Guide to Exotic Credit Derivatives Guide to Exotic Credit Derivatives

THE LEHMAN BROTHERS THE LEHMAN BROTHERS GUIDE TO EXOTIC CREDIT DERIVATIVES GUIDE TO EXOTIC CREDIT DERIVATIVES

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With over seventy professionals Structured Credit Solutions worldwide, Lehman Brothers gives you access to top quality risk-management, structuring, research and legal expertise in structured credit. The team combines local market knowledge with global co-ordinated expertise.

Lehman Brothers has designed specific solutions to our clients’ problems, including yield-enhancement, capital relief, portfolio optimisation, complex hedging and asset-liability Product Innovation management.

.Credit Default Swaps .Portfolio Swaps .Credit Index Products .Repackagings .Default Baskets .Secondary CDO trading .Customised CDO tranches .Default swaptions .Credit hybrids Leadership in Fixed Income Research For further information please contact your local sales representative or call: London: Giancarlo Saronne +44 20 7260 2745 [email protected]

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Foreword

The credit derivatives market has revolu- thetic loss tranches and default baskets cre- tionised the transfer of credit risk. Its impact ate new risk-return profiles to appeal to the has been borne out by its significant growth differing risk appetites of investors based on which has currently achieved a market notion- the tranching of portfolio credit risk. In doing al close to $2 trillion. While not directly com- so they create an exposure to default correla- parable, it is worth noting that the total tion. CDS options allow investors to express notional outstanding of global investment a view on credit spread volatility, and hybrid grade corporate bond issuance currently products allow investors to mix credit risk stands at $3.1 trillion. views with interest rate and FX risk. This growth in the credit derivatives market More recently, we have seen a stepped has been driven by an increasing realisation increase in the liquidity of these exotic credit of the advantages credit derivatives possess derivative products. This includes the devel- over the cash alternative, plus the many new opment of very liquid portfolio credit vehicles, possibilities they present to both credit the arrival of a two-way correlation market in investors and hedgers. Those investors seek- customised CDO tranches, and the developing diversification, yield pickup or new ways ment of a more liquid default swaptions mar- to take an exposure to credit are increasingly ket. To enable this growth, the market has turning towards the credit derivatives market. developed new approaches to the pricing and The primary purpose of credit derivatives is risk-management of these products. to enable the efficient transfer and repack- As a result, this book is divided into two aging of credit risk. In their simplest form, parts. In the first half, we describe how exotic credit derivatives provide a more efficient structured credit products work, their ratio- way to replicate in a derivative format the nale, risks and uses. In the second half, we credit risks that would otherwise exist in a review the models for pricing and risk manag- standard cash instrument. ing these various credit derivatives, focusing More exotic credit derivatives such as syn- on implementation and calibration issues.

Authors Dominic O'Kane Claus Pedersen T. +44 207 260 2628 T. +1 212 526 7775 E. [email protected] E. [email protected] Marco Naldi Lutz Schloegl T. +1 212 526 1728 T. +44 207 260 2113 E. [email protected] E. [email protected] Sunita Ganapati Roy Mashal T. +1 415 274 5485 T. +1 212 526 7931 E. [email protected] E. [email protected] Arthur Berd T. +1 212 526 2629 E. [email protected]

The Lehman Brothers Guide to Exotic Credit Derivatives 1

Contents

Foreword 1

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

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

References 53

2 The Lehman Brothers Guide to Exotic Credit Derivatives

Credit Derivatives Products Market overview The credit derivatives market has changed Figure 1. Market breakdown by instrument type substantially since its early days in the late 1990s, moving from a small and highly eso- Credit default teric market to a more mainstream market swaps 73% with standardised products. Initially driven by the hedging needs of bank loan managers, it has since broadened its base of users to include insurance companies, Credit linked notes Total return hedge funds and asset managers. Portfolio/ swaps 3% Options and correlation 1% The latest snapshot of the credit deriva- hybrids products 1% 22% tives market was provided in the 2003 Risk Magazine credit derivatives survey. This survey polled 12 dealers at the end of 2002, Source: Risk Magazine 2003 Credit Derivatives Survey composed of all the major players in the credit derivatives market. Although the showed that 40.1% of all reference entities reported numbers cannot be considered originate in Europe, compared with 43.8% ‘hard’, they can be used to draw fairly firm from North America. This is in stark con- conclusions about the recent direction of trast to the global credit market which has the market. a significantly smaller proportion of According to the survey, the total market European originated bonds relative to outstanding notional across all credit deriva- North America. tives products was calculated to be $2,306 The base of credit derivatives users has billion, up more than 50% on the previous been broadening steadily over the last few year. Single name CDS remain the most years. We show a breakdown of the market used instrument in the credit derivatives by end-users in Figure 2 (overleaf). Banks world with 73% of market outstanding still remain the largest users with nearly notional, as shown in Figure 1. This supports 50% share. This is mainly because of their our observation that the credit default mar- substantial use of CDS as hedging tools for ket has become more mainstream, focusing their loan books, and their active participa- on the liquid standard contracts. We believe tion in synthetic securitisations. The hedg- that this growth in CDS has been driven by ing activity driven by the issuance of hedging demand generated by synthetic synthetic CDOs (discussed later) has for CDO positions, and by hedge funds using the first time satisfied the demand to buy credit derivatives as a way to exploit capital protection coming from bank loan hedgers. structure arbitrage opportunities and to go Readers are referred to Ganapati et al (2003) outright short the credit markets. for a full discussion of the market impact. An interesting statistic from the survey is Insurance companies have also become the relatively equal representation of North an important player, mainly by investing in American and European credits. The survey investment-grade CDO tranches. As a result,

The Lehman Brothers Guide to Exotic Credit Derivatives 3

ic tranches either by issuing the full capital Figure 2. Breakdown by end users structure or hedging bespoke tranches. Since this survey was published, the credit derivatives market has continued to consoli- Banks date and innovate. The ISDA 2003 Credit (synthetic Reinsurance SPVs securitisation) Derivative Definitions were another milestone 10% 5% 10% Insurance on the road towards CDS standardisation. 14% The year 2003 has also seen a significant increase in the usage of CDS portfolio prod- Hedge ucts. There has been a stepped increase in funds Banks (other) 13% Third-party Corporates liquidity for correlation products, with daily asset managers 38% 7% 3% two-way markets for synthetic tranches now

being quoted. The credit options market, in Source: Risk Magazine 2003 Credit Derivatives Survey. particular the market for those written on CDS, has grown substantially. the insurance share of credit derivatives A number of issues still remain to be usage has increased to 14% from 9% the resolved. First, there is a need for the gener- previous year. ation of a proper term structure for credit More recently, the growth in the usage of default swaps. The market needs to build credit derivatives by hedge funds has had a greater liquidity at the long end and, espe- marked impact on the overall credit deriva- cially, the short end of the credit curve. tives market itself, where their share has Greater transparency is also needed around increased to 13% over the year. Hedge the calibration of recovery rates. Finally, the funds have been regular users of CDS espe- issue of the treatment of restructuring cially around the convertible arbitrage strate- events still needs to be resolved. Currently, gy. They have also been involved in many of the market is segregated along regional lines the ‘fallen angel’ credits where they have in tackling this issue, but it is hoped that a been significant buyers of protection. Given global standard will eventually emerge. their ability to leverage, they have substantially increased their volume of CDS con- The credit default swap tracts traded, which in many cases has been The credit default swap is the basic building disproportionate to their absolute size. block for most ‘exotic’ credit derivatives and Finally, in portfolio products, by which we hence, for the sake of completeness, we set mean synthetic CDOs and default baskets, out a short description before we explore the total notional for all types of credit more exotic products. derivatives portfolio products was $449.4 A credit default swap (CDS) is used to trans- billion. Their share has kept pace with the fer the credit risk of a reference entity (corpo- growth of the credit derivatives market at rate or sovereign) from one party to another. about 22% over the last two years. This is In a standard CDS contract one party pur- not a surprise, since there is a fundamen- chases credit protection from the other party, tally symbiotic relationship between the to cover the loss of the face value of an asset synthetic CDO and single name CDS mar- following a credit event. A credit event is a kets, caused by dealers originating synthet- legally defined event that typically includes

4 The Lehman Brothers Guide to Exotic Credit Derivatives

bankruptcy, failure to pay and restructuring. amount of deliverable obligations of the ref- Buying credit protection is economically erence entity to the protection seller in equivalent to shorting the credit risk. Equally, return for the notional amount paid in cash. selling credit protection is economically In general there are several deliverable obli- equivalent to going long the credit risk. gations from which the protection buyer can This protection lasts until some specified choose which satisfy a number of character- maturity date. For this protection, the pro- istics. Typically they include restrictions on tection buyer makes quarterly payments, to the maturity and the requirement that they the protection seller, as shown in Figure 3, be pari passu – most CDS are linked to until a credit event or maturity, whichever senior unsecured debt. occurs first. This is known as the premium If the deliverable obligations trade with dif- leg. The actual payment amounts on the pre- ferent prices following a credit event, which mium leg are determined by the CDS spread they are most likely to do if the credit event adjusted for the frequency using a basis is a restructuring, the protection buyer can convention, usually Actual 360. take advantage of this situation by buying If a credit event does occur before the and delivering the cheapest asset. The pro- maturity date of the contract, there is a pay- tection buyer is therefore long a cheapest to ment by the protection seller to the protec- deliver option. tion buyer. We call this leg of the CDS the protection leg. This payment equals the dif- Cash settlement – This is the alternative to ference between par and the price of the physical settlement, and is used less fre- assets of the reference entity on the face quently in standard CDS but overwhelming- value of the protection, and compensates the ly in tranched CDOs, as discussed later. In protection buyer for the loss. There are two cash settlement, a cash payment is made by ways to settle the payment of the protection the protection seller to the protection buyer leg, the choice being made at the initiation of equal to par minus the recovery rate of the the contract. They are: reference asset. The recovery rate is calculated by referencing dealer quotes or Physical settlement – This is the most wide- observable market prices over some period ly used settlement procedure. It requires the after default has occurred. protection buyer to deliver the notional Suppose a protection buyer purchases five-year protection on a company at a CDS spread of 300bp. The face value of the pro- Figure 3. Mechanics of a CDS tection is $10m. The protection buyer therefore makes quarterly payments approximately (we ignore calendars and day Default swap spread count conventions) equal to $10m × 0.03 (premium leg) Protection Protection

buyer seller × 0.25 = $75,000. After a short period the reference entity suffers a credit event.

Contingent payment of loss on par Assuming that the cheapest deliverable

following a credit event (protection leg) asset of the reference entity has a recovery price of $45 per $100 of face value, the payments are as follows:

The Lehman Brothers Guide to Exotic Credit Derivatives 5

■ The protection seller compensates the This can be done for long periods without protection buyer for the loss on the face assuming any repo risk. This is very use- value of the asset received by the protec- ful for those wishing to hedge current tion buyer and this is equal to $5.5m. credit exposures or those wishing to take a bearish credit view. ■ The protection buyer pays the accrued premium from the previous premium ■ CDS are unfunded so leverage is possi- payment date to the time of the credit ble. This is also an advantage for those event. For example, if the credit event who have high funding costs, because occurs after a month then the protection CDS implicitly lock in Libor funding to buyer pays approximately $10m × 300bp maturity. × 1/12 = $25,000 of premium accrued. Note that this is the standard for corpo- ■ CDS are customisable, although devia- rate reference entity linked CDS. tion from the standard may incur a liquidity cost. For severely distressed reference entities, the CDS contract trades in an up-front for- ■ CDS can be used to take a spread view mat where the protection buyer makes a on a credit, as with a bond. cash payment at trade initiation which purchases protection to some specified maturi- ■ Dislocations between cash and CDS pre- ty – there are no subsequent payments sent new relative value opportunities. unless there is a credit event in which the This is known as trading the default protection leg is settled as in a standard swap basis. CDS. For a full description of up-front CDS see O’Kane and Sen (2003). Evolution of CDS documentation Liquidity in the CDS market differs from The CDS is a contract traded within the legal the cash credit market in a number of ways. framework of the International Swaps and For a start, a wider range of credits trade in Derivatives Association (ISDA) master agree- the CDS market than in cash. In terms of ment. The definitions used by the market for maturity, the most liquid CDS is the five-year credit events and other contractual details contract, followed by the three-year, seven- have been set out in the ISDA 1999 document year and 10-year. The fact that a physical and recently amended and enhanced by the asset does not need to be sourced means ISDA 2003 document. The advantage of this that it is generally easier to transact in large standardisation of a unique set of definitions round sizes with CDS. is that it reduces legal risk, speeds up the con- firmation process and so enhances liquidity. Uses of a CDS Despite this standardisation of defini- The CDS can do almost everything that cash tions, the CDS market does not have a uni- can do and more. We list some of the main versal standard contract. Instead, there is a applications of CDS below. US, European and an Asian market standard, differentiated by the way they treat a ■ The CDS has revolutionised the credit restructuring credit event. This is the con- markets by making it easy to short credit. sequence of a desire to enhance the posi-

6 The Lehman Brothers Guide to Exotic Credit Derivatives

tion of protection sellers by limiting the For that we need a model and a discussion of value of the protection buyer’s delivery the valuation of CDS is provided on page 32. option following a restructuring credit event. A full discussion and analysis of Funded versus unfunded these different standards can be found in Credit derivatives, including CDS, can be O’Kane, Pedersen and Turnbull (2003). traded in a number of formats. The most commonly used is known as swap format, Determining the CDS spread and this is the standard for CDS. This format The premium payments in a CDS are is also termed ‘unfunded’ format because defined in terms of a CDS spread, paid peri- the investor makes no upfront payment. odically on the protected notional until Subsequent payments are simply payments maturity or a credit event. It is possible to of spread and there is no principal payment show that the CDS spread can, to a first at maturity. Losses require payments to approximation, be proxied by either (i) a par be made by the protection seller to the floater bond spread (the spread to Libor at protection buyer, and this has counterparty which the reference entity can issue a float- risk implications. ing rate note of the same maturity at a price The other format is to trade the risk in the of par) or (ii) the asset swap spread of a form of a credit linked note. This format is bond of the same maturity provided it known as ‘funded’ because the investor has trades close to par. to fund an initial payment, typically par. This Demonstrating these relationships relies par is used by the protection buyer to pur- on several assumptions that break down in chase high quality collateral. In return the pro- practice. For example, we assume a com- tection seller receives a coupon, which may mon market-wide funding level of Libor, we be floating rate, ie, Libor plus a spread, or ignore accrued coupons on default, we may be fixed at a rate above the same matu- ignore the delivery option in the CDS, and rity swap rate. At maturity, if no default has we ignore counterparty risk. Despite these occurred the collateral matures and the assumptions, cash market spreads usually investor is returned par. Any default before provide the starting point for where CDS maturity results in the collateral being sold, spreads should trade. The difference the protection buyer covering his loss and the between where CDS spreads and cash investor receiving par minus the loss. The LIBOR spreads trade is known as the protection buyer is exposed to the default risk Default Swap Basis, defined as: of the collateral rather than the counterparty.

Basis = CDS Spread – Cash Libor Spread. Traded CDS portfolio products CDS portfolio products are products that A full discussion of the drivers behind the enable the investor to go long or short the CDS basis is provided in O’Kane and credit risk associated with a portfolio of CDS McAdie (2001). A large number of in one transaction. investors now exploit the basis as a rela- In recent months, we have seen the emer- tive value play. gence of a number of very liquid portfolio Determining the CDS spread is not the products, whose aim is to offer investors a same as valuing an existing CDS contract. diverse, liquid vehicle for assuming or hedg-

The Lehman Brothers Guide to Exotic Credit Derivatives 7

ing exposure to different credit markets, one of a first-to-default (FTD) basket, n=1, and it example being the TRAC-XSM vehicle. These is the first credit in a basket of reference have added liquidity to the CDS market and credits whose default triggers a payment to also created a standard which can be used the protection buyer. As with a CDS, the con- to develop portfolio credit derivatives such tingent payment typically involves physical as options on TRAC-X. delivery of the defaulted asset in return for a The move of the CDS market from banks payment of the par amount in cash. In return towards traditional credit investors has greatly for assuming the nth-to-default risk, the pro- increased the need for a performance bench- tection seller receives a spread paid on the mark linked directly to the CDS market. As a notional of the position as a series of regular consequence, Lehman Brothers has intro- cash flows until maturity or the nth credit duced a family of global investment grade CDS event, whichever is sooner. indices which are discussed in Munves (2003). The advantage of an FTD basket is that it These consist of three sub-indices, a US enables an investor to earn a higher yield 250 name index, a European 150 name index than any of the credits in the basket. This is and a Japanese 40 name index. All names because the seller of FTD protection is lever- are corporates and the maturity of the index aging their credit risk. is maintained close to five years. Daily pric- To see this, consider that the fair-value ing of all 440 names is available on our spread paid by a credit risky asset is deter- LehmanLive website. mined by the probability of a default, times the size of the loss given default. FTD bas- Basket default swaps kets leverage the credit risk by increasing the Correlation products are based on redistribut- probability of a loss by conditioning the paying the credit risk of a portfolio of single- off on the first default among several credits. name credits across a number of different The size of the potential loss does not securities. The portfolio may be as small as increase relative to buying any of the assets five credits or as large as 200 or more credits. in the basket. The most that the investor can The redistribution mechanism is based on the lose is par minus the recovery value of the idea of assigning losses on the credit portfo- FTD asset on the face value of the basket. lio to the different securities in a specified pri- The advantage is that the basket spread ority, with some securities taking the first paid can be a multiple of the spread paid by losses and others taking later losses. This the individual assets in the basket. This is exposes the investor to the tendency of shown in Figure 4 where we have a basket assets in the portfolio to default together, ie, of five investment grade credits paying an default correlation. The simplest correlation average spread of about 28bp. The FTD bas- product is the basket default swap. ket pays a spread of 120bp. A basket default swap is similar to a CDS, More risk-averse investors can use default the difference being that the trigger is the baskets to construct lower risk assets: sec- nth credit event in a specified basket of ref- ond-to-default (STD) baskets, where n=2, erence entities. Typical baskets contain five trigger a credit event after two or more assets to 10 reference entities. In the particular case have defaulted. As such they are lower risk second-loss exposure products which will TRAC-X is a service mark of JPMorgan and Morgan Stanley pay a lower spread than an FTD basket.

8 The Lehman Brothers Guide to Exotic Credit Derivatives

■ Maturity: The effect of maturity de- Figure 4. Five-year first to default (FTD) pends on the shape of the individual basket on five credits. We show the five credit curves. year CDS spreads of the individual credits

120bp paid on $10m ■ Recovery rate: This is the expected until FTD or five-year maturity, whichever recovery rate of the nth-to-default asset is sooner following its credit event. This has only a Lehman Basket small effect on pricing since a higher Brothers investor expected recovery rate is offset by a higher implied default probability for a Contingent payment of par minus recovery given spread. However, if there is a on FTD on $10m face value default the investor will certainly prefer a

Reference portfolio higher realised recovery rate.

Coca Cola 30bp C. de Saint Gobain 30bp

Electricidade de Portugal 27bp ■ Default correlation: Increasing default Hewlett Packard 29bp Teliasonera 30bp correlation increases the likelihood of assets to default or survive together. The effect of default correlation is subtle and significant in terms of pricing. We now The basket spread discuss this is more detail. One way to view an FTD basket is as a trade in which the investor sells protection Baskets and default correlation on all of the credits in the basket with the Baskets are essentially a default correlation condition that all the other CDS cancel at product. This means that the basket spread no cost following a credit event. Such a depends on the tendency of the reference trade cannot be replicated using existing assets in the basket to default together. instruments. Valuation therefore requires It is natural to assume that assets issued a pricing model. The model inputs in order by companies within the same country and to determine the nth-to-default basket industrial sector should have a higher spread are: default correlation than those within different industrial sectors. After all, they share ■ Value of n: An FTD (n=1) is riskier than an the same market, the same interest rates STD (n=2) and so commands a higher and are exposed to the same costs. At a spread. global level, all companies are affected by the performance of the world economy. ■ Number of credits: The greater the num- We believe that these systemic sector risks ber of credits in the basket, the greater far outweigh idiosyncratic effects so the likelihood of a credit event, and so the we expect that default correlation is higher the spread. usually positive. There are a number of ways to explain how ■ Credit quality: The lower the credit quali- default correlation affects the pricing of ty of the credits in the basket, in terms of default baskets. Confusion is usually caused spread and rating, the higher the spread. by the term ‘default correlation’. The fact is

The Lehman Brothers Guide to Exotic Credit Derivatives 9

default correlation is at its maximum. In Figure 5. Loss distribution for a practice, this means that when any asset five-credit basket with 0%, 20% and 50% correlation defaults, the asset with the widest spread will always default too. As a 80 result, the risk of one default is the same ρ = 0% ρ = 20% ρ = 50% 70 as the risk of the widest spread asset 60 defaulting. Because an FTD is triggered 50 by only one credit event, it will be as risky 40 30 as the riskiest asset and the FTD basket spread should be the widest spread of Probability (%) 20 10 the credits in the basket. 0 012345 Number of defaults The best way to understand the behaviour of default baskets between these two correlation limits is to study the loss distribu- that if two assets are correlated, they will tion for the basket portfolio. See page 33 not only tend to default together, they will for a discussion of how to model the loss also tend to survive together. distribution. There are two correlation limits in which a We consider a basket of five credits with FTD basket can be priced without resorting spreads of 100bp and an assumed recovery to a model – independence and maximum rate for all of 40%. We have plotted the loss correlation. distribution for correlations of 0%, 20%, and 50% in Figure 5. The spread for an FTD bas- ■ Independence: Consider a five-credit ket depends on the probability of one or basket where all of the underlying credits more defaults which equals one minus the have flat credit curves. If the credits are probability of no defaults. We see that the all independent and never become corre- probability of no defaults increases with lated during the life of the trade, the nat- increasing correlation – the probability of ural hedge is for the basket investor to credits surviving together increases – and buy CDS protection on each of the indi- the FTD spread should fall. vidual names to the full notional. If a The risk of an STD basket depends on the credit event occurs, the CDS hedge cov- probability of two or more defaults. As corre- ers the loss on the basket and all of the lation goes up from 0–20%, the probability of other CDS hedges can be unwound at no two, three, four and five defaults increases. cost, since they should on average have This makes the STD spread increase. rolled down their flat credit curves. This The process for translating these loss dis- implies that the basket spread for tributions into a fair value spread requires a independent assets should be equal to model of the type described on page 39. the sum of the spreads of the names in Essentially we have to find the basket the basket. spread for which the present value of the protection payments equals the present ■ Maximum correlation: Consider the value of the premium payments. same FTD basket but this time where the We should not forget that in addition to the

10 The Lehman Brothers Guide to Exotic Credit Derivatives

protection leg, the premium leg of the ■ The basket can be customised to the default basket also has correlation sensitivi- investors’ exact view regarding size, ty because it is only paid for as long as the maturity, number of credits, credit selec- nth default does not occur. tion, FTD or STD. Using a model we have calculated the correlation sensitivity of the FTD and STD spread ■ Buy and hold investors can enjoy the for the five-credit basket shown in Figure 6. leveraging of the spread premium. This is At low correlation, the FTD spread is close to discussed in more detail later. 146bp, which is the sum of the spreads. At high correlation, the basket has the risk of the ■ Credit investors can use default baskets widest spread asset and so is at 30bp. The to hedge a blow-up in a portfolio of cred- STD spread is lowest at zero correlation since its more cheaply than buying protection the probability of two assets defaulting is low on the individual credits. if the assets are independent. At maximum correlation the STD spread tends towards the ■ Default baskets can be used to express a spread of the second widest asset in the bas- view on default correlation. If the ket which is also 30bp. investor’s view is that the implied correlation is too low then the investor should sell Applications FTD protection. If implied correlation is too ■ Baskets have a range of applications. high they should sell STD protection. Investors can use default baskets to leverage their credit exposure and so earn a Hedging default baskets higher yield without increasing their The issuers of default baskets need to notional at risk. hedge their risks. Spread risk is hedged by selling protection dynamically in the CDS ■ The reference entities in the basket are all market on all of the credits in the default typically investment grade and so are basket. Determining how much to sell, familiar to most credit analysts. known as the delta, requires a pricing model to calculate the sensitivity of the basket Figure 6. Correlation dependence of value to changes in the spread curve of the spread for FTD and STD basket underlying credit. Although this delta hedging should immunise the dealer’s portfolio against small 140 changes in spreads, it is not guaranteed to be 120 FTD 100 a full hedge against a sudden default. For 80 STD instance, a dealer hedging an FTD basket 60 where a credit defaults with a recovery rate of 40 20 R would receive a payment of (1-R)F from the Basket spread (bp) 0 protection seller, and will pay D(1-R)F on the 0 102030405060708090100 hedged protection, where F is the basket face Correlation (%) value and D is the delta in terms of percentage of face value. The net payment to the protection buyer is therefore (1-D)(1-R)F.

The Lehman Brothers Guide to Exotic Credit Derivatives 11

There will also probably be a loss on the Hedgers of long protection FTD baskets other CDS hedges. The expected spread are also long gamma. This means that as the widening on default on the other credits in spread of an asset widens, the delta will the basket due to their positive correlation increase and so the hedger will be selling with the defaulted asset will result in a loss protection at a wider spread. If the spread when they are unwound. The greater unwind tightens, then the delta will fall and the losses for baskets with higher correlations hedger will be buying back hedges at a will be factored into the basket spread. tighter level. So spread volatility can be ben- One way for a default basket dealer to eficial. This effect helps to offset the nega- reduce his correlation risk is by selling pro- tive carry associated with hedged FTD tection on the same or similar default bas- baskets. This is clear in the previous exam- kets. However this is difficult as it is usually ple where the income from the hedges is difficult to find protection buyers who select 211bp, lower than the 246bp paid to the FTD the exact same basket as an investor. basket investor. The alternative hedging approach is for the Different rating agencies have developed dealer to buy protection using default bas- their own model-based approaches for the kets on other orders of protection. This is rating of default baskets. We discuss these based on the observation that a dealer who on page 39. is long first, second, third up to Mth order protection on an M-credit basket has almost Synthetic CDOs no correlation risk, since this position is Synthetic collateralised debt obligations almost economically equivalent to buying (Synthetic CDOs) were conceived in 1997 as full face value protection using CDS on all M a flexible and low-cost mechanism for trans- credits in the basket. ferring credit risk off bank balance sheets. Figure 7 shows an example basket with The primary motivation was the banks’ the delta and spread for each of the five reduction of regulatory capital. credits. Note that the deltas are all very More recently, however, the fusion of cred- similar. This reflects the fact that all of the it derivatives modelling techniques and assets have a similar spread. Differences derivatives trading have led to the creation of are mainly due to our different correlation a new type of synthetic CDO, which we call assumptions. a customised CDO, which can be tailored to the exact risk appetites of different classes Figure 7. Default basket deltas for a of investors. As a result, the synthetic CDO € 10m notional five-year FTD basket on has become an investor-driven product. five credits. The FTD spread is 246bp. Overall, these different types of synthetic Reference entity CDS Spread Delta CDO have a total market size estimated by Walt Disney 62bp 6.26m the Risk 2003 survey to be close to $500 bil- Rolls Royce 60bp 6.55m lion. What is also of interest is that the deal- Sun Microsystems 60bp 6.87m er-hedging of these products in the CDS market has generated a substantial demand Eastman Chemical 60bp 7.16m to sell protection, balancing the traditional France Telecom 64bp 7.57m protection-buying demand coming from bank loan book managers.

12 The Lehman Brothers Guide to Exotic Credit Derivatives

ordination below the tranche) and width, it is Figure 8. A standard synthetic CDO possible to customise the risk profile of a tranche to the investor’s specific profile.

Senior Reference 5bp tranche pool Full capital structure synthetics €850m 100 invest- In the typical synthetic CDO structured ment grade Lehman 200bp Mezzanine names in using securitisation technology, the spon- Brothers tranche €100m CDS format €10m x 100 soring institution, typically a bank, enters 1,500bp Equity assets = 1bn into a portfolio default swap with a Special total notional tranche €50m Purpose Vehicle (SPV). This is shown in Contingent payment Figure 9 (overleaf). The SPV typically provides credit protection for 10% or less of the losses on the The performance of a synthetic CDO is reference portfolio. The SPV in turn issues linked to the incidence of default in a portfo- notes in the capital markets to cash collat- lio of CDS. The CDO redistributes this risk by eralise the portfolio default swap with the allowing different tranches to take these originating entity. The notes issued can default losses in a specific order. To see this, include a non-rated ‘equity’ piece, mezza- consider the synthetic CDO shown in Figure nine debt and senior debt, creating cash lia- 8. It is based on a reference pool of 100 bilities. The remainder of the risk, 90% or CDS, each with a €10m notional. This risk is more, is generally distributed via a senior redistributed into three tranches; (i) an equi- swap to a highly rated counterparty in an ty tranche, which assumes the first €50m of unfunded format. losses, (ii) a mezzanine tranche, which take Reinsurers, who typically have AAA/AA rat- the next €100m of losses, and (iii) the senior ings, have traditionally had a healthy appetite tranche with a notional of €850m takes all for this type of senior risk, and are the largest remaining losses. participants in this part of the capital structure The equity tranche has the greatest risk – often referred to as super-senior AAAs or and is paid the widest spread. It is typically super-senior swaps. The initial proceeds from unrated. Next is the mezzanine tranche the sale of the equity and notes are invested which is lower risk and so is paid a lower in highly rated, liquid assets. spread. Finally we have the senior tranche If an obligor in the reference pool defaults, which is protected by €150m of subordina- the trust liquidates investments in the trust tion. To get a sense of the risk of the senior and makes payments to the originating enti- tranche, note that it would require more than ty to cover default losses. This payment is 25 of the assets in the 100 credit portfolio to offset by a successive reduction in the equi- default with a recovery rate of 40% before ty tranche, then the mezzanine and finally the the senior tranche would take a principal super-seniors are called to make up losses. loss. Consequently the senior tranche is typ- See Ganapati et al (2001) for more details. ically paid a very low spread. The advantage of CDOs is that by chang- Mechanics of a synthetic CDO ing the details of the tranche in terms of its When nothing defaults in the reference port- attachment point (this is the amount of sub- folio of the CDO, the investor simply

The Lehman Brothers Guide to Exotic Credit Derivatives 13

Figure 9. The full capital structure synthetic CDO

Subordinated Super swap Issued Senior notes senior swap premium premium Special notes AAA Highly Sponsoring purpose rated bank vehicle Mezzanine notes BBB/A counterparty 10% first (SPV) $900m loss Proceeds super subordinated Equity notes(unrated) senior credit credit protection protection Credit CDS spread protection income

Reference portfolio $1bn notional

receives the Libor spread until maturity and mezzanine investor must bear the subse- nothing else changes. Using the synthetic quent losses with the corresponding reduc- CDO described earlier and shown in Figure tion in the mezzanine notional. If the losses 8, consider what happens if one of the ref- exceed €150m, then it is the senior investor erence entities in the reference portfolio who takes the principal losses. undergoes the first credit event with a 30% The mechanics of a standard synthetic recovery, causing a €7m loss. CDO are therefore very simple, especially The equity investor takes the first loss of compared with traditional cash flow CDO €7m, which is immediately paid to the orig- waterfalls. This also makes them more easi- inator. The tranche notional falls from €50m ly modelled and priced. to €43m and the equity coupon, set at 1500bp, is now paid on this smaller notion- The CDO tranche spread al. These coupon payments therefore fall The synthetic CDO spread depends on a from €7.5m to 15% times €43m = €6.45m. number of factors. We list the main ones and If traded in a funded format, the €3m describe their effects on the tranche spread. recovered on the defaulted asset is either reinvested in the portfolio or used to reduce ■ Attachment point: This is the amount of the exposure of the senior-most tranche subordination below the tranche. The (similar to early amortisation of senior higher the attachment point, the more tranches in cash flow CDOs). defaults are required to cause tranche The senior tranche notional is decreased by principal losses and the lower the tranche €3m to €847m, so that the sum of protect- spread. ed notional equals the sum of the collateral notionals which is now €990m. This has no ■ Tranche width: The wider the tranche effect on the other tranches. for a fixed attachment point, the more This process repeats following each cred- losses to which the tranche is exposed. it event. If the losses exceed €50m then the However, the incremental risk ascending

14 The Lehman Brothers Guide to Exotic Credit Derivatives

the capital structure is usually declining the portfolio loss distribution. We can expect and so the spread falls. to observe one of the two shapes shown in Figure 10. They are (i) a skewed bell curve; (ii) ■ Portfolio credit quality: The lower the a monotonically decreasing curve. quality of the asset portfolio, measured The skewed bell curve applies to the case by spread or rating, the greater the risk of when the correlation is at or close to zero. In all tranches due to the higher default this limit the distribution is binomial and the probability and the higher the spread. peak is at a loss only slightly less than the expected loss. ■ Portfolio recovery rates: The expected As correlation increases, the peak of the recovery rate assumptions have only a distribution falls and the high quantiles secondary effect on tranche pricing. This increase: the curves become monotonically is because higher recovery rates imply decreasing. We see that the probability of higher default probabilities if we keep the larger losses increases and, at the same spread fixed. These effects offset each time, the probability of smaller losses also other to first order. increases, thereby preserving the expected loss which is correlation independent (for ■ Swap maturity: This depends on the further discussion see Mashal, Naldi and shapes of the credit curves. For upward Pedersen (2003)). sloping credit curves, the tranche curve For very high levels of asset correlations will generally be upward sloping and so (hardly ever observed in practice), the distri- the longer the maturity, the higher the bution becomes U-shaped. At maximum tranche spread. default correlation all the probability mass is located at the two ends of the distribution. ■ Default correlation: If default correlation The portfolio either all survives or it all is high, assets tend to default together defaults. It resembles the loss distribution and this makes senior tranches more of a single asset. risky. Assets also tend to survive together making the equity safer. To understand Figure 10. Portfolio loss distribution this more fully we need to better under- for a large portfolio at 0%, 20% and stand the portfolio loss distribution. 95% correlation 40 The portfolio loss distribution 35 No matter what approach we use to gener- 30 ρ = 0 ate it, the loss distribution of the reference 25 ρ = 20% portfolio is crucial for understanding the risk 20 and value of correlation products. The port- ρ 15 = 95%

folio loss is clearly not symmetrically dis- Probability (%) 10 tributed: it is therefore informative to look at 5 the entire loss distribution, rather than sum- 0

marising it in terms of expected value and 0 5

10 15 47 standard deviation. We can use models of Loss (%) the type discussed on page 33 to calculate

The Lehman Brothers Guide to Exotic Credit Derivatives 15

lation, we see that most of the portfolio loss Figure 11. Equity tranche loss distribution for correlations of 20% distribution is inside the equity tranche, with and 50% about 14% beyond, as represented by the peak at 100% loss. As correlation goes to 30 50% the probability of small losses increases ρ ρ 25 = 20% = 50% while the probability of 100% losses increases only marginally. Clearly equity investors 20 benefit from increasing correlation. 15 The mezzanine tranche becomes more risky at 50% correlation. As we see in Figure 10 Probability (%) 12, the 100% loss probability jumps from 5 0.50% to 3.5%. In most cases mezzanine investors benefit from falling correlation – 0

0 they are short correlation. However, the cor- 10 20 30 40 50 60 70 80 90 100 Equity tranche loss (%) relation directionality of a mezzanine tranche depends upon the collateral and the tranche. In certain cases a mezzanine tranche with a Figure 12. Mezzanine tranche loss very low attachment point may be a long distribution for correlation of 20% and correlation position. 50%. We have eliminated the zero loss peak, which is about 86% in both cases Senior investors also see the risk of their tranche increase with correlation as more 4.0 joint defaults push out the loss tail. This is 3.5 clear in Figure 13. Senior investors are short 3.0 ρ = 20% ρ = 50% correlation. 2.5 In Figure 14 we plot the dependence of the 2.0 value of different CDO tranches on correla- 1.5 tion. As expected, we clearly see that:

Probability (%) 1.0 0.5 ■ Senior investors are short correlation. If 0.0 correlation increases, senior tranches 0

10 20 30 40 50 60 70 80 90 fall in value. Mezzanine tranche loss (%) 100 ■ Mezzanine investors are typically short How then does the shape of the portfolio correlation, although this very much loss distribution affect the pricing of tranch- depends upon the details of the tranche es? To see this we must study the tranche and the collateral. loss distribution. ■ Equity investors are long correlation. The tranche loss distribution When correlations go up, equity tranches We have plotted in Figures 11–13 the loss dis- go up in value. tributions for a CDO with a 5% equity, 10% mezzanine and 85% senior tranche for corre- In the process of rating CDO tranches, rat- lation values of 20% and 50%. At 20% corre- ing agencies need to consider all of these

16 The Lehman Brothers Guide to Exotic Credit Derivatives

risk parameters and so have adopted model Figure 13. Senior tranche loss based approaches. These are discussed on distribution for correlations of 20% and page 43. 50%. We have eliminated the zero loss peak, which is greater than 96% in Customised synthetic CDO tranches both cases Customisation of synthetic tranches has 0.8 become possible with the fusion of deriva- 0.7 tives technology and credit derivatives. 0.6 ρ = 20% ρ = 50% Unlike full capital structure synthetics, which 0.5 issue the equity, mezzanine and senior parts 0.4 of the capital structure, customised synthet- 0.3 ics may issue only one tranche. There are a Probability (%) 0.2 number of other names for customised CDO 0.1 tranches, including bespoke tranches, and 0.0 0 single tranche CDOs. 10 20 30 40 Senior tranche loss (%) The advantage of customised tranches is that they can be designed to match exactly the risk appetite and credit expertise of the Figure 14. Correlation dependence investor. The investor can choose the credits of CDO tranches in the collateral, the trade maturity, the attachment point, the tranche width, the rat- 40 Equity ing, the rating agency and the format (fund- 30 Mezzanine ed or unfunded). Execution of the trade can m) Senior € 20 take days rather than the months that full 10 capital structure CDOs require. 0 The basic paradigm has already been discussed in the context of default baskets. It –10 Tranche MTM ( is to use CDS to dynamically delta-hedge –20 the first order risks of a synthetic tranche –30 0 and to use a trading book approach to 10 20 30 40 50 60 70 80 90 Correlation (%) 100 hedge the higher order risks. This is shown in Figure 15 (overleaf). For example, consider an investor who buys a customised mezzanine tranche from Understanding delta for CDOs Lehman Brothers. We will then hedge it by For a specific credit in a CDO portfolio, the selling protection in an amount equal to the delta is defined as the notional of CDS for delta of each credit in the portfolio via the that credit which has the same mark-to- CDS market. The delta is the amount of market change as the tranche for a small protection to be sold in order to immunise movement in the credit’s CDS spread the portfolio against small changes in curve. Although the definition may be the CDS spread curve for that credit. straightforward, the behaviour of the delta Each credit in the portfolio will have its is less so. own delta. One way to start thinking about delta is to

The Lehman Brothers Guide to Exotic Credit Derivatives 17

Figure 15. Delta hedging a synthetic CDO

∆ of CDS on Name 1

∆ of CDS on Name 2

Investor ∆ of CDS on Name 3 Spread

Lehman Brothers ∆ Bespoke tranche Contingent payment

∆ of CDS on Name 99 Reference pool

100 investment grade ∆ of CDS on Name 100 names in CDS format

$10m x 100 assets = $1bn

imagine a queue of all of the credits sorted to default early and hit the equity tranche, in the order in which they should default. and also because the CDS will have a high- This ordering will depend mostly on the er spread sensitivity and so require a small- spread of the asset relative to the other er notional. credits in the portfolio and its correlation rel- To show this we take an example CDO ative to the other assets in the portfolio. If with 100 credits, each $10m notional. It has the asset whose delta you are calculating is three tranches: a 5% equity, a 10% mezza- at the front of this queue, it will be most like- nine and an 85% senior tranche. The asset ly to cause losses to the equity tranche and spreads are all 150bp and the correlation so will have a high delta for the equity between all the assets is the same. tranche. If it is at the back of the queue then The sensitivity of the delta to changing the its equity delta will be low. As it is most like- spread of the asset whose delta we are cal- ly to default after all the other asset, it will be culating is shown in Figure 16. If the single most likely to hit the senior tranche. As a asset spread is less than the portfolio aver- result the senior tranche delta will rise. This age of 150bp, then it is the least risky asset. framework helps us understand the direc- As a result, it would be expected to be the tionality of delta. last to default and so most likely to impact The actual magnitude of delta is more dif- the senior-most tranche. As the spread of ficult to quantify because it depends on the the asset increases above 150bp, it tranche notional and the contractual becomes more likely to default before the tranche spread, as well as the features of others and so impacts the equity or mezza- the asset whose delta we are examining. nine tranche. The senior delta drops and the For example the delta for a senior tranche equity delta increases. to a credit whose CDS spread has widened In Figure 17 we plot the delta of the asset will fall due to the fact that it is more likely versus its correlation with all of the other

18 The Lehman Brothers Guide to Exotic Credit Derivatives

assets in the portfolio. These all have a cor- Figure 16. Dependency of tranche relation of 20% with each other. If the asset delta on the spread of the asset is highly correlated with the other assets it is more likely to default or survive with the 8 other assets. As a result, it is more likely to 7 default en masse, and so senior and mezza- 6 Equity nine tranches are more exposed. For low 5 Mezzanine correlations, if it defaults it will tend to do so 4 Senior by itself while the rest of the portfolio tends 3 to default together. As a result, the equity Tranche delta 2 1 tranche is most exposed. 0

There is also a time effect. Through time, 0 100 200 300 400 500 600 700 800 900

senior and mezzanine tranches become 1,000 Asset spread (bp) safer relative to equity tranches since less time remains during which the subordination can be reduced resulting in principal Idiosyncratic versus systemic risk losses. This causes the equity tranche delta In terms of how they are exposed to credit, to rise through time while the mezzanine there is a fundamental difference between and senior tranche deltas fall to zero. equity and senior tranches. Equity tranches Building intuition about the delta is not triv- are more exposed to idiosyncratic risk – they ial. There are many further dependencies to incur a loss as soon as one asset defaults. be explored and we intend to describe these The portfolio effect of the CDO is only in a forthcoming paper. expressed through the fact that it may take several defaults to completely reduce the Higher order risks equity notional. This implies that equity If properly hedged, the dealer should be investors should focus less on the overall insensitive to small spread movements. properties of the collateral, and more on try- However, this is not a completely risk-free ing to choose assets which they believe will position for the dealer since there are a number of other risk dimensions that have not Figure 17. Dependency of tranche been immunised. These include correlation delta on the asset’s correlation with the sensitivity, recovery rate sensitivity, time rest of the portfolio decay and spread gamma. There is also a 10.0 risk to a sudden default which we call the 9.0 8.0 value-on-default risk (VOD). 7.0 For this reason, dealers are motivated to 6.0 5.0 do trades that reduce these higher order 4.0 3.0

risks. The goal is to flatten the risk of the Tranche delta 2.0 correlation book with respect to these high- 1.0 0.0 er order risks either by doing the offsetting 01020304050 trade or by placing different parts of the Correlation with rest of portfolio (%) capital structure with other buyers of cus- Equity Mezzanine Senior tomised tranches.

The Lehman Brothers Guide to Exotic Credit Derivatives 19

not default. As a result we would expect will usually receive a fixed coupon through- equity tranche buyers to be skilled credit out the life of the transaction and any upside investors, able to pick the right credits for or remainder in the reserve account at matu- the portfolio, or at least be able to hedge the rity. If structured to build to a predetermined credits they do not like. level, the equity tranche will usually receive On the other hand, the senior investor has a excess interest only after the reserve significant cushion of subordination to insu- account is fully funded. More details are late them from principal losses until maybe provided in Ganapati and Ha (2002). 20 or more of the assets in the collateral have Other structures incorporated features to defaulted. As a consequence, the senior share some of the excess spread with investor is truly taking a portfolio view and so mezzanine holders or to provide a step should be more concerned about the average up coupon to mezzanines if losses exceed- properties of the collateral than the quality of ed a certain level or if the tranche was any specific asset. The senior tranche is real- downgraded. Finally, over-collateralisation ly a deleveraged macro credit trade. trigger concepts were adopted from cash flow CDOs. Evolution of structures Initially full capital structure synthetic CDOs Principal protected structures had almost none of the structural features Investors who prefer to hold highly rated typically found in other securitised asset assets can do so by purchasing CDO tranch- classes and cash flow CDOs. It was only in es within a principal protected structure. 1999 that features that diverted cash flows This is designed to guarantee to return the from equity to debt holders in case of cer- investor’s initial investment of par. One par- tain covenant failures began entering the ticular variation on this theme is the Lehman landscape. The intention was to provide Brothers High Interest Principal Protection some defensive mechanism for mezzanine with Extendible Redemption (HIPER). This is holders fearing that the credit cycle would typically a 10-year note which pays a fixed affect tranche performance. Broadly, these coupon to the investor linked to the risk of a features fit into two categories – ones that CDO equity tranche. build extra subordination using excess This risk is embedded within the coupons spread, and others that use excess spread of the note such that each default causes a to provide upside participation to mezza- reduction in the coupon size. However the nine debt holders. investor is only exposed to this credit risk for The most common example of structural a first period, typically five years, and the ways to build additional subordination is the coupon paid for the remaining period is reserve account funding feature. Excess frozen at the end of year five. The coupon is spread (the difference between premium typically of the form: received from the CDS portfolio and the tranche liabilities) is paid into a reserve  Portfolio loss  Coupon =×81%max − , 0 account. This may continue throughout the  Tranche size  life of the deal or until the balance reaches a predetermined amount. If structured to accumulate to maturity, the equity tranche In Figure 18 we show the cash flows

20 The Lehman Brothers Guide to Exotic Credit Derivatives

assuming two credit events over the lifetime Figure 18. The HIPER structure of the trade. The realised return is dependent on the timing of credit events. For a 100 guaranteed given number of defaults over the trade Credit events maturity, the later they occur, the higher the final return.

Credit window Coupons not reduced Managed synthetics by defaults after maturity of credit window The standard synthetic has been based on a 100 static CDO, ie, the reference assets in the portfolio do not change. However, recently, Lehman Brothers and a number of other portfolio and so better manage downturns in dealers have managed to combine the cus- the credit cycle. tomised tranche with the ability for an asset manager or the issuer of the tranche to The CDO of CDOs manage the portfolio of reference entities. A recent extension of the CDO paradigm has This enables investors to enjoy all the bene- been the CDO of CDOs, also known as ‘CDO fits of customised tranches and the benefits squared’. Typically this is a mezzanine of a skilled asset manager. The customis- ‘super’ tranche CDO in which the collateral able characteristics include rating, rating is made up of a mixture of asset-backed agency, spread, subordination, issuance for- securities and several ‘sub’ tranches of syn- mat plus others. thetic CDOs. Principal losses are incurred if The problem with this type of structure is the sum of the principal losses on the under- that the originator of the tranche has to fac- lying portfolio of synthetic tranches exceeds tor into the spread the cost of substituting the attachment point of the super-tranche. assets in the collateral. Initially this was Looking forward, we see growing interest in based on the asset manager being told the synthetic-only portfolios. cost of substituting an asset using some black-box approach. Leveraging the spread premium More recently the format has evolved to Market spreads paid on securities bearing one where the manager can change the credit risk are typically larger than the portfolio subject to some constraints. One levels implied by the historical default rates example of such technology is Lehman for the same rating. This difference, which Brothers’ DYNAMO structure. The advan- we call the spread premium, arises tage of this approach is that it frees the man- because investors demand compensation ager to focus on the credits without having for being exposed to default uncertainty, as to worry about the cost of substitution. well as other sources of risk, such as The other advantages of such a structure spread movements, lack of liquidity or rat- for the asset manager are fees earned and ings downgrades. an increase in assets under management. Portfolio credit derivatives, such as basket For investors the incentive is to leverage the default swaps and synthetic CDO tranches, management capabilities of a credit asset offer a way for investors to take advantage manager in order to avoid blow-ups in the of this spread premium. When an investor

The Lehman Brothers Guide to Exotic Credit Derivatives 21

1. FTD baskets leverage spread premium. Figure 19. Spread premium for an FTD This makes them suitable for buy and compared with a Ba3 single-name asset hold yield-hungry investors who wish to

400 be paid a high spread but also wish to 350 minimise their default risk. 300 2. STD baskets leverage the ratio of spread 250 200 premium to the market spread. This is 150 suitable for more risk-averse investors

Spread (bp) 100 50 who wish to maximise return per unit of 0 default risk. Ba3 FTD Instrument

Actuarial spread Spread premium We therefore see that default baskets can appeal to a range of investor risk preferences. CDO tranches exhibit a similar leveraging of the premium embedded in CDS spreads. sells protection via a default basket or a The advantage of CDO of CDOs is that they CDO tranche, the note issuer passes this on provide an additional layer of leverage to by selling protection in the CDS market. This the traditional CDO. This can make leverag- hedging activity makes it possible to pass ing the spread premium arguments even this spread premium to the buyer of the more compelling. structured credit asset. For buy and hold The conclusion is that buy-and-hold corre- credit investors the spread premium paid lation investors are overcompensated for can be significant and it is possible to show, their default risk compared with single- see O’Kane and Schloegl (2003) for details of name investors. the method, that under certain criteria, these assets may be superior to single-name CDO strategies credit investments. Investors in correlation products should pri- Our results show that an FTD basket lever- marily view them as buy and hold invest- ages the spread premium such that the size ments which allow them to enjoy the spread of the spread premium is much higher for premium. This is a very straightforward an FTD basket than it is for a single-credit strategy for mezzanine and senior investors. asset paying a comparable spread. This is However, for equity investors, there are a shown in Figure 19 where we see that an number of strategies that can be employed FTD basket paying a spread of 350bp has in order to dynamically manage the idiosyn- around 290bp of spread premium. Compare cratic risk. We list some strategies below. this with a single-credit Ba3 asset also paying a spread close to 340bp. This has only 1. The investor buys CDO equity and 70bp of spread premium. hedges the full notional of the 10 or so For an STD basket we find that the spread worst names. The investor enjoys a sig- premium is not leveraged. Instead, it is the nificant positive carry and at the same ratio of spread premium to the whole time reduces his idiosyncratic default spread which goes up. There are therefore risk. The investor may also sell CDS pro- two conclusions: tection on the tightest names, using the

22 The Lehman Brothers Guide to Exotic Credit Derivatives

income to offset some of the cost of pro- Hedge funds have been the main growth tection on the widest names. user of credit options, using them for credit 2. The investor may buy CDO equity and arbitrage and also for debt-equity strategies. delta hedge. The net positive gamma They are typically buyers of volatility, hedg- makes this trade perform well in high ing in the CDS market and exploiting the spread volatility scenarios. By dynamical- positive convexity. Asset managers seeking ly re-hedging, the investor can lock in this to maximise risk-adjusted returns are convexity. The low liquidity of CDOs involved in yield-enhancing strategies such means that this hedging must continue as covered call writing. Bank loan portfolio to maturity. managers are beginning to explore default 3. The investor may use the carry from CDO swaptions as a cheaper alternative to buying equity to over-hedge the whole portfolio, outright credit protection via CDS. creating a cheap macro short position. One source of credit optionality is the cash While this is a negative carry trade, it can market. Measured by market value weight, be very profitable if the market widens 5.6% of Lehman Brothers US Credit Index dramatically or if a large number of and 54.7% of Lehman Brothers US High defaults occur. Yield Index have embedded call or put options. Hence, two strategies which have For more details see Isla (2003). been, and continue to be important in the bond options market are the repack trade Credit options and put bond stripping. Activity in credit options has grown substantially in 2003. From a sporadic market The repack trade driven mostly by one-off repackaging deals, The first active market in credit bond it has extended to an increasingly vibrant options was developed in the form of the market in both bond and spread options, repack trade, spearheaded by Lehman options on CDS and more recently options Brothers and several other dealers. Figure on portfolios and even on CDO tranches. 20 (overleaf) shows the schematic of one This growth of the credit options market such transaction. has been boosted by declines in both In a typical repack trade, Lehman Brothers spread levels and spread volatility. The purchased $32,875,000 of the Motorola reduction in perceived default risk has (MOT) 6.5% 2028 debentures and placed made hedge funds, asset managers, insur- them into a Lehman Trust called CBTC. The ers and proprietary dealer trading desks Trust then issued $25 par class A-1 more comfortable with the spread volatility Certificates to retail investors with a coupon risks of trading options and more willing to set at 8.375% – the prevailing rate for MOT in exploit their advantages in terms of lever- the retail market at the time. Since the age and asymmetric payoff. 8.375% coupon on the CBTC trust is higher The more recent growth in the market for than the coupon on the MOT Bond, the CBTC options on CDS has also been driven by the trust must be over-collateralised with enough increased liquidity of the CDS market, face value of MOT bonds to pay the 8.375% enabling investors to go long or short the coupon. An A-2 Principal Only (PO) tranche option delta amount. captures the excess principal. Both class A-1

The Lehman Brothers Guide to Exotic Credit Derivatives 23

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

8.375% (25.515mm) $25.515mm

A -1 retail

Par tranche

6.50% + par Residual principal $32.875mm CBTC $7.36mm MOT 6.50% Series A -2 PO tranche 11/15/28 2002 -14 PV of CF Market price Premium MOT call warrant Option to purchase MOT bond

and class A-2 certificates are issued with an This option can be viewed as an extension embedded call. option since by failing to exercise it, the bond This call option was sold separately to maturity is extended. In the past several investors in a form of a long-term warrant. years, the market has priced these bonds as The holder of the MOT call warrant has the though they matured on the first put date and right but not the obligation to purchase the has not given much value to the extension MOT bonds from the CBTC trust beginning option. Recently, credit investors have on 19/7/07 and thereafter at the preset call realised a way to extract this extension risk strike schedule. This strike is determined by premium via a put bond stripping strategy. the proceeds needed to pay off the A-1 cer- Essentially, an investor can buy the put tificate at par plus the A-2 certificate at the bond, and sell the call option to the first put accreted value of the PO. Because retail date at a strike price of par. Thus, the investor investors are willing to pay a premium for has a long position in the bond coupled with a the par-valued low-notional bonds of well- synthetic short forward (long put plus short known high quality issuers, the buyers of the call) with a maturity coinciding with the first call warrant can use this structure to source put date. He then hedges this position by volatility at attractive levels. asset swapping the bond to the put date, effectively eliminating all of the interest rate Put bond stripping risk and locking in the cheap volatility. Given According to Lehman Brothers’ US Credit the small amount of outstanding put bonds, Index, bonds with embedded puts consti- this strategy has led to more efficient pricing tute approximately 2.3% of the US credit of the optionality in these securities. bond market, by market value. These bonds grant the holder the right, but not the obli- Bond options gation to sell the bond back to the issuer at There are a variety of bond options traded in a predetermined price (usually par) at one or the market. The two most important ones more future dates. for investors are:

24 The Lehman Brothers Guide to Exotic Credit Derivatives

Price-based options: at exercise, the option Figure 21. Three month price call holder pays a fixed amount (strike price) and option on F 7.25% 25/10/11, struck at receives the underlying bond – the payoff is 100.76% price. proportional to the difference between the price of the bond and the strike price.

Examples of actively trading price options 1.13% premium are Brady bond options, corporate bond options, CBTC call warrants and calls on put Option Lehman bonds. See the illustration in Figure 21. buyer Brothers

Spread-based options: at exercise, the Right to buy F option holder pays an amount equal to the 7.25% 11 at 100.76 value of the underlying bond calculated using the strike spread and receives the underlying bond – the payoff is proportional to difference between the underlying spread and the strike spread (to a first order investor’s loss of upside on the price. If the of approximation). Spread options can be price is less than the strike the investor keeps structured using spreads to benchmark the bonds and the premium. Treasury bonds, default swap spreads or asset swap spreads. Naked put strategy: an investor writes an The exercise schedule can be a single date out-of-the-money put on a bond which he (European), multiple pre-specified dates does not own but would like to buy at a (Bermudan) or any date in a given range lower price. If the bond price on the expiry (American). Currently, the most active trad- date is lower than the strike price, it is deliv- ing occurs in short-term (less than 12 ered to the investor. The option premium months to expiry) European-style price compensates him for not being able to buy options on bonds. the bond more cheaply in the market. If the Two of the most common strategies using bond price is above the option strike price, price options on bonds are covered calls and the investor earns the premium. naked puts. They can be considered respec- In both of these strategies, the main objec- tively as limit orders to sell or buy the under- tive for the investor is to find a strike price at lying bond at a predetermined price (the which he is willing to buy or sell the under- option strike) on a predetermined day (the lying bond and which provides sufficient option expiry date). premium to compensate for the potential upside that he forgoes. Covered call strategy: an investor who owns the underlying bond sells an out-of-money Default swaptions and callable CDS call on the same face value, receiving an An exciting development in the credit deriva- upfront premium. If the bond price on the tives markets in the past 12 months has expiry date is greater than the strike, the been the emergence of default swaptions. investor delivers the bonds and receives the These are options on credit default swaps. strike price. The option premium offsets the The emerging terminology from this mar-

The Lehman Brothers Guide to Exotic Credit Derivatives 25

ket is that protection calls (option to buy pro- for the right, but not the obligation, to sell tection) are called payer default swaptions. CDS protection on a reference entity at a Protection puts (option to sell protection) are predetermined spread on a future date. This called receiver default swaptions. spread is the option strike. Unlike price options on bonds, the exercise We do not need to consider what happens decision for default swaptions is based on if the reference entity experiences a credit credit spread alone. As a result, they are event between trade date and expiry date as essentially a ‘pure’ credit product, with pricing they would never exercise the option in this being mostly driven by CDS spread volatility. case. As a result, there is no need for a Default swaptions give investors the oppor- knockout feature for receiver default swap- tunity to express views on the future level and tions. Consider the following example. variability of default swap spreads for a given Lehman Brothers pays 1.20% for an at- issuer. They can be traded outright or embed- the-money receiver default swaption on ded in callable CDS. The typical maturity of five-year GMAC, struck at the current five the underlying CDS is five years but can range year spread of 265bp and with three from one–10 years, and the time to option months to expiry. The investor is short the expiry is typically three months to one year. option. From the investor’s perspective, the relevant scenarios are: Payer default swaption The option buyer pays a premium to the ■ If five-year GMAC trades above 265bp in option seller for the right but not the obliga- three months, Lehman does not exercise, tion to buy CDS protection on a reference as they can sell protection for a higher entity at a predetermined spread on a future spread in the market. The investor has date. Payer default swaptions can be struc- realised an option premium of 1.20% in a tured with or without a provision for knock quarter of a year. out at no cost if there is a credit event between trade date and expiry date. If the ■ If five-year GMAC trades at 238bp in three knock out provision is included in the swap- months, the trade breaks even. (1.20% up tion, the option buyer who wishes to main- front option premium equals the payoff of tain protection over the entire maturity range (265bp–238bp)=27bp times the five-year can separately buy protection on the under- PV01 of 4.39). If five-year GMAC trades lying name until expiry of the swaption. below 238bp in three months, the loss on The relevant scenarios for this investment the exercise of the option will be greater are complementary to the ones in the case than the upfront premium and the investor of the protection put. If spreads tighten by will underperform on this trade. the expiry date, the option buyer will not exercise the right to buy protection at the Hedging default swaptions strike and the option seller will keep the Dealers hedge these default swaptions using option premium. a model of the type discussed on page 49. The underlying in a default swaption is the for- Receiver default swaption ward CDS spread from the option expiry date In a receiver default swaption, the option to the maturity date of the CDS. Theoretically, buyer pays a premium to the option seller a knock-out payer swaption should be delta

26 The Lehman Brothers Guide to Exotic Credit Derivatives

If the four-year GMAC spread in one year Figure 22. Default swaption types is less than the strike spread of 315bp, then Lehman Brothers will exercise the option and so cancel the protection, enabling us to Product Payer default Receiver default buy protection at the lower market spread. swaption swaption The investor therefore has earned 315bp for Description Option to buy Option to sell selling five-year protection on GMAC for protection protection one year. Exercised if CDS spread at CDS spread at If the four-year GMAC spread in one year expiry > strike expiry < strike is greater than 315bp, the contract contin- Credit view Short credit Long credit ues and the investor continues to earn forward forward 315bp annually. Knockout May trade with Not relevant From the perspective of the option seller, or without the callable default swap has a limited MTM upside compared with plain vanilla CDS. The additional spread of 315bp–260bp=55bp in hedged with a short protection CDS to the this example compensates the option seller final maturity of the underlying CDS and a long for the lost potential upside. protection CDS to the default swaption expiry Selling protection in callable default swap date. This combination will produce a synthet- is equivalent to a covered call strategy on ic forward CDS that knocks out at default underlying issuer spreads and is particularly before the forward date. In practice, swap- suitable as a yield-enhancement technique tions with expiry of 1 year and less are hedged for asset managers and insurers. only with CDS to final maturity due to a lack of liquidity in CDS with short maturities. We have Credit portfolio options summarised the key features of these differ- Starting in mid-2003 market participants ent swaption types in Figure 22. have been able to trade in portfolio options whose underlying asset is the TRAC-X North Callable default swaps America portfolio with 100 credits. Liquidity In addition to default swaptions, there is a is also growing in the European version. growing interest in callable default swaps. The rationale for options based on TRAC-X These are a combination of plain vanilla CDS is that the portfolio effect will reduce the with an embedded short receiver swaption option volatility and make it easier for deal- position. The seller of a callable default ers to hedge. From an investor perspective it swap is long credit exposure but this expo- presents a way to take a macro view on sure can be terminated by the option buyer spread volatility. at some strike spread on a future date. We are now seeing investors trading both Consider an example. at-the-money and out-of-money puts and Lehman Brothers buys five-year GMAC calls to maturities extending from three to protection, callable in one year, for 315bp nine months. The contracts are typically from an investor. The assumed current mid- traded with physical delivery. If the TRAC-X market spread for five-year GMAC protec- portfolio spread is wider than the strike tion is 265bp. level on the expiry date, the holder of the

The Lehman Brothers Guide to Exotic Credit Derivatives 27

payer default swaption will exercise the Clean and perfect asset swaps option and lock in the portfolio protection One important theme is the isolation of the at more favourable levels. Conversely, if the pure credit risk component in a given TRAC-X spread is tighter than the strike, the instrument. For example, a European CDO holder of the receiver swaption will benefit investor may wish to access USD collateral from exercising the option and realising the without incurring any of the associated cur- MTM gain. rency risks. Investors can monetise a view on the future Cross-currency asset swaps are the tradi- range of market spreads by trading bearish tional mechanism by which credit investors spread (buying at-the-money receiver swap- transform foreign currency fixed-rate bonds tion and selling farther out-of-money receiver into local currency Libor floaters. This has swaption) or bullish spread (buying ATM payer the benefit that it substantially reduces the swaption and selling farther out-of-money currency and interest rate risk, converting payer swaption) strategies. Other strategies the bond from an FX, interest rate and cred- include expressing views on spread changes it play into an almost pure credit play. over a given time horizon by trading calendar However, the currency risk has not been spreads (buying near maturity options and completely removed. First, note that a cross selling farther maturity options). currency asset swap is really two trades: (i) Finally, because the TRAC-X spread is less purchase of a foreign currency asset; and (ii) subject to idiosyncratic spread spikes, and entry into a cross-currency swap. In the case because of the existing two-way markets of a European investor purchasing a dollar with varying strikes, investors can express asset, the investor receives Euribor plus a their views on the direction of changes in spread paid in euros. the macro level of spread volatility by trading As long as the underlying dollar asset straddles, ie, simultaneously buying payer does not default during the life of the asset and receiver default swaptions as a way to swap there is no currency risk to the go long volatility while being neutral to the investor. However, if the asset does direction of spread changes. default, the investor loses the future dollar coupons and principal of the asset, just Hybrid products receiving some recovery amount which is Hybrid credit derivatives are those which paid in dollars on the dollar face value. As combine credit risk with other market risks the cross-currency swap is not contingent, such as interest rate or currency risk. meaning that the payments on the swap Typically, these are credit event contingent contract are unaffected by any default of instruments linked to the value of a deriva- the asset, the investor is therefore obliged tives payout, such as an interest rate swap to either continue the swap or to unwind it or an FX option. at the market value with a swap counter- There are various motivations for entering party. This unwind value can be positive or into trades which have these hybrid risks. negative – the investor can make a gain or Below, we give an overview of the economic loss – depending on the direction of move- rationale for different types of structures. We ments in FX and interest rates since the discuss the modelling of hybrid credit deriva- trade was initiated. tives in more detail on page 51. The risk is significant. We have modelled

28 The Lehman Brothers Guide to Exotic Credit Derivatives

the distribution of the swap MTM at Figure 23. Modelled distribution of the default, shown in Figure 23, for a five year swap MTM at default as a percentage of euro-dollar cross-currency swap using mar- face value for a five-year euro-dollar swap ket calibrated parameters. The downside risk is significant with possible swap relat- 6 ed losses comparable in size to the loss on 5 the defaulted credit. 4 As a result, the basic cross-currency asset swap has a default contingent inter- 3 est rate and currency risk. This can be 2 Probability (%) removed through the use of a hybrid. Two 1 variations exist. 0 –58 –28 2 32 63 93 1. In the clean asset swap, Lehman Brothers Swap mark-to-market value at default (%) takes on the default contingent swap unwind value risk. 2. In the perfect asset swap, Lehman correlations between rates, FX and credit. Brothers takes on the default contingent The cost can be amortised over the life of swap unwind value risk and guarantees the trade as a reduction in the asset swap (quantoes) the recovery rate of the default- spread paid. It is interesting to note that the ed asset in the investor’s base currency. reduction in the perfect asset swap spread may not be significant given that the two FX Both structures have featured widely in the risks to the swap MTM and the recovery rate CDO market where they have been used to are actually partially offsetting. immunise mixed-currency high-yield bond portfolios against currency risk in order to Counterparty risk hybrids allow the structure to qualify for the desired The mitigation of counterparty risk gives rating from the rating agency. rise to another type of hybrid. Consider an However, they can also be used by investor who enters into an interest rate investors to convert foreign currency assets swap with a credit risky counterparty. into their base currency. For example, Suppose also that this counterparty European investors can use the perfect asset defaults with a recovery rate of R. If the swap to take advantage of the often higher MTM of the swap is negative from the spread levels which exist for the same cred- viewpoint of the investor, the swap is its when denominated in US dollars. unwound at market value. This means that The cost of removing this default contin- in an MTM framework the investor incurs gent swap MTM risk and paying the recov- no loss from the default event; the swap ery rate in the investors domestic currency could be replaced by one with more advan- depends upon a number of factors includ- tageous terms at zero cost. However, if the ing the volatility of the FX rate, the credit value is positive, the investor loses a frac- quality of the reference credit, the shape of tion of this MTM. the Libor curves in both currencies, interest A way to mitigate the counterparty risk is rate volatilities in both currencies and the therefore to buy protection on the counter-

The Lehman Brothers Guide to Exotic Credit Derivatives 29

party, where the contingent payout is linked Hedge cost mitigation to the replacement cost of the swap. As we The final class of application concerns the discuss on page 51, the investor is effec- pure hedging of credit contingent FX or tively purchasing an interest rate swaption interest rate risks, such as those faced by which is automatically exercised upon a an international corporation in the course of default event. Clearly, similar structures can its business activities. For example, one be constructed to provide credit protection way to reduce the cost of credit hedging for other payouts. could be to purchase default protection linked to an FX rate being above or below a Yield enhancement specified threshold. In the current interest rate environment with Companies looking to hedge interest rate very low short term rates and steep yield or FX risk may be concerned with the cost of curves, coupled with the more mature credit outright hedging using vanilla derivatives. In derivatives market, there is increasing this case, a hedge which knocks out on the investor interest in credit-linked notes with default of a reference credit can provide an more exotic coupons. These include paying a adequate hedge while significantly decreas- spread over a Constant Maturity Swap (CMS) ing costs. Clearly, the hedger is implicitly rate or a particular inflation index. taking a bullish view on the reference credit.

30 The Lehman Brothers Guide to Exotic Credit Derivatives

Credit Derivatives Modelling

To be able to price and risk-manage credit ments of different maturities. They can also derivatives, we need a framework for valu- be extended to price more exotic credit ing credit risk at a single issuer level, and at derivatives. It is for these reasons that they a multi-issuer level. The growth of the credit are used for credit derivatives pricing. derivatives market has created a need for more powerful models and for a better The hazard rate approach understanding of the empirical evidence The most widely used reduced-form needed to calibrate these models. In this approach is based on the work of Jarrow section we will present a detailed overview and Turnbull (1995), who characterise a cred- of modelling approaches from a practical it event as the first event of a Poisson count- perspective, ie, we will discuss models, ing process which occurs at some time t implementation and calibration. with a probability defined as

Single credit modelling Pr[τ=t dt >+≤ λτ tt )(]| dt The world of credit modelling is divided into two main approaches, called structural and ie, the probability of a default occurring with- reduced-form. In the structural approach, the in the time interval [t,t+dt) conditional on default is characterised as the consequence surviving to time t, is proportional to some of some event such as a company’s asset time dependent function λ(t), known as the value being insufficient to cover a repayment hazard rate, and the length of the time inter- of debt. Such models are usually extensions val dt. Over a finite time period T, it is possi- of Merton’s 1974 model that used a contin- ble to show that the probability of surviving gent claims analysis for modelling default. is given by Structural models are generally used to say   T  at what spread corporate bonds should Q   ),0( ETQ exp−= ∫ λ )( dss  trade based on the internal structure of the   0  company. They therefore require information about the balance sheet of the company and The expectation is taken under the risk-neu- can be used to establish a link between pric- tral measure. A common assumption is that ing in the equity and debt markets. However, the hazard rate process is deterministic. By they are limited in a number of ways includ- extension, this assumption also implies that ing the fact that they generally lack the flex- the hazard rate is independent of interest ibility to fit exactly a given term structure of rates and recovery rates. spreads; and they cannot be easily extended to price complex credit derivatives. Pricing model for CDS In the reduced-form approach, the credit The breakeven spread in a CDS is the spread process is modelled directly via the proba- at which the present values (PV) of premium bility of the credit event itself. Reduced-form and protection legs are equal, ie models also generally have the flexibility to refit the prices of a variety of credit instru- Premium PV = Protection PV.

The Lehman Brothers Guide to Exotic Credit Derivatives 31

To determine the spread we therefore need for the risk of default per small time interval. to be able to value the protection and pre- Within this model the interest rate depen- mium legs. It is important to take into dency drops out. account the timing of the credit event Given a CDS which has a flat spread curve because this can have a significant effect on at 150bp, and assuming a 50% recovery the present value of the protection leg and rate, the implied hazard rate is 0.015 divided also the amount of premium paid on the pre- by 0.5, which implies a 3% hazard rate. The mium leg. Within the hazard rate approach implied one-year survival probability is there- we can solve this timing problem by condi- fore exp(–0.03)=97.04%. For two years it is tioning on the probability of defaulting with- exp(–0.06)=94.18%, and so on. in each small time interval [s,s+ds], given by Q(0,s)λ(s)ds, then paying (1–R) and discount- Valuation of a CDS position ing this back to today at the risk-free rate. The value of a CDS position at time t follow- Assuming that the hazard rate and risk free ing initiation at time 0 is the difference rate term structures are flat, we can write between the market implied value of the the value for the protection leg as protection and the cost of the premium payments, which have been set contractually at

T +− λ )( Tr SC. We therefore write +− λ λ 1( )( −− eR )1 − )1( ∫ λ )( sr dseR = r + λ 0 MTM(t) = ± (Protection PV –Premium PV),

The value of the premium leg is the PV of the where the sign is positive for a long protec- spread payments which are made to default tion position and negative for a short protec- or maturity. If we assume that the spread S tion position. If the current market spread is on the premium leg is paid continuously, we given by S(t) then the MTM can be written as can write the present value of the premium leg as MTM −= StSt 0()(()( )) × RPV01

T +− λ)( Tr +− λ − eS )1( ∫ )( sr dseS = where the RPV01 is the risky PV01 which is r + λ 0 given by

Equating the protection and premium legs − −+()rTtλ ()− and solving for the breakeven spread gives ()1 e RPV01() t = , ()r + λ =λ − RS )1( and where St() This relationship is known as the credit tri- λ = . − angle because it is a relationship between 1 R three variables where knowledge of any two is sufficient to calculate the third. It basical- An investor buys $10m of five-year protec- ly states that the spread paid per small time tion at 100bp. One year later, the credit interval exactly compensates the investor trades at 250bp. Assuming a recovery rate

32 The Lehman Brothers Guide to Exotic Credit Derivatives

of 40%, the value is given by substituting, expected recovery rate following a credit r=3.0%, R=40%, S(t)=0.025, S(0)=0.01 and event where the expectation is under the t=4 into the above equation to give risk-neutral measure. λ=4.17% and an MTM value of $521,661. Such expectations are only available from This is a simple yet fairly accurate model price information, and the problem in credit which works quite well when the interest is that given one price, it is difficult to separate and credit curves are flat. When this is rate the probability of default from the recov- not the case, it becomes necessary to use ery rate expectation. bootstrapping techniques to build a full term The market standard is therefore to revert to structure of hazard rates. This may be rating agency default studies for estimates of assumed to be piecewise flat or piecewise recovery rates. These typically show the aver- linear. For a description of such a model see age recovery rate by seniority and type of O’Kane and Turnbull (2003). credit instrument, and usually focus on a US corporate bond universe. Adjustments may Default probabilities be made for non-US corporate credits and for The default probabilities calculated for certain industrial sectors. pricing purposes can be quite different Problems with rating agency recovery from those calculated from historical statistics include the fact that they are back- default rates of assets with the same rat- ward looking and that they only include the ing. These real-world default probabilities default and bankruptcy credit events – are generally much lower. The reason for restructuring is not included. In their favour, this is that the credit spread of an asset one should say that, as they represent the contains not just a compensation for pure price of the defaulted asset as a fraction of default risk; it also depends on the mar- par some 30 days after the default event, ket’s risk aversion expressed through a risk they are similar to the definition of the recov- premium, as well as on supply-and- ery value in a CDS. demand imbalances. Recent work (Altman et al 2001) shows that One should also comment on the market’s there is a significant negative correlation use of Libor as a risk-free rate in pricing. between default and recovery rates. One Pricing theory shows that the price of a way to incorporate this effect is to assume derivative is the cost of replicating it in a risk- that recovery rates are stochastic. The stan- free portfolio using other securities. Since dard approach is to use a beta distribution. most market dealers are banks which fund close to Libor, the cost of funding these Modelling default correlation other securities is also close to Libor. As a By modelling correlation products, we mean consequence it is the effective risk-free rate modelling products whose pricing depends for the derivatives market. upon the joint behaviour of a set of credit assets. These include default baskets and Calibrating recovery rates synthetic CDOs. As a result of the growth in The calibration of recovery rates presents a usage of these products, this is an area of number of complications for credit deriva- pricing which has recently gained a lot of tives. Strictly speaking, the recovery rate attention, and in which we have seen a num- used in the pricing of credit derivatives is the ber of significant modelling developments.

The Lehman Brothers Guide to Exotic Credit Derivatives 33

dynamics of asset values: consistent with Figure 24. Scatterplot of 1,000 simulated asset returns with 0% and 90% corre- their descriptive approach of the default lation. Default thresholds are also shown mechanism, multivariate structural models rely on the dependence of asset returns in 10% Correlation 90% Correlation order to generate dependent default events.

VB This is shown in Figure 24 where we have 5 VB CA C 5 4 A 4 simulated 1,000 pairs of asset returns mod- 3 3 2 elled as normally distributed random vari- 2 1 i j VA VA ables, for two firms and for two different

4 2 2 4 4 2 24 asset return correlations of 10% and 90%. CB C 2 B The vertical and horizontal lines represent 3 3

4 4 Both firms default default thresholds for firms i and j respec- tively. Clearly we see that the probability of both i and j defaulting, represented by the number of points in the bottom left quad- In this section we will describe some of rant defined by the default thresholds, these current models, show how they can increases as the asset return correlation be applied to the valuation of baskets and increases. Therefore asset correlation CDOs, and towards the end discuss model leads to default correlation. calibration issues. Although the pay-offs of multi-credit default-contingent instruments such as nth- Modelling joint defaults to-default baskets and synthetic loss tranch- The valuation of default-contingent instru- es cannot be statically replicated by trading ments calls for the modelling of default in a set of single-credit contracts, the cur- mechanisms. As discussed earlier, a classical rent market practice is to value correlation dichotomy in credit models distinguishes products using standard no-arbitrage argu- between a ‘structural approach’, where ments. It follows that the valuation of these default is triggered by the market value of the multi-credit exposures boils down to the borrower’s assets (in terms of debt plus equi- computation of (risk-neutral) expectations ty) falling below its liabilities, and a ‘reduced- over all possible default scenarios. form approach’, where the default event is A number of different hybrid frameworks directly modelled as an unexpected arrival. have been proposed in the literature for mod- Although both the structural and the reduced- elling correlated defaults and pricing multi- form approaches can in principle be extended name credit derivatives. Hull and White (2001) to the multivariate case, structural models generate dependent default times by diffus- calibrated to market-implied default probabili- ing correlated asset values and calibrating ties (often called ‘hybrid’ models) have gained default thresholds to replicate a set of given favour among practitioners because of their marginal default probabilities. Multi-period tractability in high dimensions. extensions of the one-period CreditMetrics If we think of defaults as generated by paradigm are also commonly used, even if asset values falling below a given boundary, they produce the undesirable serial indepen- then the probabilities of joint defaults over a dence of the realised default rate.1 While most specified horizon must follow from the joint multi-credit models require simulation, the

34 The Lehman Brothers Guide to Exotic Credit Derivatives

need for accurate and fast computation of such as the simulation approach and the greeks has pushed researchers to look for semi-analytical framework described below, modelling alternatives. Finger (1999) and, use the dependence among asset returns to more recently, Gregory and Laurent (2003) generate joint defaults, therefore avoiding show how to exploit a low-dimensional factor the need for a direct estimation of joint structure and conditional independence to default probabilities. obtain semi-analytical solutions. Default-time simulation Asset and default event correlation Monte Carlo models generally aim at the Default event correlation (DEC) measures generation of default paths, where each the tendency of two credits to default joint- path is simply a list of default times for each ly within a specified horizon. Formally, it is of the credits in the reference portfolio defined as the correlation between two drawn at random from the joint default dis- binary random variables that indicate tribution. Knowing the time and identity of defaults, ie each default event allows for a precise valuation of any multi-credit product, no matter − how complex the contractual specification = AB ppp BA DEC of the payoff. (1 − )(1 − pppp ) A A B B In an influential paper, Li (2000) presents a simple and computationally inexpensive where pA and pB are the marginal default algorithm for simulating correlated defaults. probabilities for credits A and B, and PAB is His methodology builds on the implicit the joint default probability. Of course, pA, pB assumption that the multivariate distribution and pAB all refer to a specific horizon. Notice of default times and the multivariate distri- that default event correlation increases lin- bution of asset returns share the same early with the joint probability of default and dependence structure, which he assumes to is equal to zero if and only if the two default be Gaussian and is therefore fully charac- events are independent. Its limits are not terised by a correlation matrix. –100% to +100% but are actually a function For valuation purposes, we need to sample of the marginal probabilities themselves. from the multivariate distribution of default Default event correlations are the funda- times under the risk-neutral probability mea- mental drivers in the valuation of multi-name sure. In this case, it is common practice to credit derivatives. Unfortunately, the scarcity back out the marginal distributions of default of default data makes joint default probabili- times, which we will denote with F1,F2,…,Fd, ties, and thus default event correlations, from single-credit defaultable instruments very hard to estimate directly. As a result, (such as CDS). We then join these marginal market participants rely on alternative meth- distributions with a correlation matrix, which ods to calibrate the frequency of joint according to the stated assumptions repre- defaults within their models. Hybrid models, sents the correlation matrix of the asset returns of the reference credits. Since asset 1 Finger (2000) offers an excellent comparison of several returns are not directly observable, it is com- multivariate models in terms of the default distributions that they generate over time when calibrated to the same mon practice to proxy asset correlations marginals and first-period joint default probabilities. using equity correlations. Towards the end of

The Lehman Brothers Guide to Exotic Credit Derivatives 35

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

Cumulative normal Cumulative default probability 1.0 1.0

0.8 0.8

0.6 0.6

0.4 0.4

0.2 0.2

0 0 6 4 2 0 2 46 0 2 4 6 8 10 x Time (years) this section we will discuss whether this 1. Choleski decomposition of the correla- seems to be a reasonable approximation. tion matrix to simulate a multivariate The high-dimensionality of multi-credit Normal random vector ∈ RX d with cor- instruments means that it is not possible to relation Σ. use market prices to obtain the full depen- 2. Transform the vector into the unit hyper- dence structure. Instead, practitioners gen- cube using erally estimate the necessary correlations using historical returns, implicitly relying on = ()() () ( ) xNU 1 , N 2 ,..., xNx d the extra assumption that the correlations among asset returns remain unchanged when we move from the objective to the 3. Translate U into the corresponding pricing probability measure. default times vector t using the inverse of In the bivariate case, the joint distribution the marginal distributions: function of default times and is simply given by =τ()()()τττ = ()−1 −1 −1 () , 21 ,..., d 1 , 21 2 ,..., uFuFuF dd

P( ττ y)x, =<< N −1(F(N (x)),N −1 (y))),(F 1 2 ,2 ρ 1 2 The simulation algorithm is illustrated in Figure 25. It is easy to verify that τ has the where N2,ρ denotes the bivariate cumulative given marginals and a Normal dependence standard Normal with correlation ρ, and N–1 structure fully characterised by the correla- indicates the inverse of a cumulative stan- tion matrix Σ. dard Normal. The extension to the d-dimen- Once we know how to sample from the sional case is immediate. risk-neutral distribution of default times, it is Simulating default times from this distribu- straightforward to price a correlation trade. tion is straightforward. With d reference Generally speaking, the valuation of a given credits and correlation matrix Σ we have the correlation product always boils down to the following algorithm: computation of an expectation of the form,

36 The Lehman Brothers Guide to Exotic Credit Derivatives

τ τ τ τ τ E[f ( )] where = ( 1, 2,..., d) represents the arises whether some other numerical vector of default times and f is a function approach can be used. describing the discounted cash flows (both positive and negative) associated with the A semi-analytical approach instrument under consideration (see, eg, The recent development of correlation trad- Mashal and Naldi (2002b)). ing and the associated need for fast compu- In summary, Monte Carlo simulation tation of sensitivities have generated a great allows for accurate valuations and risk mea- deal of interest in semi-analytical models. To surements of multi-credit payoffs, even enjoy the advantages of fast pricing, one when complex path-dependencies such as needs to impose more structure on the reserve accounts, interest coverage or col- problem. One way to do this is to rely on lateralisation tests are involved. This is two basic simplifications: because both the exact default times and the identity of the defaulters are known on 1. assume a one-factor correlation structure every simulated path. Moreover, we can for asset returns, and easily expand the set of variables to be 2. discretise the time-line to allow for a finite treated as random. For example, Frye (Risk, set of dates at which defaults can hap- 2003) argues that modelling stochastic pen. These are chosen with a resolution recoveries and allowing for negative to provide a sufficient level of accuracy. correlation between recovery and default rates is essential for a proper valuation of In particular, 1) makes it possible to com- credit derivatives. pute the risk-neutral loss distribution of the The precision and flexibility of this reference portfolio for any given horizon, approach, however, come at the cost of while 2) is needed to price an instrument computational speed. The basic problem of knowing only the loss distribution of the using simulation is that defaults are rare reference portfolio at a finite set of dates. events, and a large number of simulation While these two assumptions are suffi- paths are usually required to achieve a suf- cient to price plain vanilla portfolio swaps, ficient sampling of the probability space. derivatives structures with more complex There are ways to improve the situation. path-dependencies may also require that Useful techniques include antithetic sam- we approximate the payoff function f(τ). pling, importance sampling and the use of Even in this case, it is usually possible to low-discrepancy sequences. This problem obtain reasonable approximations, so that becomes particularly significant when we the benefit of precise sensitivities can be turn our attention to the calculation of sen- retained at a relatively low cost. Moreover, sitivity measures, since for a reasonable the error can be controlled by comparing number of paths the simulation noise can the analytical solution to a Monte Carlo be similar to or greater than the price implementation. change due to the perturbation of the input We now discuss how to exploit a one-factor parameter. Once again, techniques exist to model to construct the risk neutral loss distri- alleviate the problem, but it is hard to bution. Later on, we will show how to use a achieve precise hedge ratios in a reason- sequence of loss distributions at different able amount of time. The question then horizons to price synthetic loss tranches.

The Lehman Brothers Guide to Exotic Credit Derivatives 37

A one-factor model becomes computationally intensive, making Let us start by assuming that the asset it necessary to resort to Monte Carlo simu- return of the ith issuer between today and a lation. However, substantial simplifications given horizon is described by a standard can be achieved by imposing more structure

Normal random variable Ai with mean 0 and on the model. standard deviation of 1, and is of the form The advantage of this one-factor setup is

that, conditional on ZM, the asset returns are independent. This makes it easy to compute β−ZA += 1 β 2 Z Mii ii conditional default probabilities. Conditional

on the market factor ZM, an asset defaults if where Z1,Z2,...,ZD are N(0,1) distributed independent random variables. The variable − β ZC Mii ZM describes the asset returns due to a Z ≤ i − β 2 common market factor, while Zi models the 1 i β idiosyncratic risk of the ith issuer, and i stands for the correlation of the asset return The conditional default probability pi(ZM)of of issuer i with the market. The asset return an individual issuer i is therefore given by correlation between asset i and asset j is β β given by i j.    − β ZC  Default will occur if the realised asset )( = NZp Mii Mi  − β 2  return falls below a given threshold.  1 i  Mathematically, the ith issuer defaults in the event that Ai < Ci. Given that Ai is N(0,1) distributed, it is possible to write If we assume that the loss on default for each issuer is the same unit (consistent with () =⇒= −1 pNCCNp )( i i i i all assets having the same seniority), u, then building up the portfolio loss distribution can and calibration of the marginal probabilities be done iteratively by adding assets to the is no more complicated than inverting the portfolio. The process is as follows: cumulative normal function. In the single-issuer case, this is merely an 1. Beginning with asset one, there are two exercise in calibration, as the default thresh- outcomes on the loss distribution: a loss olds are chosen to reproduce default proba- of zero with a probability 1–p1(ZM) and a bilities which re-price market instruments. loss of u with a probability p1(ZM). The concept of asset returns becomes mean- 2. Adding a second asset, we adjust each of ingful when studying the joint behaviour of the previous losses. The zero loss peak more than one credit. In this case, the returns requires that the new asset survives and are assumed to be described by a multivari- so has a probability (1–p1(ZM))(1–p2(ZM)). ate Normal distribution. Using the threshold A loss u corresponds with the previous levels determined before, it is possible to zero loss probability times the probability obtain the probabilities of joint defaults. that asset two defaults plus the previous With a general correlation structure, the u loss probability times the probability calculation of joint default probabilities asset two does not default. We arrive at a

38 The Lehman Brothers Guide to Exotic Credit Derivatives

probability (p1(ZM)+p2(ZM)–2p1(ZM)p2(ZM)). within this framework has been described in A loss of 2u can only occur if the previous detail above. Given that we know when each loss of u is multiplied by the probability asset in the portfolio defaults in each simu- that asset two also defaults to give a lation path, to calculate the fair-value spread

probability of p1(ZM)p2(ZM). we proceed as follows: 3. We continue adding on to the portfolio until all of the assets have been includ- 1. Sort default times in ascending order and τ ed. We then repeat with a different value denote the nth time to default as n(i), of ZM and integrate the loss distribu- where i is the label of the defaulted asset. tions over the N(0,1) distributed market 2. Calculate the PV of a 1bp coupon stream Z τ τ i T T factor M. paid to time *=min[ n( ), ] where is the basket maturity. τ i T The advantage of this approach is that there 3. If n( ) < then calculate the PV = B τ i R i B is no simulation noise. Other algorithms ( n( ))(1– ( )), where is the Libor dis- exist, including Fast Fourier techniques count factor and R(i) is the recovery rate for which may be more efficient. asset i. Otherwise the protection PV = 0. Both the times-to-default model and the 4. Average both the premium leg PV of 1bp, one-factor model described above are known as the basket PV01, and the pro- Gaussian copula models. This means that as tection leg PV over all paths. long as both are calibrated to the same 5. Divide the protection leg by the basket marginals, and as long as both use the same PV01 to get the fair-value spread. (one-factor) correlation matrix, both models should generate exactly the same prices. This approach is simple to implement and This is an important point – it means that the size of default baskets, typically m=5–10 models can be categorised by their copula assets means that pricing is quite fast in specification alone. It also means that differ- Monte Carlo. Monte Carlo is also flexible ent products may be priced consistently enough to enable you to introduce stochas- using different default mechanisms, as long tic recovery rates, perhaps drawn from a as they use the same copula. beta distribution. It is also quite straightforward to introduce alternative copulas, see Valuation of correlation Mashal and Naldi (2002a). products An analytical approach is also possible here. We now explain in detail how these models The main constraint is to build the basket n- may be applied to the pricing of correlation to-default probabilities while retaining the products. We start with the default basket. identity of the defaulted assets. See Gregory and Laurent (2003) for a full discussion. Pricing default baskets A default basket is inherently an idiosyncrat- Rating models for default baskets ic product in the sense that the identity of Rating agencies have developed a number the defaulted asset must be known. One of models for rating default baskets. For approach is therefore to use the times-to- example Moody’s has adopted a Monte default model via a Monte Carlo implemen- Carlo-based extension of the asset value tation. The generation of default times approach, in which an asset value is simu-

The Lehman Brothers Guide to Exotic Credit Derivatives 39

lated to multiple periods for each of the can be computationally slow if there are a assets in the basket. These asset values large number of assets in the portfolio. may be correlated internally in order to Another way is to use a semi-analytical induce a default correlation. If the asset approach which relies on the fact that a value falls below the default threshold at standard synthetic loss tranche can be the future period, the asset defaults and a priced directly from its loss distribution. recovery rate is drawn from a beta distribu- To see this, consider a portfolio CDS on a tion. The recovery amount and asset value tranche defined by the attachment point Kd can be correlated. The model is calibrated and the upper boundary Ku, expressed as using historical default statistics and the percentages of the reference portfolio assigned rating is linked to the expected notional. This is Nport so that the tranche loss of the basket to the trade maturity. notional is given by S&P has recently switched its approach from a weakest-link approach which assigns = − tranche port KKNN du )( an FTD the rating of the lowest-rated entity in the basket, to one based on a Monte Carlo model, which uses the same framework as The maturity of the portfolio swap is T, and its CDO Evaluator model. for each time t≤T we denote by L(t) the cumulative portfolio loss up to time t. The Pricing synthetic loss tranches tranche loss is therefore given by We would now like to consider in detail how we would set about valuing a loss )( = max[ KtLtL ]0,)( −− max[ − KtL ]0,)( tranche. One approach is to use the times tranche d u to default model via a Monte Carlo implementation as discussed earlier. Given that Note that this is similar to an option style we know when each asset in the portfolio payoff. Indeed it is possible to think of CDO defaults in each simulation path, we would tranches as options on the portfolio loss proceed as follows: amount. The two legs of the swap can be priced in 1. Calculate the present value of all principal the same way as a CDS if we introduce a losses on the protection leg in each path. tranche ‘default probability’ P(t) defined as 2. Calculate the PV of 1bp on the premium leg taking care to reduce the notional of Q ([ tLE )] the tranche following defaults which tP )( = 0 tranche N cause principal losses in that path. tranche 3. Average both the 1bp premium leg PV, known as the tranche PV01, and the pro- where we use the risk-neutral (pricing) mea- tection leg PV over all paths. sure for taking the expectation. 4. Divide the protection PV by the tranche Assuming that the credit, recovery rate and PV01 to compute the breakeven tranche interest rates processes are independent, spread. the contingent leg is therefore given by

This approach is simple to implement but Protection Leg PV =

40 The Lehman Brothers Guide to Exotic Credit Derivatives

thetic CDO if we have the loss distribution at T the set of future dates t and T. One way to tranche ∫ udPuBN )(),0( 0 do this is to use the one-factor model described above to calculate a loss distribu- Where B(0,u) is the Libor discount factor tion at each of the future dates required. from today, time 0 to time u. We can dis- Care must be taken to ensure that the cretise the integral by introducing the grid default threshold is re-calibrated at each points 0=t0

K An asymptotic approximation − tranche ∑ ,0( )( ii ()( tPtPtBN i−1 )) It is possible to exploit the high dimension- i=1 ality of the CDO to derive a closed form model for analysing CDO tranches. In fact To value the premium leg, we denote the we can use this approach to obtain results contractual spread on the tranche by s, and which differ from the exact approach by the coupon payment dates as 1–5% for a real CDO.

0=T0

The Lehman Brothers Guide to Exotic Credit Derivatives 41

which is given by the simple binomial the tranche notional this is given by

k    − β  max()(KL 0, max −−− KL 0, ) m  ZC M  ()= 1 2 [LP ku | Z]==  N , KKL 21  2  − KK  k   1− β  12 −    km − β ZC  −  M  If K<1, then 1 N   − β 2  () (−+≤⇔≤ )   1  , 21 1 KKKKLKKKL 12

This distribution becomes computationally This equation shows that we can easily derive intensive for large values of m, but we can the loss distribution of the tranche from that use methods of varying sophistication to of the reference portfolio. This is discontinu- approximate it. One very simple and sur- ous at the edges owing to the probability of prisingly accurate method is the so-called the portfolio loss falling outside the interval, ‘large homogeneous portfolio’ (LHP) and this discontinuity becomes more pro- approximation, originally due to Vasicek nounced when the tranche is narrowed. We (1987). Since the asset returns, conditional can further compute the expected loss of the on ZM are independent and identically dis- tranche analytically. For more details see tributed, by the law of large numbers, the O’Kane and Schloegl (2001). We then arrive at fraction of issuers defaulting will tend to the conditional probability of an asset []()= E , KKL 21 defaulting p(Z ). As a result, the condition- (− 1 ( ) −− β 2 )− (− 1 ( ) −− β 2 ) M 2 1 CKNN 1,, 2 2 CKNN 1,, − al loss is directly linked to the value of the KK 12 market factor ZM which itself is normally distributed. We can then write the probabil- This is only a one-period approach. ity of the portfolio loss being less than or However, it can easily be extended to multi- equal to some loss threshold K as ple periods as described earlier.

Correlation skew [][]−=≤ −1 KpNKLP )( It is now possible to observe tradable tranche spreads for different levels of where N(x) is the cumulative normal func- seniority. If we attempt to imply out the mar- tion. More involved calculations show that ket correlation using a simple Gaussian cop- the distribution of L actually converges to ula model fitted to observed market tranche this limit as m tends to infinity. spreads, we can observe a skew in the aver- We can use the portfolio distribution to age correlation as a function of the width derive the loss distributions of individual and attachment point of the tranche. This tranches. If L(K1,K2) is the percentage loss skew may be the consequence of a number of the mezzanine tranche with attachment of factors such as the assumption of inde- point K1 and upper loss threshold K2, then pendence of default and recovery rates. It this can be written as a function of the loss may also be due to our incorrect specifica- on the reference portfolio. As a fraction of tion of the dependence structure, as

42 The Lehman Brothers Guide to Exotic Credit Derivatives

explained later. Supply and demand imbal- Estimating the ances also play a role. dependency structure Several well-known multivariate models, Rating agency models for CDOs including the ones described earlier in this Different rating agencies have their own chapter, rely on the assumption that the models for rating CDO tranches. All of them dependence structure (or ‘copula’) of asset attempt to capture the risks of CDOs in returns is Normal. The widespread use of terms of asset quality, recovery rates, the Normal dependence structure, which is default correlation and structural features. fully characterised by a correlation matrix, Moody’s standard rating model for CDO is certainly related to its simplicity. It tranches is a multinomial extension of the remains to be seen, however, whether this Binomial Expansion Technique (BET) model. assumption is supported by empirical evi- To capture default correlation, the portfolio dence. A number of recent studies have is represented by a lower number of inde- shown that the joint behaviour of equity pendent assets, known as the diversity returns is better described by a ‘fat-tailed’ score. Roughly speaking, this is the number Student-t copula than by a Normal copula, of independent assets which have the same and that correlations are therefore not suf- width of loss distribution as the actual CDO ficient to appropriately characterise their reference portfolio of correlated assets. The dependence structure.2 The first goal of diversity score is determined using a lookup this section is to apply the same kind of table and is based on the incremental effect analysis to asset returns, and test the null of having groups of assets in the same hypothesis of Gaussian dependence industry classification. versus the alternative of ‘joint fat tails’. After calibrating to historical default data, Of course, we face a major obstacle when the model is able to generate an expected attempting to estimate the dependence loss for a tranche which takes into account structure of asset returns: asset values are the subordination. This expected loss is then not directly observable. In fact, the use of mapped to a rating category. unobservable underlying processes is one In S&P’s ratings methodology, a Monte Carlo of several criticisms that the structural simulation is used to derive a probability loss approach has received over the years. Given distribution for the underlying collateral pool the lack of observable asset returns, it has based on the total principal balance of the become customary to proxy the asset portfolio. Each corporate asset is assigned a dependence with equity dependence, and default probability based on S&P’s historical to estimate the parameters governing the default studies, dependent on its rating and joint behaviour of asset returns from equity maturity. Corporate sectors are assumed to return series.3 However, the use of equity have a correlation of 30% within a given returns to infer the joint behaviour of asset industry and 0% between industry sectors. From these inputs and the par amounts of 2 See, for example, Mashal and Naldi (2002a) and Mashal each asset, a default probability distribution is and Zeevi (2002). 3 created. Unlike Moody’s which rates on the Fitch Ratings (2003) have recently published a special report describing their methodology for constructing port- basis of expected loss, S&P rates on the basis folio loss distributions: it is based on a Gaussian copula of the probability of incurring a loss. parameterised by equity correlations.

The Lehman Brothers Guide to Exotic Credit Derivatives 43

returns is often criticised on the grounds of Figure 26. the different leverage of assets and equity. DJIA portfolio: asset and equity returns test statistics as The second goal of this section is to shed functions of null hypothesis for DoF some light on the magnitude of the error induced by using equity data as a proxy for DJIA Asset Returns Sens tivity Analysis asset returns. To provide a plausible answer to these 103 questions, we first need to ‘back out’ asset values from observable data. One way to estimate the market value of a company’s 102 assets is to implement a univariate struc- pvalue = 0.0001 tural model. Such a procedure is at the Test Statistic heart of KMV’s CreditEdge™, a popular 101 credit tool that first computes a measure of pvalue = 0.01 distance-to-default and then maps it into a default probability (EDF™) by means of a 100 101 102 103 104 105 106 4 historical analysis of default frequencies. In Null DoF what follows, we summarise recent work by Mashal, Naldi and Zeevi (2002), who use the asset value series generated by KMV’s model to study the dependence properties DJIA Equity Returns Sensitivity Analysis of asset returns. 3 10

Methodology A key observation in modelling and testing 10 2 dependencies is that any d-dimensional pvalue = 0.0001

multivariate distribution can be specified Test Statistic via a set of d marginal distributions that are 10 1 ‘knitted’ together using a copula function. pva ue = 0.01 Alternatively, a copula function can be viewed as ‘distilling’ the dependencies that 10 0 10 1 10 2 10 3 10 4 10 5 10 6 a multivariate distribution attempts to cap- Null DoF ture, by factoring out the effect of the marginals. Copulas have many important characteristics that make them a central concept in the study of joint dependencies, in our study is given by the dependence see, eg, the recent survey paper by structure underlying the multivariate Embrechts et al. (2003). Student-t distribution. The Gaussian distri- A particular copula that plays a crucial role bution lies at the heart of most financial models and builds on the concept of corre- t 4 Copyright © 2000-2002 KMV LLC. All rights reserved. KMV lation; the Student- retains the notion of and the KMV logo are registered trademarks of KMV LLC. correlation but adds an extra parameter CreditEdge and EDF are trademarks of KMV LLC. into the mix, namely, the degrees-of-free-

44 The Lehman Brothers Guide to Exotic Credit Derivatives

dom (DoF). The latter plays a crucial role in database. The reader should keep in mind, modelling and explaining extreme co- however, that equity values are observable, movements in the underlyings. while asset values have been ‘backed out’ Moreover, it is well known that the by means of KMV’s implementation of a uni- Student-t distribution is very ‘close’ to the variate Merton model. We apply our analysis Gaussian when the DoF is sufficiently large to a portfolio of 30 credits included in the (say, greater than 30); thus, the Gaussian Dow Jones Industrial Average and use daily model is nested within the t-family. The data covering the period from 31/12/00 to same statement holds for the underlying 8/11/02. The reader is referred to Mashal, dependence structures, and the DoF param- Naldi and Zeevi (2003) for more examples eter effectively serves to distinguish the two using high yield credits and different sam- models. This suggests how empirical stud- pling frequencies. ies might test whether the ubiquitous Following the methodology mentioned Gaussian hypothesis is valid or not. In par- above, we estimate the number of degrees- ticular, these studies would target the of-freedom (DoF) of a t-copula without dependence structure rather than the distri- imposing any structure on the marginal dis- butions themselves, thus eliminating the tributions of returns. Then, using a likelihood effect of marginal returns that would ‘con- ratio test statistic, we perform a sensitivity taminate’ the estimation problem in the lat- analysis for various null hypotheses of the ter case. To summarise, the t-dependence underlying tail dependence, as captured by structure constitutes an important and quite the DoF parameter. The two horizontal lines plausible generalisation of the Gaussian in Figure 26 represent significance levels of modelling paradigm, which is our main moti- 99% and 99.99%; a value of the test statis- vation for focusing on it. tic falling below these lines corresponds to a With this in mind, the key question that we value of DoF that is not rejected at the now face is how to estimate the parameters respective significance levels. of the dependence structure. Mashal, Naldi The minimal value of the test statistic is and Zeevi (2002) describe a methodology achieved at 12 DoF for asset returns and at which can be used to estimate the parame- 13 DoF for equity returns. In both cases, we ters of a t-copula without imposing any can reject any value of the DoF parameter parametric restriction on the marginal distri- outside the range [10,16] with 99% confi- butions of returns. They also construct a dence; in particular, the null assumption of likelihood ratio statistic to test the hypothe- a Gaussian copula (DoF=∞) can be rejected sis of Gaussian dependence, and compare with an infinitesimal probability of error. the dependence structures of asset and Also, the point estimates of the asset equity returns to evaluate the common prac- returns’ DoF lies within the non-rejected tice of proxying the former with the latter. In interval for the equity returns’ DoF, and vice the remainder of this section we summarise versa, indicating that the two are essential- their empirical findings. ly indistinguishable from a statistical significance viewpoint. Moreover, the difference Empirical results between the joint tail behaviour of a 12- and For the purpose of this study, asset and a 13-DoF t-copula is negligible in terms of equity values are both obtained from KMV’s any practical application.

The Lehman Brothers Guide to Exotic Credit Derivatives 45

EDL because this measure relates both to the Figure 27. Maximum likelihood agency rating (when computed under real- estimates of DoF for DJIA portfolios world probabilities) and to the fair compensation for the credit exposure (when computed Portfolio Asset returns Equity returns under risk-neutral probabilities). DoF DoF We consider a five-year deal with a refer- 30-credit DJIA 12 13 ence portfolio of 100 credits, each with $1m First 10 credits 8 9 notional. We assume i) uniform recovery Middle 10 credits 10 10 rates of 35%, for every credit in the reference portfolio; ii) 1% yearly hazard rate for each Last 10 credits 9 9 reference credit; iii) 20% asset correlation between every pair of credits; iv) a risk-free curve flat at 2%. Using a default-time simula- Figure 27 reports the point estimates of tion, Figure 28 compares the expected dis- the DoF for asset and equity returns in the counted losses for several tranches under the DJIA basket, as well as for three subsets two alternative assumptions of Gaussian consisting of the first, middle, and last 10 dependence and t dependence with 12 DoF. credits (in alphabetical order). The similari- The results show the significant impact that ties between the joint tail dependence (as the (empirically motivated) consideration of measured by the DoF) of asset and equity tail dependence has on the distribution of returns are quite striking.5 losses across the capital structure: expected Next, we compare the remaining parame- losses are clearly redistributed from the junior ters that define a t-copula, ie, the correlation to the senior tranches, as a consequence of coefficients. Using a robust estimator based the increased volatility of the overall portfolio on Kendall’s rank statistic6, we compute the loss distribution. Notice that even larger dif- two 30x30 correlation matrices from asset ferences can be observed if one compares and equity returns. The maximum absolute higher moments or tail measures of the difference (element-by-element) is 4.6%, tranches’ loss distributions. and the mean absolute difference is 1.1%, providing further evidence of the similarity The LHP with tail dependence of the two dependence structures. It is possible to incorporate a Student-t copula into the LHP model discussed above. To Example: synthetic loss tranche do so, we must change the distribution for The models described earlier in this chapter the asset returns to be multivariate Student- can be modified to account for a fat-tailed t distributed where we denote the DoF dependence structure of asset returns. Here parameter with ν and retain the one-factor we analyse the impact of a non-Normal correlation structure. This gives assumption on the expected discounted loss (EDL) of a portfolio loss tranche. We focus on ββεZ +−1 2 A = iii i W /ν 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). where W is an independent random variable

46 The Lehman Brothers Guide to Exotic Credit Derivatives

following a chi-square distribution with ν degrees of freedom. This is the simplest Figure 28. Expected discounted loss, possible way to introduce tail dependence 100K paths, standard errors in parenthesis via a Student-t copula function. Note that this is no longer a ‘factor model’ in Tranche Normal copula t copula DoF=12 Pctg the sense that the asset return is composed (%) EDL (std err %) EDL (std err %) diff of two independent and random factors. The fact that both the market and idiosyncratic 0-5 $2,256,300 (0.14) $2,012,200 (0.23) -11 terms ‘see’ the same value of W means that 5-10 $533,020 (0.63) $601,630 (0.66) 13 they are no longer independent. 10-15 $146,160 (1.37) $221,120 (1.06) 51 For this model, we have been able to cal- 15-20 $41,645 (1.70) $90,231 (1.62) 117 culate a closed-form solution for the densi- ty of the portfolio loss distribution, and show in O’Kane and Schloegl (2003) that it popular conjecture that the different lever- possesses the same tail dependent proper- age of assets and equity will necessarily ties as described above, ie, a widening of create significant differences in their joint senior spreads and a reduction in equity dynamics seems to be empirically unfound- tranche spreads. ed. From a practical point of view, these results represent good news for practition- Summary ers who only have access to equity data for Our empirical investigation of the depen- the estimation of the dependence parame- dence structure of asset returns sheds ters of their models. some light on the two main issues that were raised at the beginning of this section. Modelling credit options First, the assumption of Gaussian depen- We separate credit options into options on dence between asset returns can be reject- bonds and options on default swaps. ed with extremely high confidence in favour of an alternative ‘fat-tailed dependence.’ Pricing options on bonds Multivariate structural models that rely on Options on corporate bonds are naturally the normality of asset returns will generally divided into three groups according to how underestimate default correlations, and the exercise price is specified. The option thus undervalue junior tranches and over- can be struck on price, yield or credit value senior tranches of multi-name credit spread. The exercise price is constant for products. A fat-tailed dependence of asset options struck on price, but for options returns will produce more accurate joint struck on yield it depends on the time to default scenarios and more accurate valua- maturity of the underlying bond and is found tions. Second, the dependence structures from a standard yield-to-maturity calculation. of asset and equity returns appear to be Obviously, for European options there is no strikingly similar. The KMV algorithm that difference between specifying a strike price produces the asset values used in our anal- and a strike yield. ysis is nothing other than a sophisticated Bond options struck on spread are differ- way of de-leveraging the equity to get to the ent. For credit spread options the exercise value of a company’s assets. Therefore, the price depends both on the time to maturity

The Lehman Brothers Guide to Exotic Credit Derivatives 47

of the bond and on the term structure of interest rates at the exercise time. A credit Figure 29. Receiver default swaption spread strike is commonly specified as a yield spread to a Treasury bond or interest Default swap cash flows if exercised into rate swap, or as an asset swap spread. For short-dated European options with price (or yield) strike on long-term bonds, Default Default CDS the well-known Black-Scholes formula goes swaption swaption maturity date a long way but is not recommended settlement expiry date T T TM beyond this limited universe. An important S problem with Black-Scholes is that it does not properly account for the pull-to-par of The framework can be used to derive the bond price. This problem can be solved price/yield volatilities from return volatilities by a yield diffusion model where the bond’s and tend to give more robust estimates than yield is the underlying stochastic process. direct estimation. It is relatively easy to build a lattice for a Extending interest rate models to mod- lognormal yield, say, and price the option elling risky rates only works if we assume by standard backwards induction tech- either zero recovery on default, or assume niques. The lattice approach also allows for that recovery is paid as a fraction of the mar- easy valuation of American/Bermudan exer- ket value at default. However, creditors have cise. Yield diffusion models can be con- a claim for return of full face value in structed to fit forward bond prices but bankruptcy, so it is necessary to specifically would usually assume constant yield model the recovery at default as a fraction of volatility which is inconsistent with empiri- face value. This is especially important for cal evidence. lower credit quality issuers where the bonds The problems of pull-to-par of price and trade on price rather than yield. non-constant volatility can be solved by a full For high quality issuers, the above term structure model such as Black-Derman- approaches are less controversial for pric- Toy, Black-Karasinski or Heath-Jarrow- ing options struck on price or yield. For Morton. These models were designed for these issuers, interest rate volatility is the default-free interest rates but can be applied main driver of price action. Volatility param- analogously to credit risky issuers. Instead of eters should therefore be related to calibrating the model to Libor rates, the implied volatilities from interest rate swap- model should be calibrated to an issuer-spe- tion markets but must also incorporate the cific credit curve. negative correlation between credit It is rarely possible to calibrate volatility spreads and interest rates (see Berd and parameters because of the lack of liquid Ranguelova (2003)) which can cause yield bond option prices. Usually it is more appro- volatility on corporate bonds to be signifi- priate to base volatility parameters on histor- cantly lower than comparable interest rate ical estimates. Naldi, Chu and Wang (2002) swaption volatility. and Berd and Naldi (2002) present a multi- When pricing spread options, it is impor- factor framework for modelling the stochas- tant to specifically take into account the tic components of corporate bond returns. default risk. In the next section we discuss

48 The Lehman Brothers Guide to Exotic Credit Derivatives

how to price options to buy or sell protection = 0 are simply ignored in that case and the through CDS. Bond options struck on credit distribution of ST will be such that the proba- spread can be priced in similar fashion. bility that BT = 0 is 0. (See Harrison and Kreps (1979) for details.) Pricing default swaptions To use this result we first let A be a securi- The growing market in default swaptions has ty that at T pays 0 if default has occurred and led to a demand for models to price these otherwise pays the upfront cost (as of T) of a products. First, let us clarify terminology. An zero-premium CDS with the same maturity option to buy protection is a payer swaption as the CDS underlying the swaption. We let and an option to sell protection is a receiver B be a security that at T pays 0 if default has swaption. This terminology is analogous to occurred and otherwise pays PV01 . With T that used for interest rate swaptions. these definitions the ratio AT/BT is equal to Black’s formulas for interest rate swap- the spread ST if default has not occurred at T, tions can be modified to price European otherwise AT/BT and ST are not defined. The default swaptions. Consider a European distribution of ST should then be such that payer swaption with option expiry date T and E[ST] = A0/B0 where the probability of strike spread K. The contract is to enter into default before T is put to 0. A0/B0 is the T-for- a long protection CDS from time T to TM, and ward spread, denoted F0, for the underlying it knocks out if default occurs before T (see CDS where the forward contract knocks out Figure 29). Conditional on surviving to time if default occurs before T. T, the option payoff is The next step is to let A be the swaption,

in which case AT is 0 if default happens before T and PS otherwise. The value of the PVPS 01 max{ −⋅= KS 0, } T T T T swaption today is then  PS  where the PV01 is the value at T of a risky PS = EB T PV 01 ⋅= []max{}− KSE 0, T 00   0 T  BT  1bp annuity to time TM or default, and ST is the market spread observed at T on a CDS with maturity TM. If we make the assumption that log(ST) is Ignoring the maximum, this is the stan- normally distributed with variance σ2T, corre- dard payoff calculation for a forward start- sponding to the spread following a log-nor- ing CDS. See page 32 for a discussion of mal process with constant volatility σ, then

CDS pricing. with the requirement E(ST) = F0 (the forward From finance theory we know that for any spread), we have determined the distribution given security, say B, that only makes a pay- of ST to be used to find E[max{ST –K,0}]. It is ment at T, there is a probability distribution easy to calculate this expectation and we of the spread ST such that for any other arrive at the Black formula security, say A, that also only makes a payment at T, the ratio A /B of today’s values of 0 0 PVPS ()⋅−⋅⋅= dNKdNF ,)()(01 the securities is equal to the expectation of 0 100 2 the ratio A /B of the security payments at T. log( + σ 2TKF 2/)/ T T = 0 −= σ d1 12 Tdd The result is valid even if BT can be 0 as long σ T as AT = 0 when BT = 0. The states where BT

The Lehman Brothers Guide to Exotic Credit Derivatives 49

where N is the standard normal distribution swap maturity 20/9/2008. The valuation function. The Black formula for a receiver date is 28/8/2003. The strike is 260, which swaption is found analogously. (See Hull and is the five year CDS spread on the valuation White (2003) and Schonbucher (2003) for date. Our trading desk quoted the swaption more details.) at 2.29%/2.52%, which means that $10m

It is important that the forward spread, F0, notional can be bought for $252,000 and and PV010 are values under the knockout sold for $229,000. These are prices of non- assumption. Calculation of F and PV01 knockout options. The fair value of protec- 0 0 should be done using a credit curve that has tion until swaption maturity given the credit been calibrated to the current term structure curve on the valuation date is 0.211%, the of CDS spreads. PV01 used in the Black formula is 4.027 and To value a payer swaption that does not the forward spread is 274.7bp. These num- knock out at default, we must add the value bers imply that the bid/offer volatilities of credit protection from today until option quoted by our desk are approximately 75% maturity. Under a non-knockout payer swap- and 85%. tion, a protection payment is not made until option maturity. The value of the credit pro- Interest rates and credit risk tection is therefore less than the upfront One way to model credit spread dynamics is cost of a zero-premium CDS that matures via a hazard rate process. This provides a with the option. The knockout feature is not consistent framework for modelling spreads relevant for receiver swaptions, as they will of many different maturities. The differences never be exercised after default. between directly modelling the credit The last step in valuing the swaption is to spread and modelling the hazard rate are find the volatility σ to be used in the Black similar to the differences between mod- formula. An estimate of σ can be made from elling the yield of a bond and using a term a time series of CDS spreads. Examination structure model as discussed above. of CDS spread time series also reveals that A stochastic hazard rate model is natural- the log-normal spread assumption often is ly combined with a term structure model to inappropriate. It is not uncommon for CDS produce a unified model that can, at least in spreads to make relatively large jumps as theory, be used to price both bond options reaction to firm specific news. However cal- and credit spread options. However, a num- ibrating a mixed jump and Brownian process ber of practical complications arise in get- to spread dynamics is not easy. ting such an approach to work. Since bond To price default swaptions with and the CDS markets have their own Bermudan exercise we could construct a dynamics, usually not implying the same lattice for the forward spread, but criticism issuer curves, it may not be appropriate to analogous to that for using a yield diffusion naively price all options in the same cali- model to price bond options applies. brated model without properly adjusting for Instead, we recommend using a stochastic the basis between CDS and bonds. hazard rate model. Also, calibration of the volatility parame- To illustrate the use of the Black formula, ters of the hazard rate process, when possi- consider a payer swaption on Ford Motor ble, is less than straightforward, especially Credit with option maturity 20/12/2003 and when calibrating to bond options struck on

50 The Lehman Brothers Guide to Exotic Credit Derivatives

price or yield, or to bonds with embedded an option to enter into a receiver swap with options. Interest rate volatility parameters fixed rate k for the remaining life of the should be calibrated separately using liquid original trade at default. For simplicity, we prices of interest rate swaptions. These assume that default can only take place at parameters are then taken as given when times ti. If B denotes the price process of determining the hazard rate volatility param- the savings account, then computing the eters by fitting to time series estimates or expected discounted cash flows gives the calibrating to market prices. value V0 for the price of the default protec- Finally, it is necessary to determine the cor- tion, where relation between interest rates and hazard rates. When estimating correlations it is  n max()RS ,0 important not to use yield spread or OAS ===ti ττ[] VE0 ∑ tPtii = Bt( ) (option adjusted spread) as a proxy for the i 1 i hazard rate, because such spread measures do not properly take into account the risk of default. Using OAS is especially a problem for The interpretation of this equation is that the long maturity bonds with high default proba- value of default protection is a probability bilities and high recovery rates. For such weighted strip of receiver swaptions, where bonds there can be a significant decrease in each swaption is priced conditional on

OAS when interest rates increase even if haz- default happening at ti. ard rates remain unchanged. Correlations Representing the default protection via a should be estimated directly using bond strip of swaptions is a very useful framework prices or CDS spreads. for developing intuition around the pricing and helps us understand the importance of Modelling hybrids the shape of the interest rate curve. In terms The behaviour of hybrid credit derivatives is of volatility exposure, the protection seller driven by the joint evolution of credit has sold swaptions and is therefore short spreads and other market variables such as interest rate volatility. interest and exchange rates, which are com- If the rate and credit process are correlat- monly modelled as diffusion processes. As ed, then there will also be a spread volatility a consequence, the reduced form approach dependence. However, under the assump- is the natural framework for pricing and tion of independence, the volatility of the hedging these products. credit spread does not enter into the valua- We illustrate the main ideas via the exam- tion, only the default probabilities. ple of default protection on the MTM of an A strip decomposition, as we have shown interest rate swap. Suppose an investor above, is the basic building block for most has entered into a receiver swap with fixed hybrid credit derivatives. A cross-currency rate k with a credit risky counterparty. If the swap for example could be dealt with in

MTM of the receiver swap, RSt is positive exactly the same way, with the exchange to the investor at the time of default, this is rate as an additional state variable. For this paid by the protection seller. If RSt is nega- instrument, the exchange rate exposure is of tive, the investor receives nothing, so that prime importance. the payoff at default is max(RSt,0). This is In terms of tractability, it is important to be

The Lehman Brothers Guide to Exotic Credit Derivatives 51

able to calculate the conditional expecta- structure between credit spreads and the tions appearing in the strip decomposition. other market variables is the main challenge A further consequence of the hazard rate in modelling hybrids. The simplest approach method is that we do not actually have to is to work within a diffusion setting where condition on the realisation of the default spreads and interest rates/FX are correlated. time itself, conditioning instead on the reali- Such a model will not necessarily generate sation of the hazard rate process. This is par- levels of dependence representative of peri- ticularly advantageous for Monte Carlo ods of market stress where investment simulation, as one does not have to explicit- grade defaults are likely. As a starting point, ly simulate default times and is a useful vari- however, it appears to be reasonable. ance reduction technique. Even within this framework, the effect of The parameters needed for pricing correlation on the valuation of a hybrid hybrids are essentially volatility and depen- instrument can be marked. At this stage of dence parameters. Calibrating volatilities the market it is safe to say that this correla- for interest rates and FX is relatively tion is a ‘realised’ parameter as opposed to straightforward. Credit spread volatilities an implied one, ie pricing and hedging are somewhat more involved. Until recent- must proceed on the basis of a view on cor- ly, we have had to rely on estimates of his- relation founded on historical estimates. torical volatilities; now the growing options Going forward, it will be interesting to see markets are starting to make the calibration to what extent a market in implied correla- of implied spread volatilities feasible. tions will develop via standardised hybrid Determining the correct dependence credit derivatives.

52 The Lehman Brothers guide to exotic credit derivatives

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54 The Lehman Brothers Guide to Exotic Credit Derivatives

Approved by Lehman Brothers International Risk Waters Group and Lehman Brothers and the (Europe), which is authorised and regulated by the author have made every effort to ensure the Financial Services Authority. Rules made for the accuracy of the text; however, neither they nor any protection of investors under the UK Financial company associated with the publication can Services Act may not apply to investment business accept legal or financial responsibility for conducted at or from an office outside the UK. consequences, which may arise from errors, omissions, or any opinions given. Published by Incisive Risk Waters Group, Haymarket House, 28–29 Haymarket, London SW1Y 4RX © 2003 Lehman Brothers and Incisive RWG Ltd

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The Lehman Brothers Guide to Exotic Credit Derivatives 55

Notes

56 The Lehman Brothers Guide to Exotic Credit Derivatives

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THE LEHMANBROTHERS THE LEHMAN BROTHERS GUIDE TO EXOTIC CREDIT DERIVATIVES GUIDE TO EXOTIC CREDIT DERIVATIVES