Paper: arXiv 2510.16526

Authors: Federico Gatta, Fabrizio Lillo, Piero Mazzarisi

Abstract

We propose a new approach, termed Realized Risk Measures (RRM), to estimate Value-at-Risk (VaR) and Expected Shortfall (ES) using high-frequency financial data. It extends the Realized Quantile (RQ) approach proposed by Dimitriadis and Halbleib by lifting the assumption of return self-similarity, which displays some limitations in describing empirical data. More specifically, as the RQ, the RRM method transforms intra-day returns in intrinsic time using a subordinator process, in order to capture the inhomogeneity of trading activity and/or volatility clustering. Then, microstructural effects resulting in non-zero autocorrelation are filtered out using a suitable moving average process. Finally, a fat-tailed distribution is fitted on the cleaned intra-day returns. The return distribution at low frequency (daily) is then extrapolated via either a characteristic function approach or Monte Carlo simulations. VaR and ES are estimated as the quantile and the tail mean of the distribution, respectively. The proposed approach is benchmarked against the RQ through several experiments. Extensive numerical simulations and an empirical study on 18 US stocks show the outperformance of our method, both in terms of the in-sample estimated risk measures and in the out-of-sample risk forecasting

Complexity vs Empirical Score

  • Math Complexity: 6.5/10
  • Empirical Rigor: 8.0/10
  • Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper provides a strong combination of rigorous high-frequency stochastic processing math and empirical validation on 18 US stocks. It effectively addresses the limitations of self-similarity in previous Realized Quantile models by introducing intrinsic time subordination, MA filtering, and tail extrapolation.

Research Flowchart

  flowchart TD
    A[Goal: Improve VaR & ES Estimation using High-Frequency Data without Self-Similarity Assumption] --> B[(Inputs: High-Frequency Intra-day Stock Returns - 18 US Stocks & Simulations)]
    
    subgraph C[Realized Risk Measures Method]
        C1[1. Transform Intra-day Returns to Intrinsic Time via Subordinator] --> C2[2. Filter Microstructural Effects using MA Process]
        C2 --> C3[3. Fit Fat-Tailed Distribution on Cleaned Returns]
        C3 --> C4[4. Extrapolate Daily Return Distribution via Char. Function or Monte Carlo]
    end

    B --> C
    C --> D[Compute Daily VaR and ES Quantile & Tail Mean]
    D --> E[Outcomes: RRM outperforms Realized Quantile approach in both In-Sample Estimation & Out-of-Sample Forecasting]

Cited References (39)

  1. On the coherence of expected shortfall (2002)
  2. Expected shortfall: a natural coherent alternative to value at risk (2002)
  3. Back-testing expected shortfall (2014)
  4. Volatility and correlation forecasting (2006)
  5. Coherent measures of risk (1999)
  6. A note on quantiles in large samples (1966)
  7. Learning value-at-risk and expected shortfall (2022) — link
  8. Realizing the extremes: Estimation of tail-risk measures from a high-frequency perspective (2016)
and 31 more
  1. Realized extreme quantile: A joint model for conditional quantiles and measures of volatility with EVT refinements (2018)
  2. Statistics for long-memory processes (2017)
  3. Generalized autoregressive conditional heteroskedasticity (1986)
  4. Apparent multifractality in financial time series (2000)
  5. Algorithms for minimization without derivatives (2013)
  6. Econophysics review: I. Empirical facts (2011)
  7. Quantile forecasts of daily exchange rate returns from forecasts of realized volatility (2008)
  8. Empirical properties of asset returns: stylized facts and statistical issues (2001)
  9. A simple approximate long-memory model of realized volatility (2009)
  10. Realized quantiles (2022)
  11. Business time sampling scheme with applications to testing semi-martingale hypothesis and estimating integrated volatility (2017)
  12. An introduction to the theory of self-similar stochastic processes (2000)
  13. Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation (1982)
  14. CAViaR: Conditional autoregressive value at risk by regression quantiles (2004)
  15. Basel IV implementation: a review of the case of the European Union (2020)
  16. Higher order elicitability and Osband’s principle (2016)
  17. CAESar: Conditional Autoregressive Expected Shortfall — → 2407.06619
  18. Note on the inversion theorem (1951)
  19. Quantile prediction for Bitcoin returns using financial assets’ realized measures (2023)
  20. Modelling systemic risk using neural network quantile regression (2022)
  21. Regression quantiles (1978)
  22. Rough volatility: evidence from option prices (2018)
  23. On a measure of lack of fit in time series models (1978)
  24. A multifractal model of asset returns (1997)
  25. Estimation methods for expected shortfall (2014)
  26. Dynamic semiparametric models for expected shortfall (and value-at-risk) (2019)
  27. Quadpack: a subroutine package for automatic integration (2012)
  28. A quantile regression neural network approach to estimating the conditional density of multiperiod returns (2000)
  29. Forecasting value at risk and expected shortfall using a semiparametric approach based on the asymmetric Laplace distribution (2019)
  30. Semi-parametric conditional quantile models for financial returns and realized volatility (2015)
  31. Quantile forecasts of financial returns using realized GARCH models (2012)

1 of 39 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.