Peaks over Threshold Approach with a time-varying scale parameter and range-based volatility estimator for Value-at-Risk and Expected Shortfall estimation

Marcin Fałdziński

doi:https://doi.org/10.59139/ps.2024.01.2

Marcin Fałdziński Nicolaus Copernicus University in Toruń, Faculty of Economic Sciences and Management, Institute of Economics and Finance, ul. Gagarina 13a, 87–100 Toruń, Poland ORCID:https://orcid.org/0000-0002-6236-8500 Przegląd Statystyczny. Statistical Review, vol. 71, 2024, 1, pages: 23-77 Published online: 31 January 2025 DOI https://doi.org/10.59139/ps.2024.01.2 Citation: Fałdziński, M. (2024). Peaks over Threshold Approach with a time-varying scale parameter and range-based volatility estimator for Value-at-Risk and Expected Shortfall estimation. Przegląd Statystyczny. Statistical Review, 71(1), 23–77. https://doi.org/10.59139/ps.2024.01.2.

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ABSTRACT

Exploiting daily high-low range has become increasingly popular among volatility models due to valuable information about volatility dynamics. It has been shown in the literature that range-based volatility estimators can improve volatility and covariance forecasts, and thus models that use high and low prices can outperform standard volatility models based on closing prices solely. This paper incorporates a range-based volatility estimator in an extreme value theory framework to provide better estimates of the tails of daily asset returns. We introduce the Peaks over Threshold model with a range-based volatility estimator depicting the volatility of extreme returns that can contribute to more accurate tail risk estimation. We evaluate the proposed model based on the Monte Carlo simulation and long-period sample of the empirical financial time series by forecasting the Value-at-Risk and Expected Shortfall. We provide evidence that the proposed model can lead to better risk measure forecasts, especially for high tail probabilities.

KEYWORDS

GARCH, Value-at-Risk, Expected Shortfall, Peaks over Threshold, Extreme Value Theory

JEL

C51, C53, C58

REFERENCES

Abad, P., Benito, S., & López, C. (2014). A comprehensive review of Value at Risk methodologies. The Spanish Review of Financial Economics, 12(1), 15–32. https://doi.org/10.1016/j.srfe.2013.06.001.

Alizadeh, S., Brandt, M. W., & Diebold, F. X. (2002). Range-Based Estimation of Stochastic Volatility Models. The Journal of Finance, 57(3), 1047–1091. https://doi.org/10.1111/1540-6261.00454.

Asai, M. (2013). Heterogeneous Asymmetric Dynamic Conditional Correlation Model with Stock Return and Range. Journal of Forecasting, 32(5), 469–480. https://doi.org/10.1002/for.2252.

Bali, T. G. (2007). A Generalized Extreme Value Approach to Financial Risk Measurement. Journal of Money, Credit and Banking, 39(7), 1613–1649.

Balkema, A. A., & de Haan, L. (1974). Residual Life Time at Great Age. The Annals of Probability, 2(5), 792–804.

Basel Committee on Banking Supervision. (2011). Messages from the academic literature on risk measurement for the trading book (Working Paper No. 19). https://www.bis.org/publ/bcbs_wp19.pdf.

Basel Committee on Banking Supervision. (2019). Minimum capital requirements for market risk. Bank for International Settlements. https://www.bis.org/bcbs/publ/d457.pdf.

Bauwens, L., & Xu, Y. (2023). The contribution of realized covariance models to the economic value of volatility timing (Cardiff Economics Working Papers No. E2023/20). https://carbsecon.com/wp/E2023_20.pdf.

Bee, M., Dupuis, D. J., & Trapin, L. (2019). Realized Peaks over Threshold: A Time-Varying Extreme Value Approach with High-Frequency-Based Measures. Journal of Financial Econometrics, 17(2), 254–283. https://doi.org/10.1093/jjfinec/nbz003.

Beirlant, J., Vynckier, P., & Teugels, J. (1996). Excess functions and estimation of the extreme-value index. Bernoulli, 2(4), 293–318. https://doi.org/10.2307/3318416.

Bień-Barkowska, K. (2020). Looking at Extremes without Going to Extremes: A New Self-Exciting Probability Model for Extreme Losses in Financial Markets. Entropy, 22(7), 1–25. https://doi.org/10.3390/e22070789.

Bień-Barkowska, K. (2024). Forecasting extreme negative returns in gold and silver: A discreteduration approach to POT models. Applied Stochastic Models in Business and Industry, 40(2), 262–280. https://doi.org/10.1002/asmb.2759.

Bollerslev, T. (1986). Generalised Autoregressive Conditional Heteroskedasticity. Journal of Econometrics, 31(3), 307–327. https://doi.org/10.1016/0304-4076(86)90063-1.

Bollerslev, T. (1987). A Conditionally Heteroskedastic Time Series Model for Speculative Prices and Rates of Return. The Review of Economics and Statistics, 69(3), 542–547. https://doi.org/10.2307/1925546.

Bollerslev, T., & Wooldridge, J. M. (1992). Quasi-Maximum Likelihood Estimation and Inference in Dynamic Models with Time-Varying Covariances. Econometric Reviews, 11(2), 143–172. https://doi.org/10.1080/07474939208800229.

Brandt, M. W., & Jones, C. S. (2006). Volatility Forecasting with Range-Based EGARCH Models. Journal of Business & Economic Statistics, 24(4), 470–486. https://doi.org/10.1198/073500106000000206.

Buescu, C., Taksar, M., & Koné, F. J. (2013). An application of the method of moments to rangebased volatility estimation using daily High, Low, Opening, And Closing (HLOC) prices. International Journal of Theoretical and Applied Finance, 16(5), 1–24. https://doi.org/10.1142/S021902491350026X.

Candelon, B., Colletaz, G., Hurlin, C., & Tokpavi, S. (2011). Backtesting Value-at-Risk: A GMM Duration-based Test. Journal of Financial Econometrics, 9(2), 314–343. https://doi.org/10.1093/jjfinec/nbq025.

Candia, C., & Herrera, R. (2024). An empirical review of dynamic extreme value models for forecasting value at risk, expected shortfall and expectile. Journal of Empirical Finance, 77. https://doi.org/10.1016/j.jempfin.2024.101488.

Caporin, M. (2008). Evaluating Value-at-Risk Measures in the Presence of Long Memory Conditional Volatility. Journal of Risk, 10(3), 79–110. https://doi.org/10.21314/JOR.2008.172.

Chan, K. F., & Gray, P. (2006). Using Extreme Value Theory to Measure Value-at-Risk for Daily Electricity Spot Prices. International Journal of Forecasting, 22(2), 283–300. https://doi.org/10.1016/j.ijforecast.2005.10.002.

Chavez-Demoulin, V., Davison, A. C., & McNeil, A. J. (2005). Estimating Value-at-Risk: A Point Process Approach. Quantitative Finance, 5(2), 227–234. https://doi.org/10.1080/14697680500039613.

Chavez-Demoulin, V., & Embrechts, P. (2004). Smooth Extremal Models in Finance and Insurance. The Journal of Risk and Insurance, 71(2), 183–199. https://doi.org/10.1111/j.0022-4367.2004.00085.x.

Chavez-Demoulin, V., Embrechts, P., & Sardy, S. (2014). Extreme-Quantile Tracking for Financial Time Series. Journal of Econometrics, 181(1), 44–52. https://doi.org/10.1016/j.jeconom.2014.02.007.

Chou, R. Y. (2005). Forecasting Financial Volatilities with Extreme Values: The Conditional Autoregressive Range (CARR) Model. Journal of Money, Credit and Banking, 37(3), 561–582.

Chou, R. Y., & Cai, Y. (2009). Range-Based Multivariate Volatility Model with Double Smooth Transition in Conditional Correlation. Global Finance Journal, 20(2), 137–152. https://doi.org/10.1016/j.gfj.2008.12.001.

Chou, R. Y., Chou, H.-C., & Liu, N. (2015). Range Volatility: A Review of Models and Empirical Studies. In C. F. Lee & J. C. Lee (Eds.), Handbook of Financial Econometrics and Statistics (pp. 2029–2050). Springer. https://doi.org/10.1007/978-1-4614-7750-1_74.

Chou, R. Y., & Liu, N. (2010). The Economic Value of Volatility Timing Using a Range-Based Volatility Model. Journal of Economic Dynamics and Control, 34(11), 2288–2301. https://doi.org/10.1016/j.jedc.2010.05.010.

Chou, R. Y., Wu, C. C., & Liu N. (2009). Forecasting Time-Varying Covariance with a Range- Based Dynamic Conditional Correlation Model. Review of Quantitative Finance and Accounting, 33, 327–345. https://doi.org/10.1007/s11156-009-0113-3.

Christoffersen, P. F. (1998). Evaluating Interval Forecasts. International Economic Review, 39(4), 841–862. https://doi.org/10.2307/2527341.

Coles, S. (2001). Extremes of Non-stationary Sequences. In S. Coles (Ed.), An Introduction to Statistical Modeling of Extreme Values (pp. 105–123). Springer. https://doi.org/10.1007/978-1-4471-3675-0_6.

Creal, D., Koopman, S. J., & Lucas, A. (2013). Generalized Autoregressive Score Models with Applications. Journal of Applied Econometrics, 28(5), 777–795. https://doi.org/10.1002/jae.1279.

Danielsson, J. (1994). Stochastic Volatility in Asset Prices Estimation with Simulated Maximum Likelihood. Journal of Econometrics, 64(1–2), 375–400. https://doi.org/10.1016/0304-4076(94)90070-1.

Diebold, F. X., & Mariano R. S. (1995). Comparing Predictive Accuracy. Journal of Business & Economic Statistics, 13(3), 253–263. https://doi.org/10.1080/07350015.1995.10524599.

Du, Z., & Escanciano, J. C. (2016). Backtesting Expected Shortfall: Accounting for Tail Risk. Management Science, 63(4), 940–958. https://doi.org/10.1287/mnsc.2015.2342.

Dufour, J.-M. (2006). Monte Carlo Tests with Nuisance Parameters: A General Approach to Finite-Sample Inference and Nonstandard Asymptotics. Journal of Econometrics, 133(2), 443–477. https://doi.org/10.1016/j.jeconom.2005.06.007.

Echaust, K., & Just, M. (2020a). Value at Risk Estimation Using the GARCH-EVT Approach with Optimal Tail Selection. Mathematics, 8(1), 1–24. https://doi.org/10.3390/math8010114.

Echaust, K., & Just, M. (2020b). A Comparison of Conditional and Unconditional VaR Models. In P. Maresova, P. Jedlicka, K. Firlej & I. Soukal (Eds.), Proceedings of the International Scientific Conference Hradec Economic Days 2020 (pp. 1–12). http://dx.doi.org/10.36689/uhk/hed/2020-01-014.

Embrechts, P., Klüppelberg, C., & Mikosch, T. (2003). Modelling Extremal Events for Insurance and Finance. Springer.

Engle, R. F. (1982). Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation. Econometrica, 50(4), 987–1007. https://doi.org/10.2307/1912773.

Fałdziński, M., Fiszeder, P., & Molnár, P. (2024). Improving volatility forecasts: Evidence from range-based models. The North American Journal of Economics and Finance, 69B, 1–28. https://doi.org/10.1016/j.najef.2023.102019.

Fantazzini, D. (2011). The Intraday Analysis of Volatility, Volume and Spreads: A Review with Applications to Futures’ Markets. In G. N. Gregoriou & R. Pascalau (Eds.), Financial Econometrics Modeling: Market Microstructure, Factor Models and Financial Risk Measures (pp. 92–131). Palgrave Macmillan. https://doi.org/10.1057/9780230298101_4.

Ferro, C. A. T., & Segers, J. (2003). Inference for clusters of extreme values. Journal of the Royal Statistical Society. Series B (Statistical Methodology), 65(2), 545–556.

Fisher, R. A., & Tippett, L. H. C. (1928). Limiting Forms of the Frequency Distribution of the Largest or Smallest Member of a Sample. Mathematical Proceedings of the Cambridge Philosophical Society, 24(2), 180–190. https://doi.org/10.1017/S0305004100015681.

Fiszeder, P., & Fałdziński, M. (2019). Improving forecasts with the co-range dynamic conditional correlation model. Journal of Economic Dynamics & Control, 108, 1–16. https://doi.org/10.1016/j.jedc.2019.103736.

Fiszeder, P., Fałdziński, M., & Molnár, P. (2019). Range-based DCC models for covariance and value-at-risk forecasting. Journal of Empirical Finance, 54, 58–76. https://doi.org/10.1016/j.jempfin.2019.08.004.

Fiszeder, P., Fałdziński, M., & Molnár, P. (2023a). Modeling and forecasting dynamic conditional correlation with opening, high, low and closing prices. Journal of Empirical Finance, 70, 308–321. https://doi.org/10.1016/j.jempfin.2022.12.007.

Fiszeder, P., Fałdziński, M., & Molnár, P. (2023b). Attention to oil prices and its impact on the oil, gold and stock markets and their covariance. Energy Economics, 120, 1–10. https://doi.org/10.1016/j.eneco.2023.106643.

Fiszeder, P., & Perczak, G. (2016). Low and high prices can improve volatility forecasts during periods of turmoil. International Journal of Forecasting, 32(2), 398–410. https://doi.org/10.1016/j.ijforecast.2015.07.003.

Fuentes, F., Herrera, R., & Clements, A. (2023). Forecasting extreme financial risk: A score-driven approach. International Journal of Forecasting, 39(2), 720–735. https://doi.org/10.1016/j.ijforecast.2022.02.002.

Garman, M. B., & Klass, M. J. (1980). On the Estimation of Security Price Volatilities from Historical Data. The Journal of Business, 53(1), 67–78.

Glosten, L. R., Jagannathan, R., & Runkle, D. E. (1993). On the Relation between the Expected Value and the Volatility of the Nominal Excess Return on Stocks. The Journal of Finance, 48(5), 1779–1801. https://doi.org/10.1111/j.1540-6261.1993.tb05128.x.

Gnedenko, B. V. (1943). Sur la Distribution Limite du Terme Maximum D’Une Série Aléatoire. Annals of Mathematics, 44(3), 423–453. https://doi.org/10.2307/1968974.

Hansen, B. E. (1994). Autoregressive Conditional Density Estimation. International Economic Review, 35(3), 705–730. https://doi.org/10.2307/2527081.

Harvey, C. R., & Siddique, A. (1999). Autoregressive Conditional Skewness. Journal of Financial and Quantitative Analysis, 34(4), 465–487. https://doi.org/10.2307/2676230.

Herrera, R., & Clements, A. (2020). A marked point process model for intraday financial returns: modeling extreme risk. Empirical Economics, 58, 1575–1601. https://doi.org/10.1007/s00181-018-1600-y.

Herrera, R., & Schipp, B. (2013). Value at risk forecasts by extreme value models in a conditional duration framework. Journal of Empirical Finance, 23, 33–47. https://doi.org/10.1016/j.jempfin.2013.05.002.

Hosking, J. R. M., & Wallis, J. R. (1987). Parameter and Quantile Estimation for the Generalized Pareto Distribution. Technometrics, 29(3), 339–349. https://doi.org/10.2307/1269343.

Hosking, J. R. M., Wallis, J. R., & Wood, E. F. (1985). Estimation of the Generalized Extreme-Value Distribution by the Method of Probability-Weighted Moments. Technometrics, 27(3), 251–261. https://doi.org/10.2307/1269706.

D’Innocenzo, E., Lucas, A., Schwaab, B., & Zhang, X. (2024). Modeling Extreme Events: Time-Varying Extreme Tail Shape. Journal of Business & Economic Statistics, 42(3), 903–917. https://doi.org/10.1080/07350015.2023.2260439.

Jalal, A., & Rockinger, M. (2008). Predicting Tail-Related Risk Measures: The Consequences of Using GARCH Filters for Non-GARCH Data. Journal of Empirical Finance, 15(5), 868–877. https://doi.org/10.1016/j.jempfin.2008.02.004.

Jansen, D. W., & de Vries, C. G. (1991). On the Frequency of Large Stock Returns: Putting Booms and Busts into perspective. The Review of Economics and Statistics, 73(1), 18–24. https://doi.org/10.2307/2109682.

Jeon, J., & Taylor, J. W. (2013). Using CaViaR Models with Implied Volatility for Value-at-Risk Estimation. Journal of Forecasting, 32(1), 62–74. https://doi.org/10.1002/for.1251.

Kim, S., Shephard, N., & Chib, S. (1998). Stochastic Volatility: Likelihood Inference and Comparison with ARCH Models. Review of Economic Studies, 65(3), 361–393.

Koedijk, K. G., Schafgans, M. M. A., & de Vries, C. G. (1990). The Tail Index of Exchange Rate Returns. Journal of International Economics, 29(1–2), 93–108. https://doi.org/10.1016/0022-1996(90)90065-T.

Kuester, K., Mittnik, S., & Paolella, M. S. (2006). Value-at-Risk Prediction: a Comparison of Alternative Strategies. Journal of Financial Econometrics, 4(1), 53–89. https://doi.org/10.1093/jjfinec/nbj002.

Kupiec, P. H. (1995). Techniques for Verifying the Accuracy of Risk Measurement Models. The Journal of Derivatives, 3(2), 73–84. https://doi.org/10.3905/jod.1995.407942.

Leadbetter, M. R., Lindgren, G., & Rootzén, H. (1983). Extremes and Related Properties of Random Sequences and Processes. Springer. https://doi.org/10.1007/978-1-4612-5449-2.

Lopez, J. A. (1998). Regulatory Evaluation of Value-at-Risk Models. Journal of Risk, 1(2), 37–64. https://doi.org/10.21314/JOR.1999.005.

Massacci, D. (2016). Tail Risk Dynamics in Stock Returns: Links to the Macroeconomy and Global Markets Connectedness. Management Science, 63(9), 3072–3089. https://doi.org/10.1287/mnsc.2016.2488.

McAleer, M., Jiménez-Martín, J.-A., & Pérez-Amaral, T. (2010). What Happened to Risk Management During the 2008–2009 Financial Crisis?. In R. W. Kolb (Ed.), Lessons from the financial crisis: causes, consequences, and our economic future (pp. 307–316). Wiley. https://doi.org/10.1002/9781118266588.ch39.

McAleer, M., Jiménez-Martín, J.-Á., & Pérez-Amaral, T. (2013). International evidence on GFC--robust forecasts for risk management under the Basel Accords. Journal of Forecasting, 32(3), 267–288. https://doi.org/10.1002/for.1269.

McNeil, J. A., & Frey, F. (2000). Estimation of tail-related risk measures for heteroscedastic financial time series: an extreme value approach. Journal of Empirical Finance, 7(3–4), 271–300. https://doi.org/10.1016/S0927-5398(00)00012-8.

Melino, A, & Turnbull, S. M. (1990). Pricing Foreign Currency Options with Stochastic Volatility. Journal of Econometrics, 45(1–2), 239–265. https://doi.org/10.1016/0304-4076(90)90100-8.

Molnár, P. (2012). Properties of Range-Based Volatility Estimators. International Review of Financial Analysis, 23, 20–29. https://doi.org/10.1016/j.irfa.2011.06.012.

Molnár, P. (2016). High-low range in GARCH models of stock return volatility. Applied Economics, 48(51), 4977–4991. https://doi.org/10.1080/00036846.2016.1170929.

Nelson, D. B. (1991). Conditional Heteroskedasticity in Asset Returns: A New Approach. Econometrica, 59(2), 347–370. https://doi.org/10.2307/2938260.

Nieto, M. R., & Ruiz, E. (2016). Frontiers in VaR forecasting and backtesting. International Journal of Forecasting, 32(2), 475–501. https://doi.org/10.1016/j.ijforecast.2015.08.003.

Pagan, A. R., & Schwert, G. W. (1990). Alternative Models for Conditional Stock Volatility. Journal of Econometrics, 45(1–2), 267–290. https://doi.org/10.1016/0304-4076(90)90101-X.

Parkinson, M. (1980). The Extreme Value Method for Estimating the Variance of the Rate of Return. The Journal of Business, 53(1), 61–65. https://doi.org/10.1086/296071.

de la Pena, V. H., Rivera, R., & Ruiz-Mata, J. (2007). Quality control of risk measures: backtesting VaR models. Journal of Risk, 9(2), 39–54. http://dx.doi.org/10.21314/JOR.2007.147.

Petropoulos, F., Apiletti, D., Assimakopoulos, V., Babai, M. Z., Barrow, D. K., Taieb, S. B., Bergmeir, C., Bessa, R. J., Bijak, J., Boylan, J. E., Browell, J., Carnevale, C., Castle, J. L., Cirillo, P., Clements, M. P., Cordeiro, C., Oliveira, F. L. C., de Baets, S., Dokumentov, A., Ellison, J., Fiszeder, P., Franses, P. H., Frazier, D. T., Gilliland M., Gönül, M. S., Goodwin, P., Grossi L., Grushka-Cockayne, Y., Guidolin, M., Gunter, U., Guo, X., Guseo, R., Harvey, N., Hendry, D. F., Hollyman, R., Januschowski, T., Jeon, J., Jose, V. R. R., Kang, Y., Koehler, A. B., Kolassa, S., Kourentzes, N., Leva, S., Li, F., Litsiou, K., Makridakis, S., Marti, G. M., Martinez, A. B., Meeran, S., Modis, T., Nikolopoulos, K., Önkal, D., Paccagnini, A., Panagiotelis, A., Panapakidis, I., Pavía, J. M., Pedio, M., Pedregal, D. J., Pinson, P., Ramos, P., Rapach, D. E., Reade, J. J., Rostami-Tabar, B., Rubaszek, M., Sermpinis, G., Shang, H. L., Spiliotis, E., Syntetos, A. A., Talagala, P. D., Talagala, T. S., Tashman, L., Thomakos, D., Thorarinsdottir, T., Todini, E., Arenas, J. R. T., Wang, X., Winkler, R. L., Yusupova, A., & Ziel, F. (2022). Forecasting: Theory and Practice. International Journal of Forecasting, 38(3), 705–871. https://doi.org/10.1016/j.ijforecast.2021.11.001.

Pérignon, C., & Smith, D. R. (2008). A New Approach to Comparing VaR Estimation Method. The Journal of Derivatives, 16(2), 54–66. https://doi.org/10.3905/JOD.2008.16.2.054.

Pickands, J. (1975). Statistical Inference Using Extreme Order Statistics. Annals of Statistics, 3(1), 119–131. https://doi.org/10.1214/aos/1176343003.

Pritsker, M. (2006). The Hidden Dangers of Historical Simulation. Journal of Banking and Finance,30(2), 561–582. https://doi.org/10.1016/j.jbankfin.2005.04.013.

Rocco, M. (2014). Extreme Value Theory in Finance: a Survey. Journal of Economic Surveys, 28(1),82–108. https://doi.org/10.1111/j.1467-6419.2012.00744.x.

Rogers, L. C. G., & Satchell, S. E. (1991). Estimating Variance from High, Low and Closing Prices. The Annals of Applied Probability, 1(4), 504–512. https://doi.org/10.1214/aoap/1177005835.

Sarma, M., Thomas, S., & Shah, A. (2003). Selection of Value-at-Risk Models. Journal of Forecasting, 22(4), 337–358. https://doi.org/10.1002/for.868.

Şener, E., Baronyan, S., & Mengütürk, L. A. (2012). Ranking the predictive performance of valueat- risk estimation methods. International Journal of Forecasting, 28(4), 849–873. https://doi.org/10.1016/j.ijforecast.2011.10.002.

Shu, J., & Zhang, J. E. (2006). Testing Range Estimators of Historical Volatility. Journal of Futures Markets, 26(3), 297–313. https://doi.org/10.1002/fut.20197.

Smith, R. L. (1985). Maximum likelihood estimation in a class of nonregular cases. Biometrika, 72(1), 67–90. https://doi.org/10.1093/biomet/72.1.67.

Smith, R. L. (1987). Estimating tails of probability distributions. The Annals of Statistics, 15(3), 1174–1207. https://doi.org/10.1214/aos/1176350499.

So, M. K. P., & Yu P. L. H. (2006). Empirical analysis of GARCH models in value at risk estimation. Journal of International Financial Markets, Institutions and Money, 16(2), 180–197. https://doi.org/10.1016/j.intfin.2005.02.001.

Straumann, D. (2005). Estimation in Conditionally Heteroscedastic Time Series Models. Springer. https://doi.org/10.1007/b138400.

Su, Y. K., & Wu, C. C. (2014). A New Range-Based Regime-Switching Dynamic Conditional Correlation Model for Minimum-Variance Hedging. Journal of Mathematical Finance, 4(3), 207–219. https://doi.org/10.4236/jmf.2014.43018.

Taylor, S. (1990). Modelling Stochastic Volatility (Discussion Paper). University of Lancaster.

Tomlinson, M. F., Greenwood, D., & Mucha-Kruczyński, M. (2024). 2T-POT Hawkes model for left- and right-tail conditional quantile forecasts of financial log returns: Out-of-sample comparison of conditional EVT models. International Journal of Forecasting, 40(1), 324–347. https://doi.org/10.1016/j.ijforecast.2023.03.003.

Weiss, A. A. (1986). Asymptotic Theory for ARCH Models: Estimation and Testing. Econometric Theory, 2(1), 107–131. https://doi.org/10.1017/S0266466600011397.

White, H. (2000). A Reality Check for Data Snooping. Econometrica, 68(5), 1097–1126. https://doi.org/10.1111/1468-0262.00152.

Wu, C. C., & Liang, S. S. (2011). The Economic Value of Range-Based Covariance between Stock and Bond Returns with Dynamic Copulas. Journal of Empirical Finance, 18(4), 711–727. https://doi.org/10.1016/j.jempfin.2011.05.004.

Xie, H. (2019). Financial volatility modeling: The feedback asymmetric conditional autoregressive range model. Journal of Forecasting, 38(1), 11–28. https://doi.org/10.1002/for.2548.

Yang, D., & Zhang, Q. (2000). Drift-Independent Volatility Estimation Based on High, Low, Open and Close Prices. Journal of Business, 73(3), 477–492. https://doi.org/10.1086/209650.

Yao, X., Izzeldin, M., & Li, Z. (2019). A novel cluster HAR-type model for forecasting realized volatility. International Journal of Forecasting, 35(4), 1318–1331. https://doi.org/10.1016/j.ijforecast.2019.04.017.

Zhang, X., & Schwaab, B. (2016). Tail risk in government bond markets and ECB unconventional policies. https://www.berndschwaab.eu/papers/ScoreTailRisk.pdf.