This paper is based on the structure of one-factor copula and we use the crash-NIG copula model raised by Anna Schloesser (2009) to construct the synthetic CDO valuation model. When comparing with the valuation result of Gaussian copula, for equity tranches, nonsystematic risk factors will have larger influence to tranches than systematic risk factors do, which therefore let the credit spread increase in the valuation.
For senior tranches, we can see larger influence in systematic risk factors, so the credit spread will increase in the valuation. We can observe that all models possess higher absolute error of tranche spreads regarding the market quote during financial crisis, so we include regime-switch into our model, which can reduce the valuation error effectively when market is relatively unstable. In recently years of practical uses, the standard model for valuing CDO is one-factor Gaussian copula, but the valuation result of Gaussian copula will generate spreads that are lower then market quotes in equity tranches and senior tranches when we assumed the factor loading is a constant;
moreover, spreads of mezzanine tranches will be higher than market quotes and the absolute error between tranche spreads and market quotes will be large when using Gaussian copula to calculate during the financial crisis. If we use one-factor NIG-Regime switch copula instead to conduct the valuation of synthetic CDO, we can get the result that is closer the market quotes and it can also reduce the error in unstable market situation.
We follow the valuation model introduced by Anna Schloesser (2009), which is that we add the calculation of tranches risk besides the valuations of CDO. According to the result of this paper, the realized loss of tranches will let the risks of tranches decrease and the credit enhancement of tranches suffer loss will make the risk of traches increase. Take equity tranches for example, since they take the default loss of CDS
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portfolio first, they have higher risks in the beginning of contracts. As for the later period of contract, the loss is occurred for sure so the loss uncertainties for tranches reduce, which result in lower risk. For mezzanine tranches, equity tranches serve as protection for them, so the loss probability of mezzanine tranches will increase as time goes by. Until mezzanine tranches themselves start suffering loss, risk will decrease since partial loss is also realized. Risk of senior tranches do not change obviously during the contract period, and they still remain their characteristic of low risk as well as low return. Additionally, we also computed the hedge parameters based on the valuation model raised by Anna Schloesser (2009). Investors of synthetic CDO simultaneously buy tranches and sell existing credit index based on the hedge ratio to hedge. Although this is the way that can completely eliminate the risk of volatilities in credit spread, it may let investors of some tranches result in negative net income. For the assumed products we used in this research, the hedge cost is too high if we use credit index to hedge, except for investors of equity tranches. As a result, it may not be a feasible way to hedge. In the meantime, if the investors for mezzanine tranches and senior tranches can simultaneously buy tranches and sell single-name CDS to hedge, and the hedge parameters can use our valuation model along with CDS valuation model to generate. If the investors want to reach to better hedge effect, they can choose to adopt the single-name CDS model with larger hedge parameters to hedge because hedge parameters are actually the sensitivity of tranches regarding its changes in credit spread.
Although we can’t eliminate the risk of credit spreads of targeted portfolio completely by using single-name CDS to hedge, it possesses relatively low cost; moreover, if we can correctly select CDS with higher sensitivity regarding of tranches to hedge, they can still serve partial effects of hedge
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1. Anderson, W. (1991), Continuous-Time Markov chains, Springer Verlag.
2. Andersen, L. & Sidenius, J. (2005), Extensions to the Gaussian copula: random recovery and random factor loadings, Journal of credit risk 1(1).
3. Anna, K., Bernd, S. & Ralf, W. (2005), The Normal Inverse Gaussian Distribution for Synthetic CDO Pricing. Working paper.
4. Anna, S. & Rudi, Z. (2009), Crash-NIG copula model: regime-switching credit portfolio modeling through the crisis.
5. Barndorff-Nielsen, Ole E. (2007), Normal Inverse Gaussian Distributions and Stochastic Volatility Modeling, Journal of Statistics.
6. Bluhm, C., Laurent, J.-P. & Gregory, J. (2005), A comparative analysis of CDO pricing models. Working paper, BNP-Paribas.
7. Burtschell, X., J. Gregory & J-P. Laurent, 2005, A Comparative Analysis of CDO Pricing Models. Working paper, ISFA Actuarial School, University of Lyon &
BNP-Paribas.
8. Duffie, D. & Singleton, K. (1999), Modeling term structures of defaultable bonds, Review of Financial Studies 12, 687-720.
9. Hull, J. & White, A. (2004), Valuation of a CDO and an n-th to default CDS without a Monte Carlo simulation, Journal of Derivatives.
10. Hull, J., Predescu, M. & White, A. (2005), The valuation of correlation-dependent credit derivatives using a structural model. Working paper.
11. Jarrow, R., Lando, D. & Turnbull, S. (1997), A Markov model for the term structure of credit spread, Review of Financial Studies 10, 481-523.
12. Li, D. X. (2000), On default correlation: A copula function approach, Journal of Fixed Income 9(4), 43-54.
‧ 國
立 政 治 大 學
‧
N a tio na
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13. Max S. (2011), Calibration of multi-period single-factor Gaussian copula models for CDO pricing, University of Toronto.
14. Merton, R. (1974), On the pricing of corporate debt: The risk structure of interest rates. Journal of Finance 29(2), 449-470.
15. Nelsen, R.B. (2005), An Introduction to Copulas. Springer. Second Edition.
16. O'Kane, D. & Schloegl, L. (2003), An analytical portfolio credit model with tail dependence, Quantitative credit research. Lehman Brothers.
17. Vasicek, O. (1987), Probability of loss on loan portfolio. Memo, KMV corporation.
18. Yan ya, C. (2007), Pricing CDOs with One Factor Double Mixture Distribution Copula Model.