Chapter 2 Literature Review
The calculation of loss distributions of the portfolio of reference instruments over different time horizons is the central problem of pricing synthetic CDOs. The factor copula approach for modeling correlated defaults has become very popular. Making the additional simplifying assumption of a large homogeneous portfolio (LHP), i.e.
assuming it is possible to approximate the real reference credit portfolio with a portfolio consisting of a large number of equally weighted identical instruments (having the same term structure of default probabilities, recovery rates, and correlations to the common factor), we get a closed-form analytic synthetic CDO pricing formula. This LHP limit approximation employing the Law of Large Numbers was first proposed by Vasicek.
In the case of the one factor Gaussian copula all integrals in the pricing formulas can be computed analytically. Due to its simplicity, this model has become market standard. However, there is a fundamental problem: if we calculate the correlations that are implied by the market prices of tranches of the same CDO using the LHP approach, we do not get the same correlation over the whole structure but observe a correlation smile. The main explanation of this phenomenon is the lack of tail dependence of the Gaussian copula. After that, many authors proposed other different approaches (i.e. use other copulas with more tail dependence), like the Student t-copula in O’kane and Schloegl (2003) and the double t-copula in Hull and White (2004). Burtschell et al. (2005) compared Gaussian factor copula model with several copula models, such as the stochastic correlation extension to Gaussian copula, Student t-copula, double t-copula, Clayton copula and Marshall-Olkin copula. It showed that the results of Student-t copula and Clayton copula models were similar to the Gaussian factor copula model. The Marshall-Olkin copula leads to a dramatic
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factor loading introduced in Andersen and Sidenius (2005) leads to a correlation smile to stochastic correlation copula.Unfortunately, the integrals in the synthetic CDO pricing formula in the LHP model that is based on the double t copula cannot be computed analytically. The major problem is the instability of the Student t distribution under convolution. The calculation of the default thresholds requires a numerical root search procedure involving numerical integration that increases the computation time dramatically.
Thus, finding a different heavy tailed distribution that is similar to the Student t but stable under convolution would help to decrease the computation time tremendously.
In our opinion, the Normal Inverse Gaussian (NIG) distribution is an appropriate distribution to solve the problem. The family of NIG distributions is a special case of the generalized hyperbolic distributions (Barndorff-Nielsen). Due to their specific characteristics, NIG distributions are very interesting for applications in finance - they are a generally flexible four parameter distribution family that can produce fat tails and skewness, the class is convolution stable under certain conditions and the cumulative distribution function, density and inverse distribution functions can still be computed sufficiently fast. The distribution has been employed, e.g., for stochastic volatility modeling (Barndorff-Nielsen).
After the correlation smile was improved by using other distributions, the missing term structure still is a disadvantage of the model. Both, the Gaussian and the NIG models (as well as all analogue models with different distribution assumptions) just average the correlations and other model parameters over the complete lifetime of a CDO tranche. Thus, applying the model to the long-dated tranches is not consistent
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with the short-dated ones. The practitioners tried to fix this problem by extending the method of base correlations. In contrast to the Gaussian model, the NIG model allows for an extension into the term-structure dimension. This extension is not only helpful for the pricing of CDO tranches with different maturities, but also important for defining a consistent simulation framework for risk management applications.
From observing the trend of iTraxx spread, Anna Schloesser (2009) detects that correlations are especially high during the current sup-prime crisis. The year before the crisis in July 2007 began, the correlation was in contrast very low. For this reason, an extension of the NIG factor copula model allowing for different correlation regimes is expected to better reflect the reality than the model with the constant correlation.
Figure 2-1 Calibration with the 5-year data from iTraxx spread
It detect the second state during the market turbulences after then downgrade of Ford and General Motors in May, 2005, and during the sub-prime crisis starting in July, 2007, with a short break in September-October, 2007.(Source: Anna, Schloesser (2009). Crash-NIG copula model:
regime-switching credit portfolio modeling through the crisis)
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3.1 Valuation of Synthetic CDOs
First, we consider a synthetic CDO with a reference portfolio consisting of credit default swaps only. A protection seller of a synthetic CDO tranche receives from the protection buyer spread payments on the outstanding notional at regular payment dates (usually quarterly). If the total loss of the reference credit portfolio exceeds the notional of the subordinated tranches, the protection seller has to make compensation payments for these losses to the protection buyer.
Basically, the pricing of a synthetic CDO tranche that takes losses from K1 to K2 (with 0 ≤ K1 < K2 ≤ 1, with K1 represents tranches start taking loss and K2 represents tranches end taking loss) of the reference portfolio works in the same way as the pricing of a credit default swap. Let’s assume that
0 1 leg can be calculated according to: