Credit Derivatives: From the simple to the more advanced Jens Lund 2 March 2005

Credit Derivatives: From the simple to the more advanced2 Outline CDS Hazard Curves CDS pricing Credit Triangle Index CDS Basket credit derivatives, n-to-default, CDO Standardized iTraxx tranches, implied correlation Gaussian copula model – Correlation smile Pricing of basket credit derivatives – Implementation strategies Subjects not mentioned Conclusion

2 March 2005Credit Derivatives: From the simple to the more advanced3 CDS Cash flow Protection buyer Protection seller Protection buyer Protection seller Protection buyer Protection seller Bond 100 Spread Recovery b) physical settlement Protection leg: a) cash settlement Premium leg: Continues until maturity or default Only in the event of default

2 March 2005Credit Derivatives: From the simple to the more advanced4 Hazard curves A distribution of default times can be described by – The density f(t) – The cumulative distribution function – The survival function S(t) = 1-F(t) – The hazard (t) = f(t)/S(t) Interpretation – P(T in [t,t+dt[)  f(t)dt – P(T in [t,t+dt[|T>t)  (t)dt Conections (t) f(t)

2 March 2005Credit Derivatives: From the simple to the more advanced5 CDS Pricing Model CDS pricing models takes a lot of input – Length of contract – Risk free interest rate structure – Default probabilities of the reference entity for any given horizon – Expected recovery rate – Conventions: day count, frequency of payments, date roll etc. PV of the CDS payments: Premium payments Discount factor Survival probability Payment in the event of default Probability of default at time t Accrual factor Discount factor

2 March 2005Credit Derivatives: From the simple to the more advanced6 Credit Triangle - What Determines the Spread? Assume premium is paid continuously Assume hazard rate is constant

2 March 2005Credit Derivatives: From the simple to the more advanced7 Index CDS Simply a collection of, say, 100, single name CDS. Each name has notional 1/100 of the index CDS notional. Spread is lower than average of CDS spreads: – Intuition: the low spreads are paid for a longer time period than the high spreads. – PV01 n = value of premium leg for name n – Not correlation dependent

2 March 2005Credit Derivatives: From the simple to the more advanced8 First-to-Default Basket Basket buyer Basket seller Basket buyer Basket seller First-to-default Spread 100 – Recovery on defaulted asset Protection leg: Premium leg: Continues until the first default or until maturity Only in the event of default, and only the first default Alternative to buying protection on each name Usually cheaper than buying protection on the individual names Pays on the first (and only the first) default Spread depends on individual spreads and default correlation

2 March 2005Credit Derivatives: From the simple to the more advanced9 Standardized CDO tranches iTraxx Europe – 125 liquid names – Underlying index CDSes for sectors – 5 standard tranches, 5Y & 10Y – First to default baskets, options – US index CDX Has done a lot to provide liquidity  in structured credit Reliable pricing information available   Implied correlation information 88% Super senior 9% 3% 6% 12% 100% 3% equity Mezzanine 22%

2 March 2005Credit Derivatives: From the simple to the more advanced10 Reference Gaussian copula model N credit names, i = 1,…,N Default times: ~ curves bootstrapped from CDS quotes T i correlated through the copula:  F i (T i ) =  (X i ) with X = (X 1,…,X N ) t ~ N(0,  )  Note: F i (T i ) =  (X i )  U[0,1]   correlation matrix, variance 1, constant correlation  In model: correlation independent of product to be priced

2 March 2005Credit Derivatives: From the simple to the more advanced11 Prices in the market has a correlation smile In practice:  Correlation depends on product, 7-oct-2004, 5Y iTraxx Europe  Tranche  Maturity

2 March 2005Credit Derivatives: From the simple to the more advanced12 Why do we see the smile? Spreads not consistent with basic Gaussian copula Different investors in  different tranches have  different preferences If we believe in the Gaussian model:  Market imperfections are present and we can arbitrage! – However, we are more inclined to another conclusion: Underlying/implied distribution is not a Gaussian copula

2 March 2005Credit Derivatives: From the simple to the more advanced13 Compound correlations The correlation on the individual tranches Mezzanine tranches have low correlation sensitivity and  even non-unique correlation for given spreads No way to extend to, say, 2%-5% tranche  or bespoke tranches What alternatives exists?

2 March 2005Credit Derivatives: From the simple to the more advanced14 Base correlations Started in spring 2004 Quote correlation on all 0%-x% tranches Prices are monotone in correlation, i.e. uniqueness 2%-5% tranche calculated as: – Long 0%-5% – Short 0%-2% Can go back and forth between base and compound correlation Still no extension to bespoke tranches

2 March 2005Credit Derivatives: From the simple to the more advanced15 Base correlations Short Long

2 March 2005Credit Derivatives: From the simple to the more advanced16 Base versus compound correlations

2 March 2005Credit Derivatives: From the simple to the more advanced17 Is base correlations a real solution? No, it is merely a convenient way of describing prices on CDO tranches An intermediate step towards better models that exhibit a smile No general extension to other products No smile dynamics Correlation smile modelling, versus Models with a smile and correlation dynamics

2 March 2005Credit Derivatives: From the simple to the more advanced18 Implementation of Gaussian copula Factor decomposition:  M, Z i independent standard Gaussian,  X i low  early default FFT/Recursive: – Given T: use independence conditional on M and calculate loss distribution analyticly, next integrate over M Simulation: – Simulate T i, straight forward – Slower, in particular for risk, but more flexible – All credit risk can be calculated in same simulation run as the basic pricing

2 March 2005Credit Derivatives: From the simple to the more advanced19 Pricing of CDOs by simulation 100 names Make, say, simulations: – Simulate default times of all 100 names – Price value of cash-flow in that scenario – Do it all again, times Price = average of simulated values 88% Super senior 9% 3% 6% 12% 100% 3% equity Mezzanine

2 March 2005Credit Derivatives: From the simple to the more advanced20 Default time simulation Hazard and survival curve S = exp(-H*time)

2 March 2005Credit Derivatives: From the simple to the more advanced21 From Gaussian distribution to default time

2 March 2005Credit Derivatives: From the simple to the more advanced22 Correlation between 2 names 1000 simulations 2 names 2 dimensions – In general 100 names Gaussian/Normal distribution – Transformed to survival time

2 March 2005Credit Derivatives: From the simple to the more advanced23

2 March 2005Credit Derivatives: From the simple to the more advanced24

2 March 2005Credit Derivatives: From the simple to the more advanced25

2 March 2005Credit Derivatives: From the simple to the more advanced26

2 March 2005Credit Derivatives: From the simple to the more advanced27

2 March 2005Credit Derivatives: From the simple to the more advanced28 Default Times, Correlation = 1 Companies Have Different Spreads Spread Company A = 300 Spread Company B = 600 Correlation = 1 Note that when A defaults we always know when B defaults... …but note that they never default at the same time B always defaults earlier

2 March 2005Credit Derivatives: From the simple to the more advanced29 Correlation High correlation: – Defaults happen at the same quantile – Not the same as the same point in time! – Corr = 100% – First default time: look at the name with the highest hazard (CDS spread) Low correlation: – Defaults are independent – Corr = 0% – First default time: Multiplicate all survival times: 0.95^100 = 0.59% Default times: – Always happens as the marginal hazard describes!

2 March 2005Credit Derivatives: From the simple to the more advanced30 Copula function Marginal survival times are described by the hazard! – AND ONLY THE HAZARD – It doesn’t depend on the Gaussian distribution – We only look at the quantiles in the Gaussian distribution Copula = “correlation” describtion – Describes the co-variation among default times – Here: Gaussian multivariate distribution – Other possibilities: T-copula, Gumbel copula, general Archimedian copulas, double T, random factor, etc. – Heavier tails more “extreme observations” Copula correlation different from default time correlation etc.

2 March 2005Credit Derivatives: From the simple to the more advanced31 Subjects not mentioned Other copula/correlation models that explains the correlation smile CDO hedge amounts, deltas in different models CDO behavior when credit spreads change Details of efficient implementation strategies Flat correlation matrix or detailed correlation matrix? Etc. etc.

2 March 2005Credit Derivatives: From the simple to the more advanced32 Conclusion Still a lot of modelling to be done – In particular for correlation smiles The key is to get an efficient implementation that gives accurate risk numbers Market is evolving fast – New products – Standardized products – Documentation – Conventions – Liquidity