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

Long memory is a phenomenon that can be described as the persistence of volatility, suggesting that past observations have an impact on future values. In financial markets, this behavior is often exhibited in volatility and thus has crucial implications for forecasting and risk management. The long memory properties of financial assets have been substantially investigated and studied, with evidence indicating that volatility is indeed a long memory process [1–4].

Cryptocurrencies share numerous commonalities with traditional financial assets; however, the volatility dynamics of cryptocurrencies are often found to be higher than traditional assets like stocks [5] as well as leading fiat currencies [6]. The cryptocurrency market is decentralized with no association to a higher authority and thus making the market structure unique. The significant development of cryptocurrencies demonstrated by their growing transaction volume and market capitalization has distinguished cryptocurrencies as a revolutionary instrument in financial markets, internationally [7]. Specifically, the cryptocurrency market has grown significantly over the last decade or so, as the global market capitalization of cryptocurrencies surged from under 20 billion USD in early 2017 to over 2.5 trillion USD at its peak in late 2021, with average daily trading volumes increasing by more than 1,000% during this period [8]. In comparison to other emerging markets, this explosive growth and extreme price fluctuations have introduced new challenges for financial modeling. Additionally, their decentralized structure, sensitivity to sentiment, speculative trading behavior and continuous 24/7 trading with no market closures, contribute to the significant volatility persistence and clustering, suggesting that the long memory property has become particularly relevant for cryptocurrency modeling.

The volatility exhibited by cryptocurrencies has been studied extensively [9–13] and it is apparent that the market is highly volatile in nature. This extremity may result in distinct trends in volatility persistence and thus it is imperative to study if there are long memory influences in the volatility of these markets similar to that of other financial time series. Rambaccussing and Mazibas [14] investigated the long memory properties in the returns and volatility of five cryptocurrencies, Bitcoin, Litecoin, Ethereum, Bitcoin Cash, and Ripple of which they discovered that long memory may not explicitly be present in the returns of the cryptocurrencies, except for that of Ethereum where long memory does exist, however, long memory appears to be well prominent in the volatility of these cryptocurrencies. Jiang et al. [15] studied the impact of the dual long memory and structural break features of six highly-traded cryptorcurrencies. It was found that these cryptocurrencies does, in fact, exhibit both structural breaks and long memory properties in their returns, and are especially present in the volatility. Soylu et al. [7] explored the long memory traits of Bitcoin, Ethereum, and Ripple. The cryptocurrencies were tested for long memory using Rescaled Range Statistics (R/S), Gaussian Semi Parametric (GSP), and the Geweke and Porter-Hudak (GPH) Model Method. The squared returns of all three cryptocurrencies exhibited strong persistence indicating the presence of long memory.

Since long term dependencies of volatility exist in cryptocurrencies, the adoption of an appropriate model which has the capability to adequately capture the long memory traits, as well as the extreme volatility exhibited, is of significant importance to analyze and forecast the risks associated with these cryptocurrencies. The Generalized Autoregressive Conditional Heteroskedasticity (GARCH) model introduced by Bollerslev [16] is a conventional volatility model which can be extended to incorporate fractionally integrated models to encapsulate long memory properties. Although the use of the standard GARCH model is commonly employed for modeling volatility and long-memory properties, Davidson [17] established that this model may not be entirely ideal, as GARCH models do not adequately cater for outliers and thus may generate biased estimates. The effect of innovations on the subsequent conditional variance for a long memory process is generally that of a hyperbolic decay, and therefore the impact of outliers may be magnified resulting in a rising biased long memory estimate [18]. The Generalized Autoregressive Score (GAS) model, introduced by Creal et al. [19], is another volatility model that caters for the downfall of the GARCH model discussed above due to its unique robustness characteristics. The GAS model framework can also be enhanced to accommodate the presence of long memory in returns [20], making it a viable candidate to model cryptocurrencies. Chkili [21] investigated models which would best fit the long memory present in the volatility dynamics of the Bitcoin returns for the 2013–2020 period of which the Fractionally Integrated GARCH (FIGARCH) model was found to be the most favorable model. Gao and Shi [18] studied the long memory and regime switching in the second comment using GAS models. The robustness against outliers of the long memory GAS (LMGAS) model and the Markov switching GAS (MS-GAS) model was investigated utilizing West Texas Intermediate crude oil spot returns. The findings indicate that the GAS estimators were more robust when compared to its GARCH counterparts when catering for outliers. However, the LMGAS model still produced spurious long memory when a regular regime-switching process is fitted and thus an MS-LMGAS model was proposed which catered for both the spurious long memory and robustness against outliers.