1. New Developments in Correlation Modelling Pedro A. C. Tavares Paris, May 2005

2. Overview Gaussian-copula model Tranche and calendar correlation skew Base correlations Composite Basket Model: overview, data fit and leverages Next step: stochastic intensity modelling

3. Gaussian-Copula Model

4. Gaussian-copula Model With the choice of a single factor linear correlation model for the A i, GC allows simple and efficient implementation. However, calibration of the model is not possible across indices, maturities or even tranches written on the same portfolio. Survival probability of asset i to time T Set of correlated random variables

5. Gaussian-copula Correlation Skew The Gaussian-copula correlation structure seen in this example is a typical feature. Hence the name correlation skew (although smile would be more suitable) A GC correlation does not always exist for all market quotes as is the case for the 3-6% tranches above. iTraxx 5y tranches on 20 May 2005 (63bp) TrancheBidOfferMidGC 0-3% *38.50%40.00%39.25%15% 3-6%1.15%1.24%1.20%N/A 6-9%0.36%0.44%0.40%7% 9-12%0.20%0.29%0.25%13% 12-22%0.16%0.24%0.20%27% (*) Up-front value quoted, premium is 5%. Source: Merrill Lynch.

6. 3-6% iTraxx Tranche (20 May 2005) For this tranche we are unable to match the broker mid quote, no matter what value of correlation we use (excluding the introduction of assets with negative correlation).

7. “Calendar” Skew A useful property to have in a model would be that when all is assumed constant, then calibration parameters are also constant. In GC that doesn’t happen as the first-to-default example shows In order to recover the flat financing cost we must increase the GC correlation very aggressively This feature is important when dealing with forward trades First-loss (5 asset, 40bp) TermsPar (20%)GC (spot) 1y1.19%20.0% 1-2y2.38%91.1% 5y2.79%20.0% 5-10y4.07%66.8%

8. Base Correlation

9. Since their introduction, base correlation interpolation has become a useful quoting tool. It mostly guarantees that a correlation is found for any quote However the method does not translate well to the pricing of bespoke baskets or of exotic tranches (CDO 2, for example) Under certain conditions it introduces arbitrage opportunities (steep skew curve) It makes a poor risk-management model Single tranche “Lower” tranche “Upper” tranche

10. Base Correlation Fit And Extrapolation iTraxx Base Tranches ExhaustBase 3%15% 6%32% 9%44% 12%53% 22%72% Extrapolation on the basis of subordination alone allows no extrapolation away from the quoted baskets. However base correlation is still useful as a quoting tool. Replacing subordination with a basket sensitive quantity (spreads for example) would give us a more robust approach. -- 20% 40% 60% 80% 100% 0%3%6%9%12%15%18%21%24%

11. Composite Basket Model

12. Gaussian-copula (with flat correlation) assigns excessive value to the mezzanine tranches, and not enough to equity. We postulate that in addition to the Gaussian-copula, defaults are driven by additional exponential drivers: a global systemic shock that triggers defaults on all assets and asset specific idiosyncratic shocks that cause individual assets to default. At a given time and for a particular asset, the probability of survival can then be written as product of three terms, as above. Arbitrage requires that this product must be invariant. Systemic Correlated Idiosyncrati c

13. CBM Loss Distribution (I) Under certain conditions we can interpret this factorisation as a sum of CDS spreads: Conditioned on the systemic shock and a single factor GC one, the assets are independent. In the case of N identical assets with unit loss amounts, we can then write the loss density functions as: (We dropped the time argument for the sake of clarity)

14. CBM Loss Distribution (II) The survival probabilities then expand as postulated in the model:` With the expression above and observing that the systemic factor generates only two states: default with probability 1-p S and survival with probability p S, we can integrate that factor: Where 1 l,N is the indicator function

15. CBM Loss Distribution (III) Given a density g(X C ) we can integrate the remaining factor: We can easily generalise this to the case where assets are not identical by replacing the binomial density with the suitable convolution of each asset. Because the density of each asset (assuming known recovery) is a simple two state function (default and survival) this can be done using a recursive relation. With a little effort we can also generalise to multiple systemic and copula shocks.

16. CBM Calibration We observe that even in the current volatile environment we can find a set of parameters that fits the full set of quoted tranches. With these parameters there is little GC contribution left. Calibration to multiple indices and tenors is possible with an increase in error margin. iTraxx 5y tranches on 20 May 2005 TrancheBidOfferMidCBM 0-3% *38.50%40.00%39.25%38.67% 3-6%1.15%1.24%1.20%1.22% 6-9%0.36%0.44%0.40%0.39% 9-12%0.20%0.29%0.25%0.23% 12-22%0.16%0.24%0.20% (*) Up-front value quoted, premium is 5% Systemic0.20% Idiosyncratic1.16% Correlation79.94%

17. Tranche Leverages

18. Index tranches are most often traded together with the corresponding index hedge These results strongly suggest that market participants use GC base correlation for hedging tranches The CBM results are typical of this model: higher equity leverages, lower senior ones. iTraxx 5y tranches on 20 May 2005 TrancheBrokerGCBaseCBM (sp)CBM (id) 0-3%17.517.2117.2229.0732.24 3-6%6.5N/A5.497.776.09 6-9%2.253.371.941.380.44 9-12%1.41.811.120.530.17 12-22%0.61.170.700.00

19. Stochastic Intensity

20. Market development trends force us to consider both long term basket trades and short-term options in a consistent framework. However, popular modelling approaches offer no such framework: dealers typically price baskets with default copula models and options with intensity or spread diffusion models (Black-Scholes). Intensity or spread diffusion alone does not generate sufficient default correlation to match market prices. A combination of diffusion volatility and correlated jumps offers “easy” calibration and is able to achieve the required default correlations. Warning: stochastic intensity requires a review of CDS itself (relation between default and coupon leg changes).

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