subprime mortgage lender in the United States, and its related entities filed voluntary petitions for reorganization under the bankruptcy laws in the United States. New Century Financial Corporation listed liabilities of more than $100 million and announced that the employment of about 3,200 people, more than half the workforce, will be terminated. During the week of July 16, 2007, the Bear Stearns Company, one of the largest global investment banks and securities trading firms in the world, disclosed the two subprime hedge funds which had lost nearly all of their value amid a rapid decline in the market for subprime mortgages.
These just announced the beginning of the Subprime mortgage crisis. This crisis was started from several reasons: First, the rising of the floating subprime rate, which makes the debtors insolvent and the bursting of the US housing bubble; Second, the subprime mortgage were rolled into Mortgage Backed Securities (known as MBSs) by banks and were sold to brokerage firms. These securities were re-splited and packaged as CDOs, it caused such kinds of the structure of credit derivatives were too complicated to be aware of the risk; Third, the most important reason for the research in this article, because of these derivatives lacking of historical rating records, the mathematical pricing models became the only basis for rating; Finally the investors, such as investment banks, retirement funds, hedge funds, municipal funds and administrative funds, bought these credit derivatives according to the rate given by rating agencies. During 2006, about $100 billion subprime mortgages were packaged in nearly $ 375 billion CDOs and then were sold on the U.S. market. Many banks, mortgage lenders, real estate investment trusts (REITs), and hedge funds suffered significant losses as a result of mortgage payment defaults or mortgage asset
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devaluation. As of April 30, 2008 financial institutions had recognized subprime-related losses or write-downs exceeding U.S. $280 billion.
In this article, we hope to find a model more precise than one factor Gaussian copula model and double-t copula model to estimate the fair prices for CDOs. We first provide the one factor model with Normal Inverse Gaussian (NIG) distribution under LHP assumption. Different from normal distribution and student-t distribution, the NIG distribution includes more parameters to control the location, scale and shape of the distribution function and with more flexibility. And NIG distribution is convolution stable under certain conditions and the cumulative distribution function, density and inverse distribution functions can still be computed sufficiently fast.
However, the missing term structure still is a disadvantage of the model, and we can observe 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, we are going to generate an extension of the NIG factor copula model, which allowing for different correlation regimes, and we also try to extend the method of base correlations, which enables NIG model to be an extension of the term-structure dimension. We compare the numerical results of the one factor Gaussian copula model, Double-t copula model, NIG copula model and NIG Regime-Switch model. At last, each tranche’s fair price from NIG Regime-Switch model is more precise.
This paper also targets on different models to conduct risk analysis and hedging strategy. The expected loss of tranches is widely used by credit rating organizations as one of the important indicators for risk measurement. However, it can’t reflect the relative risk level between CDO’s tranches. Therefore, our research adopts the concept of expected loss rate, which use the proportion of expected loss to total
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principal amount to compare the relative risk of each tranche. Moreover, we also utilize the leverage of expected loss to analyze the changes of each tranche’s default risk at different point of time. When considering the spread risk, our paper also computes delta and gamma according to different models and we also use CDS index to conduct delta hedging analysis. Nevertheless, although we can reach to complete hedging when using CDS index to conduct delta hedging, high hedging cost will make mezzanine tranches and senior tranches have negative net income after hedging.
Therefore, we will encounter difficulties when we utilize it practically. As a result, we suggest that the holders of tranches can sell a single asset CDS to avoid the risk of volatile credit spread of a specific target because the spread sensitivity of tranches toward a single CDS is smaller than its own credit spread. Therefore, delta usually smaller than 1 so it may involve lower hedging cost for holders of mezzanine tranches as well as senior tranches if they adopt single asset CDS as their hedging strategy.
This article is organized as follows: In Chapter 1 we introduced some background for CDOs. We then make some brief introductions for several CDOs pricing models from early research in Chapter 2. In Chapter 3, we use the assumptions of LHP to derive the loss distribution function under different models and we also define the risk indicators and hedge parameters. In Chapter 4, we compare numerical results of pricing synthetic CDO tranches of the iTraxx with four different models and we conduct risk analysis and calculation of hedge parameters toward different models.
Finally, we give a brief conclusion in last chapter.