Credit derivatives such as CDOs have become extremely popular as they represent a straightforward vehicle for credit correlation trading. Even if an accurate and consensual modeling of credit correlation is still not available, Gaussian copula models--known also as market models--directly calibrated on single-tranche synthetic CDOs' implied correlations are in very common use.
Gaussian Copula Models: A Natural Framework
Gaussian copula models provide a natural framework using correlation matrices and Gaussian probability distributions. Under simple assumptions, such as the existence of a principal factor, it is possible to reduce dramatically the dimension of the correlation matrix calibration to a few parameters. It is even possible to rebuild a complete base correlation surface and to trade any CDOs based on this correlation surface, similar to how call spreads are traded on the equity market using the implied volatility surface. This equivalence can be drawn up to pricing, hedging and replication. The same framework can be applied to nth-to-default CDS baskets and, if traded on the base correlation surface, analogies with equity can be found using the implied volatility surface.
Challenges To Gaussian Models
These models have a major drawback, however. While they completely determine spot default probability distribution, they give no information as to the dynamics of default probability. That is, they do not provide any measure of credit volatility or of conditional default probability distributions. For linear correlation products insensitive to credit volatility such as vanilla CDOs or nthto-default CDS baskets, these models are, however, satisfactory.
Higher Order Credit Correlation And Credit Volatility
After having expanded the range of linear products to its full extent, credit correlation trading is focusing on credit options trading directly linked to higher-order credit correlation and credit volatility. Such products, if well understood, can become a much more efficient and far less risky way for investors to diversify portfolios or buy credit protection.
Higher-order credit correlation derivatives, for example CDO squareds, are a way to trade not only CDO correlation but also the correlation between different standard CDO tranches. To capture the correlation between CDO correlations is equivalent to determining the credit volatility of the underlying default dynamics. It is feasible to adapt market models by adding these dynamics to the base correlation surface. To remain manageable, most of the possible assumptions can be summarized as the assumption of market efficiency. This, however, is totally unrealistic and leads to under-estimation of credit volatility. The volatility risk premium obtained with market models is far too low.
Trading Credit Volatility Directly
The challenge becomes even more difficult when trading credit volatility directly, with options on credit-default swap indices. The prices given by market models are completely out of the market, or the market models imply a credit volatility smile which is far too skewed.
Trying to constrain externally conditional default probabilities within default probability market models is surely not the best idea. It would be much more relevant to have a model which simultaneously provides credit correlation and credit volatility as strongly dependent parameters.
The One-Factor Model
In the interest rate market, when it is necessary to correlate the different tenors of the yield curve, a one-factor model, i.e. dynamic, is built. This factor--the short term interest rate--is not tradable or observable, but the dynamic parameters can be inferred from market data. The same procedure may be used for credit. Assuming the existence of a non-tradable risk and an invisible risk factor, a random dynamic which defines the default state as soon as it crosses a predetermined trigger is a very appealing idea. This approach leads to models known as structural models where the risk factor is economically explained as being a function of the firm value of the issuer.
The dynamic of the credit-risk factor can be calibrated by concurrently using CDS index quotes, CDO correlations and credit-default swaption volatilities.
When properly parameterized, structural models relinquish the market quotes of CDS indices, standard CDO and credit default swaptions used by the calibration procedure. The implied interpolation of the default probability and the base CDO correlation surface may differ widely from those imposed by market models, however. Most of the time, the calibration of the credit risk factor dynamic has to take into account smoothing conditions or additional constraints to obtain relevant interpolations.
An Equity-Based Structural Credit Model
Trying to give a direct meaning to the credit risk factor can be very useful to analyze the risks embedded in a hybrid structure. It is well known that equity skew is explained by the default dynamics of equity which contains jumps. It is also clear that CDS payouts may be replicated with out-of-the-money equity puts. Hence, there is a strong link between equity data and credit data. To directly express the credit-risk factor as a function of the equity spot price and the equity volatility smile, it is necessary to understand the risk profiles of equity structures depending on extreme values of the underlying. For example a crash option, which pays out in the case of strong downward movement, can only be understood if valued within an equity-based structural credit model.
To adapt the calibration and credit dynamics to the structure to be priced may seem counterintuitive. If credit structures are well segmented, however, and if the data used by the calibration procedures remain homogeneous for the aggregated risk analysis such as value-at-risk, this one-by-one modeling strategy may be extremely accurate. Moreover, special attention should be paid to model arbitrage opportunities created by this kind of calibration and pricing strategy.
Modelling Credit Correlations--Future Needs
For purely credit sensitive products, the key problem is to select a dynamic rich enough to capture properly the credit market data, sufficiently flexible to reflect many different patterns for the conditional default probabilities and simple enough to remain a tractable and not over-constrained model. However, as the results obtained with structural models for non linear credit products strongly depend on the underlying credit dynamics, the implementation and use of such a model should be conservative, cautious and rely on heavy back-testing.
This week's Learning Curve was written by Jean-Noel Dordain, director of quantative research at Sophis.
