GUIDE TO EXOTIC CREDIT DERIVATIVESSecondary CDO trading.Customised CDO tranches .Default swaptions...

THE LEHMAN BROTHERS

GUIDE TO EXOTIC CREDIT DERIVATIVES

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Effective Structured Credit Solutions for our Clients

With over seventy professionalsworldwide, Lehman Brothers gives youaccess to top quality risk-management,structuring, research and legalexpertise in structured credit. The teamcombines local market knowledge withglobal co-ordinated expertise.

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Document1 06/10/2003 09:54 Page 1

The Lehman Brothers Guide to Exotic Credit Derivatives 1

The credit derivatives market has revolu-tionised the transfer of credit risk. Its impacthas been borne out by its significant growthwhich has currently achieved a market notion-al close to $2 trillion. While not directly com-parable, it is worth noting that the totalnotional outstanding of global investmentgrade corporate bond issuance currentlystands at $3.1 trillion.

This growth in the credit derivatives markethas been driven by an increasing realisationof the advantages credit derivatives possessover the cash alternative, plus the many newpossibilities they present to both creditinvestors and hedgers. Those investors seek-ing diversification, yield pickup or new waysto take an exposure to credit are increasinglyturning towards the credit derivatives market.

The primary purpose of credit derivatives isto enable the efficient transfer and repack-aging of credit risk. In their simplest form,credit derivatives provide a more efficientway to replicate in a derivative format thecredit risks that would otherwise exist in astandard cash instrument.

More exotic credit derivatives such as syn-

thetic loss tranches and default baskets cre-ate new risk-return profiles to appeal to thediffering risk appetites of investors based onthe tranching of portfolio credit risk. In doingso they create an exposure to default correla-tion. CDS options allow investors to expressa view on credit spread volatility, and hybridproducts allow investors to mix credit riskviews with interest rate and FX risk.

More recently, we have seen a steppedincrease in the liquidity of these exotic creditderivative products. This includes the devel-opment of very liquid portfolio credit vehicles,the arrival of a two-way correlation market incustomised CDO tranches, and the develop-ment of a more liquid default swaptions mar-ket. To enable this growth, the market hasdeveloped new approaches to the pricing andrisk-management of these products.

As a result, this book is divided into twoparts. In the first half, we describe how exoticstructured credit products work, their ratio-nale, risks and uses. In the second half, wereview the models for pricing and risk manag-ing these various credit derivatives, focusingon implementation and calibration issues.

Foreword

AuthorsDominic O'Kane T. +44 207 260 2628E. [email protected]

Marco NaldiT. +1 212 526 1728E. [email protected]

Sunita GanapatiT. +1 415 274 5485E. [email protected]

Arthur BerdT. +1 212 526 2629E. [email protected]

Claus PedersenT. +1 212 526 7775E. [email protected]

Lutz SchloeglT. +44 207 260 2113E. [email protected]

Roy MashalT. +1 212 526 7931E. [email protected]

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2 The Lehman Brothers Guide to Exotic Credit Derivatives

Contents

Foreword 1

Credit Derivatives ProductsMarket overview 3The credit default swap 4Basket default swaps 8Synthetic CDOs 12Credit options 23Hybrid products 28

Credit Derivatives ModellingSingle credit modelling 31Modelling default correlation 33Valuation of correlation products 39Estimating the dependency structure 43Modelling credit options 47Modelling hybrids 51

References 53

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Market overviewThe credit derivatives market has changedsubstantially since its early days in the late1990s, moving from a small and highly eso-teric market to a more mainstream marketwith standardised products. Initially drivenby the hedging needs of bank loan man-agers, it has since broadened its base ofusers to include insurance companies,hedge funds and asset managers.

The latest snapshot of the credit deriva-tives market was provided in the 2003 RiskMagazine credit derivatives survey. This sur-vey polled 12 dealers at the end of 2002,composed of all the major players in thecredit derivatives market. Although thereported numbers cannot be considered‘hard’, they can be used to draw fairly firmconclusions about the recent direction ofthe market.

According to the survey, the total marketoutstanding notional across all credit deriva-tives products was calculated to be $2,306billion, up more than 50% on the previousyear. Single name CDS remain the mostused instrument in the credit derivativesworld with 73% of market outstandingnotional, as shown in Figure 1. This supportsour observation that the credit default mar-ket has become more mainstream, focusingon the liquid standard contracts. We believethat this growth in CDS has been driven byhedging demand generated by syntheticCDO positions, and by hedge funds usingcredit derivatives as a way to exploit capitalstructure arbitrage opportunities and to gooutright short the credit markets.

An interesting statistic from the survey isthe relatively equal representation of NorthAmerican and European credits. The survey

showed that 40.1% of all reference entitiesoriginate in Europe, compared with 43.8%from North America. This is in stark con-trast to the global credit market which hasa significantly smaller proportion ofEuropean originated bonds relative toNorth America.

The base of credit derivatives users hasbeen broadening steadily over the last fewyears. We show a breakdown of the marketby end-users in Figure 2 (overleaf). Banksstill remain the largest users with nearly50% share. This is mainly because of theirsubstantial use of CDS as hedging tools fortheir loan books, and their active participa-tion in synthetic securitisations. The hedg-ing activity driven by the issuance ofsynthetic CDOs (discussed later) has forthe first time satisfied the demand to buyprotection coming from bank loan hedgers.Readers are referred to Ganapati et al (2003)for a full discussion of the market impact.

Insurance companies have also becomean important player, mainly by investing ininvestment-grade CDO tranches. As a result,

Credit Derivatives Products

Portfolio/ correlation products

22%

Credit default swaps 73%

Total return swaps

1%

Credit linkednotes3% Options and

hybrids 1%

Figure 1. Market breakdown by instrument type

Source: Risk Magazine 2003 Credit Derivatives Survey

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the insurance share of credit derivativesusage has increased to 14% from 9% theprevious year.

More recently, the growth in the usage ofcredit derivatives by hedge funds has had amarked impact on the overall credit deriva-tives market itself, where their share hasincreased to 13% over the year. Hedgefunds have been regular users of CDS espe-cially around the convertible arbitrage strate-gy. They have also been involved in many ofthe ‘fallen angel’ credits where they havebeen significant buyers of protection. Giventheir ability to leverage, they have substan-tially increased their volume of CDS con-tracts traded, which in many cases has beendisproportionate to their absolute size.

Finally, in portfolio products, by which wemean synthetic CDOs and default baskets,the total notional for all types of creditderivatives portfolio products was $449.4billion. Their share has kept pace with thegrowth of the credit derivatives market atabout 22% over the last two years. This isnot a surprise, since there is a fundamen-tally symbiotic relationship between thesynthetic CDO and single name CDS mar-kets, caused by dealers originating synthet-

ic tranches either by issuing the full capitalstructure or hedging bespoke tranches.

Since this survey was published, the creditderivatives market has continued to consoli-date and innovate. The ISDA 2003 CreditDerivative Definitions were another milestoneon the road towards CDS standardisation.The year 2003 has also seen a significantincrease in the usage of CDS portfolio prod-ucts. There has been a stepped increase inliquidity for correlation products, with dailytwo-way markets for synthetic tranches nowbeing quoted. The credit options market, inparticular the market for those written onCDS, has grown substantially.

A number of issues still remain to beresolved. First, there is a need for the gener-ation of a proper term structure for creditdefault swaps. The market needs to buildgreater liquidity at the long end and, espe-cially, the short end of the credit curve.Greater transparency is also needed aroundthe calibration of recovery rates. Finally, theissue of the treatment of restructuringevents still needs to be resolved. Currently,the market is segregated along regional linesin tackling this issue, but it is hoped that aglobal standard will eventually emerge.

The credit default swapThe credit default swap is the basic buildingblock for most ‘exotic’ credit derivatives andhence, for the sake of completeness, we setout a short description before we exploremore exotic products.

A credit default swap (CDS) is used to trans-fer the credit risk of a reference entity (corpo-rate or sovereign) from one party to another.In a standard CDS contract one party pur-chases credit protection from the other party,to cover the loss of the face value of an assetfollowing a credit event. A credit event is alegally defined event that typically includes

Hedgefunds

13%

Insurance 14%

SPVs5%

Banks (synthetic

securitisation) 10%

Banks (other)38%

Reinsurance 10%

Corporates 3%

Third-party asset managers

7%

Figure 2. Breakdown by end users

Source: Risk Magazine 2003 Credit Derivatives Survey.

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bankruptcy, failure to pay and restructuring.Buying credit protection is economicallyequivalent to shorting the credit risk. Equally,selling credit protection is economicallyequivalent to going long the credit risk.

This protection lasts until some specifiedmaturity date. For this protection, the pro-tection buyer makes quarterly payments, tothe protection seller, as shown in Figure 3,until a credit event or maturity, whicheveroccurs first. This is known as the premiumleg. The actual payment amounts on the pre-mium leg are determined by the CDS spreadadjusted for the frequency using a basisconvention, usually Actual 360.

If a credit event does occur before thematurity date of the contract, there is a pay-ment by the protection seller to the protec-tion buyer. We call this leg of the CDS theprotection leg. This payment equals the dif-ference between par and the price of theassets of the reference entity on the facevalue of the protection, and compensates theprotection buyer for the loss. There are twoways to settle the payment of the protectionleg, the choice being made at the initiation ofthe contract. They are:

Physical settlement – This is the most wide-ly used settlement procedure. It requires theprotection buyer to deliver the notional

amount of deliverable obligations of the ref-erence entity to the protection seller inreturn for the notional amount paid in cash.In general there are several deliverable obli-gations from which the protection buyer canchoose which satisfy a number of character-istics. Typically they include restrictions onthe maturity and the requirement that theybe pari passu – most CDS are linked tosenior unsecured debt.

If the deliverable obligations trade with dif-ferent prices following a credit event, whichthey are most likely to do if the credit eventis a restructuring, the protection buyer cantake advantage of this situation by buyingand delivering the cheapest asset. The pro-tection buyer is therefore long a cheapest todeliver option.

Cash settlement – This is the alternative tophysical settlement, and is used less fre-quently in standard CDS but overwhelming-ly in tranched CDOs, as discussed later. Incash settlement, a cash payment is made bythe protection seller to the protection buyerequal to par minus the recovery rate of thereference asset. The recovery rate is calcu-lated by referencing dealer quotes orobservable market prices over some periodafter default has occurred.

Suppose a protection buyer purchasesfive-year protection on a company at a CDSspread of 300bp. The face value of the pro-tection is $10m. The protection buyertherefore makes quarterly payments ap-proximately (we ignore calendars and daycount conventions) equal to $10m × 0.03× 0.25 = $75,000. After a short period thereference entity suffers a credit event.Assuming that the cheapest deliverableasset of the reference entity has a recoveryprice of $45 per $100 of face value, the pay-ments are as follows:

Contingent payment of loss on parfollowing a credit event (protection leg)

Protectionbuyer

Protectionseller

Default swap spread(premium leg)

Figure 3. Mechanics of a CDS

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■■ The protection seller compensates theprotection buyer for the loss on the facevalue of the asset received by the protec-tion buyer and this is equal to $5.5m.

■■ The protection buyer pays the accruedpremium from the previous premiumpayment date to the time of the creditevent. For example, if the credit eventoccurs after a month then the protectionbuyer pays approximately $10m × 300bp× 1/12 = $25,000 of premium accrued.Note that this is the standard for corpo-rate reference entity linked CDS.

For severely distressed reference entities,the CDS contract trades in an up-front for-mat where the protection buyer makes acash payment at trade initiation which pur-chases protection to some specified maturi-ty – there are no subsequent paymentsunless there is a credit event in which theprotection leg is settled as in a standardCDS. For a full description of up-front CDSsee O’Kane and Sen (2003).

Liquidity in the CDS market differs fromthe cash credit market in a number of ways.For a start, a wider range of credits trade inthe CDS market than in cash. In terms ofmaturity, the most liquid CDS is the five-yearcontract, followed by the three-year, seven-year and 10-year. The fact that a physicalasset does not need to be sourced meansthat it is generally easier to transact in largeround sizes with CDS.

Uses of a CDSThe CDS can do almost everything that cashcan do and more. We list some of the mainapplications of CDS below.

■■ The CDS has revolutionised the creditmarkets by making it easy to short credit.

This can be done for long periods withoutassuming any repo risk. This is very use-ful for those wishing to hedge currentcredit exposures or those wishing to takea bearish credit view.

■■ CDS are unfunded so leverage is possi-ble. This is also an advantage for thosewho have high funding costs, becauseCDS implicitly lock in Libor funding tomaturity.

■■ CDS are customisable, although devia-tion from the standard may incur a liquid-ity cost.

■■ CDS can be used to take a spread viewon a credit, as with a bond.

■■ Dislocations between cash and CDS pre-sent new relative value opportunities.This is known as trading the defaultswap basis.

Evolution of CDS documentationThe CDS is a contract traded within the legalframework of the International Swaps andDerivatives Association (ISDA) master agree-ment. The definitions used by the market forcredit events and other contractual detailshave been set out in the ISDA 1999 documentand recently amended and enhanced by theISDA 2003 document. The advantage of thisstandardisation of a unique set of definitionsis that it reduces legal risk, speeds up the con-firmation process and so enhances liquidity.

Despite this standardisation of defini-tions, the CDS market does not have a uni-versal standard contract. Instead, there is aUS, European and an Asian market stan-dard, differentiated by the way they treat arestructuring credit event. This is the con-sequence of a desire to enhance the posi-

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tion of protection sellers by limiting thevalue of the protection buyer’s deliveryoption following a restructuring creditevent. A full discussion and analysis ofthese different standards can be found inO’Kane, Pedersen and Turnbull (2003).

Determining the CDS spreadThe premium payments in a CDS aredefined in terms of a CDS spread, paid peri-odically on the protected notional untilmaturity or a credit event. It is possible toshow that the CDS spread can, to a firstapproximation, be proxied by either (i) a parfloater bond spread (the spread to Libor atwhich the reference entity can issue a float-ing rate note of the same maturity at a priceof par) or (ii) the asset swap spread of abond of the same maturity provided ittrades close to par.

Demonstrating these relationships relieson several assumptions that break down inpractice. For example, we assume a com-mon market-wide funding level of Libor, weignore accrued coupons on default, weignore the delivery option in the CDS, andwe ignore counterparty risk. Despite theseassumptions, cash market spreads usuallyprovide the starting point for where CDSspreads should trade. The differencebetween where CDS spreads and cashLIBOR spreads trade is known as theDefault Swap Basis, defined as:

Basis = CDS Spread – Cash Libor Spread.

A full discussion of the drivers behind theCDS basis is provided in O’Kane andMcAdie (2001). A large number ofinvestors now exploit the basis as a rela-tive value play.

Determining the CDS spread is not thesame as valuing an existing CDS contract.

For that we need a model and a discussion ofthe valuation of CDS is provided on page 32.

Funded versus unfundedCredit derivatives, including CDS, can betraded in a number of formats. The mostcommonly used is known as swap format,and this is the standard for CDS. This formatis also termed ‘unfunded’ format becausethe investor makes no upfront payment.Subsequent payments are simply paymentsof spread and there is no principal paymentat maturity. Losses require payments to be made by the protection seller to the protection buyer, and this has counterpartyrisk implications.

The other format is to trade the risk in theform of a credit linked note. This format isknown as ‘funded’ because the investor hasto fund an initial payment, typically par. Thispar is used by the protection buyer to pur-chase high quality collateral. In return the pro-tection seller receives a coupon, which maybe floating rate, ie, Libor plus a spread, ormay be fixed at a rate above the same matu-rity swap rate. At maturity, if no default hasoccurred the collateral matures and theinvestor is returned par. Any default beforematurity results in the collateral being sold,the protection buyer covering his loss and theinvestor receiving par minus the loss. Theprotection buyer is exposed to the default riskof the collateral rather than the counterparty.

Traded CDS portfolio products CDS portfolio products are products thatenable the investor to go long or short thecredit risk associated with a portfolio of CDSin one transaction.

In recent months, we have seen the emer-gence of a number of very liquid portfolioproducts, whose aim is to offer investors adiverse, liquid vehicle for assuming or hedg-

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ing exposure to different credit markets, oneexample being the TRAC-XSM vehicle. Thesehave added liquidity to the CDS market andalso created a standard which can be usedto develop portfolio credit derivatives suchas options on TRAC-X.

The move of the CDS market from bankstowards traditional credit investors has greatlyincreased the need for a performance bench-mark linked directly to the CDS market. As aconsequence, Lehman Brothers has intro-duced a family of global investment grade CDSindices which are discussed in Munves (2003).

These consist of three sub-indices, a US250 name index, a European 150 name indexand a Japanese 40 name index. All namesare corporates and the maturity of the indexis maintained close to five years. Daily pric-ing of all 440 names is available on ourLehmanLive website.

Basket default swapsCorrelation products are based on redistribut-ing the credit risk of a portfolio of single-name credits across a number of differentsecurities. The portfolio may be as small asfive credits or as large as 200 or more credits.The redistribution mechanism is based on theidea of assigning losses on the credit portfo-lio to the different securities in a specified pri-ority, with some securities taking the firstlosses and others taking later losses. Thisexposes the investor to the tendency ofassets in the portfolio to default together, ie,default correlation. The simplest correlationproduct is the basket default swap.

A basket default swap is similar to a CDS,the difference being that the trigger is thenth credit event in a specified basket of ref-erence entities. Typical baskets contain fiveto 10 reference entities. In the particular case

of a first-to-default (FTD) basket, n=1, and itis the first credit in a basket of referencecredits whose default triggers a payment tothe protection buyer. As with a CDS, the con-tingent payment typically involves physicaldelivery of the defaulted asset in return for apayment of the par amount in cash. In returnfor assuming the nth-to-default risk, the pro-tection seller receives a spread paid on thenotional of the position as a series of regularcash flows until maturity or the nth creditevent, whichever is sooner.

The advantage of an FTD basket is that itenables an investor to earn a higher yieldthan any of the credits in the basket. This isbecause the seller of FTD protection is lever-aging their credit risk.

To see this, consider that the fair-valuespread paid by a credit risky asset is deter-mined by the probability of a default, timesthe size of the loss given default. FTD bas-kets leverage the credit risk by increasing theprobability of a loss by conditioning the pay-off on the first default among several credits.The size of the potential loss does notincrease relative to buying any of the assetsin the basket. The most that the investor canlose is par minus the recovery value of theFTD asset on the face value of the basket.

The advantage is that the basket spreadpaid can be a multiple of the spread paid bythe individual assets in the basket. This isshown in Figure 4 where we have a basketof five investment grade credits paying anaverage spread of about 28bp. The FTD bas-ket pays a spread of 120bp.

More risk-averse investors can use defaultbaskets to construct lower risk assets: sec-ond-to-default (STD) baskets, where n=2,trigger a credit event after two or more assetshave defaulted. As such they are lower risksecond-loss exposure products which willpay a lower spread than an FTD basket.TRAC-X is a service mark of JPMorgan and Morgan Stanley

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The basket spreadOne way to view an FTD basket is as atrade in which the investor sells protectionon all of the credits in the basket with thecondition that all the other CDS cancel atno cost following a credit event. Such atrade cannot be replicated using existinginstruments. Valuation therefore requires a pricing model. The model inputs in orderto determine the nth-to-default basketspread are:

■■ Value of n: An FTD (n=1) is riskier than anSTD (n=2) and so commands a higherspread.

■■ Number of credits: The greater the num-ber of credits in the basket, the greaterthe likelihood of a credit event, and so thehigher the spread.

■■ Credit quality: The lower the credit quali-ty of the credits in the basket, in terms ofspread and rating, the higher the spread.

■■ Maturity: The effect of maturity de-pends on the shape of the individualcredit curves.

■■ Recovery rate: This is the expectedrecovery rate of the nth-to-default assetfollowing its credit event. This has only asmall effect on pricing since a higherexpected recovery rate is offset by ahigher implied default probability for agiven spread. However, if there is adefault the investor will certainly prefer ahigher realised recovery rate.

■■ Default correlation: Increasing defaultcorrelation increases the likelihood ofassets to default or survive together. Theeffect of default correlation is subtle andsignificant in terms of pricing. We nowdiscuss this is more detail.

Baskets and default correlationBaskets are essentially a default correlationproduct. This means that the basket spreaddepends on the tendency of the referenceassets in the basket to default together.

It is natural to assume that assets issuedby companies within the same country andindustrial sector should have a higherdefault correlation than those within differ-ent industrial sectors. After all, they sharethe same market, the same interest ratesand are exposed to the same costs. At aglobal level, all companies are affected bythe performance of the world economy. We believe that these systemic sector risksfar outweigh idiosyncratic effects so we expect that default correlation is usually positive.

There are a number of ways to explain howdefault correlation affects the pricing ofdefault baskets. Confusion is usually causedby the term ‘default correlation’. The fact is

Lehman Brothers

Basket investor

Contingent paymentof par minus recovery

on FTD on $10m face value

120bp paid on $10muntil FTD or five-yearmaturity, whichever

is sooner

Reference portfolioCoca Cola 30bp

C. de Saint Gobain 30bpElectricidade de Portugal 27bp

Hewlett Packard 29bpTeliasonera 30bp

Figure 4. Five-year first to default (FTD)basket on five credits. We show the fiveyear CDS spreads of the individual credits

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that if two assets are correlated, they willnot only tend to default together, they willalso tend to survive together.

There are two correlation limits in which aFTD basket can be priced without resortingto a model – independence and maximumcorrelation.

■■ Independence: Consider a five-creditbasket where all of the underlying creditshave flat credit curves. If the credits areall independent and never become corre-lated during the life of the trade, the nat-ural hedge is for the basket investor tobuy CDS protection on each of the indi-vidual names to the full notional. If acredit event occurs, the CDS hedge cov-ers the loss on the basket and all of theother CDS hedges can be unwound at nocost, since they should on average haverolled down their flat credit curves. Thisimplies that the basket spread for independent assets should be equal tothe sum of the spreads of the names inthe basket.

■■ Maximum correlation: Consider thesame FTD basket but this time where the

default correlation is at its maximum. Inpractice, this means that when any assetdefaults, the asset with the widestspread will always default too. As aresult, the risk of one default is the sameas the risk of the widest spread assetdefaulting. Because an FTD is triggeredby only one credit event, it will be as riskyas the riskiest asset and the FTD basketspread should be the widest spread ofthe credits in the basket.

The best way to understand the behaviourof default baskets between these two cor-relation limits is to study the loss distribu-tion for the basket portfolio. See page 33for a discussion of how to model the lossdistribution.

We consider a basket of five credits withspreads of 100bp and an assumed recoveryrate for all of 40%. We have plotted the lossdistribution for correlations of 0%, 20%, and50% in Figure 5. The spread for an FTD bas-ket depends on the probability of one ormore defaults which equals one minus theprobability of no defaults. We see that theprobability of no defaults increases withincreasing correlation – the probability ofcredits surviving together increases – andthe FTD spread should fall.

The risk of an STD basket depends on theprobability of two or more defaults. As corre-lation goes up from 0–20%, the probability oftwo, three, four and five defaults increases.This makes the STD spread increase.

The process for translating these loss dis-tributions into a fair value spread requires amodel of the type described on page 39.Essentially we have to find the basketspread for which the present value of theprotection payments equals the presentvalue of the premium payments.

We should not forget that in addition to the

0

10

20

30

40

50

60

70

80

0 1 2 3 4 5Number of defaults

Pro

bab

ility

(%

)

ρ = 0% ρ = 20% ρ = 50%

Figure 5. Loss distribution for a five-credit basket with 0%, 20% and50% correlation

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protection leg, the premium leg of thedefault basket also has correlation sensitivi-ty because it is only paid for as long as thenth default does not occur.

Using a model we have calculated the cor-relation sensitivity of the FTD and STD spreadfor the five-credit basket shown in Figure 6.At low correlation, the FTD spread is close to146bp, which is the sum of the spreads. Athigh correlation, the basket has the risk of thewidest spread asset and so is at 30bp. TheSTD spread is lowest at zero correlation sincethe probability of two assets defaulting is lowif the assets are independent. At maximumcorrelation the STD spread tends towards thespread of the second widest asset in the bas-ket which is also 30bp.

Applications■■ Baskets have a range of applications.

Investors can use default baskets to lever-age their credit exposure and so earn ahigher yield without increasing theirnotional at risk.

■■ The reference entities in the basket are alltypically investment grade and so arefamiliar to most credit analysts.

■■ The basket can be customised to theinvestors’ exact view regarding size,maturity, number of credits, credit selec-tion, FTD or STD.

■■ Buy and hold investors can enjoy theleveraging of the spread premium. This isdiscussed in more detail later.

■■ Credit investors can use default basketsto hedge a blow-up in a portfolio of cred-its more cheaply than buying protectionon the individual credits.

■■ Default baskets can be used to express aview on default correlation. If theinvestor’s view is that the implied correla-tion is too low then the investor should sellFTD protection. If implied correlation is toohigh they should sell STD protection.

Hedging default basketsThe issuers of default baskets need tohedge their risks. Spread risk is hedged byselling protection dynamically in the CDSmarket on all of the credits in the defaultbasket. Determining how much to sell,known as the delta, requires a pricing modelto calculate the sensitivity of the basketvalue to changes in the spread curve of theunderlying credit.

Although this delta hedging should immu-nise the dealer’s portfolio against smallchanges in spreads, it is not guaranteed to bea full hedge against a sudden default. Forinstance, a dealer hedging an FTD basketwhere a credit defaults with a recovery rate ofR would receive a payment of (1-R)F from theprotection seller, and will pay D(1-R)F on thehedged protection, where F is the basket facevalue and D is the delta in terms of percent-age of face value. The net payment to theprotection buyer is therefore (1-D)(1-R)F.

020406080

100120140

0 10 20 30 40 50 60 70 80 90 100Correlation (%)

Bas

ket

spre

ad (

bp

)

FTD

STD

Figure 6. Correlation dependence ofspread for FTD and STD basket

guide.qxd 10/10/2003 11:15 Page 11


There will also probably be a loss on theother CDS hedges. The expected spreadwidening on default on the other credits inthe basket due to their positive correlationwith the defaulted asset will result in a losswhen they are unwound. The greater unwindlosses for baskets with higher correlationswill be factored into the basket spread.

One way for a default basket dealer toreduce his correlation risk is by selling pro-tection on the same or similar default bas-kets. However this is difficult as it is usuallydifficult to find protection buyers who selectthe exact same basket as an investor.

The alternative hedging approach is for thedealer to buy protection using default bas-kets on other orders of protection. This isbased on the observation that a dealer whois long first, second, third up to Mth orderprotection on an M-credit basket has almostno correlation risk, since this position isalmost economically equivalent to buyingfull face value protection using CDS on all Mcredits in the basket.

Figure 7 shows an example basket withthe delta and spread for each of the fivecredits. Note that the deltas are all verysimilar. This reflects the fact that all of theassets have a similar spread. Differencesare mainly due to our different correlationassumptions.

Hedgers of long protection FTD basketsare also long gamma. This means that as thespread of an asset widens, the delta willincrease and so the hedger will be sellingprotection at a wider spread. If the spreadtightens, then the delta will fall and thehedger will be buying back hedges at atighter level. So spread volatility can be ben-eficial. This effect helps to offset the nega-tive carry associated with hedged FTDbaskets. This is clear in the previous exam-ple where the income from the hedges is211bp, lower than the 246bp paid to the FTDbasket investor.

Different rating agencies have developedtheir own model-based approaches for therating of default baskets. We discuss theseon page 39.

Synthetic CDOsSynthetic collateralised debt obligations(Synthetic CDOs) were conceived in 1997 asa flexible and low-cost mechanism for trans-ferring credit risk off bank balance sheets.The primary motivation was the banks’reduction of regulatory capital.

More recently, however, the fusion of cred-it derivatives modelling techniques andderivatives trading have led to the creation ofa new type of synthetic CDO, which we calla customised CDO, which can be tailored tothe exact risk appetites of different classesof investors. As a result, the synthetic CDOhas become an investor-driven product.

Overall, these different types of syntheticCDO have a total market size estimated bythe Risk 2003 survey to be close to $500 bil-lion. What is also of interest is that the deal-er-hedging of these products in the CDSmarket has generated a substantial demandto sell protection, balancing the traditionalprotection-buying demand coming frombank loan book managers.

Figure 7. Default basket deltas for a€10m notional five-year FTD basket onfive credits. The FTD spread is 246bp.

Reference entity CDS Spread Delta

Walt Disney 62bp 6.26m

Rolls Royce 60bp 6.55m

Sun Microsystems 60bp 6.87m

Eastman Chemical 60bp 7.16m

France Telecom 64bp 7.57m

guide.qxd 10/10/2003 11:15 Page 12


The performance of a synthetic CDO islinked to the incidence of default in a portfo-lio of CDS. The CDO redistributes this risk byallowing different tranches to take thesedefault losses in a specific order. To see this,consider the synthetic CDO shown in Figure8. It is based on a reference pool of 100CDS, each with a €10m notional. This risk isredistributed into three tranches; (i) an equi-ty tranche, which assumes the first €50m oflosses, (ii) a mezzanine tranche, which takethe next €100m of losses, and (iii) the seniortranche with a notional of €850m takes allremaining losses.

The equity tranche has the greatest riskand is paid the widest spread. It is typicallyunrated. Next is the mezzanine tranchewhich is lower risk and so is paid a lowerspread. Finally we have the senior tranchewhich is protected by €150m of subordina-tion. To get a sense of the risk of the seniortranche, note that it would require more than25 of the assets in the 100 credit portfolio todefault with a recovery rate of 40% beforethe senior tranche would take a principalloss. Consequently the senior tranche is typ-ically paid a very low spread.

The advantage of CDOs is that by chang-ing the details of the tranche in terms of itsattachment point (this is the amount of sub-

ordination below the tranche) and width, it ispossible to customise the risk profile of atranche to the investor’s specific profile.

Full capital structure syntheticsIn the typical synthetic CDO structuredusing securitisation technology, the spon-soring institution, typically a bank, entersinto a portfolio default swap with a SpecialPurpose Vehicle (SPV). This is shown inFigure 9 (overleaf).

The SPV typically provides credit protec-tion for 10% or less of the losses on thereference portfolio. The SPV in turn issuesnotes in the capital markets to cash collat-eralise the portfolio default swap with theoriginating entity. The notes issued caninclude a non-rated ‘equity’ piece, mezza-nine debt and senior debt, creating cash lia-bilities. The remainder of the risk, 90% ormore, is generally distributed via a seniorswap to a highly rated counterparty in anunfunded format.

Reinsurers, who typically have AAA/AA rat-ings, have traditionally had a healthy appetitefor this type of senior risk, and are the largestparticipants in this part of the capital structure– often referred to as super-senior AAAs orsuper-senior swaps. The initial proceeds fromthe sale of the equity and notes are investedin highly rated, liquid assets.

If an obligor in the reference pool defaults,the trust liquidates investments in the trustand makes payments to the originating enti-ty to cover default losses. This payment isoffset by a successive reduction in the equi-ty tranche, then the mezzanine and finally thesuper-seniors are called to make up losses.See Ganapati et al (2001) for more details.

Mechanics of a synthetic CDOWhen nothing defaults in the reference port-folio of the CDO, the investor simply

Referencepool

100 invest-ment grade names in

CDS format €10m x 100

assets = 1bn total notional

Seniortranche

€850m

5bp

Equity tranche €50m

Mezzaninetranche €100m

Lehman Brothers

200bp

1,500bp

Contingent payment

Figure 8. A standard synthetic CDO

guide.qxd 10/10/2003 11:15 Page 13


receives the Libor spread until maturity andnothing else changes. Using the syntheticCDO described earlier and shown in Figure8, consider what happens if one of the ref-erence entities in the reference portfolioundergoes the first credit event with a 30%recovery, causing a €7m loss.

The equity investor takes the first loss of€7m, which is immediately paid to the orig-inator. The tranche notional falls from €50mto €43m and the equity coupon, set at1500bp, is now paid on this smaller notion-al. These coupon payments therefore fallfrom €7.5m to 15% times €43m = €6.45m.

If traded in a funded format, the €3mrecovered on the defaulted asset is eitherreinvested in the portfolio or used to reducethe exposure of the senior-most tranche(similar to early amortisation of seniortranches in cash flow CDOs).

The senior tranche notional is decreased by€3m to €847m, so that the sum of protect-ed notional equals the sum of the collateralnotionals which is now €990m. This has noeffect on the other tranches.

This process repeats following each cred-it event. If the losses exceed €50m then the

mezzanine investor must bear the subse-quent losses with the corresponding reduc-tion in the mezzanine notional. If the lossesexceed €150m, then it is the senior investorwho takes the principal losses.

The mechanics of a standard syntheticCDO are therefore very simple, especiallycompared with traditional cash flow CDOwaterfalls. This also makes them more easi-ly modelled and priced.

The CDO tranche spreadThe synthetic CDO spread depends on anumber of factors. We list the main ones anddescribe their effects on the tranche spread.

■■ Attachment point: This is the amount ofsubordination below the tranche. Thehigher the attachment point, the moredefaults are required to cause trancheprincipal losses and the lower the tranchespread.

■■ Tranche width: The wider the tranchefor a fixed attachment point, the morelosses to which the tranche is exposed.However, the incremental risk ascending

Reference portfolio $1bn notional

CDS spread income

Equity notes (unrated)

Senior notes

AAA

Special purpose vehicle (SPV)

Credit protection

Mezzanine notes

BBB/A

Sponsoring bank

Super senior swap

premium

$900m super senior credit

protection

Highly rated

counterparty

Subordinated swap

premium

10% first loss

subordinated credit

protection

Proceeds

Issued notes

Figure 9. The full capital structure synthetic CDO

guide.qxd 10/10/2003 11:15 Page 14


the capital structure is usually decliningand so the spread falls.

■■ Portfolio credit quality: The lower thequality of the asset portfolio, measuredby spread or rating, the greater the risk ofall tranches due to the higher defaultprobability and the higher the spread.

■■ Portfolio recovery rates: The expectedrecovery rate assumptions have only asecondary effect on tranche pricing. Thisis because higher recovery rates implyhigher default probabilities if we keep thespread fixed. These effects offset eachother to first order.

■■ Swap maturity: This depends on theshapes of the credit curves. For upwardsloping credit curves, the tranche curvewill generally be upward sloping and sothe longer the maturity, the higher thetranche spread.

■■ Default correlation: If default correlationis high, assets tend to default togetherand this makes senior tranches morerisky. Assets also tend to survive togeth-er making the equity safer. To understandthis more fully we need to better under-stand the portfolio loss distribution.

The portfolio loss distributionNo matter what approach we use to gener-ate it, the loss distribution of the referenceportfolio is crucial for understanding the riskand value of correlation products. The port-folio loss is clearly not symmetrically dis-tributed: it is therefore informative to look atthe entire loss distribution, rather than sum-marising it in terms of expected value andstandard deviation. We can use models ofthe type discussed on page 33 to calculate

the portfolio loss distribution. We can expectto observe one of the two shapes shown inFigure 10. They are (i) a skewed bell curve; (ii)a monotonically decreasing curve.

The skewed bell curve applies to the casewhen the correlation is at or close to zero. Inthis limit the distribution is binomial and thepeak is at a loss only slightly less than theexpected loss.

As correlation increases, the peak of thedistribution falls and the high quantilesincrease: the curves become monotonicallydecreasing. We see that the probability oflarger losses increases and, at the sametime, the probability of smaller losses alsoincreases, thereby preserving the expectedloss which is correlation independent (forfurther discussion see Mashal, Naldi andPedersen (2003)).

For very high levels of asset correlations(hardly ever observed in practice), the distri-bution becomes U-shaped. At maximumdefault correlation all the probability mass islocated at the two ends of the distribution.The portfolio either all survives or it alldefaults. It resembles the loss distributionof a single asset.

0

5

10

15

20

25

30

35

40

0 5

10 15 47

Loss (%)

Pro

bab

ility

(%

)

ρ = 0

ρ = 20%

ρ = 95%

Figure 10. Portfolio loss distributionfor a large portfolio at 0%, 20% and95% correlation

guide.qxd 10/10/2003 11:15 Page 15


How then does the shape of the portfolioloss distribution affect the pricing of tranch-es? To see this we must study the trancheloss distribution.

The tranche loss distributionWe have plotted in Figures 11–13 the loss dis-tributions for a CDO with a 5% equity, 10%mezzanine and 85% senior tranche for corre-lation values of 20% and 50%. At 20% corre-

lation, we see that most of the portfolio lossdistribution is inside the equity tranche, withabout 14% beyond, as represented by thepeak at 100% loss. As correlation goes to50% the probability of small losses increaseswhile the probability of 100% losses increas-es only marginally. Clearly equity investorsbenefit from increasing correlation.

The mezzanine tranche becomes morerisky at 50% correlation. As we see in Figure12, the 100% loss probability jumps from0.50% to 3.5%. In most cases mezzanineinvestors benefit from falling correlation –they are short correlation. However, the cor-relation directionality of a mezzanine tranchedepends upon the collateral and the tranche.In certain cases a mezzanine tranche with avery low attachment point may be a longcorrelation position.

Senior investors also see the risk of theirtranche increase with correlation as morejoint defaults push out the loss tail. This isclear in Figure 13. Senior investors are shortcorrelation.

In Figure 14 we plot the dependence of thevalue of different CDO tranches on correla-tion. As expected, we clearly see that:

■■ Senior investors are short correlation. Ifcorrelation increases, senior tranchesfall in value.

■■ Mezzanine investors are typically shortcorrelation, although this very muchdepends upon the details of the trancheand the collateral.

■■ Equity investors are long correlation.When correlations go up, equity tranchesgo up in value.

In the process of rating CDO tranches, rat-ing agencies need to consider all of these

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0 10 20 30 40 50 60 70 80 90 100

Mezzanine tranche loss (%)

Pro

bab

ility

(%

) ρ = 20% ρ = 50%

Figure 12. Mezzanine tranche loss distribution for correlation of 20% and50%. We have eliminated the zero losspeak, which is about 86% in both cases

0

5

10

15

20

25

30

0

10 20 30 40 50 60 70 80 90 100

Equity tranche loss (%)

Pro

bab

ility

(%

)

ρ = 20% ρ = 50%

Figure 11. Equity tranche loss distribution for correlations of 20%and 50%

guide.qxd 10/10/2003 11:15 Page 16


risk parameters and so have adopted modelbased approaches. These are discussed onpage 43.

Customised synthetic CDO tranchesCustomisation of synthetic tranches hasbecome possible with the fusion of deriva-tives technology and credit derivatives.Unlike full capital structure synthetics, whichissue the equity, mezzanine and senior partsof the capital structure, customised synthet-ics may issue only one tranche. There are anumber of other names for customised CDOtranches, including bespoke tranches, andsingle tranche CDOs.

The advantage of customised tranches isthat they can be designed to match exactlythe risk appetite and credit expertise of theinvestor. The investor can choose the creditsin the collateral, the trade maturity, theattachment point, the tranche width, the rat-ing, the rating agency and the format (fund-ed or unfunded). Execution of the trade cantake days rather than the months that fullcapital structure CDOs require.

The basic paradigm has already been dis-cussed in the context of default baskets. Itis to use CDS to dynamically delta-hedgethe first order risks of a synthetic trancheand to use a trading book approach tohedge the higher order risks. This is shownin Figure 15 (overleaf).

For example, consider an investor whobuys a customised mezzanine tranche fromLehman Brothers. We will then hedge it byselling protection in an amount equal to thedelta of each credit in the portfolio via theCDS market. The delta is the amount ofprotection to be sold in order to immunisethe portfolio against small changes in the CDS spread curve for that credit. Each credit in the portfolio will have its own delta.

Understanding delta for CDOsFor a specific credit in a CDO portfolio, thedelta is defined as the notional of CDS forthat credit which has the same mark-to-market change as the tranche for a smallmovement in the credit’s CDS spreadcurve. Although the definition may bestraightforward, the behaviour of the deltais less so.

One way to start thinking about delta is to

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 10 20 30 40

Senior tranche loss (%)P

rob

abili

ty (

%)

ρ = 20% ρ = 50%

Figure 13. Senior tranche loss distribution for correlations of 20% and50%. We have eliminated the zero losspeak, which is greater than 96% in both cases

–30

–20

–10

0

10

20

30

40

0 10 20 30 40 50 60 70 80 90 100

Correlation (%)

Tra

nch

e M

TM

(€ €m

)

EquityMezzanineSenior

Figure 14. Correlation dependenceof CDO tranches

guide.qxd 10/10/2003 11:15 Page 17


imagine a queue of all of the credits sortedin the order in which they should default.This ordering will depend mostly on thespread of the asset relative to the othercredits in the portfolio and its correlation rel-ative to the other assets in the portfolio. Ifthe asset whose delta you are calculating isat the front of this queue, it will be most like-ly to cause losses to the equity tranche andso will have a high delta for the equitytranche. If it is at the back of the queue thenits equity delta will be low. As it is most like-ly to default after all the other asset, it will bemost likely to hit the senior tranche. As aresult the senior tranche delta will rise. Thisframework helps us understand the direc-tionality of delta.

The actual magnitude of delta is more dif-ficult to quantify because it depends on thetranche notional and the contractualtranche spread, as well as the features ofthe asset whose delta we are examining.For example the delta for a senior trancheto a credit whose CDS spread has widenedwill fall due to the fact that it is more likely

to default early and hit the equity tranche,and also because the CDS will have a high-er spread sensitivity and so require a small-er notional.

To show this we take an example CDOwith 100 credits, each $10m notional. It hasthree tranches: a 5% equity, a 10% mezza-nine and an 85% senior tranche. The assetspreads are all 150bp and the correlationbetween all the assets is the same.

The sensitivity of the delta to changing thespread of the asset whose delta we are cal-culating is shown in Figure 16. If the singleasset spread is less than the portfolio aver-age of 150bp, then it is the least risky asset.As a result, it would be expected to be thelast to default and so most likely to impactthe senior-most tranche. As the spread ofthe asset increases above 150bp, itbecomes more likely to default before theothers and so impacts the equity or mezza-nine tranche. The senior delta drops and theequity delta increases.

In Figure 17 we plot the delta of the assetversus its correlation with all of the other

Reference pool

100 investment grade names in CDS format

$10m x 100 assets = $1bn

Bespoke tranche

Lehman Brothers

Spread

Contingent payment

∆ of CDS on Name 1





Investor

∆

Figure 15. Delta hedging a synthetic CDO

guide.qxd 10/10/2003 11:15 Page 18


assets in the portfolio. These all have a cor-relation of 20% with each other. If the assetis highly correlated with the other assets it ismore likely to default or survive with theother assets. As a result, it is more likely todefault en masse, and so senior and mezza-nine tranches are more exposed. For lowcorrelations, if it defaults it will tend to do soby itself while the rest of the portfolio tendsto default together. As a result, the equitytranche is most exposed.

There is also a time effect. Through time,senior and mezzanine tranches becomesafer relative to equity tranches since lesstime remains during which the subordina-tion can be reduced resulting in principallosses. This causes the equity tranche deltato rise through time while the mezzanineand senior tranche deltas fall to zero.

Building intuition about the delta is not triv-ial. There are many further dependencies tobe explored and we intend to describe thesein a forthcoming paper.

Higher order risksIf properly hedged, the dealer should beinsensitive to small spread movements.However, this is not a completely risk-freeposition for the dealer since there are a num-ber of other risk dimensions that have notbeen immunised. These include correlationsensitivity, recovery rate sensitivity, timedecay and spread gamma. There is also arisk to a sudden default which we call thevalue-on-default risk (VOD).

For this reason, dealers are motivated todo trades that reduce these higher orderrisks. The goal is to flatten the risk of thecorrelation book with respect to these high-er order risks either by doing the offsettingtrade or by placing different parts of thecapital structure with other buyers of cus-tomised tranches.

Idiosyncratic versus systemic riskIn terms of how they are exposed to credit,there is a fundamental difference betweenequity and senior tranches. Equity tranchesare more exposed to idiosyncratic risk – theyincur a loss as soon as one asset defaults.The portfolio effect of the CDO is onlyexpressed through the fact that it may takeseveral defaults to completely reduce theequity notional. This implies that equityinvestors should focus less on the overallproperties of the collateral, and more on try-ing to choose assets which they believe will

0.01.02.03.04.05.06.07.08.09.0

10.0

0 10 20 30 40 50Correlation with rest of portfolio (%)

Equity Mezzanine Senior

Tra

nch

e d

elta

Figure 17. Dependency of tranchedelta on the asset’s correlation with therest of the portfolio

0

1

2

3

4

5

6

7

8

0

100

200

300

400

500

600

700

800

900

1,00

0

Asset spread (bp)

Tra

nch

e d

elta Equity

MezzanineSenior

Figure 16. Dependency of tranchedelta on the spread of the asset

guide.qxd 10/10/2003 11:15 Page 19


not default. As a result we would expectequity tranche buyers to be skilled creditinvestors, able to pick the right credits forthe portfolio, or at least be able to hedge thecredits they do not like.

On the other hand, the senior investor has asignificant cushion of subordination to insu-late them from principal losses until maybe20 or more of the assets in the collateral havedefaulted. As a consequence, the seniorinvestor is truly taking a portfolio view and soshould be more concerned about the averageproperties of the collateral than the quality ofany specific asset. The senior tranche is real-ly a deleveraged macro credit trade.

Evolution of structuresInitially full capital structure synthetic CDOshad almost none of the structural featurestypically found in other securitised assetclasses and cash flow CDOs. It was only in1999 that features that diverted cash flowsfrom equity to debt holders in case of cer-tain covenant failures began entering thelandscape. The intention was to providesome defensive mechanism for mezzanineholders fearing that the credit cycle wouldaffect tranche performance. Broadly, thesefeatures fit into two categories – ones thatbuild extra subordination using excessspread, and others that use excess spreadto provide upside participation to mezza-nine debt holders.

The most common example of structuralways to build additional subordination is thereserve account funding feature. Excessspread (the difference between premiumreceived from the CDS portfolio and thetranche liabilities) is paid into a reserveaccount. This may continue throughout thelife of the deal or until the balance reaches apredetermined amount. If structured toaccumulate to maturity, the equity tranche

will usually receive a fixed coupon through-out the life of the transaction and any upsideor remainder in the reserve account at matu-rity. If structured to build to a predeterminedlevel, the equity tranche will usually receiveexcess interest only after the reserveaccount is fully funded. More details areprovided in Ganapati and Ha (2002).

Other structures incorporated features toshare some of the excess spread withmezzanine holders or to provide a step up coupon to mezzanines if losses exceed-ed a certain level or if the tranche wasdowngraded. Finally, over-collateralisationtrigger concepts were adopted from cashflow CDOs.

Principal protected structuresInvestors who prefer to hold highly ratedassets can do so by purchasing CDO tranch-es within a principal protected structure.This is designed to guarantee to return theinvestor’s initial investment of par. One par-ticular variation on this theme is the LehmanBrothers High Interest Principal Protectionwith Extendible Redemption (HIPER). This istypically a 10-year note which pays a fixedcoupon to the investor linked to the risk of aCDO equity tranche.

This risk is embedded within the couponsof the note such that each default causes areduction in the coupon size. However theinvestor is only exposed to this credit risk fora first period, typically five years, and thecoupon paid for the remaining period isfrozen at the end of year five. The coupon istypically of the form:

In Figure 18 we show the cash flows

CouponPortfolio loss

Tranche size= × −

8 1 0% max ,

guide.qxd 10/10/2003 11:15 Page 20


assuming two credit events over the lifetimeof the trade. The realised return is depen-dent on the timing of credit events. For agiven number of defaults over the tradematurity, the later they occur, the higher thefinal return.

Managed syntheticsThe standard synthetic has been based on astatic CDO, ie, the reference assets in theportfolio do not change. However, recently,Lehman Brothers and a number of otherdealers have managed to combine the cus-tomised tranche with the ability for an assetmanager or the issuer of the tranche tomanage the portfolio of reference entities.This enables investors to enjoy all the bene-fits of customised tranches and the benefitsof a skilled asset manager. The customis-able characteristics include rating, ratingagency, spread, subordination, issuance for-mat plus others.

The problem with this type of structure isthat the originator of the tranche has to fac-tor into the spread the cost of substitutingassets in the collateral. Initially this wasbased on the asset manager being told thecost of substituting an asset using someblack-box approach.

More recently the format has evolved toone where the manager can change theportfolio subject to some constraints. Oneexample of such technology is LehmanBrothers’ DYNAMO structure. The advan-tage of this approach is that it frees the man-ager to focus on the credits without havingto worry about the cost of substitution.

The other advantages of such a structurefor the asset manager are fees earned andan increase in assets under management.For investors the incentive is to leverage themanagement capabilities of a credit assetmanager in order to avoid blow-ups in the

portfolio and so better manage downturns inthe credit cycle.

The CDO of CDOsA recent extension of the CDO paradigm hasbeen the CDO of CDOs, also known as ‘CDOsquared’. Typically this is a mezzanine‘super’ tranche CDO in which the collateralis made up of a mixture of asset-backedsecurities and several ‘sub’ tranches of syn-thetic CDOs. Principal losses are incurred ifthe sum of the principal losses on the under-lying portfolio of synthetic tranches exceedsthe attachment point of the super-tranche.Looking forward, we see growing interest insynthetic-only portfolios.

Leveraging the spread premiumMarket spreads paid on securities bearingcredit risk are typically larger than the levels implied by the historical default ratesfor the same rating. This difference, whichwe call the spread premium, arisesbecause investors demand compensationfor being exposed to default uncertainty, aswell as other sources of risk, such asspread movements, lack of liquidity or rat-ings downgrades.

Portfolio credit derivatives, such as basketdefault swaps and synthetic CDO tranches,offer a way for investors to take advantageof this spread premium. When an investor

Credit events

Credit window Coupons not reducedby defaults after maturity

of credit window 100

100

guaranteed

Figure 18. The HIPER structure

guide.qxd 10/10/2003 11:15 Page 21


sells protection via a default basket or aCDO tranche, the note issuer passes this onby selling protection in the CDS market. Thishedging activity makes it possible to passthis spread premium to the buyer of thestructured credit asset. For buy and holdcredit investors the spread premium paidcan be significant and it is possible to show,see O’Kane and Schloegl (2003) for details ofthe method, that under certain criteria, theseassets may be superior to single-name credit investments.

Our results show that an FTD basket lever-ages the spread premium such that the sizeof the spread premium is much higher foran FTD basket than it is for a single-creditasset paying a comparable spread. This isshown in Figure 19 where we see that anFTD basket paying a spread of 350bp hasaround 290bp of spread premium. Comparethis with a single-credit Ba3 asset also pay-ing a spread close to 340bp. This has only70bp of spread premium.

For an STD basket we find that the spreadpremium is not leveraged. Instead, it is theratio of spread premium to the wholespread which goes up. There are thereforetwo conclusions:

1. FTD baskets leverage spread premium.This makes them suitable for buy andhold yield-hungry investors who wish tobe paid a high spread but also wish tominimise their default risk.

2. STD baskets leverage the ratio of spreadpremium to the market spread. This issuitable for more risk-averse investorswho wish to maximise return per unit ofdefault risk.

We therefore see that default baskets canappeal to a range of investor risk preferences.

CDO tranches exhibit a similar leveragingof the premium embedded in CDS spreads.The advantage of CDO of CDOs is that theyprovide an additional layer of leverage tothe traditional CDO. This can make leverag-ing the spread premium arguments evenmore compelling.

The conclusion is that buy-and-hold corre-lation investors are overcompensated fortheir default risk compared with single-name investors.

CDO strategiesInvestors in correlation products should pri-marily view them as buy and hold invest-ments which allow them to enjoy the spreadpremium. This is a very straightforwardstrategy for mezzanine and senior investors.However, for equity investors, there are anumber of strategies that can be employedin order to dynamically manage the idiosyn-cratic risk. We list some strategies below.

1. The investor buys CDO equity andhedges the full notional of the 10 or soworst names. The investor enjoys a sig-nificant positive carry and at the sametime reduces his idiosyncratic defaultrisk. The investor may also sell CDS pro-tection on the tightest names, using the

050

100150200250300350400

Ba3 FTDInstrument

Sp

read

(b

p)

Actuarial spread Spread premium

Figure 19. Spread premium for an FTDcompared with a Ba3 single-name asset

guide.qxd 10/10/2003 11:15 Page 22


income to offset some of the cost of pro-tection on the widest names.

2. The investor may buy CDO equity anddelta hedge. The net positive gammamakes this trade perform well in highspread volatility scenarios. By dynamical-ly re-hedging, the investor can lock in thisconvexity. The low liquidity of CDOsmeans that this hedging must continueto maturity.

3. The investor may use the carry from CDOequity to over-hedge the whole portfolio,creating a cheap macro short position.While this is a negative carry trade, it canbe very profitable if the market widensdramatically or if a large number ofdefaults occur.

For more details see Isla (2003).

Credit optionsActivity in credit options has grown sub-stantially in 2003. From a sporadic marketdriven mostly by one-off repackaging deals,it has extended to an increasingly vibrantmarket in both bond and spread options,options on CDS and more recently optionson portfolios and even on CDO tranches.

This growth of the credit options markethas been boosted by declines in bothspread levels and spread volatility. Thereduction in perceived default risk hasmade hedge funds, asset managers, insur-ers and proprietary dealer trading desksmore comfortable with the spread volatilityrisks of trading options and more willing toexploit their advantages in terms of lever-age and asymmetric payoff.

The more recent growth in the market foroptions on CDS has also been driven by theincreased liquidity of the CDS market,enabling investors to go long or short theoption delta amount.

Hedge funds have been the main growthuser of credit options, using them for creditarbitrage and also for debt-equity strategies.They are typically buyers of volatility, hedg-ing in the CDS market and exploiting thepositive convexity. Asset managers seekingto maximise risk-adjusted returns areinvolved in yield-enhancing strategies suchas covered call writing. Bank loan portfoliomanagers are beginning to explore defaultswaptions as a cheaper alternative to buyingoutright credit protection via CDS.

One source of credit optionality is the cashmarket. Measured by market value weight,5.6% of Lehman Brothers US Credit Indexand 54.7% of Lehman Brothers US HighYield Index have embedded call or putoptions. Hence, two strategies which havebeen, and continue to be important in thebond options market are the repack tradeand put bond stripping.

The repack tradeThe first active market in credit bondoptions was developed in the form of therepack trade, spearheaded by LehmanBrothers and several other dealers. Figure20 (overleaf) shows the schematic of onesuch transaction.

In a typical repack trade, Lehman Brotherspurchased $32,875,000 of the Motorola(MOT) 6.5% 2028 debentures and placedthem into a Lehman Trust called CBTC. TheTrust then issued $25 par class A-1Certificates to retail investors with a couponset at 8.375% – the prevailing rate for MOT inthe retail market at the time. Since the8.375% coupon on the CBTC trust is higherthan the coupon on the MOT Bond, the CBTCtrust must be over-collateralised with enoughface value of MOT bonds to pay the 8.375%coupon. An A-2 Principal Only (PO) tranchecaptures the excess principal. Both class A-1

guide.qxd 10/10/2003 11:15 Page 23


and class A-2 certificates are issued with anembedded call.

This call option was sold separately toinvestors in a form of a long-term warrant.The holder of the MOT call warrant has theright but not the obligation to purchase theMOT bonds from the CBTC trust beginningon 19/7/07 and thereafter at the preset callstrike schedule. This strike is determined bythe proceeds needed to pay off the A-1 cer-tificate at par plus the A-2 certificate at theaccreted value of the PO. Because retailinvestors are willing to pay a premium forthe par-valued low-notional bonds of well-known high quality issuers, the buyers of thecall warrant can use this structure to sourcevolatility at attractive levels.

Put bond strippingAccording to Lehman Brothers’ US CreditIndex, bonds with embedded puts consti-tute approximately 2.3% of the US creditbond market, by market value. These bondsgrant the holder the right, but not the obli-gation to sell the bond back to the issuer ata predetermined price (usually par) at one ormore future dates.

This option can be viewed as an extensionoption since by failing to exercise it, the bondmaturity is extended. In the past severalyears, the market has priced these bonds asthough they matured on the first put date andhas not given much value to the extensionoption. Recently, credit investors haverealised a way to extract this extension riskpremium via a put bond stripping strategy.

Essentially, an investor can buy the putbond, and sell the call option to the first putdate at a strike price of par. Thus, the investorhas a long position in the bond coupled with asynthetic short forward (long put plus shortcall) with a maturity coinciding with the firstput date. He then hedges this position byasset swapping the bond to the put date,effectively eliminating all of the interest raterisk and locking in the cheap volatility. Giventhe small amount of outstanding put bonds,this strategy has led to more efficient pricingof the optionality in these securities.

Bond optionsThere are a variety of bond options traded inthe market. The two most important onesfor investors are:

CBTC Series

2002 -14

$25.515mm A -1 retail tranche

6.50% + par

$32.875mm MOT 6.50% 11/15/28

$7.36mm A -2 PO tranche

MOT call warrant

8.375% (25.515mm)

Residual principal

Option to purchase MOT bond

Par

PV of CF

Market price Premium

Figure 20. Mechanics of MOT 6.5% 15/11/28 repack transaction

guide.qxd 10/10/2003 11:15 Page 24


Price-based options: at exercise, the optionholder pays a fixed amount (strike price) andreceives the underlying bond – the payoff isproportional to the difference between theprice of the bond and the strike price.Examples of actively trading price optionsare Brady bond options, corporate bondoptions, CBTC call warrants and calls on putbonds. See the illustration in Figure 21.

Spread-based options: at exercise, theoption holder pays an amount equal to thevalue of the underlying bond calculatedusing the strike spread and receives theunderlying bond – the payoff is proportion-al to difference between the underlyingspread and the strike spread (to a first orderof approximation). Spread options can bestructured using spreads to benchmarkTreasury bonds, default swap spreads orasset swap spreads.

The exercise schedule can be a single date(European), multiple pre-specified dates(Bermudan) or any date in a given range(American). Currently, the most active trad-ing occurs in short-term (less than 12months to expiry) European-style priceoptions on bonds.

Two of the most common strategies usingprice options on bonds are covered calls andnaked puts. They can be considered respec-tively as limit orders to sell or buy the under-lying bond at a predetermined price (theoption strike) on a predetermined day (theoption expiry date).

Covered call strategy: an investor who ownsthe underlying bond sells an out-of-moneycall on the same face value, receiving anupfront premium. If the bond price on theexpiry date is greater than the strike, theinvestor delivers the bonds and receives thestrike price. The option premium offsets the

investor’s loss of upside on the price. If theprice is less than the strike the investor keepsthe bonds and the premium.

Naked put strategy: an investor writes anout-of-the-money put on a bond which hedoes not own but would like to buy at alower price. If the bond price on the expirydate is lower than the strike price, it is deliv-ered to the investor. The option premiumcompensates him for not being able to buythe bond more cheaply in the market. If thebond price is above the option strike price,the investor earns the premium.

In both of these strategies, the main objec-tive for the investor is to find a strike price atwhich he is willing to buy or sell the under-lying bond and which provides sufficientpremium to compensate for the potentialupside that he forgoes.

Default swaptions and callable CDSAn exciting development in the credit deriva-tives markets in the past 12 months hasbeen the emergence of default swaptions.These are options on credit default swaps.

The emerging terminology from this mar-

Option

buyer

LehmanBrothers

1.13% premium

Right to buy F

7.25% 11 at 100.76

Figure 21. Three month price calloption on F 7.25% 25/10/11, struck at100.76% price.

guide.qxd 10/10/2003 11:15 Page 25


ket is that protection calls (option to buy pro-tection) are called payer default swaptions.Protection puts (option to sell protection) arecalled receiver default swaptions.

Unlike price options on bonds, the exercisedecision for default swaptions is based oncredit spread alone. As a result, they areessentially a ‘pure’ credit product, with pricingbeing mostly driven by CDS spread volatility.

Default swaptions give investors the oppor-tunity to express views on the future level andvariability of default swap spreads for a givenissuer. They can be traded outright or embed-ded in callable CDS. The typical maturity ofthe underlying CDS is five years but can rangefrom one–10 years, and the time to optionexpiry is typically three months to one year.

Payer default swaptionThe option buyer pays a premium to theoption seller for the right but not the obliga-tion to buy CDS protection on a referenceentity at a predetermined spread on a futuredate. Payer default swaptions can be struc-tured with or without a provision for knockout at no cost if there is a credit eventbetween trade date and expiry date. If theknock out provision is included in the swap-tion, the option buyer who wishes to main-tain protection over the entire maturity rangecan separately buy protection on the under-lying name until expiry of the swaption.

The relevant scenarios for this investmentare complementary to the ones in the caseof the protection put. If spreads tighten bythe expiry date, the option buyer will notexercise the right to buy protection at thestrike and the option seller will keep theoption premium.

Receiver default swaptionIn a receiver default swaption, the optionbuyer pays a premium to the option seller

for the right, but not the obligation, to sellCDS protection on a reference entity at apredetermined spread on a future date. Thisspread is the option strike.

We do not need to consider what happensif the reference entity experiences a creditevent between trade date and expiry date asthey would never exercise the option in thiscase. As a result, there is no need for aknockout feature for receiver default swap-tions. Consider the following example.

Lehman Brothers pays 1.20% for an at-the-money receiver default swaption onfive-year GMAC, struck at the current fiveyear spread of 265bp and with threemonths to expiry. The investor is short theoption. From the investor’s perspective, therelevant scenarios are:

■■ If five-year GMAC trades above 265bp inthree months, Lehman does not exercise,as they can sell protection for a higherspread in the market. The investor hasrealised an option premium of 1.20% in aquarter of a year.

■■ If five-year GMAC trades at 238bp in threemonths, the trade breaks even. (1.20% upfront option premium equals the payoff of(265bp–238bp)=27bp times the five-yearPV01 of 4.39). If five-year GMAC tradesbelow 238bp in three months, the loss onthe exercise of the option will be greaterthan the upfront premium and the investorwill underperform on this trade.

Hedging default swaptionsDealers hedge these default swaptions usinga model of the type discussed on page 49.The underlying in a default swaption is the for-ward CDS spread from the option expiry dateto the maturity date of the CDS. Theoretically,a knock-out payer swaption should be delta

guide.qxd 10/10/2003 11:15 Page 26


hedged with a short protection CDS to thefinal maturity of the underlying CDS and a longprotection CDS to the default swaption expirydate. This combination will produce a synthet-ic forward CDS that knocks out at defaultbefore the forward date. In practice, swap-tions with expiry of 1 year and less are hedgedonly with CDS to final maturity due to a lack ofliquidity in CDS with short maturities. We havesummarised the key features of these differ-ent swaption types in Figure 22.

Callable default swapsIn addition to default swaptions, there is agrowing interest in callable default swaps.These are a combination of plain vanilla CDSwith an embedded short receiver swaptionposition. The seller of a callable defaultswap is long credit exposure but this expo-sure can be terminated by the option buyerat some strike spread on a future date.Consider an example.

Lehman Brothers buys five-year GMACprotection, callable in one year, for 315bpfrom an investor. The assumed current mid-market spread for five-year GMAC protec-tion is 265bp.

If the four-year GMAC spread in one yearis less than the strike spread of 315bp, thenLehman Brothers will exercise the optionand so cancel the protection, enabling us tobuy protection at the lower market spread.The investor therefore has earned 315bp forselling five-year protection on GMAC forone year.

If the four-year GMAC spread in one yearis greater than 315bp, the contract contin-ues and the investor continues to earn315bp annually.

From the perspective of the option seller,the callable default swap has a limited MTMupside compared with plain vanilla CDS. Theadditional spread of 315bp–260bp=55bp inthis example compensates the option sellerfor the lost potential upside.

Selling protection in callable default swapis equivalent to a covered call strategy onunderlying issuer spreads and is particularlysuitable as a yield-enhancement techniquefor asset managers and insurers.

Credit portfolio optionsStarting in mid-2003 market participantshave been able to trade in portfolio optionswhose underlying asset is the TRAC-X NorthAmerica portfolio with 100 credits. Liquidityis also growing in the European version.

The rationale for options based on TRAC-Xis that the portfolio effect will reduce theoption volatility and make it easier for deal-ers to hedge. From an investor perspective itpresents a way to take a macro view onspread volatility.

We are now seeing investors trading bothat-the-money and out-of-money puts andcalls to maturities extending from three tonine months. The contracts are typicallytraded with physical delivery. If the TRAC-Xportfolio spread is wider than the strikelevel on the expiry date, the holder of the

Product Payer default Receiver defaultswaption swaption

Description Option to buy Option to sell protection protection

Exercised if CDS spread at CDS spread at expiry > strike expiry < strike

Credit view Short credit Long credit forward forward

Knockout May trade with Not relevant or without

Figure 22. Default swaption types

guide.qxd 10/10/2003 11:15 Page 27


payer default swaption will exercise theoption and lock in the portfolio protectionat more favourable levels. Conversely, if theTRAC-X spread is tighter than the strike, theholder of the receiver swaption will benefitfrom exercising the option and realising theMTM gain.

Investors can monetise a view on the futurerange of market spreads by trading bearishspread (buying at-the-money receiver swap-tion and selling farther out-of-money receiverswaption) or bullish spread (buying ATM payerswaption and selling farther out-of-moneypayer swaption) strategies. Other strategiesinclude expressing views on spread changesover a given time horizon by trading calendarspreads (buying near maturity options andselling farther maturity options).

Finally, because the TRAC-X spread is lesssubject to idiosyncratic spread spikes, andbecause of the existing two-way marketswith varying strikes, investors can expresstheir views on the direction of changes inthe macro level of spread volatility by tradingstraddles, ie, simultaneously buying payerand receiver default swaptions as a way togo long volatility while being neutral to thedirection of spread changes.

Hybrid productsHybrid credit derivatives are those whichcombine credit risk with other market riskssuch as interest rate or currency risk.Typically, these are credit event contingentinstruments linked to the value of a deriva-tives payout, such as an interest rate swapor an FX option.

There are various motivations for enteringinto trades which have these hybrid risks.Below, we give an overview of the economicrationale for different types of structures. Wediscuss the modelling of hybrid credit deriva-tives in more detail on page 51.

Clean and perfect asset swapsOne important theme is the isolation of thepure credit risk component in a giveninstrument. For example, a European CDOinvestor may wish to access USD collateralwithout incurring any of the associated cur-rency risks.

Cross-currency asset swaps are the tradi-tional mechanism by which credit investorstransform foreign currency fixed-rate bondsinto local currency Libor floaters. This hasthe benefit that it substantially reduces thecurrency and interest rate risk, convertingthe bond from an FX, interest rate and cred-it play into an almost pure credit play.

However, the currency risk has not beencompletely removed. First, note that a crosscurrency asset swap is really two trades: (i)purchase of a foreign currency asset; and (ii)entry into a cross-currency swap. In the caseof a European investor purchasing a dollarasset, the investor receives Euribor plus aspread paid in euros.

As long as the underlying dollar assetdoes not default during the life of the assetswap there is no currency risk to theinvestor. However, if the asset doesdefault, the investor loses the future dollarcoupons and principal of the asset, justreceiving some recovery amount which ispaid in dollars on the dollar face value. Asthe cross-currency swap is not contingent,meaning that the payments on the swapcontract are unaffected by any default ofthe asset, the investor is therefore obligedto either continue the swap or to unwind itat the market value with a swap counter-party. This unwind value can be positive ornegative – the investor can make a gain orloss – depending on the direction of move-ments in FX and interest rates since thetrade was initiated.

The risk is significant. We have modelled

guide.qxd 10/10/2003 11:15 Page 28


the distribution of the swap MTM atdefault, shown in Figure 23, for a five yeareuro-dollar cross-currency swap using mar-ket calibrated parameters. The downsiderisk is significant with possible swap relat-ed losses comparable in size to the loss onthe defaulted credit.

As a result, the basic cross-currencyasset swap has a default contingent inter-est rate and currency risk. This can beremoved through the use of a hybrid. Twovariations exist.

1. In the clean asset swap, Lehman Brotherstakes on the default contingent swapunwind value risk.

2. In the perfect asset swap, LehmanBrothers takes on the default contingentswap unwind value risk and guarantees(quantoes) the recovery rate of the default-ed asset in the investor’s base currency.

Both structures have featured widely in theCDO market where they have been used toimmunise mixed-currency high-yield bondportfolios against currency risk in order toallow the structure to qualify for the desiredrating from the rating agency.

However, they can also be used byinvestors to convert foreign currency assetsinto their base currency. For example,European investors can use the perfect assetswap to take advantage of the often higherspread levels which exist for the same cred-its when denominated in US dollars.

The cost of removing this default contin-gent swap MTM risk and paying the recov-ery rate in the investors domestic currencydepends upon a number of factors includ-ing the volatility of the FX rate, the creditquality of the reference credit, the shape ofthe Libor curves in both currencies, interestrate volatilities in both currencies and the

correlations between rates, FX and credit.The cost can be amortised over the life of

the trade as a reduction in the asset swapspread paid. It is interesting to note that thereduction in the perfect asset swap spreadmay not be significant given that the two FXrisks to the swap MTM and the recovery rateare actually partially offsetting.

Counterparty risk hybridsThe mitigation of counterparty risk givesrise to another type of hybrid. Consider aninvestor who enters into an interest rateswap with a credit risky counterparty.Suppose also that this counterpartydefaults with a recovery rate of R. If theMTM of the swap is negative from theviewpoint of the investor, the swap isunwound at market value. This means thatin an MTM framework the investor incursno loss from the default event; the swapcould be replaced by one with more advan-tageous terms at zero cost. However, if thevalue is positive, the investor loses a frac-tion of this MTM.

A way to mitigate the counterparty risk istherefore to buy protection on the counter-

0

1

2

3

4

5

6

–58 –28 2 32 63 93

Pro

bab

ility

(%

)

Swap mark-to-market value at default (%)

Figure 23. Modelled distribution of theswap MTM at default as a percentage offace value for a five-year euro-dollar swap

guide.qxd 10/10/2003 11:15 Page 29


party, where the contingent payout is linkedto the replacement cost of the swap. As wediscuss on page 51, the investor is effec-tively purchasing an interest rate swaptionwhich is automatically exercised upon adefault event. Clearly, similar structures canbe constructed to provide credit protectionfor other payouts.

Yield enhancement In the current interest rate environment withvery low short term rates and steep yieldcurves, coupled with the more mature creditderivatives market, there is increasinginvestor interest in credit-linked notes withmore exotic coupons. These include paying aspread over a Constant Maturity Swap (CMS)rate or a particular inflation index.

Hedge cost mitigationThe final class of application concerns thepure hedging of credit contingent FX orinterest rate risks, such as those faced byan international corporation in the course ofits business activities. For example, oneway to reduce the cost of credit hedgingcould be to purchase default protectionlinked to an FX rate being above or below aspecified threshold.

Companies looking to hedge interest rateor FX risk may be concerned with the cost ofoutright hedging using vanilla derivatives. Inthis case, a hedge which knocks out on thedefault of a reference credit can provide anadequate hedge while significantly decreas-ing costs. Clearly, the hedger is implicitlytaking a bullish view on the reference credit.

guide.qxd 10/10/2003 11:15 Page 30


To be able to price and risk-manage creditderivatives, we need a framework for valu-ing credit risk at a single issuer level, and ata multi-issuer level. The growth of the creditderivatives market has created a need formore powerful models and for a betterunderstanding of the empirical evidenceneeded to calibrate these models. In thissection we will present a detailed overviewof modelling approaches from a practicalperspective, ie, we will discuss models,implementation and calibration.

Single credit modellingThe world of credit modelling is divided intotwo main approaches, called structural andreduced-form. In the structural approach, thedefault is characterised as the consequenceof some event such as a company’s assetvalue being insufficient to cover a repaymentof debt. Such models are usually extensionsof Merton’s 1974 model that used a contin-gent claims analysis for modelling default.

Structural models are generally used to sayat what spread corporate bonds shouldtrade based on the internal structure of thecompany. They therefore require informationabout the balance sheet of the company andcan be used to establish a link between pric-ing in the equity and debt markets. However,they are limited in a number of ways includ-ing the fact that they generally lack the flex-ibility to fit exactly a given term structure ofspreads; and they cannot be easily extendedto price complex credit derivatives.

In the reduced-form approach, the creditprocess is modelled directly via the proba-bility of the credit event itself. Reduced-formmodels also generally have the flexibility torefit the prices of a variety of credit instru-

ments of different maturities. They can alsobe extended to price more exotic creditderivatives. It is for these reasons that theyare used for credit derivatives pricing.

The hazard rate approachThe most widely used reduced-formapproach is based on the work of Jarrowand Turnbull (1995), who characterise a cred-it event as the first event of a Poisson count-ing process which occurs at some time twith a probability defined as

ie, the probability of a default occurring with-in the time interval [t, t+dt) conditional onsurviving to time t, is proportional to sometime dependent function λ(t), known as thehazard rate, and the length of the time inter-val dt. Over a finite time period T, it is possi-ble to show that the probability of survivingis given by

The expectation is taken under the risk-neu-tral measure. A common assumption is thatthe hazard rate process is deterministic. Byextension, this assumption also implies thatthe hazard rate is independent of interestrates and recovery rates.

Pricing model for CDSThe breakeven spread in a CDS is the spreadat which the present values (PV) of premiumand protection legs are equal, ie

Premium PV = Protection PV.

−= ∫

TQ dssETQ

0

)(exp),0( λ

dtttdtt )(]|Pr[ λττ =>+≤

Credit Derivatives Modelling

guide.qxd 10/10/2003 11:15 Page 31


To determine the spread we therefore needto be able to value the protection and pre-mium legs. It is important to take intoaccount the timing of the credit eventbecause this can have a significant effect onthe present value of the protection leg andalso the amount of premium paid on the pre-mium leg. Within the hazard rate approachwe can solve this timing problem by condi-tioning on the probability of defaulting with-in each small time interval [s,s+ds], given byQ(0,s)λ(s)ds, then paying (1–R) and discount-ing this back to today at the risk-free rate.Assuming that the hazard rate and risk freerate term structures are flat, we can writethe value for the protection leg as

The value of the premium leg is the PV of thespread payments which are made to defaultor maturity. If we assume that the spread Son the premium leg is paid continuously, wecan write the present value of the premiumleg as

Equating the protection and premium legsand solving for the breakeven spread gives

This relationship is known as the credit tri-angle because it is a relationship betweenthree variables where knowledge of any twois sufficient to calculate the third. It basical-ly states that the spread paid per small timeinterval exactly compensates the investor

for the risk of default per small time interval.Within this model the interest rate depen-dency drops out.

Given a CDS which has a flat spread curveat 150bp, and assuming a 50% recoveryrate, the implied hazard rate is 0.015 dividedby 0.5, which implies a 3% hazard rate. Theimplied one-year survival probability is there-fore exp(–0.03)=97.04%. For two years it isexp(–0.06)=94.18%, and so on.

Valuation of a CDS positionThe value of a CDS position at time t follow-ing initiation at time 0 is the differencebetween the market implied value of theprotection and the cost of the premium pay-ments, which have been set contractually atSC. We therefore write

MTM(t) = ± (Protection PV –Premium PV),

where the sign is positive for a long protec-tion position and negative for a short protec-tion position. If the current market spread isgiven by S(t) then the MTM can be written as

where the RPV01 is the risky PV01 which isgiven by

and where

An investor buys $10m of five-year protec-tion at 100bp. One year later, the credittrades at 250bp. Assuming a recovery rate

S t

R1= ( )

−λ .

RPV te

r

r T t

011

( ) =−( )

+( )

− +( ) −( )λ

λ,

−= StStMTM ))0()(()( × RPV01

)1( RS −= λ

∫ +−=

+−+−

T Trsr

r

eSdseS

0

)()( )1(

λ

λλ

∫ +−−=−

+−+−

T Trsr

r

eRdseR

0

)()( )1)(1(

)1(λ

λλλ

λ

guide.qxd 10/10/2003 11:15 Page 32


of 40%, the value is given by substituting,r=3.0%, R=40%, S(t)=0.025, S(0)=0.01 andt=4 into the above equation to giveλ=4.17% and an MTM value of $521,661.

This is a simple yet fairly accurate modelwhich works quite well when the interestrate and credit curves are flat. When this isnot the case, it becomes necessary to usebootstrapping techniques to build a full termstructure of hazard rates. This may beassumed to be piecewise flat or piecewiselinear. For a description of such a model seeO’Kane and Turnbull (2003).

Default probabilitiesThe default probabilities calculated forpricing purposes can be quite differentfrom those calculated from historicaldefault rates of assets with the same rat-ing. These real-world default probabilitiesare generally much lower. The reason forthis is that the credit spread of an assetcontains not just a compensation for puredefault risk; it also depends on the mar-ket’s risk aversion expressed through a riskpremium, as well as on supply-and-demand imbalances.

One should also comment on the market’suse of Libor as a risk-free rate in pricing.Pricing theory shows that the price of aderivative is the cost of replicating it in a risk-free portfolio using other securities. Sincemost market dealers are banks which fundclose to Libor, the cost of funding theseother securities is also close to Libor. As aconsequence it is the effective risk-free ratefor the derivatives market.

Calibrating recovery ratesThe calibration of recovery rates presents anumber of complications for credit deriva-tives. Strictly speaking, the recovery rateused in the pricing of credit derivatives is the

expected recovery rate following a creditevent where the expectation is under therisk-neutral measure.

Such expectations are only available fromprice information, and the problem in creditis that given one price, it is difficult to sepa-rate the probability of default from the recov-ery rate expectation.

The market standard is therefore to revert torating agency default studies for estimates ofrecovery rates. These typically show the aver-age recovery rate by seniority and type ofcredit instrument, and usually focus on a UScorporate bond universe. Adjustments maybe made for non-US corporate credits and forcertain industrial sectors.

Problems with rating agency recoverystatistics include the fact that they are back-ward looking and that they only include thedefault and bankruptcy credit events –restructuring is not included. In their favour,one should say that, as they represent theprice of the defaulted asset as a fraction ofpar some 30 days after the default event,they are similar to the definition of the recov-ery value in a CDS.

Recent work (Altman et al 2001) shows thatthere is a significant negative correlationbetween default and recovery rates. Oneway to incorporate this effect is to assumethat recovery rates are stochastic. The stan-dard approach is to use a beta distribution.

Modelling default correlationBy modelling correlation products, we meanmodelling products whose pricing dependsupon the joint behaviour of a set of creditassets. These include default baskets andsynthetic CDOs. As a result of the growth inusage of these products, this is an area ofpricing which has recently gained a lot ofattention, and in which we have seen a num-ber of significant modelling developments.

guide.qxd 10/10/2003 11:15 Page 33


In this section we will describe some ofthese current models, show how they canbe applied to the valuation of baskets andCDOs, and towards the end discuss modelcalibration issues.

Modelling joint defaultsThe valuation of default-contingent instru-ments calls for the modelling of defaultmechanisms. As discussed earlier, a classicaldichotomy in credit models distinguishesbetween a ‘structural approach’, wheredefault is triggered by the market value of theborrower’s assets (in terms of debt plus equi-ty) falling below its liabilities, and a ‘reduced-form approach’, where the default event isdirectly modelled as an unexpected arrival.Although both the structural and the reduced-form approaches can in principle be extendedto the multivariate case, structural modelscalibrated to market-implied default probabili-ties (often called ‘hybrid’ models) have gainedfavour among practitioners because of theirtractability in high dimensions.

If we think of defaults as generated byasset values falling below a given boundary,then the probabilities of joint defaults over aspecified horizon must follow from the joint

dynamics of asset values: consistent withtheir descriptive approach of the defaultmechanism, multivariate structural modelsrely on the dependence of asset returns inorder to generate dependent default events.

This is shown in Figure 24 where we havesimulated 1,000 pairs of asset returns mod-elled as normally distributed random vari-ables, for two firms i and j for two differentasset return correlations of 10% and 90%.The vertical and horizontal lines representdefault thresholds for firms i and j respec-tively. Clearly we see that the probability ofboth i and j defaulting, represented by thenumber of points in the bottom left quad-rant defined by the default thresholds,increases as the asset return correlationincreases. Therefore asset correlationleads to default correlation.

Although the pay-offs of multi-creditdefault-contingent instruments such as nth-to-default baskets and synthetic loss tranch-es cannot be statically replicated by tradingin a set of single-credit contracts, the cur-rent market practice is to value correlationproducts using standard no-arbitrage argu-ments. It follows that the valuation of thesemulti-credit exposures boils down to thecomputation of (risk-neutral) expectationsover all possible default scenarios.

A number of different hybrid frameworkshave been proposed in the literature for mod-elling correlated defaults and pricing multi-name credit derivatives. Hull and White (2001)generate dependent default times by diffus-ing correlated asset values and calibratingdefault thresholds to replicate a set of givenmarginal default probabilities. Multi-periodextensions of the one-period CreditMetricsparadigm are also commonly used, even ifthey produce the undesirable serial indepen-dence of the realised default rate.1 While mostmulti-credit models require simulation, the

-4

-3

-2

-1

0

1

2

3

4

5

-4 -2 0 2 4

-4

-3

-2

-1

0

1

2

3

4

5

-4 -2 0 2 4

CA

CB

CA

CB

VA

VB VB

VA

Both firms default

10% Correlation 90% Correlation

Figure 24. Scatterplot of 1,000 simulat-ed asset returns with 0% and 90% corre-lation. Default thresholds are also shown

guide.qxd 10/10/2003 11:15 Page 34


need for accurate and fast computation ofgreeks has pushed researchers to look formodelling alternatives. Finger (1999) and,more recently, Gregory and Laurent (2003)show how to exploit a low-dimensional factorstructure and conditional independence toobtain semi-analytical solutions.

Asset and default event correlationDefault event correlation (DEC) measuresthe tendency of two credits to default joint-ly within a specified horizon. Formally, it isdefined as the correlation between twobinary random variables that indicatedefaults, ie

where pA and pB are the marginal defaultprobabilities for credits A and B, and PAB isthe joint default probability. Of course, pA, pBand pAB all refer to a specific horizon. Noticethat default event correlation increases lin-early with the joint probability of default andis equal to zero if and only if the two defaultevents are independent. Its limits are not–100% to +100% but are actually a functionof the marginal probabilities themselves.

Default event correlations are the funda-mental drivers in the valuation of multi-namecredit derivatives. Unfortunately, the scarcityof default data makes joint default probabili-ties, and thus default event correlations,very hard to estimate directly. As a result,market participants rely on alternative meth-ods to calibrate the frequency of jointdefaults within their models. Hybrid models,

such as the simulation approach and thesemi-analytical framework described below,use the dependence among asset returns togenerate joint defaults, therefore avoidingthe need for a direct estimation of jointdefault probabilities.

Default-time simulationMonte Carlo models generally aim at thegeneration of default paths, where eachpath is simply a list of default times for eachof the credits in the reference portfoliodrawn at random from the joint default dis-tribution. Knowing the time and identity ofeach default event allows for a precise valu-ation of any multi-credit product, no matterhow complex the contractual specificationof the payoff.

In an influential paper, Li (2000) presents asimple and computationally inexpensivealgorithm for simulating correlated defaults.His methodology builds on the implicitassumption that the multivariate distributionof default times and the multivariate distri-bution of asset returns share the samedependence structure, which he assumes tobe Gaussian and is therefore fully charac-terised by a correlation matrix.

For valuation purposes, we need to samplefrom the multivariate distribution of defaulttimes under the risk-neutral probability mea-sure. In this case, it is common practice toback out the marginal distributions of defaulttimes, which we will denote with F1,F2,…,Fd,from single-credit defaultable instruments(such as CDS). We then join these marginaldistributions with a correlation matrix, whichaccording to the stated assumptions repre-sents the correlation matrix of the assetreturns of the reference credits. Since assetreturns are not directly observable, it is com-mon practice to proxy asset correlationsusing equity correlations. Towards the end of

( ) ( )BBAA

BAAB

pppp

pppDEC

−−

−=11

1 Finger (2000) offers an excellent comparison of severalmultivariate models in terms of the default distributions thatthey generate over time when calibrated to the samemarginals and first-period joint default probabilities.

guide.qxd 10/10/2003 11:15 Page 35


this section we will discuss whether thisseems to be a reasonable approximation.

The high-dimensionality of multi-creditinstruments means that it is not possible touse market prices to obtain the full depen-dence structure. Instead, practitioners gen-erally estimate the necessary correlationsusing historical returns, implicitly relying onthe extra assumption that the correlationsamong asset returns remain unchangedwhen we move from the objective to thepricing probability measure.

In the bivariate case, the joint distributionfunction of default times and is simplygiven by

where N2,ρ denotes the bivariate cumulativestandard Normal with correlation ρ, and N–1

indicates the inverse of a cumulative stan-dard Normal. The extension to the d-dimen-sional case is immediate.

Simulating default times from this distribu-tion is straightforward. With d referencecredits and correlation matrix Σ we have thefollowing algorithm:

1. Choleski decomposition of the correla-tion matrix to simulate a multivariateNormal random vector with cor-relation Σ.

2. Transform the vector into the unit hyper-cube using

3. Translate U into the correspondingdefault times vector t using the inverse ofthe marginal distributions:

The simulation algorithm is illustrated inFigure 25. It is easy to verify that τ has thegiven marginals and a Normal dependencestructure fully characterised by the correla-tion matrix Σ.

Once we know how to sample from therisk-neutral distribution of default times, it isstraightforward to price a correlation trade.Generally speaking, the valuation of a givencorrelation product always boils down to thecomputation of an expectation of the form,

( ) ( ) ( ) ( )( )ddd uFuFuF 12

121

1121 ,...,,,...,, −−−== ττττ

( ) ( ) ( )( )dxNxNxNU ,...,, 21=

dRX ∈

(y))),(F(x)),N(F(NNy)x, P( 21

11

,221−−=<< ρττ

0

0.2

0.4

0.6

0.8

1.0 Cumulative normal

0 2 4 6 8 100

0.2

0.4

0.6

0.8

Time (years)

Cumulative default probability

46x

2 0 2 4 6

1.0

Figure 25. Mapping a normal random variable to a default time

guide.qxd 10/10/2003 11:15 Page 36


E[f (τ)] where τ = (τ1,τ2,...,τd) represents thevector of default times and f is a functiondescribing the discounted cash flows (bothpositive and negative) associated with theinstrument under consideration (see, eg,Mashal and Naldi (2002b)).

In summary, Monte Carlo simulationallows for accurate valuations and risk mea-surements of multi-credit payoffs, evenwhen complex path-dependencies such asreserve accounts, interest coverage or col-lateralisation tests are involved. This isbecause both the exact default times andthe identity of the defaulters are known onevery simulated path. Moreover, we caneasily expand the set of variables to betreated as random. For example, Frye (Risk,2003) argues that modelling stochasticrecoveries and allowing for negativecorrelation between recovery and defaultrates is essential for a proper valuation ofcredit derivatives.

The precision and flexibility of thisapproach, however, come at the cost ofcomputational speed. The basic problem ofusing simulation is that defaults are rareevents, and a large number of simulationpaths are usually required to achieve a suf-ficient sampling of the probability space.There are ways to improve the situation.Useful techniques include antithetic sam-pling, importance sampling and the use oflow-discrepancy sequences. This problembecomes particularly significant when weturn our attention to the calculation of sen-sitivity measures, since for a reasonablenumber of paths the simulation noise canbe similar to or greater than the pricechange due to the perturbation of the inputparameter. Once again, techniques exist toalleviate the problem, but it is hard toachieve precise hedge ratios in a reason-able amount of time. The question then

arises whether some other numericalapproach can be used.

A semi-analytical approachThe recent development of correlation trad-ing and the associated need for fast compu-tation of sensitivities have generated a greatdeal of interest in semi-analytical models. Toenjoy the advantages of fast pricing, oneneeds to impose more structure on theproblem. One way to do this is to rely ontwo basic simplifications:

1. assume a one-factor correlation structurefor asset returns, and

2. discretise the time-line to allow for a finiteset of dates at which defaults can hap-pen. These are chosen with a resolutionto provide a sufficient level of accuracy.

In particular, 1) makes it possible to com-pute the risk-neutral loss distribution of thereference portfolio for any given horizon,while 2) is needed to price an instrumentknowing only the loss distribution of thereference portfolio at a finite set of dates.

While these two assumptions are suffi-cient to price plain vanilla portfolio swaps,derivatives structures with more complexpath-dependencies may also require thatwe approximate the payoff function f(τ).Even in this case, it is usually possible toobtain reasonable approximations, so thatthe benefit of precise sensitivities can beretained at a relatively low cost. Moreover,the error can be controlled by comparingthe analytical solution to a Monte Carloimplementation.

We now discuss how to exploit a one-factormodel to construct the risk neutral loss distri-bution. Later on, we will show how to use asequence of loss distributions at differenthorizons to price synthetic loss tranches.

guide.qxd 10/10/2003 11:15 Page 37


A one-factor modelLet us start by assuming that the assetreturn of the ith issuer between today and agiven horizon is described by a standardNormal random variable Ai with mean 0 andstandard deviation of 1, and is of the form

where Z1,Z2,...,ZD are N(0,1) distributedindependent random variables. The variableZM describes the asset returns due to acommon market factor, while Zi models theidiosyncratic risk of the ith issuer, and βistands for the correlation of the asset returnof issuer i with the market. The asset returncorrelation between asset i and asset j isgiven by βiβj.

Default will occur if the realised assetreturn falls below a given threshold.Mathematically, the ith issuer defaults in theevent that Ai < Ci. Given that Ai is N(0,1) dis-tributed, it is possible to write

and calibration of the marginal probabilitiesis no more complicated than inverting thecumulative normal function.

In the single-issuer case, this is merely anexercise in calibration, as the default thresh-olds are chosen to reproduce default proba-bilities which re-price market instruments.The concept of asset returns becomes mean-ingful when studying the joint behaviour ofmore than one credit. In this case, the returnsare assumed to be described by a multivari-ate Normal distribution. Using the thresholdlevels determined before, it is possible toobtain the probabilities of joint defaults.

With a general correlation structure, thecalculation of joint default probabilities

becomes computationally intensive, makingit necessary to resort to Monte Carlo simu-lation. However, substantial simplificationscan be achieved by imposing more structureon the model.

The advantage of this one-factor setup isthat, conditional on ZM, the asset returns areindependent. This makes it easy to computeconditional default probabilities. Conditionalon the market factor ZM, an asset defaults if

The conditional default probability pi(ZM) ofan individual issuer i is therefore given by

If we assume that the loss on default foreach issuer is the same unit (consistent withall assets having the same seniority), u, thenbuilding up the portfolio loss distribution canbe done iteratively by adding assets to theportfolio. The process is as follows:

1. Beginning with asset one, there are twooutcomes on the loss distribution: a lossof zero with a probability 1–p1(ZM) and aloss of u with a probability p1(ZM).

2. Adding a second asset, we adjust each ofthe previous losses. The zero loss peakrequires that the new asset survives andso has a probability (1–p1(ZM))(1–p2(ZM)).A loss u corresponds with the previouszero loss probability times the probabilitythat asset two defaults plus the previousu loss probability times the probabilityasset two does not default. We arrive at a

−−=

21)(

i

MiiMi

ZCNZp

ββ

21 i

Miii

ZCZ

ββ−

−≤

( ) )(1iiii pNCCNp −=⇒=

iiMii ZZA 21 ββ −+=

guide.qxd 10/10/2003 11:15 Page 38


probability (p1(ZM)+p2(ZM)–2p1(ZM)p2(ZM)).A loss of 2u can only occur if the previousloss of u is multiplied by the probabilitythat asset two also defaults to give aprobability of p1(ZM)p2(ZM).

3. We continue adding on to the portfoliountil all of the assets have been includ-ed. We then repeat with a different valueof ZM and integrate the loss distribu-tions over the N(0,1) distributed marketfactor ZM.

The advantage of this approach is that thereis no simulation noise. Other algorithmsexist, including Fast Fourier techniqueswhich may be more efficient.

Both the times-to-default model and theone-factor model described above areGaussian copula models. This means that aslong as both are calibrated to the samemarginals, and as long as both use the same(one-factor) correlation matrix, both modelsshould generate exactly the same prices.This is an important point – it means thatmodels can be categorised by their copulaspecification alone. It also means that differ-ent products may be priced consistentlyusing different default mechanisms, as longas they use the same copula.

Valuation of correlation productsWe now explain in detail how these modelsmay be applied to the pricing of correlationproducts. We start with the default basket.

Pricing default baskets A default basket is inherently an idiosyncrat-ic product in the sense that the identity ofthe defaulted asset must be known. Oneapproach is therefore to use the times-to-default model via a Monte Carlo implemen-tation. The generation of default times

within this framework has been described indetail above. Given that we know when eachasset in the portfolio defaults in each simu-lation path, to calculate the fair-value spreadwe proceed as follows:

1. Sort default times in ascending order anddenote the nth time to default as τn(i),where i is the label of the defaulted asset.

2. Calculate the PV of a 1bp coupon streampaid to time τ*=min[τ

n(i),T ] where T is the

basket maturity.3. If τ

n(i) < T then calculate the PV =

B(τn(i))(1–R(i)), where B is the Libor dis-

count factor and R(i) is the recovery rate forasset i. Otherwise the protection PV = 0.

4. Average both the premium leg PV of 1bp,known as the basket PV01, and the pro-tection leg PV over all paths.

5. Divide the protection leg by the basketPV01 to get the fair-value spread.

This approach is simple to implement andthe size of default baskets, typically m=5–10assets means that pricing is quite fast inMonte Carlo. Monte Carlo is also flexibleenough to enable you to introduce stochas-tic recovery rates, perhaps drawn from abeta distribution. It is also quite straightfor-ward to introduce alternative copulas, seeMashal and Naldi (2002a).

An analytical approach is also possible here.The main constraint is to build the basket n-to-default probabilities while retaining theidentity of the defaulted assets. See Gregoryand Laurent (2003) for a full discussion.

Rating models for default basketsRating agencies have developed a numberof models for rating default baskets. Forexample Moody’s has adopted a MonteCarlo-based extension of the asset valueapproach, in which an asset value is simu-

guide.qxd 10/10/2003 11:15 Page 39


lated to multiple periods for each of theassets in the basket. These asset valuesmay be correlated internally in order toinduce a default correlation. If the assetvalue falls below the default threshold atthe future period, the asset defaults and arecovery rate is drawn from a beta distribu-tion. The recovery amount and asset valuecan be correlated. The model is calibratedusing historical default statistics and theassigned rating is linked to the expectedloss of the basket to the trade maturity.

S&P has recently switched its approachfrom a weakest-link approach which assignsan FTD the rating of the lowest-rated entityin the basket, to one based on a Monte Carlomodel, which uses the same framework asits CDO Evaluator model.

Pricing synthetic loss tranchesWe would now like to consider in detailhow we would set about valuing a losstranche. One approach is to use the timesto default model via a Monte Carlo imple-mentation as discussed earlier. Given thatwe know when each asset in the portfoliodefaults in each simulation path, we wouldproceed as follows:

1. Calculate the present value of all principallosses on the protection leg in each path.

2. Calculate the PV of 1bp on the premiumleg taking care to reduce the notional ofthe tranche following defaults whichcause principal losses in that path.

3. Average both the 1bp premium leg PV,known as the tranche PV01, and the pro-tection leg PV over all paths.

4. Divide the protection PV by the tranchePV01 to compute the breakeven tranchespread.

This approach is simple to implement but

can be computationally slow if there are alarge number of assets in the portfolio.Another way is to use a semi-analyticalapproach which relies on the fact that astandard synthetic loss tranche can bepriced directly from its loss distribution.

To see this, consider a portfolio CDS on atranche defined by the attachment point Kdand the upper boundary Ku, expressed aspercentages of the reference portfolionotional. This is Nport so that the tranchenotional is given by

The maturity of the portfolio swap is T, andfor each time t≤T we denote by L(t) thecumulative portfolio loss up to time t. Thetranche loss is therefore given by

Note that this is similar to an option stylepayoff. Indeed it is possible to think of CDOtranches as options on the portfolio lossamount.

The two legs of the swap can be priced inthe same way as a CDS if we introduce atranche ‘default probability’ P(t) defined as

where we use the risk-neutral (pricing) mea-sure for taking the expectation.

Assuming that the credit, recovery rate andinterest rates processes are independent,the contingent leg is therefore given by

Protection Leg PV =

tranche

trancheQ

N

tLEtP

)]([)( 0=

]0,)(max[]0,)(max[)( udtranche KtLKtLtL −−−=

)( duporttranche KKNN −=

guide.qxd 10/10/2003 11:15 Page 40


Where B(0,u) is the Libor discount factorfrom today, time 0 to time u. We can dis-cretise the integral by introducing the gridpoints 0=t0<t1<…<tK=T where greateraccuracy is obtained by having a highernumber of grid points. In practice wewould generally assume that monthlyintervals would be sufficient. There are anumber of ways of evaluating an integralon this interval, the simplest being firstorder differences:

Protection Leg PV =

To value the premium leg, we denote thecontractual spread on the tranche by s, andthe coupon payment dates as0=T0<T1<…<Tk=T. The accrual factor forthe ith payment is denoted by ∆i. The PV ofthe premium leg is then given by

Premium Leg PV=

The PV of the tranche from the perspectiveof the investor who is receiving the spread istherefore given by

Tranche PV = Premium Leg PV – ProtectionLeg PV.

As a result we can value a standard syn-

thetic CDO if we have the loss distribution atthe set of future dates t and T. One way todo this is to use the one-factor modeldescribed above to calculate a loss distribu-tion at each of the future dates required.Care must be taken to ensure that thedefault threshold is re-calibrated at eachhorizon date such that the marginal distribu-tion is correctly recovered.

This approach can be more efficient thanMonte Carlo since it is uses the loss distri-bution directly and there are fast ways ofcalculating this. To emphasise this point,one can actually exploit the large numberof assets in a synthetic CDO to derive aclosed-form analytical solution to calculat-ing the loss distribution of a portfolio.

An asymptotic approximationIt is possible to exploit the high dimension-ality of the CDO to derive a closed formmodel for analysing CDO tranches. In factwe can use this approach to obtain resultswhich differ from the exact approach by1–5% for a real CDO.

To begin with, let us assume that the port-folio is homogeneous, ie, that each asset’s βand default probability are the same. Hence,all assets have the same default threshold C.As a result, the conditional default probabili-ty of any individual issuer in the referenceportfolio is given by

If we also assume that the loss exposure toeach issuer is of the same notional amountu, the probability that the percentage loss Lof the portfolio is ku is equal to the proba-bility that exactly k of the m issuers default,

−

−=

21)(

β

β MM

ZCNZp .

∑=

−∆n

jjjjtranche TBTPsN

1

),0())(1(

∑=

−−K

iiiitranche tPtPtBN

11 ))()()(,0(

∫T

tranche udPuBN0

)(),0(

guide.qxd 10/10/2003 11:15 Page 41


which is given by the simple binomial

This distribution becomes computationallyintensive for large values of m, but we canuse methods of varying sophistication toapproximate it. One very simple and sur-prisingly accurate method is the so-called‘large homogeneous portfolio’ (LHP)approximation, originally due to Vasicek(1987). Since the asset returns, conditionalon ZM are independent and identically dis-tributed, by the law of large numbers, thefraction of issuers defaulting will tend tothe conditional probability of an assetdefaulting p(ZM). As a result, the condition-al loss is directly linked to the value of themarket factor ZM which itself is normallydistributed. We can then write the probabil-ity of the portfolio loss being less than orequal to some loss threshold K as

where N(x) is the cumulative normal func-tion. More involved calculations show thatthe distribution of L actually converges tothis limit as m tends to infinity.

We can use the portfolio distribution toderive the loss distributions of individualtranches. If L(K1,K2) is the percentage lossof the mezzanine tranche with attachmentpoint K1 and upper loss threshold K2, thenthis can be written as a function of the losson the reference portfolio. As a fraction of

the tranche notional this is given by

If K<1, then

This equation shows that we can easily derivethe loss distribution of the tranche from thatof the reference portfolio. This is discontinu-ous at the edges owing to the probability ofthe portfolio loss falling outside the interval,and this discontinuity becomes more pro-nounced when the tranche is narrowed. Wecan further compute the expected loss of thetranche analytically. For more details seeO’Kane and Schloegl (2001). We then arrive at

This is only a one-period approach.However, it can easily be extended to multi-ple periods as described earlier.

Correlation skewIt is now possible to observe tradabletranche spreads for different levels ofseniority. If we attempt to imply out the mar-ket correlation using a simple Gaussian cop-ula model fitted to observed market tranchespreads, we can observe a skew in the aver-age correlation as a function of the widthand attachment point of the tranche. Thisskew may be the consequence of a numberof factors such as the assumption of inde-pendence of default and recovery rates. Itmay also be due to our incorrect specifica-tion of the dependence structure, as

( )[ ]( )( ) ( )( )

12

22

12

21

12

21

1,,1,,

,

KK

CKNNCKNN

KKLE

−−−−−−−−

=−− ββ

( ) ( )12121, KKKKLKKKL −+≤⇔≤

( ) ( ) ( )12

2121

0,max0,max,

KK

KLKLKKL

−−−−=

[ ] [ ])(1 KpNKLP −−=≤

[ ]km

M

k

M

ZCN

ZCN

k

mZkuLP

−

−

−−

−

−

==

2

2

11

1|

β

β

β

β

guide.qxd 10/10/2003 11:15 Page 42


explained later. Supply and demand imbal-ances also play a role.

Rating agency models for CDOsDifferent rating agencies have their ownmodels for rating CDO tranches. All of themattempt to capture the risks of CDOs interms of asset quality, recovery rates,default correlation and structural features.

Moody’s standard rating model for CDOtranches is a multinomial extension of theBinomial Expansion Technique (BET) model.To capture default correlation, the portfoliois represented by a lower number of inde-pendent assets, known as the diversityscore. Roughly speaking, this is the numberof independent assets which have the samewidth of loss distribution as the actual CDOreference portfolio of correlated assets. Thediversity score is determined using a lookuptable and is based on the incremental effectof having groups of assets in the sameindustry classification.

After calibrating to historical default data,the model is able to generate an expectedloss for a tranche which takes into accountthe subordination. This expected loss is thenmapped to a rating category.

In S&P’s ratings methodology, a Monte Carlosimulation is used to derive a probability lossdistribution for the underlying collateral poolbased on the total principal balance of theportfolio. Each corporate asset is assigned adefault probability based on S&P’s historicaldefault studies, dependent on its rating andmaturity. Corporate sectors are assumed tohave a correlation of 30% within a givenindustry and 0% between industry sectors.From these inputs and the par amounts ofeach asset, a default probability distribution iscreated. Unlike Moody’s which rates on thebasis of expected loss, S&P rates on the basisof the probability of incurring a loss.

Estimating the dependency structureSeveral well-known multivariate models,including the ones described earlier in thischapter, rely on the assumption that thedependence structure (or ‘copula’) of assetreturns is Normal. The widespread use ofthe Normal dependence structure, which isfully characterised by a correlation matrix,is certainly related to its simplicity. Itremains to be seen, however, whether thisassumption is supported by empirical evi-dence. A number of recent studies haveshown that the joint behaviour of equityreturns is better described by a ‘fat-tailed’Student-t copula than by a Normal copula,and that correlations are therefore not suf-ficient to appropriately characterise theirdependence structure.2 The first goal ofthis section is to apply the same kind ofanalysis to asset returns, and test the nullhypothesis of Gaussian dependence versus the alternative of ‘joint fat tails’.

Of course, we face a major obstacle whenattempting to estimate the dependencestructure of asset returns: asset values arenot directly observable. In fact, the use ofunobservable underlying processes is oneof several criticisms that the structuralapproach has received over the years. Giventhe lack of observable asset returns, it hasbecome customary to proxy the assetdependence with equity dependence, andto estimate the parameters governing thejoint behaviour of asset returns from equityreturn series.3 However, the use of equityreturns to infer the joint behaviour of asset

2 See, for example, Mashal and Naldi (2002a) and Mashaland Zeevi (2002).3 Fitch Ratings (2003) have recently published a specialreport describing their methodology for constructing port-folio loss distributions: it is based on a Gaussian copulaparameterised by equity correlations.

guide.qxd 10/10/2003 11:15 Page 43


returns is often criticised on the grounds ofthe different leverage of assets and equity.The second goal of this section is to shedsome light on the magnitude of the errorinduced by using equity data as a proxy forasset returns.

To provide a plausible answer to thesequestions, we first need to ‘back out’ assetvalues from observable data. One way toestimate the market value of a company’sassets is to implement a univariate struc-tural model. Such a procedure is at theheart of KMV’s CreditEdge™, a popularcredit tool that first computes a measure ofdistance-to-default and then maps it into adefault probability (EDF™) by means of ahistorical analysis of default frequencies.4 Inwhat follows, we summarise recent workby Mashal, Naldi and Zeevi (2002), who usethe asset value series generated by KMV’smodel to study the dependence propertiesof asset returns.

MethodologyA key observation in modelling and testingdependencies is that any d-dimensionalmultivariate distribution can be specifiedvia a set of d marginal distributions that are‘knitted’ together using a copula function.Alternatively, a copula function can beviewed as ‘distilling’ the dependencies thata multivariate distribution attempts to cap-ture, by factoring out the effect of themarginals. Copulas have many importantcharacteristics that make them a centralconcept in the study of joint dependencies,see, eg, the recent survey paper byEmbrechts et al. (2003).

A particular copula that plays a crucial role

in our study is given by the dependencestructure underlying the multivariateStudent-t distribution. The Gaussian distri-bution lies at the heart of most financialmodels and builds on the concept of corre-lation; the Student-t retains the notion ofcorrelation but adds an extra parameterinto the mix, namely, the degrees-of-free-

101 102 103 104 105 106100

101

102

103

Null DoF

Tes

t Sta

tistic

DJIA Asset Returns Sensitivity Analysis

pvalue = 0.01

pvalue = 0.0001

Figure 26. DJIA portfolio: asset andequity returns test statistics as functions of null hypothesis for DoF

101 102 103 104 105 106100

101

102

103

Null DoF

Tes

t Sta

tistic

DJIA Equity Returns Sensitivity Analysis

pvalue = 0.01

pvalue = 0.0001

4 Copyright © 2000-2002 KMV LLC. All rights reserved. KMVand the KMV logo are registered trademarks of KMV LLC.CreditEdge and EDF are trademarks of KMV LLC.

guide.qxd 10/10/2003 11:15 Page 44


dom (DoF). The latter plays a crucial role inmodelling and explaining extreme co-movements in the underlyings.

Moreover, it is well known that theStudent-t distribution is very ‘close’ to theGaussian when the DoF is sufficiently large(say, greater than 30); thus, the Gaussianmodel is nested within the t-family. Thesame statement holds for the underlyingdependence structures, and the DoF param-eter effectively serves to distinguish the twomodels. This suggests how empirical stud-ies might test whether the ubiquitousGaussian hypothesis is valid or not. In par-ticular, these studies would target thedependence structure rather than the distri-butions themselves, thus eliminating theeffect of marginal returns that would ‘con-taminate’ the estimation problem in the lat-ter case. To summarise, the t-dependencestructure constitutes an important and quiteplausible generalisation of the Gaussianmodelling paradigm, which is our main moti-vation for focusing on it.

With this in mind, the key question that wenow face is how to estimate the parametersof the dependence structure. Mashal, Naldiand Zeevi (2002) describe a methodologywhich can be used to estimate the parame-ters of a t-copula without imposing anyparametric restriction on the marginal distri-butions of returns. They also construct alikelihood ratio statistic to test the hypothe-sis of Gaussian dependence, and comparethe dependence structures of asset andequity returns to evaluate the common prac-tice of proxying the former with the latter. Inthe remainder of this section we summarisetheir empirical findings.

Empirical resultsFor the purpose of this study, asset andequity values are both obtained from KMV’s

database. The reader should keep in mind,however, that equity values are observable,while asset values have been ‘backed out’by means of KMV’s implementation of a uni-variate Merton model. We apply our analysisto a portfolio of 30 credits included in theDow Jones Industrial Average and use dailydata covering the period from 31/12/00 to8/11/02. The reader is referred to Mashal,Naldi and Zeevi (2003) for more examplesusing high yield credits and different sam-pling frequencies.

Following the methodology mentionedabove, we estimate the number of degrees-of-freedom (DoF) of a t-copula withoutimposing any structure on the marginal dis-tributions of returns. Then, using a likelihoodratio test statistic, we perform a sensitivityanalysis for various null hypotheses of theunderlying tail dependence, as captured bythe DoF parameter. The two horizontal linesin Figure 26 represent significance levels of99% and 99.99%; a value of the test statis-tic falling below these lines corresponds to avalue of DoF that is not rejected at therespective significance levels.

The minimal value of the test statistic isachieved at 12 DoF for asset returns and at13 DoF for equity returns. In both cases, wecan reject any value of the DoF parameteroutside the range [10,16] with 99% confi-dence; in particular, the null assumption ofa Gaussian copula (DoF=∞) can be rejectedwith an infinitesimal probability of error.Also, the point estimates of the assetreturns’ DoF lies within the non-rejectedinterval for the equity returns’ DoF, and viceversa, indicating that the two are essential-ly indistinguishable from a statistical signif-icance viewpoint. Moreover, the differencebetween the joint tail behaviour of a 12- anda 13-DoF t-copula is negligible in terms ofany practical application.

guide.qxd 10/10/2003 11:15 Page 45


Figure 27 reports the point estimates ofthe DoF for asset and equity returns in theDJIA basket, as well as for three subsetsconsisting of the first, middle, and last 10credits (in alphabetical order). The similari-ties between the joint tail dependence (asmeasured by the DoF) of asset and equityreturns are quite striking.5

Next, we compare the remaining parame-ters that define a t-copula, ie, the correlationcoefficients. Using a robust estimator basedon Kendall’s rank statistic6, we compute thetwo 30x30 correlation matrices from assetand equity returns. The maximum absolutedifference (element-by-element) is 4.6%,and the mean absolute difference is 1.1%,providing further evidence of the similarityof the two dependence structures.

Example: synthetic loss trancheThe models described earlier in this chaptercan be modified to account for a fat-taileddependence structure of asset returns. Herewe analyse the impact of a non-Normalassumption on the expected discounted loss(EDL) of a portfolio loss tranche. We focus on

EDL because this measure relates both to theagency rating (when computed under real-world probabilities) and to the fair compensa-tion for the credit exposure (when computedunder risk-neutral probabilities).

We consider a five-year deal with a refer-ence portfolio of 100 credits, each with $1mnotional. We assume i) uniform recoveryrates of 35%, for every credit in the referenceportfolio; ii) 1% yearly hazard rate for eachreference credit; iii) 20% asset correlationbetween every pair of credits; iv) a risk-freecurve flat at 2%. Using a default-time simula-tion, Figure 28 compares the expected dis-counted losses for several tranches under thetwo alternative assumptions of Gaussiandependence and t dependence with 12 DoF.The results show the significant impact thatthe (empirically motivated) consideration oftail dependence has on the distribution oflosses across the capital structure: expectedlosses are clearly redistributed from the juniorto the senior tranches, as a consequence ofthe increased volatility of the overall portfolioloss distribution. Notice that even larger dif-ferences can be observed if one compareshigher moments or tail measures of thetranches’ loss distributions.

The LHP with tail dependenceIt is possible to incorporate a Student-t cop-ula into the LHP model discussed above. Todo so, we must change the distribution forthe asset returns to be multivariate Student-t distributed where we denote the DoFparameter with ν and retain the one-factorcorrelation structure. This gives

where W is an independent random variable

21

/i i i

i

ZA

W

β β εν

+ −=

5 The range of accepted DoF is very narrow in each case,exhibiting similar behaviour to that displayed in Figure 26. 6 See Lindskog (2000).

Figure 27. Maximum likelihood estimates of DoF for DJIA portfolios

Portfolio Asset returns Equity returns DoF DoF

30-credit DJIA 12 13

First 10 credits 8 9

Middle 10 credits 10 10

Last 10 credits 9 9

guide.qxd 10/10/2003 11:15 Page 46


following a chi-square distribution with νdegrees of freedom. This is the simplestpossible way to introduce tail dependencevia a Student-t copula function.

Note that this is no longer a ‘factor model’ inthe sense that the asset return is composedof two independent and random factors. Thefact that both the market and idiosyncraticterms ‘see’ the same value of W means thatthey are no longer independent.

For this model, we have been able to cal-culate a closed-form solution for the densi-ty of the portfolio loss distribution, andshow in O’Kane and Schloegl (2003) that itpossesses the same tail dependent proper-ties as described above, ie, a widening ofsenior spreads and a reduction in equitytranche spreads.

Summary Our empirical investigation of the depen-dence structure of asset returns shedssome light on the two main issues thatwere raised at the beginning of this section.First, the assumption of Gaussian depen-dence between asset returns can be reject-ed with extremely high confidence in favourof an alternative ‘fat-tailed dependence.’Multivariate structural models that rely onthe normality of asset returns will generallyunderestimate default correlations, andthus undervalue junior tranches and over-value senior tranches of multi-name creditproducts. A fat-tailed dependence of assetreturns will produce more accurate jointdefault scenarios and more accurate valua-tions. Second, the dependence structuresof asset and equity returns appear to bestrikingly similar. The KMV algorithm thatproduces the asset values used in our anal-ysis is nothing other than a sophisticatedway of de-leveraging the equity to get to thevalue of a company’s assets. Therefore, the

popular conjecture that the different lever-age of assets and equity will necessarilycreate significant differences in their jointdynamics seems to be empirically unfound-ed. From a practical point of view, theseresults represent good news for practition-ers who only have access to equity data forthe estimation of the dependence parame-ters of their models.

Modelling credit optionsWe separate credit options into options onbonds and options on default swaps.

Pricing options on bondsOptions on corporate bonds are naturallydivided into three groups according to howthe exercise price is specified. The optioncan be struck on price, yield or creditspread. The exercise price is constant foroptions struck on price, but for optionsstruck on yield it depends on the time tomaturity of the underlying bond and is foundfrom a standard yield-to-maturity calculation.Obviously, for European options there is nodifference between specifying a strike priceand a strike yield.

Bond options struck on spread are differ-ent. For credit spread options the exerciseprice depends both on the time to maturity

Figure 28. Expected discounted loss,100K paths, standard errors in parenthesis

Tranche Normal copula t copula DoF=12 Pctg

(%) EDL (std err %) EDL (std err %) diff

0-5 $2,256,300 (0.14) $2,012,200 (0.23) -11

5-10 $533,020 (0.63) $601,630 (0.66) 13

10-15 $146,160 (1.37) $221,120 (1.06) 51

15-20 $41,645 (1.70) $90,231 (1.62) 117

guide.qxd 10/10/2003 11:15 Page 47


of the bond and on the term structure ofinterest rates at the exercise time. A creditspread strike is commonly specified as ayield spread to a Treasury bond or interestrate swap, or as an asset swap spread.

For short-dated European options withprice (or yield) strike on long-term bonds,the well-known Black-Scholes formula goesa long way but is not recommendedbeyond this limited universe. An importantproblem with Black-Scholes is that it doesnot properly account for the pull-to-par ofthe bond price. This problem can be solvedby a yield diffusion model where the bond’syield is the underlying stochastic process.It is relatively easy to build a lattice for alognormal yield, say, and price the optionby standard backwards induction tech-niques. The lattice approach also allows foreasy valuation of American/Bermudan exer-cise. Yield diffusion models can be con-structed to fit forward bond prices butwould usually assume constant yieldvolatility which is inconsistent with empiri-cal evidence.

The problems of pull-to-par of price andnon-constant volatility can be solved by a fullterm structure model such as Black-Derman-Toy, Black-Karasinski or Heath-Jarrow-Morton. These models were designed fordefault-free interest rates but can be appliedanalogously to credit risky issuers. Instead ofcalibrating the model to Libor rates, themodel should be calibrated to an issuer-spe-cific credit curve.

It is rarely possible to calibrate volatilityparameters because of the lack of liquidbond option prices. Usually it is more appro-priate to base volatility parameters on histor-ical estimates. Naldi, Chu and Wang (2002)and Berd and Naldi (2002) present a multi-factor framework for modelling the stochas-tic components of corporate bond returns.

The framework can be used to deriveprice/yield volatilities from return volatilitiesand tend to give more robust estimates thandirect estimation.

Extending interest rate models to mod-elling risky rates only works if we assumeeither zero recovery on default, or assumethat recovery is paid as a fraction of the mar-ket value at default. However, creditors havea claim for return of full face value inbankruptcy, so it is necessary to specificallymodel the recovery at default as a fraction offace value. This is especially important forlower credit quality issuers where the bondstrade on price rather than yield.

For high quality issuers, the aboveapproaches are less controversial for pric-ing options struck on price or yield. Forthese issuers, interest rate volatility is themain driver of price action. Volatility param-eters should therefore be related toimplied volatilities from interest rate swap-tion markets but must also incorporate thenegative correlation between creditspreads and interest rates (see Berd andRanguelova (2003)) which can cause yieldvolatility on corporate bonds to be signifi-cantly lower than comparable interest rateswaption volatility.

When pricing spread options, it is impor-tant to specifically take into account thedefault risk. In the next section we discuss

Default swaption

expiry date T

CDS maturity date TM

Default swaption settlement

TS

Default swap cash flows if exercised into

Figure 29. Receiver default swaption

guide.qxd 10/10/2003 11:15 Page 48


how to price options to buy or sell protectionthrough CDS. Bond options struck on creditspread can be priced in similar fashion.

Pricing default swaptionsThe growing market in default swaptions hasled to a demand for models to price theseproducts. First, let us clarify terminology. Anoption to buy protection is a payer swaptionand an option to sell protection is a receiverswaption. This terminology is analogous tothat used for interest rate swaptions.

Black’s formulas for interest rate swap-tions can be modified to price Europeandefault swaptions. Consider a Europeanpayer swaption with option expiry date T andstrike spread K. The contract is to enter intoa long protection CDS from time T to TM, andit knocks out if default occurs before T (seeFigure 29). Conditional on surviving to timeT, the option payoff is

where the PV01T

is the value at T of a risky1bp annuity to time TM or default, and ST isthe market spread observed at T on a CDSwith maturity TM.

Ignoring the maximum, this is the stan-dard payoff calculation for a forward start-ing CDS. See page 32 for a discussion ofCDS pricing.

From finance theory we know that for anygiven security, say B, that only makes a pay-ment at T, there is a probability distributionof the spread ST such that for any othersecurity, say A, that also only makes a pay-ment at T, the ratio A0/B0 of today’s values ofthe securities is equal to the expectation ofthe ratio AT/BT of the security payments at T.The result is valid even if BT can be 0 as longas AT = 0 when BT = 0. The states where BT

= 0 are simply ignored in that case and thedistribution of ST will be such that the proba-bility that BT = 0 is 0. (See Harrison andKreps (1979) for details.)

To use this result we first let A be a securi-ty that at T pays 0 if default has occurred andotherwise pays the upfront cost (as of T) of azero-premium CDS with the same maturityas the CDS underlying the swaption. We letB be a security that at T pays 0 if default hasoccurred and otherwise pays PV01

T. With

these definitions the ratio AT/BT is equal tothe spread ST if default has not occurred at T,otherwise AT/BT and ST are not defined. Thedistribution of ST should then be such thatE[ST ] = A0/B0 where the probability ofdefault before T is put to 0. A0/B0 is the T-for-ward spread, denoted F0, for the underlyingCDS where the forward contract knocks outif default occurs before T.

The next step is to let A be the swaption,in which case AT is 0 if default happensbefore T and PST otherwise. The value of theswaption today is then

If we make the assumption that log(ST) isnormally distributed with variance σ2T, corre-sponding to the spread following a log-nor-mal process with constant volatility σ, thenwith the requirement E(ST) = F0 (the forwardspread), we have determined the distributionof ST to be used to find E[max{ST –K,0}]. It iseasy to calculate this expectation and wearrive at the Black formula

( ),)()(01 21000 dNKdNFPVPS ⋅−⋅⋅=

TddT

TKFd σ

σσ

−=+

= 12

20

1 2/)/log(

{ }[ ]0,max01000 KSEPVB

PSEBPS T

T

T −⋅=

=

{ }0,max01 KSPVPS TTT −⋅=

guide.qxd 10/10/2003 11:15 Page 49


where N is the standard normal distributionfunction. The Black formula for a receiverswaption is found analogously. (See Hull andWhite (2003) and Schonbucher (2003) formore details.)

It is important that the forward spread, F0,and PV010 are values under the knockoutassumption. Calculation of F0 and PV01

0

should be done using a credit curve that hasbeen calibrated to the current term structureof CDS spreads.

To value a payer swaption that does notknock out at default, we must add the valueof credit protection from today until optionmaturity. Under a non-knockout payer swap-tion, a protection payment is not made untiloption maturity. The value of the credit pro-tection is therefore less than the upfrontcost of a zero-premium CDS that matureswith the option. The knockout feature is notrelevant for receiver swaptions, as they willnever be exercised after default.

The last step in valuing the swaption is tofind the volatility σ to be used in the Blackformula. An estimate of σ can be made froma time series of CDS spreads. Examinationof CDS spread time series also reveals thatthe log-normal spread assumption often isinappropriate. It is not uncommon for CDSspreads to make relatively large jumps asreaction to firm specific news. However cal-ibrating a mixed jump and Brownian processto spread dynamics is not easy.

To price default swaptions withBermudan exercise we could construct alattice for the forward spread, but criticismanalogous to that for using a yield diffusionmodel to price bond options applies.Instead, we recommend using a stochastichazard rate model.

To illustrate the use of the Black formula,consider a payer swaption on Ford MotorCredit with option maturity 20/12/2003 and

swap maturity 20/9/2008. The valuationdate is 28/8/2003. The strike is 260, whichis the five year CDS spread on the valuationdate. Our trading desk quoted the swaptionat 2.29%/2.52%, which means that $10mnotional can be bought for $252,000 andsold for $229,000. These are prices of non-knockout options. The fair value of protec-tion until swaption maturity given the creditcurve on the valuation date is 0.211%, thePV01 used in the Black formula is 4.027 andthe forward spread is 274.7bp. These num-bers imply that the bid/offer volatilitiesquoted by our desk are approximately 75%and 85%.

Interest rates and credit riskOne way to model credit spread dynamics isvia a hazard rate process. This provides aconsistent framework for modelling spreadsof many different maturities. The differencesbetween directly modelling the creditspread and modelling the hazard rate aresimilar to the differences between mod-elling the yield of a bond and using a termstructure model as discussed above.

A stochastic hazard rate model is natural-ly combined with a term structure model toproduce a unified model that can, at least intheory, be used to price both bond optionsand credit spread options. However, a num-ber of practical complications arise in get-ting such an approach to work. Since bondand the CDS markets have their owndynamics, usually not implying the sameissuer curves, it may not be appropriate tonaively price all options in the same cali-brated model without properly adjusting forthe basis between CDS and bonds.

Also, calibration of the volatility parame-ters of the hazard rate process, when possi-ble, is less than straightforward, especiallywhen calibrating to bond options struck on

guide.qxd 10/10/2003 11:15 Page 50


price or yield, or to bonds with embeddedoptions. Interest rate volatility parametersshould be calibrated separately using liquidprices of interest rate swaptions. Theseparameters are then taken as given whendetermining the hazard rate volatility param-eters by fitting to time series estimates orcalibrating to market prices.

Finally, it is necessary to determine the cor-relation between interest rates and hazardrates. When estimating correlations it isimportant not to use yield spread or OAS(option adjusted spread) as a proxy for thehazard rate, because such spread measuresdo not properly take into account the risk ofdefault. Using OAS is especially a problem forlong maturity bonds with high default proba-bilities and high recovery rates. For suchbonds there can be a significant decrease inOAS when interest rates increase even if haz-ard rates remain unchanged. Correlationsshould be estimated directly using bondprices or CDS spreads.

Modelling hybridsThe behaviour of hybrid credit derivatives isdriven by the joint evolution of creditspreads and other market variables such asinterest and exchange rates, which are com-monly modelled as diffusion processes. Asa consequence, the reduced form approachis the natural framework for pricing andhedging these products.

We illustrate the main ideas via the exam-ple of default protection on the MTM of aninterest rate swap. Suppose an investorhas entered into a receiver swap with fixedrate k with a credit risky counterparty. If theMTM of the receiver swap, RSt is positiveto the investor at the time of default, this ispaid by the protection seller. If RSt is nega-tive, the investor receives nothing, so thatthe payoff at default is max(RSt,0). This is

an option to enter into a receiver swap withfixed rate k for the remaining life of theoriginal trade at default. For simplicity, weassume that default can only take place attimes ti. If B denotes the price process ofthe savings account, then computing theexpected discounted cash flows gives thevalue V0 for the price of the default protec-tion, where

The interpretation of this equation is that thevalue of default protection is a probabilityweighted strip of receiver swaptions, whereeach swaption is priced conditional ondefault happening at ti.

Representing the default protection via astrip of swaptions is a very useful frameworkfor developing intuition around the pricingand helps us understand the importance ofthe shape of the interest rate curve. In termsof volatility exposure, the protection sellerhas sold swaptions and is therefore shortinterest rate volatility.

If the rate and credit process are correlat-ed, then there will also be a spread volatilitydependence. However, under the assump-tion of independence, the volatility of thecredit spread does not enter into the valua-tion, only the default probabilities.

A strip decomposition, as we have shownabove, is the basic building block for mosthybrid credit derivatives. A cross-currencyswap for example could be dealt with inexactly the same way, with the exchangerate as an additional state variable. For thisinstrument, the exchange rate exposure is ofprime importance.

In terms of tractability, it is important to be

( )( ) [ ]0

1

max ,0i

nt

i ii i

RSV E t P t

B tτ τ

=

= = =

∑

guide.qxd 10/10/2003 11:15 Page 51

52 The Lehman Brothers guide to exotic credit derivatives

able to calculate the conditional expecta-tions appearing in the strip decomposition.A further consequence of the hazard ratemethod is that we do not actually have tocondition on the realisation of the defaulttime itself, conditioning instead on the reali-sation of the hazard rate process. This is par-ticularly advantageous for Monte Carlosimulation, as one does not have to explicit-ly simulate default times and is a useful vari-ance reduction technique.

The parameters needed for pricinghybrids are essentially volatility and depen-dence parameters. Calibrating volatilitiesfor interest rates and FX is relativelystraightforward. Credit spread volatilitiesare somewhat more involved. Until recent-ly, we have had to rely on estimates of his-torical volatilities; now the growing optionsmarkets are starting to make the calibrationof implied spread volatilities feasible.

Determining the correct dependence

structure between credit spreads and theother market variables is the main challengein modelling hybrids. The simplest approachis to work within a diffusion setting wherespreads and interest rates/FX are correlated.Such a model will not necessarily generatelevels of dependence representative of peri-ods of market stress where investmentgrade defaults are likely. As a starting point,however, it appears to be reasonable.

Even within this framework, the effect ofcorrelation on the valuation of a hybridinstrument can be marked. At this stage ofthe market it is safe to say that this correla-tion is a ‘realised’ parameter as opposed toan implied one, ie pricing and hedgingmust proceed on the basis of a view on cor-relation founded on historical estimates.Going forward, it will be interesting to seeto what extent a market in implied correla-tions will develop via standardised hybridcredit derivatives.

guide.qxd 10/10/2003 11:15 Page 52


Altman E, Resti A and Sirone A, 2001Analyzing and explaining default recovery ratesReport submitted to ISDA, Stern School ofBusiness, New York University, DecemberBerd A and Naldi M, 2002New estimation options for the LehmanBrothers risk modelQuantitative Credit Research Quarterly,Lehman Brothers, SeptemberBerd A and Ranguelova E, 2003The co-movement of interest rates andspreads: implications for credit investorsHigh Grade Research, Lehman Brothers, Embrechts P, Lindskog F and McNeal A, 2003Modelling dependence with copulas andapplications to risk managementIn Handbook of Heavy Tailed Distributions inFinance, Elsevier, New York, pages 329–384Finger C, 1999Conditional approaches for CreditMetricsportfolio distributionsRiskMetrics Group, New YorkFinger C, 2000A comparison of stochastic default rate modelsRiskMetrics Group, New YorkFitch Ratings, 2003Default correlation and its effect on portfolios of credit riskStructured Finance, Credit Products SpecialReport, FebruaryFrye J, 2003A false sense of securityRisk August, pages 95–99

Ganapati S, Ha P and O’Kane D, 2001Synthetic CDOs – how are they structured?how do they work?CDO Monthly Update, Lehman Brothers,NovemberGanapati S and Ha P, 2002Evolution of synthetic arbitrage CDOsStructured Credit Strategies, LehmanBrothers, NovemberGanapati S, Berd A, Ha P and Ranguelova E, 2003The synthetic CDO bid and basis conver-gence – which sector is next?Structured Credit Strategies, LehmanBrothers, MarchGregory J and Laurent J-P, 2003I will surviveRisk June, pages 103–107Harrison J and Kreps D, 1979Martingales and arbitrage in multiperiodsecurities marketsJournal of Economic Theory 20, pages381–408Hull J, 2000Options, futures and other derivativesFourth edition, Prentice HallHull J and White A, 2001Valuing credit default swaps II: modellingdefault correlationsJournal of Derivatives 8, pages 12–22Hull J and White A, 2003The valuation of credit default swap optionsWorking paper, University of TorontoIsla L, 2003Hedged CDO equity strategiesFixed Income Research, Lehman Brothers,September

References

guide.qxd 10/10/2003 11:15 Page 53


Jarrow R and Turnbull S, 1995Pricing derivatives on financial securities subject to credit riskJournal of Finance 50, pages 53–85Li D, 2000On default correlation: a copula function approachJournal of Fixed Income 9, pages 43–54Lindskog F, 2000Linear correlation estimationWorking paper, RiskLab, ETH ZurichMashal R and Naldi M, 2002aExtreme events and default basketsRisk June, pages 119–122Mashal R and Naldi M, 2002b Beyond CADR: searching for value in theCDO marketQuantitative Credit Research Quarterly,Lehman Brothers, September 2002 Mashal R, Naldi M and Pedersen C, 2003Leverage and correlation risk of syntheticloss tranchesQuantitative Credit Research Quarterly,Lehman Brothers, AprilMashal R, Naldi M and Zeevi A, 2002The dependence structure of asset returnsQuantitative Credit Research Quarterly.Lehman Brothers, DecemberMashal R, Naldi M and Zeevi A, 2003Extreme events and multi-name credit derivativesIn Credit Derivatives The Definitive Guide,Risk, LondonMashal R and Zeevi A, 2002Beyond correlation: extreme co-movementsbetween financial assetsWorking paper, Columbia University, New York

Munves D, Berd A, O’Kane D and Desclee R, 2003The Lehman Brothers CDS indicesFixed Income Research, Lehman Brothers,SeptemberNaldi M, Chu K and Wang G, 2002The new Lehman Brothers credit risk modelQuantitative Credit Research Quarterly,Lehman Brothers, MayO’Kane D and McAdie R, 2001Explaining the basis: cash versus default swapsLehman Brothers, MarchO’Kane D, Pedersen C and Turnbull S, 2003Valuing the restructuring clause in CDSLehman Brothers, JuneO’Kane D and Schloegl L, 2001Modelling credit: theory and practiceLehman Brothers, FebruaryO’Kane D and Schloegl L, 2003Leveraging the spread premium with correlation productsQuantitative Credit Research Quarterly,Lehman Brothers, JulyO’Kane D and Sen S, 2003Upfront credit default swapsQuantitative Credit Research Quarterly,Lehman Brothers, JulyO’Kane D and Turnbull S, 2003Valuation of credit default swapsQuantitative Credit Research Quarterly,Lehman Brothers, JuneSchonbucher P, 2003A note on survival measures and the pricingof options on credit default swapsWorking paper, ETH ZurichVasicek O, 1987Probability of loss on loan portfolioWorking paper, KMV Corporation

guide.qxd 10/10/2003 11:15 Page 54


Approved by Lehman Brothers International(Europe), which is authorised and regulated by theFinancial Services Authority. Rules made for theprotection of investors under the UK FinancialServices Act may not apply to investment businessconducted at or from an office outside the UK.

Published by Incisive Risk Waters Group,Haymarket House, 28–29 Haymarket, London SW1Y 4RX© 2003 Lehman Brothers and Incisive RWG Ltd

All rights reserved. No part of this publication maybe reproduced, stored in or introduced into anyretrieval system, or transmitted, in any form or byany means, electronic, mechanical, photocopying,recording, or otherwise, without the prior writtenpermission of the copyright owners.

Risk Waters Group and Lehman Brothers and theauthor have made every effort to ensure the accuracy of the text; however, neither they nor anycompany associated with the publication canaccept legal or financial responsibility for consequences, which may arise from errors, omissions, or any opinions given.

guide.qxd 10/10/2003 11:15 Page 55


Notes

guide.qxd 10/10/2003 11:15 Page 56

The Lehman Brothers Global Fixed Income ResearchGroup has been consistently recognised as a top-rankedresearch department.

.For the fourth year in a row, the Group was ranked #1 in the Institutional Investor 2003 All-America Fixed Income Research Team Survey . Institutional Investor ranked Lehman Brothers #1 for Fixed Income Research in 10 out of the last 14 years.Lehman Brothers website, LehmanLive - winner of 8 “Best Site” categories in the Euromoney.com Awards.LehmanLive - #1 dealer website on the Bond.Hub portal .Dominant global marketshare in Fixed Income Indices and Index Analytics - available on our POINTTM

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We have an established tradition of providing the highestquality research services to our clients. With over 375dedicated professionals worldwide covering the wholespectrum of Fixed Income products, we provide strategicadvice and analysis to our institutional clients.

All of our research and analysis is accessible through ourwebsite, LehmanLive.

For further information please contact the Global FixedIncome Research Group or visit www.lehmanlive.com

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Document1 06/10/2003 09:59 Page 1

THE LEHMAN BROTHERS

GUIDE TO EXOTIC CREDIT DERIVATIVES

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