Quantitative Comparison of Historical VaR Models 

Master Thesis in "Mathematical Finance", Oxford (2013)


In this study, different models for the Value-at-Risk (VaR) calculation for short and intermediate time horizons are quantitavely compared. VaR is the most important and most regularly used portfolio risk measure in financial industry. It gives an estimate of the maximum portfolio loss over a specific time horizon, which is not exceeded in a given confidence level.
The time horizons used in this thesis are 1 and 5 business days, corresponding to the usual needs in Banks and Clearing Houses. The studied risk models are several variations and improvements of the historical simulation method. 

The Asymetric t-Copula with Individual Degrees of Freedom 

Master Thesis in "Mathematical Finance", Oxford (2012)


This thesis investigates asymmetric dependence structures of multivariate asset returns. Evidence of such asymmetry for equity returns has been reported in the literature. In order to model the dependence structure, a new t-copula approach is proposed called the skewed t-copula with individual degrees of freedom (SID t-copula). This copula provides the flexibility to assign an individual degree-of-freedom parameter and an individual skewness parameter to each asset in a multivariate setting. Applying this approach to GARCH residuals of bivariate equity index return data and using maximum likelihood estimation, we find significant asymmetry. By means of the Akaike information criterion, it is demonstrated that the SID t-copula provides the best model for the market data compared to other copula approaches without explicit asymmetry parameters. In addition, it yields a better fit than the conventional skewed t-copula with a single degree-of-freedom parameter. In a model impact study, we analyse the errors which can occur when modelling asymmetric multivariate SID-t returns with the symmetric multivariate Gauss or standard t-distribution. The comparison is done in terms of the risk measures value-at-risk and expected shortfall. We find large deviations between the modelled and the true VaR/ES of a spread position composed of asymmetrically distributed risk factors. Going from the bivariate case to a larger number of risk factors, the model errors increase. 

Aspects of asset pricing under fictitious scenarios in risk management applications

Master Thesis in "Mathematical Finance", Oxford (2012)


The present thesis surveys and classifies difficulties sometimes encountered when theoretical pricing formulas are employed in practical risk management applications. To this end, we formulate selected necessary consistency conditions and describe possible solution strategies ensuring their satisfaction. This discussion focuses especially on the condition that the probability measure defining the pricing model must dominate the probability measure used for generating risk factor scenarios. The notions of risk and probability measures as well as key concepts of measure theory are briefly reviewed as needed in order to keep the discussion self contained.
Some prominent interest rate models tacitly assume that certain values must be positive. This concerns especially rates underlying derivatives, but also parameters such as the mean reversion speed in the Hull White model. These assumptions do not always hold in the real world, much less in risk factor scenarios, and are a principal cause of practical pricing difficulties. Based on historical precedents and actual time series, one factor interest rate models (namely the Vasicek and Hull White models) serve as quantitative examples to illustrate the use of the abstract solution strategies mentioned above, and to connect them with practical applications.
In the conclusions we point out some possibilities for future work on the subject of pricing difficulties. Two selected examples are given to highlight that the concepts presented herein are applicable more generally to asset classes other than interest rate derivatives, e.g. equity, commodity, and credit derivatives.

Causal Bayesian Nets in Risk Management

Master Thesis in "Mathematical Finance", Oxford (2012)


The value-at-risk is the most common way to measure risk. This purely statistical
number relies on historic time series and correlations. During a crisis these correla-
tions can change dramatically and historical returns and volatilities are no longer an appropriate estimate of the future. The value-at-risk cannot capture this change because the time series used for calculating the statistics is much larger than the typical scale of the changes. Stress testing is a method to measure risk for a certain (unlikely) event. It becomes more and more popular among regulators and institutions. In this framework the robustness of a portfolio against market shocks or extreme events is analyzed. The main task in this approach is to construct such events and estimate the market behavior. A useful tool to help assessing this information are causal Bayesian nets. This is a graphical representation of states or (sub-)events with a causal link. The idea is that adding causal information leads to a more robust treatment of shocks without having to use historic examples. Conditional probabilities for the states of the net have to be assigned by experts. In practice using large Bayesian nets leads to serious problems. The number of conditional probabilities grows exponentially with the number of nodes. This makes an effective use impossible as the experts cannot assign all necessary probabilities in a reasonable amount of time. This thesis is looking at these practical problems for a causal Bayesian net implementation. Techniques to reduce the number of input probabilities such as maximum entropy, causal independence, combining two Bayesian nets or Monte Carlo on the net are examined. The techniques are applied for the special case of risk management applications.


Incremental Risk Charge Modelling within a Merton-Style Factor Model

Master Thesis in "Mathematical Finance", Oxford (2011)


This thesis deals with the computation of the incremental risk charge (IRC) with a modified Mertonstyle portfolio model.The incremental risk charge is an additional capital buffer for a bank’s trading book and covers risks arising from rating migration and default events. Modelling the IRC is currently a top priority topic in market risk management since the charge becomes effective by 2012 and since it heavily increases the overall regulatory capital to be hold for the trading book. The IRC models that are currently developed by major banks combine both market- and credit risk components and hence IRC is a first step to integrate market and credit risk modelling. In this thesis we firstly present the motivation, the regulatory evolution and the final regulatory requirements on IRC. Secondly, we develop a fully-fledged IRC model based on a Merton-style credit portfolio that is compliant with the regulatory requirements. In the third chapter, we specify three enhancements of the basic IRC model that improve its risk sensitivity, namely how to incorporate active short term management of trading products (constant level of risk assumption), the modelling of a stochastic recovery rate and the consideration of the default risk of hedge counterparties. Fourthly, we implement the specified IRC model and conduct a wide range of quantitative studies to assess the effect of model assumptions, calibration parameters and portfolio composition of real bank portfolios with different risk profiles as well as different compositions of traded bonds and related hedge positions. The numerical results are presented in chapter 5. One major feature of this thesis is the consistent integration of a wide range methodologies from market- and credit risk: to model correlated migration- and default events a Merton-style portfolio model commonly used to compute economic capital for banking books is used, liquidity aspects are incorporated by a multistep extension of this model and the reevaluation of positions given rating migration is based on the standard spread curve model for general- and specific market risk. The repricing of positions with modified credit spread curves is based on the standard pricing formulae. Therefore, we elaborate in detail the pricing theory for bond- and CDS positions. Another major feature of the thesis is the practical relevance of its quantitative results: The comprehensive test calculations have been conducted on real world (sub) portfolios with realistic parameterizations and condense the experience gained on several IRC consulting projects. Altogether, we believe that this work presents interesting results regarding the severity of the IRC, the materiality of rating migration events (non-default) for the IRC, the IRC’s sensitivities to correlation parameters, the impact of the constant-level-of-risk assumption and the effects implied by stochastic recovery and the modelling of hedge counterparty defaults. 

Calibrating and validating real-world interest rate scenarios on the 1y-time horizon

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


The Solvency II directive has brought the attention of both researchers and practitioners towards real-world projections on the 1y-time horizon. Clearly, calibration and validation on this time frame requires tools very different from those employed on the 1d- to 1w-horizon typically employed in a banking context. Interest rate scenarios pose a special challenge, since the projected yield curves should both statistically match observed data and represent future real-world yield curves realistically. This work describes a model specifically aiming at economic scenario generation for the Solvency II context. This interest rate model is characterized by its accessibility and parsimoniousness. By simulating the stochastic evolution of each zero rate tenor individually using a Vasicek- type process, the real-world 1y-projection of the yield curve shows a shape diversity very close to that found in historical time series. Both its accessibility and the high exibility in producing yield curve diversity render this model useful for scenario generation for Solvency II. Along with the model a range of validation tools suited for systematically assessing scenario quality is introduced and subsequently employed to the model.  

A Model for VaR Calculation Using Extreme Value Theory 

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


The present thesis investigates heavy tail models for the calculation of value-at-risk (VaR) for single equity risk factors. This is motivated by the fact that modeling of heavy tails within market risk models has become more and more important in the recent time, especially since the new risk charge Incremental Risk has been proposed as an extension of the Basel II market risk framework. Besides other requirements, models for the calculation of Incremental Risk must obtain VaR figures for market risks at a confidence level of 99,9% on a risk horizon of one year and hence need to be able to capture the extreme tails of the risk factors well. Representation of the extreme tails of risk factor time series is achieved in this thesis by applying Extreme Value Theory (EVT). In particular, the log-returns of equity prices are modeled as a generalised autoregressive conditional heteroskedastic (GARCH) process where the innovations are described using techniques from extreme value theory (EVT). The tail behaviour of the GARCH innovations are modeled by Generalised Pareto Distributions (GPD). However, EVT is an asymptotic theory which captures only the extreme tails of probability distributions. Hence a probability distribution is constructed which incorporates Pareto behaviour in its tails and which can also describe the non-extreme behaviour of risk factors. This probability distribution is used within Monte Carlo simulations for the calculation of VaR figures for single risk factors at high confidence levels and it will be seen that the distribution performs well in forecasting the VaR for a single risk factor.