Measuring Risk in Complex Stochastic Systems

Complex dynamic processes of life and sciences generate risks that have to be taken. The need for clear and distinctive definitions of different kinds of risks, adequate methods and parsimonious models is obvious. The identification of important risk factors and the quantification of risk stemming from an interplay between many risk factors is a prerequisite for mastering the challenges of risk perception, analysis and management successfully. The increasing complexity of stochastic systems, especially in finance, have catalysed the use of advanced statistical methods for these tasks. The methodological approach to solving risk management tasks may, however, be undertaken from many different angles. A financial insti­ tution may focus on the risk created by the use of options and other derivatives in global financial processing, an auditor will try to evalu­ ate internal risk management models in detail, a mathematician may be interested in analysing the involved nonlinearities or concentrate on extreme and rare events of a complex stochastic system, whereas a statis­ tician may be interested in model and variable selection, practical im­ plementations and parsimonious modelling. An economist may think about the possible impact of risk management tools in the framework of efficient regulation of financial markets or efficient allocation of capital.

1 Allocation of Economic Capital in loan portfolios.- 1.1 Introduction.- 1.2 Credit portfolios.- 1.2.1 Ability to Pay Process.- 1.2.2 Loss distribution.- 1.3 Economic Capital.- 1.3.1 Capital allocation.- 1.4 Capital allocation based on Var/Covar.- 1.5 Allocation of marginal capital.- 1.6 Contributory capital based on coherent risk measures.- 1.6.1 Coherent risk measures.- 1.6.2 Capital Definition.- 1.6.3 Contribution to Shortfall-Risk.- 1.7 Comparision of the capital allocation methods.- 1.7.1 Analytic Risk Contribution.- 1.7.2 Simulation procedure.- 1.7.3 Comparison.- 1.7.4 Portfolio size.- 1.8 Summary.- 2 Estimating Volatility for Long Holding Periods.- 2.1 Introduction.- 2.2 Construction and Properties of the Estimator.- 2.2.1 Large Sample Properties.- 2.2.2 Small Sample Adjustments.- 2.3 Monte Carlo Illustrations.- 2.4 Applications.- 2.5 Conclusion.- 3 A Simple Approach to Country Risk.- 3.1 Introduction.- 3.2 A Structural No-Arbitrage Approach.- 3.2.1 Structural versus Reduced-Form Models.- 3.2.2 Applying a Structural Model to Sovereign Debt.- 3.2.3 No-Arbitrage vs Equilibrium Term Structure.- 3.2.4 Assumptions of the Model.- 3.2.5 The Arbitrage-Free Value of a Eurobond.- 3.2.6 Possible Applications.- 3.2.7 Determination of Parameters.- 3.3 Description of Data and Parameter Setting.- 3.3.1 DM-Eurobonds under Consideration.- 3.3.2 Equity Indices and Currencies.- 3.3.3 Default-Free Term Structure and Correlation.- 3.3.4 Calibration of Default-Mechanism.- 3.4 Pricing Capability.- 3.4.1 Test Methodology.- 3.4.2 Inputs for the Closed-Form Solution.- 3.4.3 Model versus Market Prices.- 3.5 Hedging.- 3.5.1 Static Part of Hedge.- 3.5.2 Dynamic Part of Hedge.- 3.5.3 Evaluation of the Hedging Strategy.- 3.6 Management of a Portfolio.- 3.6.1 Set Up of the Monte Carlo Approach.- 3.6.2 Optimality Condition.- 3.6.3 Application of the Optimality Condition.- 3.6.4 Modification of the Optimality Condition.- 3.7 Summary and Outlook.- 4 Predicting Bank Failures in Transition.- 4.1 Motivation.- 4.2 Improving “Standard” Models of Bank Failures.- 4.3 Czech banking sector.- 4.4 Data and the Results.- 4.5 Conclusions.- 5 Credit Scoring using Semiparametric Methods.- 5.1 Introduction.- 5.2 Data Description.- 5.3 Logistic Credit Scoring.- 5.4 Semiparametric Credit Scoring.- 5.5 Testing the Semiparametric Model.- 5.6 Misclassification and Performance Curves.- 6 On the (Ir) Relevancy of Value-at-Risk Regulation.- 6.1 Introduction.- 6.2 VaR and other Risk Measures.- 6.2.1 VaR and Other Risk Measures.- 6.2.2 VaR as a Side Constraint.- 6.3 Economic Motives for VaR Management.- 6.4 Policy Implications.- 6.5 Conclusion.- 7 Backtesting beyond VaR.- 7.1 Forecast tasks and VaR Models.- 7.2 Backtesting based on the expected shortfall.- 7.3 Backtesting in Action.- 7.4 Conclusions.- 8 Measuring Implied Volatility Surface Risk using PCA.- 8.1 Introduction.- 8.2 PCA of Implicit Volatility Dynamics.- 8.2.1 Data and Methodology.- 8.2.2 The results.- 8.3 Smile-consistent pricing models.- 8.3.1 Local Volatility Models.- 8.3.2 Implicit Volatility Models.- 8.3.3 The volatility models implementation.- 8.4 Measuring Implicit Volatility Risk using VaR.- 8.4.1 VaR: Origins and definition.- 8.4.2 VaR and Principal Components Analysis.- 9 Detection and estimation of changes in ARCH processes.- 9.1 Introduction.- 9.2 Testing for change-point in ARCH.- 9.2.1 Asymptotics under null hypothesis.- 9.2.2 Asymptotics under local alternatives.- 9.3 Change-point estimation.- 9.3.1 ARCH model.- 9.3.2 Extensions.- 10 Behaviour of Some Rank Statistics for Detecting Changes.- 10.1 Introduction.- 10.2 Limit Theorems.- 10.3 Simulations.- 10.4 Comments.- 10.5 Acknowledgements.- 11 A stable CAPM in the presence of heavy-tailed distributions.- 11.1 Introduction.- 11.2 Empirical evidence for the stable Paretian hypothesis.- 11.2.1 Empirical evidence.- 11.2.2 Univariate und multivariate ?-stable distributions.- 11.3 Stable CAPM and estimation for ?-coefficients.- 11.3.1 Stable CAPM.- 11.3.2 Estimation of the ?-coefficient in stable CAPM.- 11.4 Empirical analysis of bivariate symmetry test.- 11.4.1 Test for bivariate symmetry.- 11.4.2 Estimates for the ?-coefficient in stable CAPM.- 11.5 Summary.- 12 A Tailored Suit for Risk Management: Hyperbolic Model.- 12.1 Introduction.- 12.2 Advantages of the Proposed Risk Management Approach.- 12.3 Mathematical Definition of the P & L Distribution.- 12.4 Estimation of the P & L using the Hyperbolic Model.- 12.5 How well does the Approach Conform with Reality.- 12.6 Extension to Credit Risk.- 12.7 Application.- 13 Computational Resources for Extremes.- 13.1 Introduction.- 13.2 Computational Resources.- 13.2.1 XploRe.- 13.2.2 Xtremes.- 13.2.3 Extreme Value Analysis with XploRe and Xtremes.- 13.2.4 Differences between XploRe and Xtremes.- 13.3 Client/Server Architectures.- 13.3.1 Client/Server Architecture of XploRe.- 13.3.2 Xtremes CORBA Server.- 13.4 Conclusion.- 14 Confidence intervals for a tail index estimator.- 14.1 Confidence intervals for a tail index estimator.- 15 Extremes of alpha-ARCH Models.- 15.1 Introduction.- 15.2 The model and its properties.- 15.3 The tails of the stationary distribution.- 15.4 Extreme value results.- 15.4.1 Normalizing factors.- 15.4.2 Computation of the extremal index.- 15.5 Empirical study.- 15.5.1 Distribution of extremes.- 15.5.2 Tail behavior.- 15.5.3 The extremal index.- 15.6 Proofs.- 15.7 Conclusion.