In the last decade, a considerable growth has been added to the volume of the credit risk derivatives market. This growth has been followed by the current financial market turbulence. These two periods have outlined how significant and important are the credit derivatives market and its products. Modelling-wise, this growth has parallelised by more complicated and assembled credit derivatives products such as mth to default Credit Default Swaps (CDS), m out of n (CDS) and collateralised debt obligation (CDO). In this thesis, the Lévy process has been proposed to generalise and overcome the Credit Risk derivatives standard pricing model's limitations, i.e. Gaussian Factor Copula Model. One of the most important drawbacks is that it has a lack of tail dependence or, in other words, it needs more skewed correlation. However, by the Lévy Factor Copula Model, the microscopic approach of exploring this factor copula models has been developed and standardised to incorporate an endless number of distribution alternatives those admits the Lévy process. Since the Lévy process could include a variety of processes structural assumptions from pure jumps to continuous stochastic, then those distributions who admit this process could represent asymmetry and fat tails as they could characterise symmetry and normal tails. As a consequence they could capture both high and low events' probabilities. Subsequently, other techniques those could enhance the skewness of its correlation and be incorporated within the Lévy Factor Copula Model has been proposed, i.e. the 'Stochastic Correlated Lévy Factor Copula Model' and 'Lévy Random Factor Loading Copula Model'. Then the Lévy process has been applied through a number of proposed Pricing Basket CDS&CDO by Lévy Factor Copula and its skewed versions and evaluated by V-FFT limiting and mixture cases of the Lévy Skew Alpha-Stable distribution and Generalized Hyperbolic distribution. Numerically, the characteristic functions of the mth to default CDS's and (n/m) th to default CDS's number of defaults, the CDO's cumulative loss, and loss given default are evaluated by semi-explicit techniques, i.e. via the DFT's Fast form (FFT) and the proposed Very Fast form (VFFT). This technique through its fast and very fast forms reduce the computational complexity from O(N2) to, respectively, O(N log2 N ) and O(N ).

近年來，信用衍生性金融商品蓬勃發展，市場上陸續出現不同特色的信用擔保憑證。過去評價信用衍生性金融商品多採用Hull & White (2004)年所發表的因子相關結構型模型(factor copula approach)。由於因子相關模型在描述違約事件，可降低處理維度，使得計算更容易處理，更方便建立出損失分配，讓評價工作更順利進行。但是，降低維度的便利，卻犧牲了違約時點的動態描述，在因子模型中，我們無法掌握損失分配的期間結構，所以只能處理單一到期日的信用衍生性金融商品。 但市場上逐漸出現具有時間相關性的信用金融商品，例如：遠期生效型信用擔保憑證(Forward-starting CDO)、信用擔保憑證分券選擇權(Option on CDO tranches)、重設型信用擔保憑證等。其中遠期生效契約的特色在於，在生效日之前，標的資產若違約，並不構成損失的發生，只會將此商品從投資標的中剔除。故投資人在生效日之前，受到一層信用保護，所以相較於同天到期的信用擔保憑證，會使遠期契約的信用價差會比較低，可降低發行商的成本。在加上近年來，信用曲線出現越來越陡峭的情形，代表到期日相差越長，報酬差異越大，所以投資較長天期的商品，相對報酬提高較多。而次順位分券信用價差近年來下降許多，不少投資人為了達到報酬目標，轉而投資較長天期的信用投資產品。而且信用曲線過於陡峭，投資人預期未來違約環境會呈現平緩或變佳的趨勢，可以透過購買遠期契約，來獲得投資利潤。 由於我們不想放棄因子相關性結構模型在使用上簡便的優勢，所以試圖將跨期相關因子引入因子模型，將期間之間的相關性考慮進去，讓遠期生效信用擔保憑證的評價工作得以運行。除此之外，我們分析各分券對參數的敏感性，並加以探討其中的經濟意涵，最後以討論遠期信用擔保憑證避險的策略作結。

In the thesis we perform analysis of systemic risk in the financial and energy sector in Europe. As the econometric tool for estimating dependencies across the subjects we employ factor copula model with GAS dynamics of Oh & Patton (2013b). We apply this model to daily CDS spreads. Based on the estimated results we perform Monte Carlo simulations in order to obtain future values of CDS spreads and measure probability of systemic events. We conclude that substantially higher systemic risk is present within the financial sector. We also find that the most systemic companies from both sectors come from Spain. JEL Classification C53, C55, C58, G17 Keywords Credit Default Swap, Energy Sector, Factor Copula, Financial Sector, Generalized Autore- gressive Score Model, Systemic Risk Author's e-mail matej.nevrla@gmail.com Supervisor's e-mail barunik@fsv.cuni.cz

在市場上最常被用來評價抵押債務債券(Collateralized Debt Obligation, CDO)的分析方法即為應用大樣本同質性資產組合(Large Homogeneous Portfolio, LHP)假設之單因子關聯結構模型(One Factor Copula Model)。由過去文獻指出，自2008年起，抵押債務債券的商品結構已漸漸出現改變，而目前所延伸之各種單因子關聯結構模型在新型商品的評價結果中皆仍有改善空間。 在本文中使用標準化偏斜Student-t分配(Standardized Skew Student-t distribution, SSTD)取代傳統的高斯分配進行抵押債務債券之分券的評價，此分配擁有控制分配偏態與峰態的參數。但是與Student-t分配相同，SSTD同樣不具備穩定的摺積(convolution)性質，因此在評價過程中會額外消耗部分時間。而在實證分析中，以單因子SSTD關聯結構模型評價擔保債務債券新型商品之分券時得到了較佳的結果，並且比單因子高斯關聯結構模型擁有更多參數以符合實際需求。 / The most widely used method for pricing collateralized debt obligation(CDO) is the one factor copula model with Large Homogeneous Portfolio assumption. Based on the literature of discussing, the structure of CDO had been changed gradually since 2008. The effects for pricing new type CDO tranches in the current extended one factor copula models are still improvable. In this article, we substitute the Gaussian distribution with the Standardized Skew Student-t distribution(SSTD) for pricing CDO tranches, and it has the features of heavy-tail and skewness. However, similar to the Student-t distribution, the SSTD is not stable under convolution as well. For this reason, it takes extra time in the pricing process. The empirical analysis shows that the one factor SSTD copula model has a good effect for pricing new type CDO tranches, and furthermore it brings more flexibility to the one factor Gaussian copula model.

根據以往探討評價合成型抵押擔保債務憑證之文獻研究，最廣泛使用的方法應為大樣本一致性資產組合(large homogeneous portfolio portfolio；LHP)假設之單因子常態關聯結構模型來評價，但會因為常態分配的厚尾度及偏斜性造成與市場報價間的差異過大，且會造成相關性微笑曲線現象。故像是Kalemanova et al.在2007年提出之應用LHP假設的單因子Normal Inverse Gaussian(NIG)關聯結構模型以及邱嬿燁(2007)提出NIG及Closed Skew Normal(CSN)複合分配之單因子關聯結構模型(MIX模型)皆是為了改善其在各分劵評價時能達到更佳的評價結果 ，然而過去的文獻在評價合成型抵押擔保債務憑證時，需要將CDS價差、各分劵真實報價之資訊導入模型，並藉由此兩種資訊進而得到相關係數及報價，故靜態模型大多為事後之驗證，在靜態模型方面，我們嘗試使用不同概念之CDS取法以及相對到期日期數遞減之概念來比較此兩種不同方法與原始的關聯結構模型進行比較分析，在動態模型方面，我們應用與時間序列相關之方法套入以往的評價模型，針對不同商品結構的合成型抵押擔保債券評價，並由實證分析來比較此兩種模型，而在最後，我們利用時間序列模型來對各分劵進行預測。

根據以往探討評價合成型抵押擔保債券之文獻研究，最廣為使用的方法應用大樣本一致性資產組合(large homogeneous portfolio portfolio ; LHP)假設之單因子常態關聯結構模型來評價，但會造成合成型抵押擔保債券憑證與市場報價間的差異過大，且會造成相關性微笑曲線現象。由文獻顯示，單因子關聯結構模型若能加入厚尾度或偏斜性能夠改善以上問題，且對於分券評價時也會有較好的效果，像是Kalemanova et al. (2007) 提出應用LHP假設之單因子Normal Inverse Gaussian(NIG)關聯結構模型以及邱嬿燁(2007)提出NIG及Closed Skew Normal(CSN)複合分配之單因子關聯結構模型(MIX模型)在實證分析中得到極佳的評價結果。自2008年起，合成型抵押擔保債券商品結構開始出現變化，而以往評價合成型抵押擔保債券價格時，商品結構皆為同一種型式。本文將利用常態分配、NIG分配、CSN分配以及NIG與CSN複合分配作為不同的單因子關聯結構模型，藉由絕對誤差極小化方法，針對不同商品結構的合成型抵押擔保債券評價，並進行模型比較分析。由最後實證分析結果顯示，單因子NIG(2)關聯結構模型優於其他模型，也證明NIG分配的第二個參數 β 能夠帶來改善的評價效果，此項證明與過去文獻結論有所不同，但 MIX模型則為唯一一個符合LHP假設的模型。 / Based on the literature of discussing the approach for pricing synthetic CDOs, the most widely used methods used application of Large Homogeneous Portfolio (LHP) assumption of the one factor Gaussian copula model, however , it fails to fit the prices of synthetic CDOs tranches and leads to the implied correlation smile. The literature shows that one factor copula model adding the heavy-tail or skew can improve the above problem, and also has a good effect for pricing tranches such as Kalemanova et al (2007) proposed the application of LHP assumption of one factor NIG copula model and Qiu Yan Ye (2007) proposed the application of LHP assumption of one factor NIG and CSN copula model. This article found that the structure of synthetic CDOs began to change since 2008. The past of pricing synthetic CDOs, the structure of synthetic CDOs are the same type, so this article will use different one factor copula model for pricing different structure of synthetic CDOs by using the absolute error minimization. This article will observe whether the above model can be applied in the new synthetic CDOs and implement of different type model for comparative analysis. The last empirical analysis shows that one factor NIG (2) copula model is superior to other models, more meeting the actual market demand, also proving the second parameter β of the NIG distribution able to bring about improvements in pricing results. This proving is different for the past literature conclusions. However, the MIX model is the only one in line with the LHP assumptions.

In den letzten Jahren gab es ein drastisches Wachstum in verfügbaren Finanzdaten. Finanzmärkte haben starke und oft nicht ganz vorhersagbare Änderungen ihrer Dynamik erlebt. Diese Tendenz hat dazu geführt, dass die Methoden der Risikomodellierung sowohl das Problem der hohen Dimensionalität als auch dynamische nicht Gaußsche Strukturen behandeln müssen. Das Ziel dieser Dissertation ist es, Methoden der Risikomodellierung vorzuschlagen, die gleichzeitig Reduzierung der Dimensionalität und dynamische Struktur in drei Anwendungen erlauben: 1) Asset Allocation und Hedging, 2) stochastische Modellierung von multivariaten Prozessen, 2) Messung der systemischen Risiken. Die vorgeschlagenen Methoden demonstrieren gute Ergebnisse im Vergleich mit den existierenden Methoden der Risikomodellierung und führen neue Verfahren zur Erkennung der extremen Risiken und Anomalien auf Finanzmärkten sowie zur deren Management. / Over the recent years, there have been a significant increase in financial data availability. On the other hand, financial markets have experienced sharp and often unforeseen changes in their dynamics. This tendency has caused the need for risk modeling approaches addressing both high dimensionality problem and accustoming for dynamic non Gaussian structure. The primary aim of this dissertation is to propose several risk modeling approaches which allow for simultaneous dimension reduction and dynamic structures in three setups: 1) asset allocation and hedging, 2) stochastic surface modeling and 3) systemic risk determination. Proposed models demonstrate good performance when compared to existing approaches for risk modeling and introduce new flexible ways to detect extreme risks and anomalies on financial markets as well as methods for their modeling and management.

依據之前的文獻研究，市場上主要是在LHP (Large Homogeneous Portfolio) 假設下利用單因子常態關聯結構模式(One factor double Gaussian copula model) 評價擔保債權憑證 (Collateralized debt obligation, CDO)。但這會造成擔保債權憑證的評價與市場報價的差距過大，且會造成base correlation偏斜的情況。Kalemanova et al. (2007) 提出用Normal inverse Gaussian (NIG) 取代常態分配評價擔保債權憑證，此模型不但計算快速而且可以準確估計權益分券 (equity tranche) 的價格，但是它也過於高估了其它的分券的價格。 在本文中使用多變量封閉常態分配(Closed skew normal, 簡稱CSN) 分配取代NIG分配作擔保債權憑證分券的評價，CSN分配具有常態分配的性質，其線性組合仍具有封閉性的特質，且具有較多的參數以控制分配的偏態與峰態。但是與單因子常態關聯結構模式相同，多變量封閉常態分配的單因子關聯結構模式仍然無法估計的很準確，僅有在最高等級分券(senior tranche)的評價上有明顯的改進。 因此在本文中我們使用NIG與CSN複合分配之單因子關聯結構模式評價擔保債權憑證分券，在實例分析時得到極佳的評價結果，並且比單因子常態關聯結構模型具有更多的的參數以使模型更符合實際的需求。 / This article extends the Large Homogeneous Portfolio (LHP) and one factor double Gaussian copula approach for pricing CDOs. In the literature, the one factor double Gaussian copula model under LHP assumption fails to fit the prices of CDO tranches, moreover, it leads to the implied base correlation skew. Some researchers proposed using one factor double NIG copula model to price CDO tranches. It not only economizes on time but also fits the equity tranches exactly, but NIG models do not price other tranches well simultaneously. On the other hand, we substitute the NIG distribution with the Closed Skew normal (CSN) distribution. This family also has properties similar to the normal distribution, which is closure under convolution, and has extra parameters to control the shape. By using this model we get a better fit in the senior tranches, but it seriously overprices subordinate tranches. Thus we consider a mixture distribution of NIG and CSN distributions. The employments of this mixture distribution are comparatively well, and furthermore it brings more flexibility to the dependence structure.

本文的主旨在評價二次擔保債權憑證。在條件獨立機率的假設下，我們使用factor copula的方法去刻劃違約事件間的相關係數，並提供了一個有效率的迴圈演算法去建構損失分配。本方法同時考慮違約數目及違約位置，同時亦可解決重疊性的問題。本文所建構的是Hull and White(2004)的延申模型。我們也對各參數作敏感度分析，以求得其對分券價差的影響。文中亦主張一些風險衝量指標，以量化重疊性的程度等風險議題。 / In this paper we address the pricing issues of CDO of CDOs. Underlying the conditional indepdence assumption we use the factor copula approach to characterize the correlation of defaults events. We provide an efficient recursive algorithm that constructs the loss distribution. Our algorithm accounts for the number of defaults, the location of defaults among inner CDOs, and in addition the degree of overlapping between inner CDOs. Our algorithm is a natural extension of the probability bucketing method of Hull and White (2004). We analyze the sensitivity of different parameters on the tranche spreads of a CDO-squared, and in order to characterize the risk-reward profiles of CDO-squared tranches, we introduces appropriate risk measures that quantify the degree of overlapping among the inner CDOs. Hull and White (2004) presents a recursive scheme known as probability bucketing approach to construct conditional loss distribution of CDO. However, this approach is insufficient to capture the complexities of CDO². In the case of the modeling of CDO, we are concerned for the probabilities of different number of defaults upon a time horizon t, e.g., the probabilities of 3 defaults happened within a year. With the mentioned probabilities, we can then calculate the expected loss within the time horizon, which enables us to figure out the spreads of CDO. However, in the modeling of CDO², an appropriate valuation should be able to overcome two more difficulties: (1) the overlapping structure of the underlying CDOs, and (2) the location where defaults happened, in order to get the fair spreads of CDO².