Modeling default contagion effects in stochastic credit basket models  

Master Thesis in "Mathematical Finance", Oxford (2014)


While a whole zoo of derivative products on stocks, commodities and other assets has been used in the financial sector for decades, credit risk remained rather difficult to assess. With the advent of newer models for credit risk, which allowed the pricing of credit insurances (CDS), and the evaluation of risks associated with different tranches of collateralized debt obligations (CDOs), the interest in these products grew rapidly. However, the financial crisis in 2008 has demonstrated that the understanding of the risks associated with these products was incomplete. Certain aspects like counterparty risk and insufficent consideration of correlation effects were some weaknesses of these models. Furthermore, it turned out that default contagion may be relevant in getting the risk estimates right. Default contagion means that the default probability of individual obligors may not only depend on systematic factors like the overall economic situation and idiosynchratic risks like bad management but also on the default of entities which are tightly connected to the obligor. This thesis will analyse different ways how to include the effect of default contagion into a structural credit risk model. The model uses a Multilevel Monte Carlo algorithm. Effects of the default contagion onto the numerical efficiency of the solution algorithm will be included into the discussion.


A new approach in dynamic credit default modeling

Master Thesis in "Quantitative Finance", Frankfurt School of Finance & Management (2014)


Utility Indifference Pricing of CDOs in a Variance Gamma Models  

Master Thesis in "Mathematical Finance", Oxford (2010)


Using the methodology of utility indifference pricing in intensity based models of default we define a multivariate Varinace Gamma model with common jumps. On the basis of this model we derive the systems of HJB PIDE to valuate CDOs. We study first the simplified problem with constant intensities which are essentially explicitly solvable. Here we observe that the value functions both are dependend on the risk aversion parameter in contrast to the diffusion case. Then we also derive the pricing equations for the situation featuring stochastic intensities. The numerical solution of the systems of PIDE, however, turns out to be rather complex and could not successfully be implemented.


Valuation of CDOs using Common Poisson Shock Models  

Master Thesis in "Mathematical Finance", Oxford (2009)


The scope of this thesis is to analyse the theoretical framework of the common Poisson shock model and to empirically investigate its abilitiy to value CDO market tranches. Theoretical details of the common Poisson shock model are presented and an analytical solution as well as two approximations of the portfolio loss distribution are derived and their different charcteristics ara analysed. As the loss distribution usually has several modes, we show how the modes and their values are determined by the model parameters.
To investigate the ability to value liquid market tranche quotes the three solutions of the common Poisson shock model as well as the one factor Gaussian copula model were implemented and applied to iTraxx tranche data. Starting from a base case, the sensitivities of the model losses ans spreads to the input parameters are discussed. Results show that the model offers enough flexibility to fit market data, even in the current distressed market environment. However, senior tranche spreads can only be fitted using special parameter values such as assuming all common risk to be due to one inter-sector type.

Valuation of CDO sensitivities and the dynamics of the iTraxx index before and after the financial crisis  

Master Thesis in "Mathematical Finance", Oxford (2009)


In this thesis, sensitivities and dynamical properties of CDO tranches are assessed. First a general introduction is given, in which CDO contracts are characterized and a benchmark index, the iTraxx, is presented. In the following chapter, fundamental mathematical concepts are introduced which form the basis for the following discussion. Then, the theory of CDO pricing is first introduced in a general setting. In the limiting case of a large homogenous CDS portfolio, analytical formulae are derived for the expected loss and the CDO tranche spread. The validity of the approximation as well as limiting cases are discussed. The first major result of this thesis are the derivation of analytical formulae for the sensitivity of the tranche spread to changes of the underlying default probability and correlation. These formulae have not been published before. They allow for a fast and accurate risk assessment of CDO tranches and can be readily implemented e.g. in risk or trading systems. In the following chapter the concepts of tranche and base correlation are introduced and discussed. It is shown that the tranche correlation exhibits a non-monotic behaviour for mezzanine tranches, leading to the observation that different tranche correlations may result in the same tranche spread. This ambiguity is resolved by considering the base correlation instead, which exhibits a monotonic behaviour. In the last chapter, the dynamics of the tranche spread and base correlation of the iTraxx Europe index before and after the financial crisis is characterized. This analysis compromises the second focus of this thesis. By applying a principal component analysis, it is shown that both before and after the financial crisis, both the tranche spread and base correlation are characterized by a strong co-movement across tranches. In addition, it is shown that before and after the financial crisis, the tranche spread and base correlation as a function of the detachment point exhibit a universal shape which changed during the financial crisis. This type of analysis has not been presented so far in the literature.

The Hidden Correlation of Collateralized Debt Obligations  

Master Thesis in "Mathematical Finance", Oxford (2009)


We propose a model for the correlation structure of reference portfolios of collateralized debt obligations. The model is capable of exhibiting typical characteristics of the implied correlation smile (skew, respectively) observed in the market. Moreover, it features a simple economic interpretation and is computationally inexpensive as it naturally integrates into the factor model framework. 

Assessment and Recognition of Counterparty Risk in Derivatives 

Master Thesis in "Quantitative Finance", Frankfurt School of Finance & Management (2009)


This document is presented as the Master Thesis in the framework of the program "Master of Quantitative Finance" at the Frankfurt School of Finance and Management (FSFM). The aim is to arrive at a well founded proposition of best practice in credit risk assessment and recognition for derivatives portfolios. To this end an overview of the relevant regulatory requirements, available market information and mathematical tools is given. It is shown that the most significant limitation of precision in the assessment of counterparty risk in derivatives stems from the extraction of probabilities of default from market data. The main result of the thesis is a quantitative estimate of the precision achieved in various valuation steps.

The findings of the thesis are comprised in three main chapters. A qualitative part summarizes the regulatory and statutory specifications, gives an overview over the economic background and outlines the market information available. Subsequently, the theoretical and methodological background of the assessment of counter party risk related quantities is described. In a quantitative part example applications are given and analyzed with respect to their mathematical precision and applicability.

Approaches to Models of Default Dependence with a View on Basket Default Swaps and CDOs

Master Thesis in "Quantitative Finance", Frankfurt School of Finance & Management (2008)


This work is devoted to various approaches to models of default dependence used in the treatment of products associated with portfolios of credit risk.

In a brief introduction to copula functions we cover dependence measures, the connection of extreme events and so-called tail dependence and a family of copula functions with radial asymmetry.

In the main part of this thesis we present the ingredients necessary to price basket CDS and CDO tranches in the context of factor models. We give a survey of different factor model specifications and show how these allow computing the individual probabilities of default. An overview of various useful techniques to compute the conditional loss distribution as well as an introduction to the computation of credit spread sensitivities in factor models is also provided. 

The factor models discussed here are essentially static one-period models primarily useful to derive the loss distribution of a given portfolio for a single fixed maturity. While this is often sufficient for the valuation of plain vanilla contracts, it becomes a drawback when one would like to consider more sophisticated products. We therefore consider attempts to go beyond static factor models. Starting our discussion with full-fledged structural models, we conclude this survey with simplifying approaches trying to retain some dynamical properties while keeping the attractive features of static models. 

Modelling Collateralized Debt Obligations using Variance Gamma Processes 

Master Thesis in "Mathematical Finance", Oxford (2008)


This thesis deals with the valuation of synthetic Collateralized Debt Obligation tranches with a Variance Gamma factor copula model. After a brief review of the valuation techniques for synthetic CDO tranches, a two and a three parameter factor copula model are introduced.

Both model variants involve the combination of marginal Variance Gamma distributions via a simple correlation structure. Calibrating these models to iTraxx tranche spreads shows a convincing fit using one parameter set only. Consistent pricing of non-standard tranches is thus possible. Furthermore, the behaviour of the tranche spreads as a function of the model parameters is intuitive and the calibrated model parameters are stable through time.

Though, during the subprime crisis one observes fluctuating parameters reflecting strong market dynamics. The usage of the Variance Gamma model for synthetic CDO valuation finally allows for a powerful parameter-based risk management.