This section focuses on recent and relevant literature, and is divided into three subsections. The first subsection focuses on cryptocurrency indices, and the second reviews relevant literature based on GARCH models applied to cryptocurrencies. Finally, studies based on cryptocurrency volatility indices are considered.

#### 2.1. Cryptocurrency Indices

According to

Chu et al. (

2017), with the exception of Bitcoin, there is not much literature focused on the application of GARCH models to cryptocurrencies. Therefore, the CRIX is also considered in this paper. According to

Abboud (

2017), there have been several attempts to construct a cryptocurrency index. Most cryptocurrency index attempts make use of empirical models from traditional financial markets with arbitrary parameters fitted to cryptocurrencies. The indices include capitalisation weighted indices like CRIX, Bletchley, TaiFu30, Crypto30, LBI, and Smith + Crown SCI. Furthermore, capped capitalisation indices include: CRYPTO20, CCX30, and BIT20. Finally, the smoothed capitalisation weighted index, such as the CCI30.

According to

Kim et al. (

2019), the CRIX is comparable to the S&P 500 index (reflection of the current state of the US market) because it gives an indication of the current state of the cryptocurrency market. Furthermore,

Kim et al. (

2019) explain that the CRIX provides a statistically backed (the number of constituents is determined by the explanatory power of each cryptocurrency has over market movements, this is based on the Akaike information criterion) market measure, which distinguishes it from other cryptocurrency indices. Therefore, the CRIX is used in this study to give an indication of the volatility of the cryptocurrency market as a whole. The CRIX was also used as a proxy for the cryptocurrency market by

Elendner et al. (

2018);

Hafner (

2020);

Klein et al. (

2018).

#### 2.2. GARCH Models Applied to Cryptocurrencies

When it comes to the topic of time-varying volatility, most financial modelling researchers and practitioners will agree that the GARCH model is the most popular. GARCH models applied to cryptocurrencies have gained a lot of attention in recent years (as mentioned previously, most of this work has been based on Bitcoin). In an attempt to forecast Bitcoin risk,

Agyarko et al. (

2019) made use of univariate symmetric and asymmetric GARCH models. Their empirical results indicate that the symmetric GARCH(1,1) model provides the best fit. This is also consistent with the argument by

Hansen and Lunde (

2005), that it is difficult to find a model that consistently outperforms the GARCH(1,1) model because it is highly robust and parsimonious. With regard to forecasting risk,

Agyarko et al. (

2019) explain that no model clearly emerged as superior. Therefore, the study indicates that it is reliable to use the best fitted model when forecasting volatility (symmetric GARCH, in the case of Bitcoin).

Chen et al. (

2018) performed an econometric analysis of the CRIX for portfolio investment. The empirical analysis included the application of autoregressive integrated moving average (ARIMA), univariate GARCH, and multivariate GARCH models. Their empirical results illustrate that the GARCH(1,1) model is sufficient to explain the heteroskedasticity of the CRIX.

Chen et al. (

2018) also consider alternate GARCH specifications. To capture the leverage effect (negative relationship between return shocks and subsequent shocks to volatility), the exponential GARCH (EGARCH) model was estimated. However,

McAleer and Hafner (

2014) show that leverage is not possible for the EGARCH model.

Chen et al. (

2018) conclude that the symmetric GARCH(1,1) model with a Student-

t error distribution is the best performing univariate model when applied to the CRIX.

In order to determine the effect weather has on the cryptocurrency market,

Kathiravan et al. (

2019) made use of a GARCH(1,1) model, Johansen cointegration, and a Granger causality test. The Coinbase index was used as a proxy for the cryptocurrency market in this study. The GARCH analysis showed that temperature is the only weather factor that is statistically significant when modelling cryptocurrency volatility.

To give an indication of the best performing volatility model when applied to the cryptocurrencies market (not focused on Bitcoin only),

Chu et al. (

2017) applied twelve GARCH models (eight different error distributions) to the seven most popular cryptocurrencies. The models were compared based on the goodness of fit, forecasting performance, and acceptability of value-at-risk estimates. Their empirical results indicate that the normal distribution provides the best fitting GARCH model in most cases. Furthermore, the symmetric integrated GARCH(1,1) (IGARCH(1,1)) model with normal innovation was the best fitting model for most cryptocurrencies.

In a recent study,

Hafner (

2020) made use of GARCH models to test for the existence of speculative bubbles in the cryptocurrency market. The empirical analysis made use of eleven of the largest cryptocurrencies and the CRIX. The estimated parameters of the GARCH models indicate that volatility clustering is important and significant when modelling cryptocurrency volatility and, unlike equities, cryptocurrencies do not have asymmetric news impact curves. More specifically, the asymmetry terms of asymmetric GARCH models are generally statistically insignificant when applied to cryptocurrencies; this is consistent with other findings in the literature (see, e.g.,

Baur and Dimpfl 2018). According to

Hafner (

2020), there is general evidence that speculative bubbles exist in cryptocurrency markets.

Gyamerah (

2019) made use of the symmetric GARCH(1,1), threshold-GARCH(1,1) (TGARCH(1,1)), and IGARCH(1,1) models to model the volatility of Bitcoin returns. With regard to the error distribution,

Gyamerah (

2019) considered the Student-

$t,$ generalised error, and normal inverse Gaussian distributions. The different models were compared based on the Akaike and Bayesian information criteria. Their empirical results indicate that the asymmetric TGARCH(1,1) model with a normal inverse Gaussian error distribution is the best fitting model when modelling volatility of Bitcoin returns. This implies that incorporating asymmetry in the GARCH model specification, and skewness and kurtosis in the error distribution, can improve the fit of a GARCH model when applied to Bitcoin.

In order to determine the best performing model when forecasting exchange rate and cryptocurrency (Bitcoin, Ethereum, and Dash) volatility,

Peng et al. (

2018) made use of the following univariate GARCH models: GARCH(1,1), EGARCH(1,1) and the Glosten, Jagannathan and Runkle GARCH(1,1) (GJR-GARCH(1,1)) model. Three different error distributions were considered: normal, Student-

t, and skewed Student-

t distributions. In addition, a support vector regression (SVR) GARCH(1,1) model was also estimated. Their empirical results show that the SVR-GARCH(1,1) model is superior when compared to the other models considered. Furthermore, when the traditional GARCH models are compared, the GJR-GARCH(1,1) performed slightly better when compared to the symmetric GARCH(1,1) and EGARCH(1,1) models. The different error distributions yielded similar results. This illustrates that different GARCH specifications can offer better results when applied to exchange rate and cryptocurrency volatility.

#### 2.3. Cryptocurrency Volatility Indices

Studies based on cryptocurrency volatility indices are limited; this is because there is not a well established cryptocurrency derivatives market. Volatility indices are used based on implied volatility obtained from the option market (e.g., the CBOE VIX).

Alexander and Imeraj (

2019) constructed a Bitcoin volatility index by making use of the VIX methodology (geometric variance swap), Bitcoin option data were obtained from the Deribit exchange. In addition,

Alexander and Imeraj (

2019) note that Bitcoin prices tend to jump, therefore the fair value of geometric variance swaps are underestimated using this method. Hence, the method based on arithmetic variance swaps was also employed.

Alexander and Imeraj (

2019) recommend the use of the arithmetic index for horizons of one month or more. However, the volatility index based on arithmetic or geometric (VIX methodology) variance swaps is dependent on an established derivatives market, this is not the case for all cryptocurrencies and therefore a different approach is required.

In a recent study,

Kim et al. (

2019) construct a cryptocurrency volatility index based on the CRIX. The purpose of the index is to offer a forecast for the mean annualised volatility of the next month. Due to the shortcomings of the cryptocurrency derivatives market,

Kim et al. (

2019) make use of a proxy for implied volatility, therefore rolling volatility is used; this is based on historical volatility of the underlying. To get forward looking estimates (for the next 30 days) of rolling volatility,

Kim et al. (

2019) made use of GARCH family models, the Heterogeneous Auto-Regressive (HAR) model, and a neural network-based Long short-term memory cell; the performance of the different models was compared based on the mean squared error and the mean absolute error. Their empirical results show that the HAR model is the best performing model when forecasting rolling volatility of the CRIX. However, rolling volatility is based on historical volatility and not risk-neutral volatility. Therefore, the GARCH option pricing model is used in this study in order to estimate the implied volatility index (risk-neutral) in the absence of a well established derivatives market.