Modelling volatility dynamics of cryptocurrencies: A comparison of garch and stochastic volatility models

Abstract

Cryptocurrencies have gained popularity since the invention of Bitcoin by Nakamoto. Although they lack regulation and carry a high risk of financial loss, they offer significant advantages, such as accessibility to the public, private transactions, decentralization, and diversified investment portfolios. One of the most distinguishing characteristics of cryptocurrencies is their high price volatility, which is significantly greater than that observed in traditional financial markets such as stocks or bonds. High volatility presents both opportunities and challenges for various stakeholders, including researchers, investors, financial analysts, risk managers, and policymakers. Moreover, as of March 11, 2025, the growing significance of cryptocurrencies in the global financial system is underscored by their combined market capitalization, which has exceeded 2.67 trillion US dollars. Hence, understanding and accurately modeling cryptocurrency volatility is crucial for making good investment decisions and developing robust risk management strategies. In this study, we focus on the volatility of the two largest cryptocurrencies by market capitalization: Bitcoin and Ethereum. We compare seven Generalized Autoregressive Conditional Heteroskedasticity (GARCH) models and seven Stochastic Volatility (SV) models using a Bayesian approach, based on the daily closing prices of Bitcoin and Ethereum. We compute daily returns using closing prices sourced from cryptoQuotes, which provides open-access cryptocurrency market data, sentiment indicators, and interactive charts. The data covers the period from "19.10.2017 03:00" to "28.02.2025 03:00." for both Bitcoin and Ethereum. For each cryptocurrency, a total of 2683 observations are considered. We consider the following models: the standard GARCH(1,1) and SV with first order autoregressive (AR(1)) log-volatility process, as well as their more flexible variants-including jump components (GARCH-J, SV-J), volatility-in mean versions (GARCH-M, SV-M), versions with moving average innovations (GARCH-MA,SV-MA), versions with t-distributed innovations (GARCH-t, SV-t), and version thatallow for leverage effects (GARCH-GJR, SV-L). We also consider GARCH(2,1) and SV with second order autoregressive (AR(2)) log-volatility process (GARCH-2, SV-2). In this thesis, we aim to compare the performance of these volatility models across two dimensions: estimation accuracy and forecasting performance. For the estimation part, we evaluate the the goodness of fits of the GARCH and SV models to the daily closing returns of Bitcoin and Ethereum. Using a Bayesian technique, we compute the log-marginal likelihood of each model and calculate the posterior means of the model parameters. Our results show that SV-t is the best model for both Bitcoin and Ethereum, with SV models generally outperforming GARCH counterparts. GARCH-t ranks second, showing the benefit of t-innovations. While the jump component enhances the estimation performance of the standard GARCH model for both Bitcoin and Ethereum, it does not have a significant impact on the standard SV model for Ethereum. For Bitcoin, the AR(2) specification underperforms compared to AR(1) process, whereas for Ethereum, AR(2) process demonstrates superior performance. The moving average component improves the estimation performance of Ethereum's SV model. The volatility feedback parameter in the volatility in mean models is statistically insignificant. Incorporating the leverage effect in the GARCH-GJR model enhances the performance over the standard GARCH specification, while it remains largely irrelevant in the SV models. Likewise, for Ethereum price returns, the leverage effect parameter is found to be statistically insignificant. In our forecasting evaluation, we employ the log-predictive score to assess model performance. The results reveal distinct patterns across cryptocurrencies: the GARCH-t model demonstrates superior forecasting accuracy for Bitcoin, while GARCH-J performs best for Ethereum. We find that incorporating an AR(2) specification improves the Stochastic Volatility (SV) model's performance exclusively for Bitcoin. The inclusion of jump components significantly enhances forecasting accuracy for both standard GARCH and SV models in Bitcoin price returns. For Ethereum, jump components substantially improve the standard GARCH model while providing more modest gains to the baseline SV specification. The volatility feedback channel, as incorporated in both GARCH-M and SV-M models, plays a significant role in forecasting Bitcoin price returns, whereas it proves to be unnecessary for forecasting Ethereum price returns. The moving average innovations enhance the forecasting performance of the standard SV model for Bitcoin. Additionally, they yield minor improvements in the forecast accuracy of GARCH models for both cryptocurrencies and SV models for Ethereum. The inclusion of t-distributed innovations contributes to enhanced forecasting accuracy. Leverage effect benefits Bitcoin in GARCH and SV models but is insignificant for Ethereum.