1 / 32

Credit Derivatives: From the simple to the more advanced

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

prince
Download Presentation

Credit Derivatives: From the simple to the more advanced

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


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

  2. 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 Credit Derivatives: From the simple to the more advanced

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

  4. 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) Credit Derivatives: From the simple to the more advanced

  5. 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: Payment in the event of default Discount factor Probability of default at time t Premium payments Accrual factor Discount factor Survival probability Credit Derivatives: From the simple to the more advanced

  6. Credit Triangle - What Determines the Spread? Assume hazard rate is constant Assume premium is paid continuously Credit Derivatives: From the simple to the more advanced

  7. 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. • PV01n = value of premium leg for name n • Not correlation dependent Credit Derivatives: From the simple to the more advanced

  8. First-to-Default Basket • 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 Premium leg: Basket buyer Basket seller First-to-default Spread Continues until the first default or until maturity Protection leg: Basket buyer Basket seller 100 – Recovery on defaulted asset Only in the event of default, and only the first default Credit Derivatives: From the simple to the more advanced

  9. Standardized CDO tranches 100% • 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 22% Mezzanine 12% 9% 6% 3% 3% equity Credit Derivatives: From the simple to the more advanced

  10. Reference Gaussian copula model • N credit names, i = 1,…,N • Default times: ~ •  curves bootstrapped from CDS quotes • Ti correlated through the copula: • Fi(Ti) = (Xi) with X = (X1,…,XN)t ~ N(0,) • Note: Fi(Ti) = (Xi)  U[0,1] •  correlation matrix, variance 1, constant correlation  • In model: correlation independent of product to be priced Credit Derivatives: From the simple to the more advanced

  11. Prices in the market has a correlation smile • In practice: • Correlation depends on product, 7-oct-2004, 5Y iTraxx Europe • Tranche • Maturity Credit Derivatives: From the simple to the more advanced

  12. 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 Credit Derivatives: From the simple to the more advanced

  13. 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? Credit Derivatives: From the simple to the more advanced

  14. 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 Credit Derivatives: From the simple to the more advanced

  15. Base correlations Short Long Credit Derivatives: From the simple to the more advanced

  16. Base versus compound correlations Credit Derivatives: From the simple to the more advanced

  17. 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 Credit Derivatives: From the simple to the more advanced

  18. Implementation of Gaussian copula • Factor decomposition: • M, Zi independent standard Gaussian, • Xi low  early default • FFT/Recursive: • Given T: use independence conditional on M and calculate loss distribution analyticly, next integrate over M • Simulation: • Simulate Ti, straight forward • Slower, in particular for risk, but more flexible • All credit risk can be calculated in same simulation run as the basic pricing Credit Derivatives: From the simple to the more advanced

  19. 100% 12% 9% 6% 3% Pricing of CDOs by simulation • 100 names • Make, say, 100000 simulations: • Simulate default times of all 100 names • Price value of cash-flow in that scenario • Do it all again, 100000 times • Price = average of simulated values 88% Super senior Mezzanine 3% equity Credit Derivatives: From the simple to the more advanced

  20. Default time simulationHazard and survival curve S = exp(-H*time) Credit Derivatives: From the simple to the more advanced

  21. From Gaussian distribution to default time Credit Derivatives: From the simple to the more advanced

  22. 1000 simulations 2 names 2 dimensions In general 100 names Gaussian/Normal distribution Transformed to survival time Correlation between 2 names Credit Derivatives: From the simple to the more advanced

  23. Credit Derivatives: From the simple to the more advanced

  24. Credit Derivatives: From the simple to the more advanced

  25. Credit Derivatives: From the simple to the more advanced

  26. Credit Derivatives: From the simple to the more advanced

  27. Credit Derivatives: From the simple to the more advanced

  28. Default Times, Correlation = 1Companies 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 Credit Derivatives: From the simple to the more advanced

  29. 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! Credit Derivatives: From the simple to the more advanced

  30. 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. Credit Derivatives: From the simple to the more advanced

  31. 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. Credit Derivatives: From the simple to the more advanced

  32. 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 Credit Derivatives: From the simple to the more advanced

More Related