Paper: arXiv 2408.02401

Authors: Sören Bettels, Stefan Weber

Abstract

Distortion risk measures play a critical role in quantifying risks associated with uncertain outcomes. Accurately estimating these risk measures in the context of computationally expensive simulation models that lack analytical tractability is fundamental to effective risk management and decision making. In this paper, we propose an efficient important sampling method for distortion risk measures in such models that reduces the computational cost through machine learning. We demonstrate the applicability and efficiency of the Monte Carlo method in numerical experiments on various distortion risk measures and models.

Complexity vs Empirical Score

  • Math Complexity: 7.5/10
  • Empirical Rigor: 6.5/10
  • Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper provides a strong theoretical framework combining importance sampling, machine learning surrogate models, and distortion risk measures. It includes an algorithmic specification (Algorithm 1) and empirical case studies, though the excerpt focuses heavily on the underlying measure-theoretic foundations.

Research Flowchart

  flowchart TD
    A[Goal: Efficient Estimation of Distortion Risk Measures in Expensive Black-Box Models] --> B[Input: Multi-Asset Inputs & Simulation Models]
    B --> C[Method: Train Machine Learning Model to Approximate Black-Box System]
    C --> D[Method: Construct Importance Sampling PDF via ML Surrogate]
    D --> E[Compute: Perform Variance-Reduced Monte Carlo Simulation]
    E --> F[Outcome: Significant Computational Cost Reduction & High Estimation Accuracy]

Cited References (78)

  1. Importance Sampling for Monte Carlo Estimation of Quantiles (1996)
  2. Large Sample Behavior of the CTE and VaR Estimators under Importance Sampling (2011) — link
  3. A General Framework of Importance Sampling for Value-at-Risk and Conditional Value-at-Risk (2009)
  4. Coherent Measures of Risk (1999) — link
  5. Premium Calculation by Transforming the Layer Premium Density (1996)
  6. Spectral Measures of Risk: A Coherent Representation of Subjective Risk Aversion (2002)
  7. Stochastic Finance (2016) — link
  8. Remarks on Quantiles and Distortion Risk Measures (2012) — link
and 70 more
  1. New Measures for Performance Evaluation (2008) — link
  2. Solvency II , or How to Sweep the Downside Risk under the Carpet (2018) — link
  3. Risk Measures with Comonotonic Subadditivity or Convexity and Respecting Stochastic Orders (2009) — link
  4. On the Calibration of Distortion Risk Measures to Bid-Ask Prices (2014) — link
  5. Parameter Uncertainty and Residual Estimation Risk (2015) — link
  6. Worst-Case Range Value-at-Risk with Partial Information (2018) — link
  7. Insurance Pricing and Increased Limits Ratemaking by Proportional Hazards Transforms (1995) — link
  8. Distortion Risk Measures for Nonnegative Multivariate Risks (2018) — link
  9. A Synthesis of Risk Measures for Capital Adequacy (1999) — link
  10. Stochastic Dominance, Mean Variance, and Gini’s Mean Difference (1982)
  11. Robustifying Convex Risk Measures for Linear Portfolios: A Nonparametric Approach (2014)
  12. Spectral Risk Measures: Properties and Limitations (2008) — link
  13. Distortion Risk Measure or the Transformation of Unimodal Distributions into Multimodal Functions (2014)
  14. Methods for Generating Coherent Distortion Risk Measures (2018) — link
  15. A Class of Distortion Operators for Pricing Financial and Insurance Risks (2000)
  16. A Risk Measure that Goes Beyond Coherence (2001)
  17. Extreme Versions of Wang Risk Measures and Their Estimation for Heavy-Tailed Distributions (2017)
  18. Multinomial Backtesting of Distortion Risk Measures (2024)
  19. Real Analysis: Modern Techniques and Their Applications (1999)
  20. Stochastic Simulation: Algorithms and Analysis (2007)
  21. Convex Measures of Risk and Trading Constraints (2002) — link
  22. Putting Order in Risk Measures (2002) — link
  23. Bayesian Estimates of Equation System Parameters: An Application of Integration by Monte Carlo (1978) — link
  24. Monte Carlo Methods in Financial Engineering (2003) — link
  25. Rare Event Simulation Methodologies in Systems Biology (2009) — link
  26. The Axiomatic Approach to Risk Measures for Capital Determination (2015) — link
  27. Theory of Capacities (1954) — link
  28. Non-Additive Measure and Integral (1994) — link
  29. Risk Measures and Comonotonicity: A Review (2006) — link
  30. An Overview of Representation Theorems for Static Risk Measures (2009) — link
  31. The Representation of Two Types Functionals on $L^\infty(\Omega, \mathcal F )$ and $L^\infty(\Omega, \mathcal F , P)$ (2006) — link
  32. Simulation Methods for Robust Risk Assessment and the Distorted Mix Approach (2022) — link
  33. On Law Invariant Coherent Risk Measures (2001) — link
  34. Rare Event Simulation using Monte Carlo Methods (2009)
  35. Introduction to Rare Event Simulation (2004)
  36. Efficient Rare-Event Simulation for the Maximum of Heavy-Tailed Random Walks (2008) — link
  37. Importance Sampling, Large Deviations, and Differential Games (2004)
  38. Large Deviations for Weighted Empirical Measures Arising in Importance Sampling (2016) — link
  39. Rare Events Simulation for Heavy-Tailed Distributions (2000) — link
  40. Rare-Event Simulation Techniques: An Introduction and Recent Advances (2006) — link
  41. Efficient Calculation of Risk Measures by Importance Sampling–The Heavy Tailed Case (2009) — link
  42. Portfolio Value-at-Risk with Heavy-Tailed Risk Factors (2002) — link
  43. Efficient Monte Carlo Methods for Convex Risk Measures in Portfolio Credit Risk Models (2007)
  44. Estimation of Distortion Risk Measures (2013) — link
  45. Estimation of Spectral Risk Measures (2021) — link
  46. Linear Functions of Order Statistics with Smooth Weight Functions (1974) — link
  47. Approximation Theorems of Mathematical Statistics (1980) — link
  48. Estimating Conditional Tail Expectation with Actuarial Applications in View (2008) — link
  49. A Modified Functional Delta Method and Its Application to the Estimation of Risk Functionals (2010) — link
  50. Certifiable Deep Importance Sampling for Rare-Event Simulation of Black-Box Systems (2021)
  51. Regression (2013) — link
  52. An Introduction to Kernel and Nearest-Neighbor Nonparametric Regression (1992) — link
  53. Understanding Machine Learning: From Theory to Algorithms (2014)
  54. The Nature of Statistical Learning Theory (1999)
  55. Support Vector Regression Machines (1996)
  56. Working Set Selection Using Second Order Information for Training Support Vector Machines (2005)
  57. On the Calibration of Distortion Risk Measures to Bid-Ask Prices (2014)
  58. Measure Theory (2007)
  59. Sequential Minimal Optimization: A Fast Algorithm for Training Support Vector Machines (1998)
  60. Deep Learning in Finance and Banking: A Literature Review and Classification (2020)
  61. Foundations of Machine Learning (2018)
  62. Market Consistent Embedded Value – Eine Praxisorientierte E inf "u hrung (2014)
  63. Optimal Risk Sharing in Insurance Networks (2019)
  64. MATLAB Program for Quadrature in 2D (2008)
  65. Kernel Based Algorithms for Mining Huge Data Sets (2006)
  66. The perceptron: A probabilistic model for information storage and organization in the brain (1958)
  67. A tutorial on Support Vector Regression (2004) — link
  68. Conditional value-at-risk for general loss distributions (2002)
  69. Optimization of Conditional Value-at-Risk (2000)
  70. An Old-New Concept of Convex Risk Measures: The Optimized Certainty Equivalent (2007) — link

0 of 78 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.