lévy base correlation

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WILMOTT Journal | Volume 1 Number 2Published online in Wiley InterScience | DOI: 10.1002/wilj.6 (www.interscience.wiley.com)

1 IntroductionSince the introduction of the one-factor Gaussian copula modelfor pricing synthetic Collateralized Debt Obligation (CDO)tranches by Andersen et al. (2003) correlation is seen as anexogenous parameter used to match observed market quotes.First the market adopted the concept of implied compound cor-relation. In tandem with the concept of volatility in the BlackScholes option pricing framework compound correlation wasthe parameter to be put in the model to match observed marketprices of tranches. One of the problems of this approach residesin its unsuitability for interpolation. Given the implied com-pound correlations for the [3%–6%] and [6%–9%] tranches of aliquid index such as iTraxx Europe Main1 it is not clear whichvalue to use for a nonstandard tranche such as e.g. [5%–8%].Besides these interpolation problems we also point out that dur-ing some market events non-meaningful values may be found forimplied compound correlation, e.g. during the auto crisis of May2005. The current widespread market approach is to use the con-cept of base correlation (BC). In the base correlation method-ology only equity or base tranches2 are defined. The price of atranche [A—D] is calculated using the two equity tranches with Aand D as detachment points. Using BC it is quite straightforward

to bootstrap between standard attachment points. Additionallythe BC concept is quite adapted to interpolation for nonstandardtranches. Hence the [5%–8%] tranche would be priced by inter-polating the BC curve for values at 5% and 8% respectively. Themethodology however has some weaknesses. First of all it is verysensitive to the interpolation technique used. Even worse, themethodology may not be arbitrage-free. Finally the methodologydoes not provide any guidance on how to extrapolate the curve,especially below the 3% attachment point. These problems havebeen addressed by Garcia and Goossens (2008). The solutionresides in using arbitrage-free interpolation techniques in thebase correlation framework, interpolating expected losses ofbase tranches instead of interpolating base correlation curvesdirectly.

In this paper we introduce Lévy base correlation and com-pare it to the classical Gaussian copula. The remainder of thispaper is organised as follows. In section 2 we review the genericone-factor model for valuation of CDO tranches and introduceLévy base correlation. The historical study is outlined in sec-tion 3. We start by looking at base correlation in these twomodels. First we show the evolution over time and second we

Lévy Base CorrelationJoão GarciaDexia Group, BelgiumSerge GoossensDexia Bank, BelgiumViktoriya MasolEURANDOM, The NetherlandsWim SchoutensK.U.Leuven, Department of Mathematics, Belgium, e-mail: [email protected]

AbstractIn this paper we investigate one-factor models that extend the classical Gaussian copula model for pricing tranches of CDOs. We introduce

Lévy base correlation and compare it to the classical Gaussian copula. The results of a historical study of both models on the iTraxx Europe

Main dataset are presented. Our focus is on the base correlation surface and on the deltas of the tranches with respect to the index. We

observe that the Lévy base correlation curve is much flatter than the corresponding Gaussian one. With respect to the deltas we conclude

that the Lévy model produces larger deltas for the equity tranche and smaller deltas for the senior tranches.

KeywordsCDO models, Levy processes, correlation, securitization, credit derivatives

The opinions expressed in this paper are those of the authors and do not necessarily reflect those of their employers.

1. The standard attachment points are 3%, 6%, 9%, 12% and 22% for iTraxx Europe Main and 3%, 7%, 10%, 15% and 30% for CDX.NA.IG.

2. Base or equity tranches are tranches with attachment point 0.

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96 | Wilmott Journal

WILMOTT Journal | Volume 1 Number 2 Published online in Wiley InterScience | DOI: 10.1002/wilj (www.interscience.wiley.com)

consider the behaviour across maturity and look at the basecorrelation surface. Next we compare hedge parameters in thedifferent models. We focus on the deltas of the tranches withrespect to the index. This is also done across maturity. Finallyour conclusions are presented in section 4.

2 Generic One-Factor ModelThe one-factor Gaussian copula model using the so calledrecursion algorithm was first introduced by Andersen et al.(2003) and is in widespread use by market participants. InAlbrecher et al. (2006) the Gaussian case was extended to ageneral Lévy setting. In what follows we give a brief descriptionof this generic one-factor algorithm. Consider a portfolio of Nfirms and fix a time horizon T. It is standard market practiceto assume the default process to follow an inhomogeneousPoisson process and as such for any 0 ≤ t ≤ T the default timesτi and default intensities λi(t), i = 1, . . . , N, satisfy

P (τi > t) = exp

(−

∫ t

0λi(u)du

)

where P is the risk-neutral probability measure. In a one-factormodel of portfolio defaults, a single systemic factor X is intro-duced, conditional upon which all default probabilities areindependent. The single name survival probabilities P(τi > t)are typically implied from the credit default swap (CDS) market.The fair spread of a CDS balances the present value of thecontingent leg C, that is the present value (PV) of losses in caseof defaults, given by

C = NCDS(1 − R)

n∑i=1

d(ti) (PS(ti−1) − PS(ti)) (1)

and the present values of the fee leg F, given by

F = NCDSS

(n∑

i=1

PS(ti)d(ti)�ti

)+ AD, (2)

where NCDS is the CDS notional and AD is the accrual on default

AD = 1

2

n∑i=1

NCDS d(ti) (PS(ti−1) − PS(ti)) �ti. (3)

In these equations the summations run over the payment dates, S isthe spread premium on a yearly basis, PS(ti) is the survival probabil-ity at time ti, R is the recovery rate, d(t) is the risk-free discount fac-tor and �ti = ti − ti−1 is the year fraction. The key step in valuingCDO tranches is to compute the joint loss distribution. In the recur-sion algorithm one computes a discretised version of the condition-al loss distribution by means of a simple recursion formula. A lossunit u is chosen so that within a certain tolerance, losses can be rep-resented by integers. For the iTraxx Europe Main portfolio with anassumed uniform recovery rate of 40%, the loss unit is 0.48%. Wedenote by P(i)(l, t|X) the probability of l losses (in terms of the lossunit u) at time t with i names conditional on the market factor X.Recalling that conditional on X all default probabilities areindependent, we can write that P(i)(l, t|X) is the sum of two terms:

P(i)(l, t|X) = P(i−1)(l, t|X)P (τi > t|X)

+P(i−1)(l − ω(i), t|X)(1 − P (τi > t|X)), (4)

where ω(i) is the number of loss units incurred by a default ofthe ith name. The unconditional loss distribution is found byintegrating over the market factor

P(l, t) =∫

�X

P(l, t|X)f (X)dX, (5)

where f (X) is the density of the probability distribution of themarket factor X. Analogous to the CDS case the fair spread of aCDO tranche balances the present value of the fee leg F, given by

F = Sn∑

i=1

(N(Tr) − E [L(Tr)i ])d(ti)�ti (6)

and the present value of the contingent leg C, given by

C =n∑

i=1

d(ti+ 12)

(E [L

(Tr)i ] − E [L

(Tr)i−1 ]

). (7)

In these equations the summations run over the paymentdates, S is the spread premium on a yearly basis, d(t) is therisk-free discount factor, �ti = ti − ti−1 is the year fraction,E [L

(Tr)i ] is the expected loss on the tranche at time ti and N(Tr)

is the tranche size.3 In the base correlation framework, theexpected loss on a tranche is computed as the difference of theexpected loss of two equity tranches

E L[A—D] = E L[0—D;ρD ] − E L[0—A;ρA ].

A distribution is said to be infinitely divisible, if for every posi-tive integer n, the characteristic function φ(u) = E [iuX] is alsothe nth power of a characteristic function. Given an infinitelydivisible distribution, a stochastic process, X = {Xt, t ≥ 0}, canbe constructed. This so-called Lévy process starts at zero X0 = 0,has independent and stationary increments and the distribu-tion of the increments Xt+s − Xs , has (φ(u))t as characteristicfunction. The cumulative distribution function of Xt and itsinverse are denoted by Ht and H[−1]

t respectively. We normalisethe distribution so that E [X1] = 0 and Var[X1] = 1. Hence onehas Var[Xt ] = t. For more details on Lévy processes we refer toBertoin (1996), Sato (2000) and Kyprianou (2006), and toSchoutens (2003) for applications of Lévy processes in finance.

In the so-called latent variable model default occurs when a cer-tain (latent) variable Ai (usually the return) falls below a certainthreshold Ki that is implied from CDS prices. The market or systemicfactor X and the idiosyncratic factor X(i) are random variables whosefunctional form depends on model assumptions. In the generic onefactor Lévy model the latent variable is represented as

Ai = Xρ + X(i)1−ρ , i = 1, . . . , N, (8)

where X and X(i) are independent and identically distributedLévy processes. Hence, each Ai has the same infinitely divisible

3. The [0–3%] equity tranche is quoted as an upfront payment plus 500 basis points running.



WILMOTT Journal | Volume 1 Number 2Published online in Wiley InterScience | DOI: 10.1002/wilj (www.interscience.wiley.com)

distribution function H1 . Note that for i �= j, we haveCorr[Ai, Aj] = ρ . The threshold implied from the CDS risk neutralprobability of default pi(t) = P (τi < t) is given by

Ki(t) = H[−1]1 (pi(t)). (9)

The conditional default probability of firm i given the value yfor the systemic factor is given by

P(τi < t|Xρ = y) = pi(y; t) = H1−ρ (Ki(t) − y). (10)

We consider two choices for the distributions of the latent variables.First, note that the classical Gaussian copula model is a special caseof this generic one-factor model, in which the normal distributionis used. Second, we use a shifted Gamma distribution and setXt = √

at − Gt, in which Gt follows a Gamma(at,√

a) distribution sothat E [X1] = 0 and Var[X1] = 1. The density of a Gamma(a, b) dis-tribution is given by

f (x; a, b) = ba

(a)xa−1 exp (−bx) , x > 0.

Both the cumulative distribution function Ht(x; a) of Xt, and itsinverse H[−1]

t (y; a), can easily be obtained from the Gamma cumu-lative distribution function and its inverse. Lévy base correlation isdefined as the base correlation in the shifted Gamma model withfixed a = 1. Hence the Exponential(1) distribution is used for eachAi. The motivation for the use of the exponential distribution liesin the fact that it has a fatter tail behaviour than the normal dis-tribution; the logarithm of the density decays linearly, in contrastwith a quadratic decay for the normal density. Note that theparameter a could be left free to offer more flexibility in order toobtain a smooth base correlation curve. In that case we need toimpose an additional constraint or set an additional optimisationtarget. Several authors have described one-factor models using dis-tributions other than the standard normal distribution, Baxter(2007), Hooda (2006), Guégan and Houdain (2005), Kalemanova etal. (2001) and Moosbrucker (2006). A general generic model wasdescribed in Albrecher et al. (2006). Several Lévy settings and basecorrelation choices are compared in Masol and Schoutens (2008);there, it is concluded that the performance of the exponentialmodel is completely in line with other Lévy choices. However, theexponential model is much more tractable. Important to note isthat from a numerical point of view, the most computer intensivepart in the algorithm is the calculation of the unconditional lossdistribution as in (5). For example calculating the probability of kdefaults out of a group of n equals under the Exponential model ∫ +∞

−∞�k

n,ydHρ (y; a = 1) =∫ +∞

0�k

n,ρ−z

1

(ρ)zρ−1 exp(−z)dz,

where �kn,y denotes the conditional probability of having k

defaults out of a group on n, given the common factor is y.These integrals can be calculated using Gauss-Laguerre quadra-ture. However Gauss-Laguerre quadrature can lead in some caseto small numerical errors, because the number of effectivenodes is typically small. In Dobránszky (2008), alternative

numerical quadratures are detailed and compared, a simplesplitting up of the integration problem in combination with amidpoint rule quadrature applied on the probability space isscalable, robust and accurate.

3 Historical StudyIn this section we briefly describe the historical study. For moredetails we refer to the study by Garcia and Goossens (2007).Another study has been done by Masol and Schoutens (2008).The dataset used is in this paper is the iTraxx Europe Main datafrom April 20th, 2005 until March 16th, 2007. We look at indexand tranche spreads for the 5, 7 and 10 year maturity contracts.Figure 1 shows the historical evolution of the spreads for the 5year maturity. The key parameters in both models are the basecorrelation curves. First we show the evolution over time andsecond we consider the behaviour across maturity and look atthe base correlation surface. For trading purposes it is impor-tant to understand the hedge parameters in the two models. Wepresent a comparison between the two models and focus on thedeltas of the tranches with respect to the index.

3.1 Base Correlation Evolution over TimeAs mentioned above both models are parametrised with a basecorrelation curve. We have compared the evolution over time ofthe base correlation curves in both models and found that theirbehaviour is very similar. The evolution over time for the 12–22tranche for the Gaussian and Lévy base correlation models isshown in Figure 2. We present the results for the 12–22 tranchesince this one is the last one to be computed in the bootstrappingprocedure and consequently it is sensitive to all previous com-putations. Lévy base correlation and Gaussian base correlationclearly behave in the same way, just on a different scale. The sit-uation on some particular days can be found in Figure 3. Basecorrelation curve is flatter, as well on a normal day as on a crisisday (auto crisis in May 2005). A standard technique to price non-standard tranches is by interpolating on the base correlationcurve. It is however well know that this from time to time canlead to arbitrageable prices. We believe the observed arbitrageopportunity under the Gaussian model appear due to the linear

Figure 1: iTraxx Europe Main On the run 5y Spreads.




interpolation. As the Lévy base correlation is flatter this arbi-trage due to linear interpolation is less probable (see Figure 4).

3.2 Base Correlation across MaturityWe now turn to the base correlation surface, that is we considerbase correlation as a function of the attachment point and thematurity. Figure 5 shows the Gaussian copula base correlationsurface. For every maturity we observe the typical upward slop-ing curve at the standard attachment points. The behaviouracross maturity for a given attachment point is less uniform. Thiscurve can be smiling, upward or downward sloping. Looking atthese curves it should be clear that there is no widely accepted

standard approach for interpolating a base correlation surfacefor a nonstandard attachment point or for a nonstandard matu-rity. Moreover interpolation schemes may not be arbitrage-free.For a detailed discussion on a more efficient way to interpolatein the base correlation framework, we refer to Garcia andGoossens (2008). The Lévy base correlation surface is shown inFigure 6. Contrary to the Gaussian case, the behaviour acrossmaturity for a given attachment point is more uniform. There isan upward trend with maturity. We also see the typical upwardsloping curve at the standard attachment points, except for the10 year maturity, where we see a smile. In fact this is expectedand the explanation is given in Garcia and Goossens (2007).

3.3 Hedge ParametersThe evolution over time for the delta of the 0–3 tranche withrespect to the index for the Gaussian and Lévy base correlationmodels is shown in Figure 7. The values for this sensitivityparameter produced by the Gaussian and Lévy models clearly

Figure 3: iTraxx Europe Main 5y Gaussian and Lévy BaseCorrelation on 04-May-2006 and on 19-May-2005 (“auto crisis”).

0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.120

50

100

150

200

250

Tranchlets – 19–05–2005

Gaussian, splineGaussian, linearGamma(1) linearGamma(1) spline

Figure 4: iTraxx Europe Main 5y Gaussian and Lévy BaseCorrelation Tranchlet prices on 19-May-2005.

Figure 5: Gaussian Copula Base Correlation.

Figure 2: iTraxx Europe Main 5y Gaussian and Lévy Base Correlationfor 12–22 tranche.



WILMOTT Journal | Volume 1 Number 2Published online in Wiley InterScience | DOI: 10.1002/wilj (www.interscience.wiley.com)

behave in the same way, just on a different scale. In order toquantify this similarity, a regression of the delta produced bythe Lévy model on the delta produced by the Gaussian mod-els is performed. This is done for all tranches and all matu-rities. The model is

δ(L)Tr, T = βTr, Tδ

(G)Tr, T (11)

where δ(L)Tr, T is the delta of tranche Tr with respect to the under-

lying index, for a maturity T, produced by the Lévy model andδ(G)Tr, T is the delta produced by the Gaussian model. We now

turn to the quality of these regressions. Table 1 shows theregression coefficient, the standard error and the coefficient ofdetermination R2 for each tranche for the 5 year maturity. TheR2 statistic confirms that the model (11) is a good fit. The regres-sion results for the 7 year maturity are shown in Table 2. In thiscase the R2 statistic also confirms that the model (11) is a goodfit. Table 3 shows the regression results for the 10 year maturi-ty. Also here the R2 statistic confirms that the model (11) is agood fit. These results show that compared to the Gaussianmodel, the Lévy model produces a delta that is larger for theequity tranches and smaller for the senior tranches. Roughly

speaking we can say that the Lévy model delta is approximate-ly 20% larger for the equity tranche and 40% smaller for thesenior tranches. A similar story holds for hedging with respect

Table 3: Regression of Lévy delta on Gaussian delta for the 10year maturity tranches.

Tr β Std Err R2

0–3 1.182632 0.002417 0.99813–6 1.337550 0.002275 0.9987 6–9 0.895605 0.005092 0.9852 9–12 0.640572 0.001964 0.9956 12–22 0.622463 0.006962 0.9449

20 40 60 80 100 120 140 160 180 2000

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

CDS spread, bp

Del

ta

Deltas w.r.t Single Name CDSs

0–3, Gaussian0–3 Gamma3–6 Gaussian3–6 Gamma6–9 Gaussian6–9 Gamma

Figure 8: iTraxx Europe Main 5y Gaussian and Lévy BaseCorrelation deltas with regard to individual CDSs.

Figure 7: iTraxx Europe Main 5y Delta for 0–3 tranche Gaussianand Lévy model.


Tr βTr, T Std Err R2

0–3 1.198891 0.001143 0.9996 3–6 0.761988 0.002592 0.9946 6–9 0.567959 0.001864 0.995 9–12 0.527024 0.004981 0.9601 12–22 0.49989 0.01036 0.8334


Tr β Std Err R2

0–3 1.25069 0.00104 0.9997 3–6 1.059253 0.004181 0.9928 6–9 0.638850 0.002096 0.995 9–12 0.567896 0.003036 0.9869 12–22 0.557808 0.009115 0.8893

Figure 6: Lévy Base Correlation.




to the individual underlying CDSs (see Figure 8). Individual CDSdeltas are also systematically different and the ratio with forthe Lévy base correlation model compared with Gaussian caseis inline with ratio detailed above for hedging with the index.

4 ConclusionsIn this paper we considered generic one-factor models and intro-duced Lévy base correlation. The results of a historical studycomparing Lévy base correlation to the classical Gaussian copulamodel on the iTraxx Europe Main dataset have been presented.We have compared the evolution over time of the base correla-tion surfaces in both models and found that their behaviour isvery similar. Hedge parameters in the different models have alsobeen studied. We have focused on the deltas of the tranches withrespect to the index and CDSs found the values for this sensitivityparameter produced by the Gaussian and Lévy models clearlybehave in the same way. This is illustrated by the fact that aregression of one on the other results in a very good fit. Roughlyspeaking we can say that the Lévy model delta is approximately20% larger for the equity tranche and 40% smaller for the seniortranches.

João Garcia is the Head of the Credit Modelling team at the Treasury andFinancial Markets of Dexia Group in Brussels. His current interest includescredit derivatives, structured products, correlation mapping of credit portfo-lios in indexes, developing strategies and trading signals for credit deriva-tives indexes and pricing distressed assets. Before that he worked for fouryears on the construction of a grid system for strategic credit portfolio man-agement of the credit portfolio of the whole Dexia Group. The system hadbeen in place already in June 2006 and had been designed for the institu-tion to survive the credit crunch that began in June 2007. Additionally hehas experience on methodologies to rate and price cash flow CDOs for secu-ritization and credit derivatives instruments and for the allocation of crediteconomic capital besides the pricing of exotic interest rate derivatives. He isan Electronic Eng. from Instituto Tecnológico de Aeronáutica (ITA, Brazil),with a M.Sc. in Physics (UFPe, Brazil) and a Ph.D. in Physics (UA, Belgium).Some of his recent work can be consulted at www.sergeandjoao.com.Serge Goossens is a Senior Quantitative Analyst working on credit deriva-tives and correlation modelling in the Treasury and Financial Markets divi-sion of Dexia Bank Belgium. He has a vast experince in building pricingalgorithms of credit instruments for securitization, structuring, rating, hedg-ing and trading purposes. From his previous positions he has extensiveexpertise in parallel large scale numerical simulation of complex systems,ranging from computational fluid dynamics (CFD) to electronics. Serge holdsan M.Sc.in Engineering and a Ph.D. from the Faculty of Engineering of theK.U.Leuven and a Master of Financial and Actuarial Engineering degreeobtained from the Leuven School of Business and Economics. He has pub-lished a number of papers and he has presented at conferences worldwide.Some of his recent work can be consulted at www.sergeandjoao.com.Viktoriya Masol is a Risk Advisor at Risk Management of KBC Group. Inthis function she is responsible for the validation of market and trading riskmodels. Prior to joining KBC Group Viktoriya was a Research Fellow at

EURANDOM, an independent research institute sited in Eindhoven, TheNetherlands. At EURANDOM, she worked together with Wim Schoutens(K.U.Leuven) on multivariate risk modelling and credit derivatives pricingunder jump-driven Levy models. Viktoriya holds a Ph.D. in Mathematicsobtained at Kyiv National Taras Shevchenko University.Wim Schoutens is a research professor in the Department of Mathematicsat the Catholic University of Leuven, Belgium. He is a regular independentconsultant and trainer to the banking industry on equity modeling, struc-tured products, credit derivatives, and other financial engineering problems.Wim is author of the Wiley books ‘‘Lévy Processes in Finance: PricingFinancial Derivatives’’ and ‘‘Lévy Processes in Credit Risk’’. He is also editor(together with A.E. Kyprianou and Paul Wilmott) of the Wiley book ‘‘ExoticOption Pricing and Advanced Lévy Models’’. He is Managing Editor of theInternational Journal of Theoretical and Applied Finance and AssociateEditor of Mathematical Finance and Review of Derivatives Research. He cur-rently teaches several courses related to financial engineering in differentMaster programs and is a regular lecturer for the financial industry.

REFERENCESAlbrecher, H., Ladoucette, S. and Schoutens, W. A Generic one-factor LévyModel for pricing synthetic CDOs. Birkhauser, Dec 2006. In: Advances inMathematical Finance. Fu, M.C., Jarrow, R.A., Yen, J.Y. and Elliott, R.J. (eds.).Andersen, L., Sidenius, J. and Basu, S. All your hedges in one basket. Risk, Nov 2003.Baxter, M. Gamma process dynamic modelling of credit. Risk Magazine, Oct 2007.Bertoin, J. Lévy Processes. Cambridge University Press, 1996.Dobránszky, P. Numerical Quadrature to calculate Lévy Base Correlation.Technical Report 2008-2, K.U. Leuven, Section of Statistics, 2008.Garcia, J. and Goossens, S. Lévy base correlation explained. Workingpaper, Credit Modelling, Treasury and Financial Markets, Dexia Bank,Aug 2007.Garcia, J. and Goossens, S. Explaining the Lévy base correlation smile.Risk, pages 84–88, July 2008.Guégan, D. and Houdain, J. Collateralized Debt Obligations pricing and factormodels: A new methodology using normal inverse Gaussian distribution.Techncial Report IDHE-MORA 007-2005, ENS Cashan, Section of Statistics,2005.Hooda, S. Explaining base correlation skew using NG (normal gamma) process.Working paper, Nomura, 2006.Kalemanova, A. and Schmid, B. and Werner, R. The normal inverseGaussian distribution for synthetic CDO pricing. Journal of Derivatives,1144(3), 2001.Kyprianou, A. Introductory Lectures on Fluctuations of Lévy Processes withApplications. Springer, 2006.Moosbrucker, T. Pricing CDOs with correlated variance gamma distributions.Journal of Fixed Income, 1166(2): 62-75, Fall 2006.Masol, V. and Schoutens, W. Comparing some alternative Lévy base correlationmodels for pricing and hedging CDO tranches. Technical Report 2008-1, K.U. Leuven, Section of Statistics, 2008.Sato, K. Lévy Processes and Infinitely Divisible Distributions. Cambridge UniversityPress, 2000.Schoutens, W. Lévy Processes in Finance: Pricing Financial Derivatives.John Wiley & Sons, 2003.


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