Market Risk

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.

Risk Aggregation
Master Thesis in "Mathematical Finance", Oxford (2005)
Following a directive of the European Parliament, regulations have been proposed or passed in many member states of the European Union, that will require corporations with a major business unit that sells financial services to institute an aggregated risk management. This poses the question what one can do to assess risk across such diverse
categories as market risk, credit risk, commodity risk, and risk of fluctuation in income, or across several levels of hierarchy within a corporation. One of the most widely used measures for market risk is the Value at Risk. There are a number of methods to arrive at approximations for the Value at Risk for market risk, but recently using Monte Carlo simulation to obtain direct estimates for the Value at Risk has become computationally feasible. In this thesis we present and analyze a particular method of short term risk assessment suggested by P�ezier: While not directly observable, all the risks should contribute to the earnings of the corporation. These earnings are observable in the form of profit and loss numbers, which many corporations collect in a monthly fashion. At the level of the business units, linear regression is used to determine the impact of each risk factor. Actual profit and loss numbers will be diluted by accounting practices, but even if one takes care to remove these effects, they may not exactly represent the effect of risk factors on the earnings. Therefore we investigate the stability of our method with respect to measurement error. Our method is capable of generating an approximation to the distribution of losses by Monte Carlo simulation. Although this makes other risk measures available, we concentrate on the traditional Value at Risk in this thesis, giving an analytic approximation formula and investigating three corporation models numerically. We find that in the cases we have examined, the uncertainty introduced by fitting to model generated pseudo market data is larger than the regression error resulting from reasonable error terms in the profit and loss numbers. We demonstrate that correlation between the error terms and the underlying risk factors introduces systematic bias to the Value at Risk estimates.

Extreme Value Theory and L�vy Processes
Master Thesis in "Mathematical Finance", Oxford (2004)
In this thesis an empirical investigation of extreme events of Normal Inverse Gaussian (NIG) L�vy processes is presented. Random numbers following a NIG distribution are simulated with a method introduced by Rydberg. By fitting the simulated probability densities to market data of stock and stock index log returns, suitable parameter ranges for the NIG distribution are identified. Using QQ-plots we show that not only the center, but also the tails of the market data distributions are well described by NIG distributed random numbers. Extreme Value Theory is applied to the data sets by studying the mean excess function and by estimating the shape parameter that characterizes the extreme value distribution. In all cases, shape parameters significantly larger than zero are found, indicating a Fr�chet like behaviour of the extreme value distribution. In some cases this is confirmed by the mean excess function results, in other cases there is evidence for the distribution belonging to the Maximum Domain of Attraction of the Gumble distribution. The latter is confirmed by an approximation of the NIG distribution tails, which is shown to be the product of a polynomial and an exponential. We conclude that the inner part of the distribution tail is dominated by the polynomial and that therefore mean excess function and extreme value distribution show properties of a Fr�chet distribution. The outer part of the tails on the other hand decays more like an exponential and therefore Gumble-like behaviour is observed in this range. The Peaks Over Threshold model is used to estimate Value at Risk and expected shortfall. By applying this method to simple test portfolios of simulation and market data, we show that VaR and expected shortfall based on extreme value theory is far better suited to assess the risk of unexpected loss than classical methods. Since very similar results are obtained for simulations and market data we conclude that NIG L�vy processes are well able to model extreme events in a risk management context.

The Time Decay in Calculating Value-at-Risk
Master Thesis in "Mathematical Finance", Oxford (2003)
There are several approaches to calculate the Value--at--Risk (VaR) of a portfolio exploiting an approximation of the portfolio value. Such an approximation can be obtained by expanding into a Taylor series and then neglecting the higher orders. It can be observed in the literature that often merely the terms including the first and second derivatives with respect to the underlying values (Delta and Gamma) are considered and the derivative with respect to the time - the so called time decay - is left out. This might not be consistent because this term is of the same order as the Gamma term. In this thesis the effect of considering the time decay in the approximations employed in calculating Value--at--Risk is investigated by means of sample portfolios. As a reference method the Monte Carlo method is used.

Improving Value at Risk Calculations by Using Copulas and Non-Gaussian Margins
Master Thesis in "Mathematical Finance", Oxford (2002)
Value at risk (VaR) is of central importance in modern financial risk management. Of the various methods that exist to compute the VaR, the most popular are historical simulation, the variance-covariance method and Monte Carlo (MC) simulation. While historical simulation is not based on particular assumptions as to the behaviour of the risk factors, the two other methods assume some kind of multinormal distribution of the risk factors. Therefore the dependence structure between different risk factors is described by the covariance or correlation between these factors. It is shown in Embrechts et al. (1999, 2002) that the concept of correlation entails several pitfalls. As a consequence, copulas are proposed to describe the dependence between n variables with arbitrary marginal distributions. A copula is a function C: [0,1]^n -> [0,1] with certain special properties (see Nelsen (1999)), so that the joint distribution can be written as Prob(R_1 <= r_1, ..., R_n <= r_n) = C(F_1(r_1), ..., F_n(r_n)). F_1, ..., F_n denote the cumulative probability functions of the n variables. In general, a copula C depends on one or more parameters p_1, ..., p_k that determine the dependence between the variables r_1, ..., r_n. In this sense, these parameters assume the role of correlations. The second pitfall that arises from the multinormal distribution ansatz is the fat tail problem of the margins (see Duffie and Pan (1997), Zichenko (2001)). One way of taking extreme values better into account is to assume that the risk factors obey Student distribution patterns instead of Gaussian patterns. In this thesis we investigate two risk factors only. We model each risk factor independently using a Student distribution and describe their dependence by both the Frank copula (see Frank (1979)) and the Gumbel-Hougaard copula (see Hutchinson and Lai (1990)). We present algorithms to estimate the parameters of the margins and the copulas and to generate pseudo random numbers due to a copula dependence. Making use of historical data spanning a period of nineteen years, we compute the VaR using a copula-modified MC algorithm. To see the advantage of this method, we compare our results with VaR results obtained from the three standard methods mentioned at the beginning. Based on backtesting results, we find that the copula method is more reliable than both traditional MC simulation and the variance-covariance method and about as good as historical simulation.

Comparison of different methods for calculation of delta-gamma value at risk
Diploma Thesis in "Mathematical Finance", Oxford (2002)
There are several approaches to calculate value at risk (VaR) under the assumption
of a second order approximation to the payoff profile (delta-gamma approximation).
This project compares the VaR numbers for simple options strategies calculated using
Cornish Fisher approximations of different order to the exact delta-gamma solution
obtained using fourier inversion. To judge the accuracy of the results also Monte
Carlo simulations with full evaluation of the portfolio are included.

Value at Risk estimation based on incomplete market data
Master Thesis in "Mathematical Finance", Oxford (2002)