**4. Conclusions**

It is now a widely accepted view that risk models should account for the stylized facts of the data in order to be successfully validated. Estimating risk was mainly performed on many financial asset markets but not on the emerging cryptocurrency market, which has been proven to be extremely volatile. Typical volatility models may not adequately provide an accurate representation of the cryptocurrencies volatility process for successful risk management purposes. In particular, risk models must be able to capture the cryptocurrencies volatility process that includes stochastic volatility, persistence in volatility, and jump process. All these stylized features are critical for capturing unpredictable and large movements in the price process and for accurately predicting tail risk and expected shortfall. There is limited research on this topic despite the fact that investors are exploring how cryptocurrencies can be integrated into a portfolio along with other traditional assets such as stocks, bonds, currencies, and commodities. Choosing a proper model that provides a parsimonious representation of the distribution of the return-generating process is the first step.

In this paper, we identified risk models for the cryptocurrency market and evaluated their performance for validation purposes. We evaluated models based on stochastic volatility with co-jumps in returns and volatility (SVCJ), threshold GARCH volatility (TGARCH), and RiskMetrics. Backtesting methods using the conditional and unconditional coverage were performed to test the validity of the models, and the regulatory loss function was applied to choose the most accurate model.

The validation results reveal that, although the models considered in this paper are effective for fitting the cryptocurrency returns, the SVCJ model more accurately forecasts risk in a VaR and ES sense, and the reality check proves its superiority over TGARCH and RiskMetrics models. Therefore, incorporating jumps in the cryptocurrency volatility model improves the forecasting ability of risk in terms of VaR and ES. This is important for risk-averse investors and for speculative investors who are particularly interested in hedging their risk in a VaR sense. It is, therefore, recommended to use a model that accounts for jumps, leptokurtosis, and leverage effects when dealing with cryptocurrency market data. Such a model improves risk forecasting in terms of VaR and Expected Shortfall.

The results in this study have several implications for applying the SVCJ model to other assets including commodities, foreign currencies, and stock market indices, especially in times of stress. The global financial market has seen unprecedented volatility in recent days, given falling oil prices and concerns related to the COVID-19 pandemic. It would be interesting to see if such wild swings in the market can be studied using the SVCJ model to incorporate the co-jumps in returns and volatility affecting the measurement of VaR and Expected Shortfalls in the contagion like period that we now have. We leave that for a future study.

**Author Contributions:** R.N. collected the data, carried the statistical analysis, and drafted the manuscript. J.S. carried the literature review and helped with the discussion. All authors have read and agreed to the published version of the manuscript.

**Acknowledgments:** R.N. would like to acknowledge the financial support provided by Applied Science University (Grant: Admin1/2019).

**Conflicts of Interest:** The authors declare no conflict of interest.

**Appendix A**






insignificance

cryptocurrencies.

 at 5% and 1% levels. Higher

LogLikelihood

 and lower AIC indicate the best fit model for
