4.1. GJR-GARCH and GARCH Models for Cryptocurrency Volatilities
First, we present descriptive statistics of the ten cryptocurrencies in
Table 5. Sharpe ratio is the average return earned in excess of the risk-free rate per unit of volatility or total risk. Intuitively, it measures the excess return earned per unit of risk. In our sample, Chainlink has the highest Sharpe ratio of 0.046, whereas Bitcoin has the second-highest Sharpe ratio of 0.029 in our sample. Some cryptocurrencies have negative Sharpe ratios, which signify that their average return is less than the risk-free rate in our sample.
Table 5 shows that cryptocurrency returns exhibit fat-tail distributions. Generally, prior literature has used the
GARCH model over the autoregressive conditional heteroscedastic (
ARCH) model. However, to control for asymmetric responses of volatility to innovation fluctuations, we used the
GJR-GARCH model. From
Table 5, we observe all positive values of excess kurtosis, which means all of the returns have leptokurtic distributions. This refers to a heavy degree of risks with extreme return values. Moreover, except Bitcoin SV and TETHER, all the other cryptocurrencies show skewness with negative values, which are compensated with high future returns for higher volatility. These two higher moments play with investors’ sentiments.
Figure 5 depicts the comparison between the
GARCH(p,q) and
GJR-GARCH(p,q) models to estimate volatility using daily data. The top left panel of this Figure shows the time-series and the right panel shows the scatter plot of the volatility estimate. The middle panel shows a comparison in volatility forecasting. The bottom panel shows the news impact curve. We used the daily data for Bitcoin and we found similar results for all others. Therefore, to avoid redundancy and to show volatility model selection, we only show results for a single cryptocurrency. We also use the lowest AIC value for determining the best
p and
q for a robust volatility
GJR-GARCH(p,q) model for each cryptocurrency. Furthermore, there is a valid reason for choosing
GJR-GARCH over the
GARCH model: it is empirically found that negative cryptocurrency returns at time
have a stronger impact on the volatility at time
t than positive cryptocurrency returns. This asymmetric process is known as the leverage effect. The increment of the risk is realized to come from the increased leverage induced by the negative returns of cryptos
Glosten et al. (
1993).
In
Figure 6, we forecast the volatility for the ten cryptocurrencies. It shows daily data where the blue line indicates predicted volatility. If we examine the historical return series, it does not show conditional mean offset and thus exhibits volatility clustering.
However, we initially forecast the volatility of Bitcoin using both
GARCH and
GJR-GARCH in
Figure 5 and found that
GJR-GARCH provides the better estimate. That is why we implemented the
GJR-GARCH(p,q) model to forecast the return and volatility for our selected cryptocurrencies. In
Figure 6, we forecast the conditional variance (vs. absolute value of returns) for all ten cryptocurrencies.
4.2. Monte Carlo Simulations of the Cryptocurrencies’ Volatility Using GJR-GARCH Model
Next, we simulated the conditional variance of the cryptocurrency’s returns from a fully specified
GJR-GARCH model based on historical data. In
Figure 7, we plotted the average and the
and
percentiles of the simulated paths and compared the simulation statistics to the original data for each cryptocurrency. The red band indicates the confidence bound, and all of our selected cryptocurrencies are embedded in the simulated Monte Carlo paths. We followed the
GJR-GARCH(p,q) model based on a
confidence level and daily frequency to estimate VaR. We have presented our results both in
Table 5 and
Figure 7. Our results show that Bitcoin, Bitcoin Cash, Etherium, and Litecoin suffer the most from tail-risk whereas TETHER has the least tail-risk.
In
Figure 8, the jagged red line running across the bottom of the plot indicates the portfolio’s (negative) one-day
value-at-risk. For any instance of a cryptocurrency falling below that line, there will occur an exceedance risk. This would predict a
value-at-risk measure to experience approximately very few exceedances in under our horizon. We observe that Bitcoin, Bitcoin Cash, EOS, Ethereum, and Litecoin are highly risky assets, because these have many exceedance of jagged lines. Referring to
Figure 8, Bitcoin has at least four exceedances of jagged lines. Thus, it supports the hypothesis that Bitcoin is one of the highly volatile cryptocurrencies.
4.3. ARIMA and ANN for Forecasting Cryptocurrencies’ Prices
Figure 9 presents a window plot of the predicted price of each cryptocurrency with their actual prices. From close observation we see that
ARIMA models also give higher accuracy for the next-cryptocurrency price prediction.
We use ANN models for forecasting the future price of ten cryptocurrencies. However, we choose to show the complete process of forecasting future prices using ANN for only Chainlink. After training, the network may be converted to closed loop form. We choose daily price data of Chainlink randomly and divide into three different category named training set (70%), testing set (15%) and Validation set (15%).
In
Figure 10, we find that we don’t have any overfit of the model. In
Figure 10a, we evaluate regression fit by the value of regression coefficient.
Figure 10b allows us to check on the training progress of our Neural Network. The
Figure 10b shows
MSE and epochs of the Neural Network for training and test set. To observe whether our
ANN is training well, we look for training set’s Loss and Accuracy and whether they converge as the number of epochs increases. As our Loss and Accuracy from both sets are diverging from each other, there is no sign of overfitting from our
ANN model. It shows a good fit for Chainlink. We also observe similar output for all other nine cryptocurrencies.
In our case, we show it for 9 epochs in
Figure 10c. All types of data sets are approximately overlapping at 3 epochs. Therefore, we do not have to change our
ANN structure. Next, we tested our model against actual data to see how well it performed. We found that for all of the ten cryptocurrencies, our
ANN model could predict the price, which means it is performing quite well. Here
Figure 10,
R is actually
; the coefficient of the model.
Table 6 has been given for the
MSE of each model. In the studied supervised machine learning technique, we evaluated the model by using
MSE. For some mentioned cryptocurrencies, we obtained less
MSE for
ANN predictions, and for other cryptocurrencies, we see
ARIMA perform better. As we know, minimal or least
MSE is considered the best fit. Therefore, from this study, it is not possible to say that the
ANN is the best model but one can implement the
ANN as an alternative procedure. For Chainlink and all other nine cryptocurrencies, K-fold validations are compared using the mean square error (
MSE) metric. We observe that the MSE of the fitted model is less than cross-validation MSE for all ten cryptocurrencies. For our example, the cross-validation MSE of Chainlink is 0.53 approximately, which is obviously less than the MSE of
ARIMA and
ANN prediction.
The window plot in
Figure 11 presents only a one-month prediction output. All of the models are showing good output. In most of the cases, the
ANN is performing better than
ARIMA.
Overall, in this paper we have contributed two separate studies for cryptocurrencies. One is for volatility/risk and another is for return. An investor may think about these two properties of the financial market.
Cheikh et al. (
2020) investigated the presence of asymmetric volatility dynamics in Bitcoin, Ethereum, Ripple, and Litecoin using threshold
GARCH models. We have extended this group of literature. We compared
GARCH and
GJR-GARCH volatility estimates for estimating volatility considering the normal inverse Gaussian (
NIG) distribution. Then, we showed that to estimate volatility for cryptocurrencies, the
GJR-GARCH model under
NIG is better. Similarly, for predicting prices of cryptocurrencies, we applied an
ANN and showed the comparison with
ARIMA instead of ordinary least squares (
OLS) regression, support vector regression (
SVR), and the least absolute shrinkage and selection operator (
LASSO) of
Bouri et al. (
2021). Our findings have implications for money managers. Money managers can implement an
ANN for predicting the prices of cryptocurrencies.