Half-Life Volatility Measure of the Returns of Some Cryptocurrencies

This paper explores the half-life volatility measure of three cryptocurrencies (Bitcoin, Litecoin and Ripple). Two GARCH family models were used (PGARCH (1, 1) and GARCH (1, 1)) with the student-t distribution. It was realised that, the PGARCH (1, 1) was the most appropriate model. Therefore, it was used in determining the half-life of the three returns series. The results revealed that, the half-life was 3 days, 6 days and 4 days for Bitcoin, Litecoin and Ripple respectively. This shows that, the three coins have strong mean reversion and short half-life and that it takes the respective days for volatility in each of coin to return half way back without further volatility.


Introduction
In recent years, a new type of currencies, a synthetic one, emerged. This new type of currency is named as "synthetic" because it is not the decision of a nation or state, nor represents any underlying asset or tangible wealth source. It appears as a new tradable asset resulting from a private agreement and facilitated by the anonymity of internet. Among this synthetic currencies, Bitcoin (BTC) emerges as the most important one, with a market capitalization of 17 billion, as of June 2018. There are other cryptocurrencies, based on block chain technology, such as Litecoin (LTC), Ethereum (ETH), and Ripple (XRP) among others. Cryptocurrency is an asset derived from mathematical cryptography; it is based on a new technology called the block chain (Bradbury, 2013;Ali et al., 2014). Its other fundamental characteristics are: being decentralised, and having a fixed total One open question today is if cryptocurrencies are in fact, or may be considered as a currency. Until now, we cannot observe that cryptocurrencies fulfils the main properties of a standard currency. They are barely accepted as a medium of exchange (e.g. to buy some products online); it is not used as unit of account (there are no financial statements valued in any cryptocurrency), and we can hardly believe that, given the great swings in price, anyone can consider any cryptocurrency as a suitable option to store value. Given these characteristics, none of the cryptocurrencies could be considered as an ideal asset for speculative purposes. There are no underlying assets to relate their value to and there is an open platform to operate round the clock.
Most of the existing studies focused on Bitcoin returns. For example, Baur et al. (2017) show that Bitcoin returns are essentially uncorrelated with traditional asset classes such as stocks or bonds, which points to diversification possibilities.
Others investigated the determinants of Bitcoin returns. The findings of Li Xin and Chong Alex Wang (2017), among others, suggest that measures of financial and macroeconomic activity are drivers of Bitcoin returns . Kristoufek Ladislav (2015) considered financial uncertainty, Bitcoin trading volume in Chinese Yuan and Google trends as potential drivers of Bitcoin returns. The inclusion of Google trends as some sort of proxy for sentiment or interest is fairly common within the literature (see, for example, Polasik et al. (2015)). A recurrent theme in the literature is the question to which asset class Bitcoin belongs, with many comparing it to gold; others to precious metals or to speculative assets (see, among others, Baur et al. (2017); or Bouri et al. (2017)). Some have classified Bitcoin as something in between a currency and a commodity (see, for example, Dehrberg Anne Haubo (2016)). For other recent contributions, see Cheah et al. Some literatures try to model Bitcoin volatility. Among the first papers was Balcilar et al. (2017), who analysed the causal relation between trading volume and Bitcoin returns and volatility. They found that volume cannot help predict the volatility of Bitcoin returns. Dehrberg Anne Haubo (2016) explored Bitcoin volatility using GARCH models. The models estimated in Dehrberg Anne Haubo More recently, Conrad and Kleen (2018) used the GARCH-MIDAS model to extract the long and short term volatility component of cryptocurrencies. They considered measures of volatility and risk in the US stock market as well as a measure of global activity. It was realized that, S & P 500 volatility had a negative and highly significant effect on long term Bitcoin volatility. They also found the, the S & P 500 volatility risk premium had a significantly positive effect on long term Bitcoin volatility and that there is a positive association between Baltic dry index and long-term Bitcoin volatility. Salisu et al. (2018) exploited several condition heteroskedasticity models with various supported distributions in order to find the best distribution as well as the best GARCH-type model that could be used to model volatility of Bitcoin returns. They established that, pre-testing the residuals of the Bitcoin returns for the best distribution could help to identify the appropriate distribution when modelling with GARCH-type models regardless of the data frequency.
The purpose of this paper is to investigate the half-life volatility measure of some cryptocurrencies. This is to provide players on the cryptocurrency market with information pertaining to the half-life measure and volatility persistence of some of the cryptocurrencies so as to make informed choice on their investments.

Methods of Data Analysis
The daily closing prices were converted into compound returns given by; where t r is the continuous compound returns at time t, t p is the current closing coin price at time t and 1 t p − is the previous closing coin price.

Volatility Modelling
The error term of the GARCH-type model is fitted into the following distributions: Gaussian, Student's-t and Generalized Error.
For flexibility, the first lag is allowed in the relevant variables both in the mean and variance equations. An ARMA (1, 1) is the mean equation common to the GARCH-type models given by; where, 0 β ≠ ; The respective variance equation of the latter varies for the two GARCH-type models (GARCH (1, 1) and PGARCH (1, 1)) but with a common lag combination (1, 1) for the ARCH and GARCH components respectively.
The GARCH-type models considered are: 1) The Generalized ARCH (GARCH) Model-GARCH (1, 1); 2) The Power GARCH (PGARCH) Model-PGARCH (1, 1); ( ) where, o α is a constant, 1 α and 1 β are the standard ARCH and GARCH parameters, γ is the leverage parameter and δ is the parameter for the power term, and 0 δ > , 1 1 γ ≤ . When 2 δ = , the above equation becomes a classic GARCH model that allows for leverage effects and when 1 δ = , the conditional standard deviation will be estimated. Also, the flexibility of the PGARCH model could be increased by considering δ as another coefficient that must be estimated.

Mean Reversion
Mean reversion means that current information has no influence on the long run forecast of the volatility. Persistence dynamics in volatility is generally captured in the GARCH coefficient(s) of a stationary GARCH-type model. In stationary GARCH-type models, the volatility mean reverts to its long run level, at a rate given by the sum of ARCH and GARCH coefficients, which is usually close to one (1) for financial time series. The average number of time periods for the volatility to revert to its long run level is measured by the half-life of the volatility shock. The mean reverting form of the basic GARCH (1, 1) model is given by;

Half-Life Measure of Volatility
One measure of volatility persistence is the volatility half-life τ, Engle and Patton (2001) defined half-life as the time required for the volatility to move half way back towards its unconditional mean. More precisely, τ is the smallest k such that where k is the number of days, | t k t σ + is the conditional expected value of volatility k days into the future and 2 σ is the unconditional long run level of volatility ( the mean level to which the unconditional variance eventually reverts).
Also, the GARCH (1, 1) process is mean reverting if ( ) Thus, the forecast conditional variance reverts to the unconditional variance as the forecast horizon increases.
For k ≥ 2 and a GARCH (1, 1) process, the value of | t k t σ + is given by; From Equation (6) and Equation (7), the number of days k for a GARCH (1, 1) process is given by; Therefore, the half-life of a GARCH (1, 1) process is given by;  The three mean returns recorded kurtosis ranging from 11.1786 to 29.9421 meaning the three mean returns are leptokurtic thus, highly volatile.     respectively. It could be seen that, there is evidence of volatility clustering and long memory. That is, high returns tend to be followed by high returns and low returns followed by low returns. Also, the nature of volatility of the returns decays slowly. Table 2 shows the normality and autocorrelation test of the return series. It was realized that, the Jarque-Bera test for normality was significant at the 5% level of significance for the three return series meaning the return series were not normally distributed. The Ljung-Box Q-statistics for the return and squared returns show evidence of autocorrelation in both the return and squared return series since Q (30) and Q 2 (30) were significant at the 5% level of significance.

Further Analysis
To confirm that the returns series were not normally distributed, the Quan-    In testing for stationarity, the ADF test was employed. It is evidence from Table 3 that, the three returns series showed evidence of stationarity at the 5% level of significance.
In volatility modelling, it very paramount that one check for the presence of ARCH effects in the returns series. To examine the returns series for ARCH effects, the ARCH-LM test was employed. It is evidence from Table 4 that, all the three returns series exhibited ARCH effects at the 5% level of significance at lag 10, 20 and 30 since the P-values are all less than 0.05 significance level.
It is always necessary to know the error distribution to use when performing ized Error Distribution (GED)). Table 5, reports on the AIC and BIC after fitting the residuals to a GARCH model. It was realized that, all the three returns    Salisu et al. (2018) who used the pretesting of residual in selecting the appropriate error distribution.
A symmetric univariate GARCH model was employed to handle the magnitude of the returns. From Table 6, it is evidence that all the returns series were not stationary since their summation of α and β are all greater than one, meaning volatility in these returns series using GARCH (1, 1) model is an explosive process and that, volatility will not mean revert. Nevertheless, even though the model exhibited non-stationarity, the ACRH-LM test was not significant for all the returns series meaning further conditional heteroscedasticity was remove from the returns series.
Since the GARCH (1, 1) exhibited non-stationarity, there was the need to employ another volatility model. Hence, an asymmetric volatility model; PGARCH was considered.
An asymmetric GARCH model was employed to model the magnitude and sign of volatility in the returns series and also since the symmetric GARCH model was not stationary for all the returns series. From Table 7, it could be seen that, the PGARCH (1, 1) was stationary for all the three returns series since      some systematic trends in them with Bitcoin exhibiting much systematic trend followed by Litecoin and then Ripple.
The PGARCH (1, 1) model was employed to investigate the half-life volatility measure of the returns series. From Table 8 it is evidence that, all the three returns series exhibited volatility persistence and long memory since all the three return series had their ARCH and GARCH components less 1. The half-life of Bitcoin, Litecoin and Ripple were 3 days, 6 days and 4 days respectively. This means that, they all have strong mean reverting rate and short half-life. Again, a shock in Bitcoin will take 3 days for it to return half way back to its volatility. A shock in the returns of Litecoin will take 6 days for it to mean revert without any further volatility. Volatility in Ripple will last for 4 days after which it will return half way back to its mean without further volatility. For investors on the coin market, it will be prudent to stay with coins with strong mean reversion and thus short half-life so as not to suffer much volatility with your portfolios. Therefore, in this paper, it advisable to stay with Bitcoin since it has the strongest mean reversion rate and the shortest half-life, followed by Ripple and then Litecoin.

Conclusion
This paper determined the half-life volatility measure of three cryptocurrencies the volatility modelling, a pre-testing of the residuals was done in which the student's-t distribution was selected. A symmetric GARCH (1, 1) model was employed but it was realized that; it was non-stationary for all the returns series.
Therefore another GARCH-family model was employed; the PGARCH (1, 1) which exhibited stationarity in all the three returns series. The PGARCH (1, 1) was then considered in determining the half-life of the three returns series. The results revealed that, the half-life was 3 days, 6 days and 4 days for Bitcoin, Litecoin and Ripple respectively. This shows that, the three coins had strong mean reversion and short half-life. It is therefore prudent for investors to stay with Bitcoin since it has the shortest half-life.