It is well-known that the volatility of asset returns varies over time. While considerable research has examined the time-series relation between realised volatility and expected returns for traditional asset classes, no systematic study has been conducted as far as cryptocurrencies are concerned.
In this report, we aim at constructing a Realised Volatility (RV) measure for each of the top 25 cryptocurrencies by market capitalisation. In addition, we construct aggregate measures of realised volatility by aggregating crypto-specific RVs based on both a value-weighting scheme and a volume-weighting scheme. In this respect, we create both a value-weighted and a volume-weighted volatility index for the cryptocurrency market as a whole.
We first define the intraday log returns for each cryptocurrency. On day t, the ith intraday return is given by:
here p is the natural logarithm of the price and N is the number of returns observations in a trading day. Notice that a trading day in cryptocurrency markets spans over 24 hours as there is no official market closure. A well-known realised volatility measure is obtained by summing squares of intraday high-frequency returns:
As is standard, we do not estimate the mean of the high-frequency returns because mean estimates are dominated by the variance at this frequency. The reason why we focus on RV measures is that they are model-free in nature, which makes them appealing compared to other estimation methods. Moreover, as discussed in existing academic literature, realised variance converges to a well-defined quadratic variation limit as the sample frequency N increases. We construct measures for each day t and each cryptocurrency, so that we have a collection. By aggregating the RV measures, we can obtain aggregate market-wide measures of risk. In particular, market RV can be constructed as:
Equation 1: Realised Volatility
The weights are calculated in three different ways; Equation 1 by taking a simple equal-weight average, and then Equation 2 by looking at the relative market capitalisation of a given cryptocurrency, that is:
with MV as the market-value of cryptocurrency j in day t, and Equation 3 by taking the relative amount of transaction volume for a given cryptocurrency, that is:
with TradingVolume as the daily average trading volume in US dollars for cryptocurrency j at day t.
We first report the dynamics of the realised volatility for the market for different weighting schemes. The left panel of Figure 1 shows the daily realised volatility (RV) indexes calculated as a weighted average of the square root of the sum of intra-day squared returns for the top 25 (in terms of market capitalization) cryptocurrencies. The sample period goes from January 2017 to May 2019. The blue line represents an equal-weight RV index, whereas the red and the green lines represent a value-weight and a volume-weight RV index, respectively.
Figure 1: Daily realised volatility
A few interesting facts emerge. First, it is clear that there is a downward trend in volatility in cryptocurrency markets. This is true regardless the weighting scheme, that is, reduction in aggregate volatility is not a phenomenon that is due to value effects or a stabilisation in trading volume. As a matter of fact, increasingly low realised volatility is evident even for the equal-weight measure. Second, by taking a simple equal-weight average of the individual RV measures, the aggregate market volatility is massively overestimated. This is due to the fact that by assuming we implicitly overstate the effect of smaller cryptocurrencies while underweight (in relative terms) the contribution of larger ones. However, smaller cryptocurrencies tend to be more volatile, which explains the highest magnitude of the equal-weight volatility index. Third, the dynamics of volatility in cryptocurrency markets is highly similar regardless if the weighting scheme is value-based or volume-based. In this respect, the dynamics of the value-weighted and the volume-weighted indexes are virtually the same.
The right panel shows a smoother version of the RV indexes constructed by taking the 20-day moving average of the daily RV measure. Such smoothed dynamics should give a clearer understanding of the mismatching between equal-weight and alternative weighting schemes, as well as the aggregate downward trend in the dynamics of volatility in cryptocurrency markets.
Individual Realised Volatility Measures
Figure 1 reports the aggregate dynamics of realised volatility in cryptocurrency markets. However, by definition, this is an average and does not consider crypto-specific dynamics for volatilities. We now investigate the dynamics of volatility for a subset of major cryptocurrencies. Figure 2 reports the daily realised volatility (RV) measure for four major cryptocurrencies, namely Bitcoin, Ethereum, Litecoin, and Ripple.
Figure 2: Realised Volatility of bitcoin, Ether, Litecoin and XRP
In each panel, the figure reports both the raw measure (light-blue line) and a smoothed RV measure (red line). Except few nuances, there is some commonality in the dynamics of volatility across major cryptocurrencies. All cryptos under investigation showed increasing volatility around July 2017, the period of the massive price increase, as well as in the early 2018, the overall collapse of the market. Such spikes in volatility can be seen more clearly by the smoothed RV measures. In all cases there is an increasing trend in volatility towards mid-2017 and early-2018.
Next, we report some descriptive statistics of the RV for all the major cryptocurrencies under investigation. In particular, Figure 3 shows the average RV over the sample period, January 2017 to May 2019. The figure shows that there is a substantial heterogeneity in the cross-section of RVs, with, for instance, Verge (XVG) that has a daily average volatility as high as 12%, whereas Bitcoin (BTC) has a daily average volatility as low as 2%. Such heterogeneity is possibly due to a variety of reasons, such as the diffusion of a given crypto, its maturity either as investment and / or method of payment, as well as the status of its development.
Figure 3: Average daily realised volatility for top cryptocurrencies
We also investigate the relationship between current volatility and future returns in the cross-section of cryptocurrencies. In order to investigate such a relationship, on average, we estimate a panel regression of the following form:
where Returns represents the one-day ahead return on the cryptocurrency i = 1, ..., N at time t = 1, ..., T, sigma the realised volatility measure for the cryptocurrency i, x a set of control variables, such as lagged returns and realised volatility, alpha an individual fixed effect, theta a time fixed-effect, and nu an idiosyncratic error term.
Table 1: Risk-return trade-off
Table 1 reports the estimation results. The key parameter of interest is the effect of volatility on future returns. Few interesting aspects emerge. First, there is no evidence of a significant relationship between current realised volatility and returns. This is true regardless of the regression specification. Second, there mild evidence of a short-term reversal of returns. That is, there is a weakly significant autocorrelation of daily returns (see columns 6 and 7). Third, there is strong evidence of a leading effect of current realised volatility on one-day ahead realised returns. That means that the higher the volatility the higher the risk premium for a given crypto, on average. In other words, there seems to be a quite strong predictive content in current volatility for future cryptocurrency returns, at least in the short term.
Delving further into the aggregate volatility in cryptocurrency markets we now investigate the existence of spill over effects between Bitcoin volatility and the returns variability of alternative major cryptocurrencies. In order to investigate the significance of such spill over effects we estimate the following panel regression:
where omega is the realised volatility of cryptocurrency i = 1, ..., N at time t = 1, ..., T, omega is the daily realised volatility of Bitcoin in a given day, x a set of control variables, such as lagged returns and realised volatility, alpha an individual fixed effect, theta a time fixed-effect, and nu an idiosyncratic error term.
Table 2: Volatility spill over between BTC and alternative coins
Table 2 reports the estimation results. The empirical evidence clearly shows that there is a strong contemporaneous spill over effect of Bitcoin volatility on the volatility of other cryptocurrencies, on average. Changes in the volatility of BTC not only have a contemporaneous relationship with the volatility of other cryptocurrencies, but also show some leading effects. As a matter of fact, the coefficient on omega (BTC, t-1) is highly statistically significant. The results are robust to the inclusion of lagged volatilities, i.e., omega (i, t-1), which in turn show some high time series dependence.
In this report we show that (1) the realised volatility of cryptocurrencies is highly time varying, (2) there has been a downward trend in the aggregate market volatility from early 2018 onward, (3) there is a positive and significant relationship between lagged volatility and current returns, and (4) there are substantial spill over effects between Bitcoin and alternative cryptocurrencies when it comes to the variability of returns.