Value at Risk long memory volatility models with heavy-tailed distributions for cryptocurrencies

Abstract

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This paper investigates the volatility dynamics and underlying long memory features of four major cryptocurrencies—Bitcoin, Ethereum, Litecoin, and Ripple—which were selected due to their high liquidity, large trading volumes, and historical significance in the digital asset market. The long-range dependence exhibited in cryptocurrency markets is often overlooked. However, based on the strong evidence of persistent dependence in the return series, we adopt advanced volatility models that are capable of accommodating high volatility and heavy-tails, as well as the long memory properties of cryptocurrencies. Specifically, we employ long-memory extensions of the GAS (Long memory GAS) and GARCH (Fractionally Integrated Asymmetric Power ARCH) models, integrating heavy-tailed innovation distributions: the Generalized Hyperbolic Distribution (GHD) and Generalized Lambda Distribution (GLD). Standard GARCH and GAS models are included as benchmarks. The performance of the models are assessed using Value-at-Risk (VaR) estimation, backtesting (in-sample and out-of-sample) and volatility forecasting metrics. The results indicate that long memory models, particularly the FIAPARCH model, consistently outperforms the standard GAS and GARCH models in capturing tail risk and the volatility persistence. These findings emphasize the critical role of long memory in modeling the risk of cryptocurrencies, indicating that accounting for volatility persistence can significantly enhance the accuracy of risk estimates and strengthen risk management practices.

Keywords

• cryptocurrency
• Generalized Autoregressive Conditional Heteroskedasticity (GARCH)
• generalized autoregressive score (GAS)
• long memory (LM)
• Value-at-Risk (VaR)