, De Gruyter, Berlin. Giesecke, Kay & Baeho Kim (2007), Estimating tranche spreads by loss process simulation, in S. G. Henderson, B. Biller, M.-H. Hsieh, J. Shortle, J. D. Tew & R. R. Barton, eds, Proceed- ings of the 2007 Winter Simulation Conference’, IEEE Press, pp. 967–975. Giesecke, Kay, Hossein Kakavand & Mohammad Mousavi (2008), Simulating point processes by intensity projection, in S. J. Mason, R. R. Hill, L. Moench & O. Rose, eds, Proceedings of the 2008 Winter Simulation Conference’, IEEE Press. Giesecke, Kay & Lisa Goldberg (2005), A top down approach to multi-name credit. Working Pa- per, Stanford University. Karatzas, Ioannis & Steven E. Shreve (1988), Brownian Motion and Stochastic Calculus, Graduate Texts in Mathemathics; 113, Springer-Verlag, New York. Leippold, Markus & Liuren Wu (2002), Asset pricing under the quadratic class’, Journal of Finan- cial and Quantitative Analysis 37, 271–295. Meyer, Paul-Andr (1971), Dmonstration simplife d’un thore`me de Knight, in Sminaire de Probabilits V, Lecture Note in Mathematics 191’, Springer-Verlag Berlin, pp. 191–195. Mortensen, Allan (2005), Semi-analytical valuation of basket credit derivatives in intensity-based models. Working Paper, Copenhagen Business School. Protter, Philip (2004), Stochastic Integration and Differential Equations, Springer-Verlag, New York. Sarig, Oded & Arthur Warga (1989), Some empirical estimates of the risk structure of interest rates’, Journal of Finance 44, 1351–1360. Staum, Jeremy (2008), Incomplete markets, in J. Birge & V. Linetsky, eds, Handbooks in Opera- tions Research and Management Science’, Vol. 15, Elsevier, chapter 12.

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Henri Cartan,

ein franzsischer Klaus Hulek Freund

Klaus Hulek1 und 2 Cartan (1985) – Foto: G. Fischer Thomas Peternell Thomas Peternell

Abstract

- Mathematics Subject Classification: 01A60 - Keywords and Phrases: Geschichte der Mathematik, Henri Cartan, Bourbaki

Letztes Jahr verstarb Henri Cartan, Ehrenmitglied der DMV, im Alter von 104 Jahren. In diesem Aufsatz gehen wir vor allem dem Zusammenwirken zwischen Cartan und deutschen Mathematikern nach. Die Kontakte zu Cartan waren vor allem in den ersten Nachkriegsjahren fr die Mathematik in Deutschland von grßter Bedeutung. Zum einen gelangten auf diese Weise die neuesten Ideen aus Paris nach Deutschland, zum anderen ffneten diese Kontake fr die deutschen Mathematiker die Tr, ihre internationale Isolation zu berwinden.

1 Gottfried Wilhelm Leibniz Universitt Hannover, Institut fr Algebraische Geometrie, Welfen- garten 1, D-30167 Hannover, [email protected] 2 Mathematisches Institut, Universitt Bayreuth, D-95440 Bayreuth, [email protected]

JAHRESBERICHT DER DMV Eingegangen: 16.02.2009 Vieweg+Teubner 2009

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Am 13. August 2008 verstarb der bedeutende franzsische Mathematiker Henri Cartan im Alter von 104 Jahren. Wir mchten in diesem Artikel – neben einer kurzen Darstel- lung seiner mathematischen Leistungen – den Fokus auf eine Wrdigung seiner Bezie- hungen zu Deutschland und seiner Verdienste zur franzsisch-deutschen Ausshnung nach dem zweiten Weltkrieg legen. Cartan war einer der ersten, die nach dem Ende der Nazi-Herrschaft wieder Kontakte nach Deutschland knpften, trotz eines schweren familiren Verlustes – sein Bruder Louis war als Mitglied der Rsistance in Deutschland ermordet worden. Henri Cartan wurde am 8. Juli 1904 als Sohn von Elie Cartan, der selbst fundamen- tale Beitrge zur Lietheorie und Differentialgeometrie leistete, in Nancy geboren. Nach dem Studium in Paris wurde er 1928 an der E´ cole Normale Supe´rieure promoviert. Er hatte zunchst Stellen in Lille und Straßburg inne (die dortige Universitt wurde 1939 nach Clermont-Ferrand evakuiert) und kehrte dann 1940 an die E´ cole Normale Supe´- rieure nach Paris zurck; berhmt wurde hier sein „Sminaire Cartan“, das er seit 1948 leitete. Nach einem Vierteljahrhundert wissenschaftlicher Arbeit an der ENS wechselte Cartan nach Orsay an die Universit Paris-Sud und wurde dort 1975 emeritiert. Cartan war einer der wichtigsten Mathematiker des zwanzigsten Jahrhunderts; un- terstrichen wird dies durch die Verleihung des Wolf-Preises 1980 und durch zahlreiche Ehrendoktorwrden und Akademiemitgliedschaften, etwa bei der Acadmie des Sci- ences und der Royal Society. Von 1964–1967 war er Vizeprsdident, von 1967–1970 Prsident der Internationalen Mathematischen Union und 1970 Prsident des Fields- Komitees sowie Prsident des „steering committees“ des Europischen Mathematik- Kongresses 1992 in Paris. Neben seinen Ttigkeiten in internationalen Wissenschafts- organisationen war Cartan auch politisch aktiv: Der berzeugte Europer kandidierte fr das Europaparlament, und er engagierte sich im Comit des Mathematiciens fr politisch verfolgte Wissenschaftler. Die franzsische Zeitung Le Figaro beschreibt ihn in einem Nachruf als „militant de la libert“. Im Folgenden gehen wir auf die Rolle Henri Cartans fr die Entwicklung der Ma- thematik im Deutschland der Nachkriegszeit ein. Im Sommer 1931 besuchte Cartan – er war damals 26 Jahre alt – auf Einladung von Heinrich Behnke die Universitt Mns- ter. Zu dieser Zeit waren solche Besuche im Vergleich zu heute eher selten. Die beiden Mathematiker hatten jedoch viele gemeinsame Interessen, vor allem ihr Arbeitsgebiet, nmlich das damals noch sehr junge Gebiet der Funktionentheorie mehrerer Variablen. Sie interessierten sich zu dieser Zeit insbesondere fr so genannte Kreisgebiete, also Ge- biete in C2; die den Ursprung enthalten und eine Operation der Kreislinie S1 zulassen. Beide hatten sie, unabhngig voneinander, Resultate ber die Abbildung zwischen Kreisgebieten bewiesen: Es gilt nmlich, dass Abbildungen zwischen beschrnkten Kreisgebieten notwendigerweise linear sind. Aber auch Konvexittstheorie und die Existenz bestimmter Holomorphiegebiete waren Gegenstnde der Diskussionen. Wh- rend seines Aufenthalts in Mnster hielt Cartan insgesamt fnf Vortrge, davon vier auf Deutsch. Zahlreiche Spaziergnge und eine gemeinsame Wanderung frderten weiter den Ideenaustausch. Dies war der Beginn einer ber Jahrzehnte anhaltenden mathematischen Zusammenarbeit zwischen Cartan und Behnke, die spter auf die „Behnke- schule“ ausgedehnt wurde, ebenso wie einer tiefen persnlichen Freundschaft.

86 JB 111. Band (2009), Heft 2 Klaus Hulek und Thomas Peternell: Henri Cartan, ein franzsischer Freund

Unter anderem schloss Cartan auch mit Peter Thullen Freundschaft, der damals Behnkes Assistent war und spter einen sehr ungewhnlichen Lebensweg nehmen sollte. Als berzeugter Katholik musste er bereits 1935 Nazi-Deutschland verlassen und ging ber Italien nach Ekuador, wo er zunchst eine Professur in Quito innehatte und spter das Sozialversicherungssystem Ekuadors mit aufbaute. Nach dem Krieg wechselte Thullen als Chefmathematiker zum Internationalen Arbeitsamt in Genf, wirkte aber nach seiner Pensionierung wieder als Professor, unter anderem an der ETH Zrich. Die Diskussionen Cartans mit Behnke und Thullen flossen insbesondere auch in den berhmten, im Jahr 1934 im Springer-Verlag erschienen, Ergebnisbericht „Theorie der Funktionen mehrerer komplexer Vernderlichen“ von Behnke und Thullen [BT] ein. Dies blieb fr viele Jahre das Standardwerk auf diesem Gebiet der mehr-dimensionalen Funktionentheorie. Dieses Buch behandelt neben den Grundlagen auch das Levi-Pro- blem (das man allerdings damals noch nicht so nannte), die Frage nach der Existenz von regulren (holomorphen) und meromorphen Funktionen mit vorgegebenen Null- und Polstellenmengen (Cousinprobleme), Holomorphiegebiete und ein langes Kapitel zur Abbildungstheorie. Dort werden unter anderem die Cartanschen Krper bespro- chen, eine Verallgemeinerung der Kreisgebiete, die im Mittelpunkt des Besuchs von 1932 gestanden waren. Das Literaturverzeichnis belegt die enge Zusammenarbeit der Schule um Behnke mit Cartan, es werden 10 Arbeiten von Henri Cartan und eine gemeinsame Arbeit von Cartan und Thullen zitiert. Behnke und Cartan trafen sich bereits 1932 wieder, und zwar beim Internationalen Mathematikerkongress in Zrich. 1937 fuhr Behnke zu einem privaten Besuch zu Cartan nach Frankreich und 1938 besuchte Cartan nochmals Mnster. Inzwischen hatte sich die Atmosphre dramatisch verndert, und Cartans Besuch stand ganz im Schat- ten des nationalsozialistischen Regimes am Vorabend des zweiten Weltkriegs. Nichts- destoweniger war auch dieser Besuch mathematisch fruchtbar. Unter anderem kam es zu ausgedehnten Diskussionen von Cartan mit Karl Stein, einem jngeren Mitglied der Behnke-Schule. Die Kontakte zwischen Cartan und deutschen Mathematikern kamen selbst mit Ausbruch des zweiten Weltkriegs nicht vollstndig zum Erliegen. Behnke versuchte, einige Manuskripte Cartans aus dessen Straßburger Wohnung zu retten (Cartan musste ja zwangsweise nach Clermont-Ferrand umziehen). Nach einem zunchst vergeblichen Anlauf gelang Behnke dies auch, Cartan erhielt die Papiere schließlich nach dem Ende des zweiten Weltkriegs. Darunter befand sich unter anderem das Protokoll der Sitzung jener Gruppe junger franzsischer Mathematiker, zu denen auch Chevalley, Dieudonn und Weil gehrten und aus der kurze Zeit danach die Gruppe Bourbaki wurde. Im Jahr 1941 leitete Behnke Cartan einen Brief des japanischen Mathematikers Oka, der sich mit dem Leviproblem befasst hatte, zu. Die Arbeiten von Oka verwendeten unter anderem Ergebnisse von Behnke, Cartan, Stein und Thullen. Besonders tragisch aber war das Schicksal von Cartans Bruder Louis. Dieser hatte sich der franzssischen Rsistance angeschlossen und war im Februar 1943 nach Deutschland verschleppt worden. Dort wurde er im Dezember 1943 von den Nazis ermordet. Cartan wandte sich an einige deutsche Kollegen, darunter auch Heinrich Behn- ke, mit der Bitte um Hilfe. Diese versuchten, allerdings vergeblich, Informationen ber

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das Schicksal von Louis Cartan zu erhalten. Erst 1945 erfuhr Henri Cartan vom Tod seines Bruders. Umso erstaunlicher ist es, dass Cartan bereits im November 1946 wieder nach Deutschland reiste, und zwar nach Oberwolfach, wo er auch Behnke erneut traf. Das Gstebuch in Oberwolfach verzeichnet einen Vortrag von Cartan ber die Galoistheo- rie nicht-kommutativer Krper, ebenso wie ein gemeinsames Konzert. Weitere Besuche in Oberwolfach folgten 1950 und 1957. Bei diesen Treffen trug Cartan ber homologische Algebra und reell-analytische Untervarietten vor.

Eintragung Cartans im Vortragsbuch 1946

Im Dezember 1949 fand der erste Nachkriegsbesuch von Henri Cartan in Mnster statt. Neben Karl Stein, der jetzt Dozent in Mnster war, waren mehrere jngere Ma- thematiker zu der Gruppe um Heinrich Behnke gestoßen, insbesondere Hans Grauert, Friedrich Hirzebruch und Reinhold Remmert. Die Besuche von Cartan waren fr die Mnsteraner Schule, und in weiterem Sinn fr die Mathematik in Deutschland, von

88 JB 111. Band (2009), Heft 2 Klaus Hulek und Thomas Peternell: Henri Cartan, ein franzsischer Freund großer Bedeutung. Nach dem zweiten Weltkrieg waren die deutschen Mathematiker zunchst international isoliert, man hatte den Anschluss an wichtige internationale Ent- wicklungen verloren. Hier half Cartan in mehrfacher Weise. Insbesondere berichtete er ber die neuesten Ergebnisse, vor allem der gerade in Frankreich einsetzenden str- mischen Entwicklung. Speziell durch die Arbeiten von Cartan, Serre und spter auch Grothendieck hatte eine Neuorientierung eingesetzt, die die Funktionentheorie mehrerer Vernderlichen, ebenso wie die gesamte algebraische Geometrie, auf eine vllig neue Grundlage stellte. Ein entscheidender Einschnitt war dabei die Einfhrung der Garben- theorie. Die Garbentheorie hat verschiedene historische Wurzeln, in jedem Fall zu nen- nen sind die Arbeiten von J. Leray, der diese Technik bei der Behandlung partieller Dif- ferentialgleichungen benutzte. Es war Henri Cartan, der diese Theorie in der Funktio- nentheorie einfhrte: Im Jahr 1948 erschien erstmals im Sminaire Cartan eine Abhandlung ber Garbentheorie. Die Anwendung der neuen Techniken wurde dann von Cartan und Serre (Theoreme A und B, Endlichkeitssatz, s. u., Serre-Dualitt) weiter vorangetrieben und schließlich von Grothendieck nochmals durch das Konzept der Topoi erweitert. Auch in der algebraischen Topologie, die ebenfalls in dieser Zeit große Fortschritte machte, so wurde etwa die Theorie der Faserbndel entwickelt, spielte die Garbentheorie eine wichtige Rolle (lokale Systeme). Obwohl in erster Linie ein Forma- lismus, erffnete das Konzept der Garbentheorie enorme technische Mglichkeiten und konzeptionell neue Zugnge. Zum einen fhrte dies zu starken Verallgemeinerungen, so wurden etwa die bis dahin untersuchten affinen oder projektiven Varietten durch den sehr viel umfassenderen Begriff der Schemata ersetzt, dem in der komplexen Analysis die komplexen Rume entsprechen. Zum anderen ergab sich die Mglichkeit, bisher nicht miteinander in Beziehung stehende Gebiete der Mathematik miteinander zu ver- binden und unter gemeinsamen Aspekten zu behandeln. Ohne diesen Zugang sind we- der die moderne algebraische Geometrie noch weite Teile der modernen Zahlentheorie, vor allem der arithmetischen Geometrie, welche sehr stark die Sprache und Konzepte der Schemata benutzt, denkbar. Diese Zeit der frhen 50er Jahre wurde mitunter auch als „franzsische Revolution“ bezeichnet, deren Motto „il faut faisceautiser“ gewesen war. Es war das große Glck der jungen Mathematiker in Mnster, diese Entwicklung durch die Besuche von Cartan aus erster Hand kennen zu lernen. Die Mitglieder der Behnke-Schule ließen sich von den modernen Methoden begeistern, nahmen diese schnell auf und entwickelten sie spter selbstndig weiter. So formulierte schließlich Grauert nach Arbeiten von Behnke, Cartan, Remmert, Serre und Stein, die um die rich- tige Verallgemeinerung des Begriffs der Riemannschen Flche auf hhere Dimensionen gerungen hatten, die allgemeine, heute gltige Definiton eines komplexen Raums, der auch nilpotente Elemente zulsst; wir gehen darauf noch etwas genauer ein. In der algebraischen Geometrie wurde das entsprechende Konzept von Grothendieck einge- fhrt. Neben den Informationen, die auf diese Weise nach Deutschland flossen, befreiten die Besuche von Cartan zudem die deutschen Mathematiker aus der internationalen Isolation. Insbesondere bemhte sich Cartan auch darum, dass außer ihm weitere franzsische Mathematiker schon frh an Tagungen in Oberwolfach teilnahmen. So kam bereits im August 1949 Dieudonn in Begleitung von Braconnier, Serre und Thom zu ei-

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ner Zahlentheorietagung nach Oberwolfach. Umgekehrt berichteten Cartan und Serre bereits im Dezember 1953 im Sminaire Bourbaki ber Arbeiten von Hirzebruch und machten so dessen Ergebnisse im Ausland einem grßeren Kreis zugnglich. Spter folgten weitere Bourbaki-Berichte, unter anderem auch ber die Arbeiten von Grauert. Ein dominantes Thema der frhen 50er Jahre in Mnster war die Theorie der holomorph-vollstndigen Mannigfal- tigkeiten bzw. holomorph-vollstndigen komplexen Rume. Es war Serre, der hierfr den Begriff der „Varits de Stein“ (deutsch Steinsche Mannigfaltig- keiten bzw. Steinsche Rume) prgte. In einem Kolloquium in Brssel im Jahr 1953 berichtete Cartan ber die Theo- reme A und B fr Steinsche Mannigfal- tigkeiten, die er gemeinsam mit Serre bewiesen hatte – wir gehen darauf unten noch etwas genauer ein. Dieser Vortrag machte einen außerordentlichen Ein- druck auf die Anwesenden. Die dort pr- sentierten Ergebnisse stellen sicher einen Hhepunkt der Entwicklung der spten 40er und frhen 50er Jahre dar. Die Be- nennung nach Karl Stein, der die ent- scheidenden Eigenschaften dieser Rume beschrieben hatte, zeigte auch, dass die Mnsteraner Schule im Ausland zuneh- mend an Reputation gewann. Diese Auf- H. Behnke und H. Cartan 1957 bruchphase der komplexen Analysis in Foto L. Danzer Deutschland und der Mnsteraner Schu- le im Besonderen ging in den 60er Jahren zu Ende. Im Jahre 1968 erschien die zweite Auflage des Ergebnisberichts von Peter Thullen, herausgegeben von R. Remmert. Neben einem Abdruck des ursprnglichen Texts haben verschiedene Autoren Kapitel zu den neuesten Entwicklungen geschrieben. Der historisch interessierte Leser findet hier eine gute Quelle, um die Entwicklung bis 1968 nachzuverfolgen. Die Deutschen waren aber nicht nur von dem Mathematiker Cartan begeistert. Auch als Mensch machte er einen großen Eindruck auf sie. Nach dem Vortrag von Cartan in Mnster im Dezember 1949 gab es einen Empfang. Dabei brachte Cartan einen Toast „ l’Europe“ aus – fr die meisten der anwesenden deutschen Mathematiker zu diesem Zeitpunkt bestenfalls eine khne Vision. Noch in seinen „Journe en l’hon- neur d’Henri Cartan“ aus dem Jahr 1997 erinnert sich Reinhold Remmert, welchen gro- ßen Eindruck dies auf die damals Anwesenden machte.

90 JB 111. Band (2009), Heft 2 Klaus Hulek und Thomas Peternell: Henri Cartan, ein franzsischer Freund

Cartan besuchte weiterhin regelmßig Deutschland. Im Jahr 1952 erhielt er von der Universitt Mnster die Ehrendoktorwrde zugesprochen. Demnach htte er 2002 sein goldenes Ehrendoktorjubilum feiern knnen – vermutlich ein sehr seltenes Ereignis. Zu erwhnen ist ferner etwa seine Teilnahme an der Antrittsvorlesung von Hirzebruch in Mnster im Jahre 1955. Im Jahr 1973 kam Cartan nach Mnchen, wo er anlsslich des 60. Geburtstags von Karl Stein einen Kolloquiumsvortrag hielt. Dort berichtete er ber die Arbeiten von Stein, unter anderem auch ber die Steinsche Lsung der Cousin- probleme. Auch das Mathematische Forschungsinstitut Oberwolfach war weiterhin das Ziel von Reisen von Cartan. In den Jahren 1984 und 1994 sprach er Grußworte anlsslich des 40- bzw. 50-jhrigem Jubilums. Die letzte Teilnahme an einer wissenschaftlichen Tagung war 1981 bei einer von O. Forster und K. Stein veranstalteten Tagung ber Komplexe Analysis. Bei dieser Gelegenheit hatten auch die Autoren das Privileg, Henri Cartan zu treffen.

Cartan in Oberwolfach 1971 – Foto: K. Peters

Im Jahr 1994 verlieh die DMV Henri Cartan anlsslich seines 90. Geburtstags die Ehrenmitgliedschaft. In seinem Anschreiben wrdigte der damalige DMV-Vorsitzende Martin Grtschel insbesondere auch die Rolle, die Cartan fr die Mathematik in Deutschland in der Nachkriegszeit gespielt hatte. Er schrieb, dass Cartan vielen, vor allem jngeren, Mathematikern das Gefhl gegeben habe, dass sie „trotz des Unheils, das Deutschland ber die Welt gebracht hatte, nicht aus der internationalen Gemeinschaft der Mathematiker ausgeschlossen waren“ und dass dies „Mut und Kraft fr viele in der ungewissen Zeit nach dem Kriege“ gespendet habe. Im letzten Jahr wrdigte der Pr- sident der DMV, Gnter Ziegler, die Verdienste von Henri Cartan in einem Beitrag fr die Onlineausgabe der Wochenzeitschrift Die Zeit.

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Wir gehen nun noch etwas auf das mathematische Werk Cartans ein. Eine eingehen- de und umfassende Wrdigung des mathematischen Werks von Henri Cartan ist in einem solchen Artikel unmglich. Wir sprechen daher nur wenige Punkte an und konzen- trieren uns vor allem auf eines seiner Hauptarbeitsgebiete, nmlich die komplexe Ana- lysis, denn gerade hier war die Zusammenarbeit mit deutschen Mathematikern besonders befruchtend. Cartan gilt zu Recht als einer der Pioniere der modernen komplexen Analysis, die nach 1945 den bergang von der Funktionentheorie mehrerer Va- riablen zur Theorie der komplexen Mannigfaltigkeiten und der komplexen Rume voll- zog. Insbesondere die Klrung des Begriffs des komplexen Raumes stand dabei im Vor- dergrund; Cartan und Serre fhrten dabei eine Version ein, die heute als normaler komplexer Raum bezeichnet wird. Diese normalen komplexen Rume sind dadurch charakterisiert, dass der zweite Riemannsche Hebbarkeitssatz gilt, dass sich nmlich holomorphe Funktionen ber analytische Mengen der Kodimension 2 fortsetzen. Serre fhrte dann 1955 reduzierte komplexe Rume ein; z. B. ist die Neilsche Parabel oder jede singulre komplexe Kurve ein reduzierter, aber nicht normaler komplexer Raum. Lokal ist ein reduzierter komplexer Raum einfach das Nullstellengebilde endlich vieler holomorpher Funktionen in einer offenen Menge in CN : Die finale Definition wurde dann 1960 von Grauert gegeben: Hier sind auch nilpotente holomorphe Funktionen zu- gelassen. Dies ist z. B. wichtig, um Fasern holomorpher Abbildungen studieren zu knnen, oder Familien komplexer Untermannigfaltigkeiten. So degenerieren glatte Kegel- schnitte im 2-dimensionalen komplex-projektiven Raum in Doppelgeraden, die auf natrliche Weise komplexe Rume mit nilpotenten Elementen sind. Wesentlich fr die Begriffsbildung des komplexen Raumes waren dabei neue Techniken der Garbentheorie und deren Kohomologie, die dann, wie schon beschrieben, von Cartan nach Deutsch- land, insbesondere Mnster, exportiert wurden. Hier ist insbesondere der Begriff der Kohrenz einer Garbe von zentraler Bedeutung. Der Begriff der Kohrenz erlaubt den bergang vom Punktalen zum Lokalen, u. a. kann man alle Halme in einer offenen Umgebung U eines beliebigen Punktes simultan durch Schnitte auf U erzeugen. In diesem Zusammenhang sind zwei Stze von fundamentaler Bedeutung: der Okasche Ko- hrenzsatz und der Cartan-Okasche Kohrenzsatz: Kohrenzsatz von Oka. Ist X eine komplexe Mannigfaltigkeit (oder ein komplexer Raum), so ist die Strukturgarbe OX der holomorphen Funktionen kohrent (als OX -Mo- dulgarbe). Kohrenzsatz von Cartan-Oka. Ist X ein komplexer Raum und A X eine analytische Menge, so ist die Idealgarbe I A bestehend aus den holomorphen Funktionen, die auf A verschwinden, kohrent (als OX -Modulgarbe). Von zentralem geometrischen Interesse waren in den 1940er und 1950er Jahren die sog. Steinschen Mannigfaltigkeiten bzw. Rume. Eine Steinsche Mannigfaltigkeit kann durch folgende zwei Eigenschaften charakterisiert werden:

- holomorphe Separabilitt (d. h., die globalen holomorphen Funktionen sind punk- tetrennend); - Holomorphiekonvexitt.

92 JB 111. Band (2009), Heft 2 Klaus Hulek und Thomas Peternell: Henri Cartan, ein franzsischer Freund

Der Begriff der Holomorphiekonvexitt ist eine natrliche Verallgemeinerung des Kon- vexittsbegriffs: eine offene Menge A Cn ist konvex, wenn fr jede kompakte Menge K A die konvexe Hlle in A wieder kompakt ist. Ersetzt man beim Bilden der Hlle der linearen Funktionen durch holomorphe Funktionen in einem komplexen Raum, so erhlt man den Begriff der Holomorphiekonvexitt. Insbesondere ist eine offene konvexe Menge in Cn holomorph-konvex, hingegen ist C2 nf0g wegen des Riemannschen Fortsetzungssatzes nicht holomorph-konvex. Die wichtigsten Stze dieser Theorie, die berhmten Theoreme A und B von Cartan- Serre, lauten nun: Theorem A. Ist X eine Steinsche Mannigfaltigkeit (oder ein Steinscher Raum) und ist F eine kohrente Garbe auf X, so erzeugen die globalen Schnitte der Garbe F jeden Halm F xðx 2 XÞ: Theorem B. Alle Kohomologiegruppen vom Grad q 1 einer kohrenten Garbe F auf einem Steinschen Raum X verschwinden: HqðX; FÞ ¼ 0fuur€ q 1: Tatschlich ist Theorem A eine direkte Konsequenz aus Theorem B, und es sei ange- merkt, dass Theorem B die Steinschen Rume charakterisiert. Unmittelbare Anwen- dungen sind z. B. der Satz von Mittag-Leffler oder die Fortsetzbarkeit einer holomorphen Funktion von einer Untermannigfaltigkeit auf die umgebende Steinsche Mannig- faltigkeit. Noch in der vorkohrenten Zeit bewiesen Behnke und Stein 1943 (nicht in dieser Notation natrlich, publiziert erst 1948), dass jede nicht-kompakte Riemannsche Flche eine Steinsche Mannigfaltigkeit ist, insbesondere, dass berhaupt eine nicht- konstante holomorphe Funktion existiert. Von ebenso fundamentaler Bedeutung ist der Endlichkeitssatz von Cartan-Serre fr kompakte Rume: Theorem. Sei X ein kompakter komplexer Raum und F eine kohrente Garbe auf X. Dann sind alle Kohomologiegruppen HqðX; FÞendlichdimensional. Ist X eine Mannigfaltigkeit und F lokal-frei, so kann dies auch mit Hilfe der Theorie el- liptischer Differentialoperatoren bewiesen werden. Darber hinaus lieferte Cartan auch entscheidende Beitrge zu anderen Gebieten der Mathematik. Hier sind vor allem die algebraische Topologie und die homologische Algebra zu erwhnen. Ein Ergebnis der Zusammenarbeit mit Samuel Eilenberg ist das Buch ber homologische Algebra [C], das auch heute noch gelesen und zitiert wird. Weiter hat Cartan entscheidend zur Entwicklung der Theorie der Faserbndel beigetra- gen. Zu erwhnen sind weiterhin Verffentlichungen zur Potentialtheorie, vor allem auf homogenen Rumen. Maßgeblichen Einfluss auf die Entwicklung der Mathematik hatte Cartan auch als einer der Grndervter des Unternehmens „Bourbaki“. Gemeinsam mit Mathemati- kern wie C. Chevalley, J. Dieudonn, Ren de Possel, Szolem Mandelbrojt und Andr Weil grndete er 1935 diese Gruppe, die fortan unter dem Pseudonym Bourbaki ver- ffentlichte. Der Anspruch dieser Gruppe, und dies drckt sich auch in dem Titel der

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Buchreihe „E´ lments de Mathmatique“ aus, war hoch: Man wollte die Mathematik auf ein neues Fundament stellen und eine einheitliche Darstellung schaffen. Unzweifel- haft hat der „Bourbaki-Stil“ die Mathematik des 20. Jahrhunderts ganz wesentlich be- einflusst und geprgt. Seine Bedeutung als akademischer Lehrer wird am besten durch die Liste seiner Schler belegt, die u. a. J. P. Serre, R. Thom und A. Douady enthlt. Betrachtet man das Lebenswerk von Henri Cartan, so kann man mit Recht sagen, dass es auch seinem Wirken zu verdanken ist, dass die franzsische Mathematik nach dem zweiten Weltkrieg international eine fhrende Rolle spielt. Danksagung. Unser Dank gilt Herrn Stephan Klaus und Frau Ivonne Vetter vom Mathematischen Institut in Oberwolfach fr Recherchen und die Zurverfgungstellung von Bildern. Ebenso danken wir Herrn Prof. Ludwig Danzer (Dortmund), Herrn Prof. Gerd Fischer (Mnchen) und Herrn Dr. Klaus Peters (Boston) fr die freundliche Er- laubnis, ihre Fotos verwenden zu drfen.

References

[BT] Behnke, H.; Thullen, P.: Theorie der Funktionen mehrerer komplexer Vernderlichen, Er- gebnisse der Mathematik und ihrer Grenzgebiete, Band III, Heft 3, Springer-Verlag 1934. Zweite erweiterte Fassung, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 51, Springer-Verlag 1970. [C] Cartan, H.; Eilenberg, S.: Homological algebra. Princeton University Press, Princeton, N. J., 1956. [C2] Cartan, H.: Œuvres. Vol. I, II, III. Edited by Reinhold Remmert and Jean-Pierre Serre. Springer-Verlag, Berlin-New York, 1979. xxix+1469 pp. [J] Jackson, A.: Interview with H. Cartan, Notices of the AMS Vol. 46, Number 7, 782–788 (1999).

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