Simulation methods for robust risk assessment and the distorted mix approach

Uncertainty requires suitable techniques for risk assessment. Combining stochastic approximation and stochastic average approximation, we propose an efficient algorithm to compute the worst case average value at risk in the face of tail uncertainty. Dependence is modelled by the distorted mix method that flexibly assigns different copulas to different regions of multivariate distributions. We illustrate the application of our approach in the context of financial markets and cyber risk.

[1]  Paul Embrechts,et al.  The AEP algorithm for the fast computation of the distribution of the sum of dependent random variables , 2011, 1106.2920.

[2]  Stefan Weber,et al.  Optimal risk sharing in insurance networks , 2020 .

[3]  Jörn Dunkel,et al.  Efficient Monte Carlo methods for convex risk measures in portfolio credit risk models , 2007, 2007 Winter Simulation Conference.

[4]  G. Pflug.  Some Remarks on the Value-at-Risk and the Conditional Value-at-Risk , 2000 .

[5]  Jie Xu,et al.  An importance sampling method for portfolio CVaR estimation with Gaussian copula models , 2010, Proceedings of the 2010 Winter Simulation Conference.

[6]  Paul Embrechts,et al.  The GAEP algorithm for the fast computation of the distribution of a function of dependent random variables , 2012 .

[7]  Paul Glasserman,et al.  Robust risk measurement and model risk , 2014 .

[8]  Phillipp Meister,et al.  Stochastic Recursive Algorithms For Optimization Simultaneous Perturbation Methods , 2016 .

[9]  Berç Rustem,et al.  Robust portfolio optimization with copulas , 2014, Eur. J. Oper. Res..

[10]  Haiyan Liu,et al.  Quantile-Based Risk Sharing , 2017, Oper. Res..

[11]  Steven Kou,et al.  External Risk Measures and Basel Accords , 2013, Math. Oper. Res..

[12]  B. Acciaio,et al.  Are law-invariant risk functions concave on distributions? , 2013 .

[13]  Gilles Pagès,et al.  Computing VaR and CVaR using stochastic approximation and adaptive unconstrained importance sampling , 2008, Monte Carlo Methods Appl..

[14]  P. Embrechts,et al.  Quantitative Risk Management: Concepts, Techniques, and Tools , 2005 .

[15]  M. KarthyekRajhaaA.,et al.  Quantifying Distributional Model Risk via Optimal Transport , 2019, Math. Oper. Res..

[16]  A. McNeil,et al.  Archimedean Copulas in High Dimensions: Estimators and Numerical Challenges Motivated by Financial Applications , 2012 .

[17]  Ruodu Wang,et al.  Risk aggregation with dependence uncertainty , 2014 .

[18]  Samuel Drapeau,et al.  Computational aspects of robust optimized certainty equivalents and option pricing , 2017 .

[19]  Stanislav Uryasev,et al.  Conditional Value-at-Risk for General Loss Distributions , 2002 .

[20]  I. Csiszár,et al.  MEASURING DISTRIBUTION MODEL RISK , 2016 .

[21]  Laurent Condat,et al.  A Fast Projection onto the Simplex and the l 1 Ball , 2015 .

[22]  Harold J. Kushner,et al.  Stochastic Approximation Algorithms and Applications , 1997, Applications of Mathematics.

[23]  Kam Chuen Yuen,et al.  Distorted Mix Method for constructing copulas with tail dependence , 2014 .

[24]  P. Embrechts,et al.  Model Uncertainty and VaR Aggregation , 2013 .

[25]  F. Hampel.  A General Qualitative Definition of Robustness , 1971 .

[26]  O. Bardou,et al.  CVaR HEDGING USING QUANTIZATION‐BASED STOCHASTIC APPROXIMATION ALGORITHM , 2016 .

[27]  H. Föllmer,et al.  Stochastic Finance: An Introduction in Discrete Time , 2002 .

[28]  Jörn Dunkel,et al.  Stochastic Root Finding and Efficient Estimation of Convex Risk Measures , 2010, Oper. Res..

[29]  J. Kiefer,et al.  Stochastic Estimation of the Maximum of a Regression Function , 1952 .

[30]  A. Shapiro.  Monte Carlo Sampling Methods , 2003 .

[31]  Stefan Weber,et al.  Solvency II, or how to sweep the downside risk under the carpet , 2017, Insurance, Mathematics & Economics.

[32]  S. Weber,et al.  DISTRIBUTION‐INVARIANT RISK MEASURES, INFORMATION, AND DYNAMIC CONSISTENCY , 2006 .

[33]  M. Teboulle,et al.  AN OLD‐NEW CONCEPT OF CONVEX RISK MEASURES: THE OPTIMIZED CERTAINTY EQUIVALENT , 2007 .

[34]  Vartan Choulakian,et al.  Goodness-of-Fit Tests for the Generalized Pareto Distribution , 2001, Technometrics.

[35]  Michael C. Fu,et al.  Chapter 19 Gradient Estimation , 2006, Simulation.

[36]  Hui Shao,et al.  Worst-Case Range Value-at-Risk with Partial Information , 2017, SIAM J. Financial Math..

[37]  Carl Scarrott,et al.  A Review of Extreme Value Threshold Estimation and Uncertainty Quantification , 2012 .

[38]  Rama Cont,et al.  Robustness and sensitivity analysis of risk measurement procedures , 2010 .

[39]  Claudia Czado,et al.  Selecting and estimating regular vine copulae and application to financial returns , 2012, Comput. Stat. Data Anal..

[40]  Zhaolin Hu,et al.  Kullback-Leibler divergence constrained distributionally robust optimization , 2012 .

[41]  M. Goh,et al.  Stochastic Optimization Problems with CVaR Risk Measure and Their Sample Average Approximation , 2010 .

[42]  Valeria Bignozzi,et al.  On elicitable risk measures , 2015 .

[43]  R. Pasupathy,et al.  A Guide to Sample Average Approximation , 2015 .

[44]  Masao Fukushima,et al.  Worst-Case Conditional Value-at-Risk with Application to Robust Portfolio Management , 2009, Oper. Res..

[45]  S. Vanduffel,et al.  A New Approach to Assessing Model Risk in High Dimensions , 2015 .

[46]  R. Rockafellar,et al.  Optimization of conditional value-at risk , 2000 .

[47]  C. Czado,et al.  Modeling dependent yearly claim totals including zero claims in private health insurance , 2012 .

[48]  Ludger Rüschendorf,et al.  Reduction of Value-at-Risk bounds via independence and variance information , 2015 .

[49]  Mark Broadie,et al.  General Bounds and Finite-Time Improvement for the Kiefer-Wolfowitz Stochastic Approximation Algorithm , 2011, Oper. Res..

[50]  Soumyadip Ghosh,et al.  Robust Analysis in Stochastic Simulation: Computation and Performance Guarantees , 2015, Oper. Res..

[51]  Ludger Rüschendorf,et al.  Risk Bounds and Partial Dependence Information , 2017 .

[52]  Shie Mannor,et al.  Optimizing the CVaR via Sampling , 2014, AAAI.

[53]  Ling Hu.  Dependence patterns across financial markets: a mixed copula approach , 2006 .

[54]  Volker Krätschmer,et al.  Comparative and qualitative robustness for law-invariant risk measures , 2012, Finance Stochastics.

[55]  Yong Wang,et al.  Asymptotic Analysis of Sample Average Approximation for Stochastic Optimization Problems with Joint Chance Constraints via Conditional Value at Risk and Difference of Convex Functions , 2014, J. Optim. Theory Appl..

[56]  Martin Eling,et al.  Copula approaches for modeling cross-sectional dependence of data breach losses , 2018, Insurance: Mathematics and Economics.