Correlation Skew

As noted in section 2.4, the latent variables X ’s can be thought of as the asset _i values of companies. Therefore, the correlation parameter ρ in (3.1) can be viewed as the correlation between asset values of two companies. Since the asset value of a company cannot be directly observed in the market, it is often assumed that the correlation between their equities equals the correlation between their asset values. If we allow the correlation parameter ρ to be different for each company, then the standard market model can be rewritten as follows:

1 , 0 1, ~ (0,1),{ }ⁿ1~ (0,1), 1, 2,...,

i i i i i i i

X = ρ M + −ρ Z ≤ρ ≤ M N Z ₌ N i= n (3.2)

Since the correlation between X and M is _i ρ_i , the parameter ρ_i for each company can be estimated by the correlation between the equity return of the company and the return of a market index, i.e., the beta coefficient of the company’s equity.

However, as Walker (2005) points out, the default correlation under risk-neutral measure can be very different from the default correlation under real world measure.

Hence, the use of the equity correlations in real world measure is inadequate for pricing basket credit derivatives..

As the credit derivatives market grows, it becomes possible to calibrate risk-neutral correlation ρ in (3.1) from observed market prices of credit products directly, which avoids the hypothesis that the risk-neutral default correlation is the same as the real world default correlation. Since index tranches are standardized credit products and have been actively traded, dependence calibration from credit derivatives has been massively used with index tranches. Because index tranches involve a large number of

been assumed to be the same among all reference entities in the portfolio, i.e., , 1, 2, ,

i i n

ρ ρ= = . This leads to formula (3.1).

However, when the correlation parameter is implied through inverting the standard market model, different correlation parameter is needed to match each tranche spreads even if those tranches have the same underlying index portfolio. The correlation found by matching the model generated spreads to market quoted spreads is called implied correlation or compound correlation.

iTraxx Europe 5 Year, Series 6 on 4 January 2007 Source: Bloomberg

Tranche 0-3% 3-6% 6-9% 9-12% 12-22% Index

Spread/Upfront fee 10% 44 bp 12 bp 4 bp 1 bp 22 bp Table 3.1: Market quotes of iTraxx Europe 5 year on January 4, 2007.

0.00 0.05 0.10 0.15 0.20 0.25

Correlation 0.1535 0.0824 0.1846 0.1822 0.1987

0-3% 3-6% 6-9% 9-12% 12-22%

Tranche

Figure 3.1: Compound correlations of iTraxx Europe 5 year on January 4, 2007.¹ Consider the market quotes of iTraxx Europe 5 year on January 4, 2007 shown in Table 3.1. The compound correlations of index tranches are presented in Figure 3.1.

Because the mezzanine tranche typically has a lower compound correlation than the

1 When searching the compound correlations of 3-6% tranche and 6-9% tranche, there are two values such that the marked-to-market values of these two tranches have zero values. The values are 0.0824 and 0.9646 for 3-6% tranche and 0.1846 and 0.9393 for 6-9% tranche. Because 0.9646 and 0.9393 are unreasonably high compared to the compound correlations of other tranches, they are ruled out and only 0.0824 and 0.1846 are reported.

equity or senior tranche, this phenomenon is called correlation smile or correlation skew.

Although the standard market model is simplistic in that the default correlations are assumed to be the same among all reference entities, the correlation parameter ρ should not depend on the attachment point and the detachment point of the tranche priced in the model. Theoretically, one would expect an almost flat implied correlation curve among tranches. However, the existence of correlation smile shows that there must be some problems in the standard market model.

To investigate the meanings of correlation smile further, it is necessary to

0-3% Tranche

-0.40 -0.20 0.00 0.20 0.40 0.60 0.80 1.00

0.0 0.2 0.4 0.6 0.8 1.0

Correlation

Market Value

3-6% Tranche

-0.15 -0.10 -0.05 0.00 0.05 0.10

0 0.2 0.4 0.6 0.8 1

Correlation

Market Value

6-9% Tranche

-0.10 -0.08 -0.06 -0.04 -0.02 0.00 0.02 0.04

0 0.2 0.4 0.6 0.8 1

Correlation

Market Value

9-12% Tranche

-0.08 -0.06 -0.04 -0.02 0.00 0.02

0 0.2 0.4 0.6 0.8 1

Correlation

Market Value

12-22% Tranche

-0.25 -0.20 -0.15 -0.10 -0.05 0.00 0.05

0 0.2 0.4 0.6 0.8 1

Correlation

Market value

understand the relationships between the market values of tranches and the correlation parameter in formula (3.1). The relationships between the market value of each tranche and the correlation parameter are plotted in Figure 3.2. Formula (2.1) is used to determine the market values of tranches under the standard market model with the market spreads or the upfront fee presented in Table 3.1 applied. The total notional is assumed to be one unit of currency, and the recovery rate is taken as 40% for all of reference entities in Figure 3.2. From Figure 3.2, it is observed that when the correlation increases, the market value of the equity tranche (0-3%) will rise. It is caused by the fact that higher correlation leads to higher probability of joint default occurrence. In other words, high correlation implies that there will be either few defaults or many and thus have positive effect on the market value of equity tranche. This observation holds true generally for the equity tranche. That’s why market participants often call investing equity tranche as long correlation.

If the one-factor Gaussian copula model uses the compound correlation of the mezzanine tranche to price the equity tranche, the market value of the equity tranche will be underestimated. It is because the compound correlation of the equity tranche is higher than the compound correlation of the mezzanine tranche and the value of the equity tranche is an increasing function of correlation. Therefore, the shape of correlation smile means essentially that Gaussian copula model underestimates the chance of observing a very high or very low number of defaults. Since a fat-tailed distribution has a higher probability to observe extreme events than the normal distribution, it means that the market implied loss distribution is fat-tailed.

Nonetheless, the subprime mortgage crisis not only has negative impact on CDOs market but also changes the shape of correlation smile as well. Taking the market quotes in Table 2.1 for example, we plot the corresponding compound correlations in Figure 3.3. From Figure 3.3, it is observed that the shape and the level of correlation smile have changed a lot compared to Figure 3.1 after the subprime mortgage crisis.

Actually, the 6-9% tranche has two values of compound correlations and they are 0.0466 and 0.9753; the 9-12% tranche have two values of compound correlations and they are 0.1540 and 1.0000. However, even if we take the multiple values into account, it cannot change the fact that the compound correlation of equity tranche and senior tranche increase a lot, especially for the 0-3% tranche. Thus, the subprime mortgage

crisis must cause some structural changes in the CDO markets and raise doubts about the applicability of the current CDO pricing models.

0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90

Correlation 0.4716 0.8476 0.0466 0.1540 0.2495

0-3% 3-6% 6-9% 9-12% 12-22%

Tranche

Figure 3.3: Compound correlations of iTraxx Europe 5 year on April 7, 2008.