This paper studies the benefits of timing Bitcoin returns by upside and downside volatilities. Standard volatility management implicitly treats volatility spikes as signals of adverse states, reducing exposure when total volatility increases. However, in Bitcoin, volatility spikes are frequently due to price rallies, which typically indicate subsequent positive returns. We show that semivolatility timing rules that account for both downside and upside risk concerns yield substantially stronger risk-adjusted performance than buy-and-hold and volatility-managed strategies. This stems from the fact that high upside-driven volatility states in Bitcoin are disproportionately associated with positive returns in the next period.
This paper measures price differences between Hegic option quotes on Arbitrum and a model-based benchmark built on Black--Scholes model with regime-sensitive volatility estimated via a two-regime MS-AR-(GJR)-GARCH model. Using option-level feasible GLS, we find benchmark prices exceed Hegic quotes on average, especially for call options. The price spread rises with order size, strike, maturity, and estimated volatility, and falls with trading volume. By underlying, wrapped Bitcoin options show larger and more persistent spreads, while Ethereum options are closer to the benchmark. The framework offers a data-driven analysis for monitoring and calibrating on-chain option pricing logic.
Abstract This paper considers option valuation under finite mixture models in a discrete-time economy. Specifically, the Esscher transform is employed to select a pricing kernel. Novel finite mixture models with negative-shifted Gamma and negative-shifted inverse Gaussian distributions are developed. A hybrid finite mixture model that allows different parametric forms for component distributions is introduced to incorporate model uncertainty. An empirical characteristic function estimation method is employed to estimate the finite mixture models. Closed-form pricing formulas for a European call option are obtained for some finite mixture models. Empirical examples using data on the Bitcoin-USD prices are provided to illustrate an application of the proposed models to value Bitcoin options.
The aim of this thesis is to examine the pricing and efficiency of Bitcoin options. It reviews theories of market efficiency and considers how effectively these frameworks apply to cryptocurrency markets. The thesis examines multiple option pricing models by comparing their performance for pricing Bitcoin options. Bitcoin’s high volatility and the relatively young age of its market development highlight the need to analyze how these characteristics influence both option pricing and overall market efficiency. In addition, the thesis examines the characteristics of Bitcoin options. The study provides guidelines for future research and market development, helping to build trust and support the integration of cryptocurrency derivatives into the broader financial system. Tämän opinnäytetyön tavoitteena on tarkastella Bitcoin-optioiden hinnoittelua ja markkinoiden tehokkuutta. Työssä käydään läpi markkinatehokkuuden teorioita ja arvioidaan, kuinka hyvin nämä viitekehykset soveltuvat kryptovaluuttamarkkinoihin. Opinnäytetyössä tarkastellaan useita optioiden hinnoittelumalleja vertailemalla niiden toimi- vuutta Bitcoin-optioiden hinnoittelussa. Bitcoinin korkea volatiliteetti ja sen markkinoiden suhteellisen varhaisessa kehitysvaiheessa oleva tila korostavat tarvetta analysoida, miten nämä ominaisuudet vaikuttavat sekä optioiden hinnoitteluun että markkinoiden yleiseen tehokkuuteen. Lisäksi työssä tarkastellaan Bitcoin-optioiden erityispiirteitä. Tutkimus tarjoaa suuntaviivoja tu- levalle tutkimukselle ja markkinoiden kehittämiselle, ja sen tavoitteena on lisätä luottamusta sekä tukea kryptovaluuttajohdannaisten integroitumista laajempaan finanssijärjestelmään.
This Master's thesis investigates the application of machine learning methods to cryptocurrency market prediction and the development of hybrid trading strategies that combine predictive signals with decentralized finance yield components. The study addresses how machine learning models can predict directional shifts in cryptocurrency markets and whether integrating DeFi yield elements can improve risk-adjusted portfolio returns compared to traditional buy-and-hold approaches. The empirical investigation examined multiple machine learning architectures for binary directional forecasting of Bitcoin price movements. Models were trained on data spanning January 2018 to August 2024 using walk-forward validation. LightGBMRegressor achieved 53 % directional accuracy, while Random Forest reached 52 % accuracy. Other tested models, including LSTM networks and MLP, performed within the 51-56 % accuracy range. These results indicate that while machine learning methods demonstrate potential for market direction prediction when combined with properly formatted datasets and appropriate technical indicators, achieving high prediction accuracy remains challenging. A composed trading strategy was developed that integrated LSTM predictions with real-world DeFi yield rates from liquidity pools. The strategy utilized actual yield data to provide realistic performance assessment. Despite modest directional prediction accuracy of 53 %, the hybrid approach reduced drawdown by 50 % compared to the benchmark buy-and-hold strategy. The DeFi yield component compensated for imperfect directional signals, demonstrating that yield-enhanced strategies can achieve adequate risk-adjusted returns even without superior prediction accuracy. The study also examined structural differences between decentralized and traditional financial systems. DeFi offers global accessibility, programmable infrastructure, and fast settlement, but faces challenges including security vulnerabilities and regulatory uncertainty. However, the primary contribution lies in demonstrating that hybrid strategies combining machine learning signals with DeFi yield mechanisms represent a viable approach to portfolio management, when effective risk management is implemented.
Portfolio optimization is a cornerstone of modern financial decision-making, tradition-ally based on the mean–variance model introduced by Markowitz. However, this framework relies on restrictive assumptions—such as normally distributed returns and symmetric risk preferences—that often fail in real-world markets, particularly in volatile and non-Gaussian environments such as cryptocurrencies. To address these limitations, this paper proposes a novel multi-objective model that combines expected return max-imization, mean absolute deviation (MAD) minimization, and entropy-based diversifi-cation into a unified optimization structure: the Mean–Deviation–Entropy (MDE) model. The MAD metric offers a robust alternative to variance by capturing the average mag-nitude of deviations from the mean without inflating extreme values, while entropy serves as an information-theoretic proxy for portfolio diversification and uncertainty. Three entropy formulations are considered—Shannon entropy, Tsallis entropy, and cumulative residual Sharma–Taneja–Mittal entropy (CR-STME)—to explore different notions of uncertainty and structural diversity. The MDE model is formulated as a tri-objective optimization problem and solved via scalarization techniques, enabling flexible trade-offs between return, deviation, and en-tropy. The framework is empirically tested on a cryptocurrency portfolio composed of Bitcoin (BTC), Ethereum (ETH), Solana (SOL), and Binance Coin (BNB), using daily data over a 12-month period. The empirical setting reflects a high-volatility, high-skewness regime, ideal for testing entropy-driven diversification. Comparative outcomes reveal that entropy-integrated models yield more robust weightings, particularly when tail risk and regime shifts are present. Comparative results against classical mean–variance and mean–MAD models indicate that the MDE model achieves improved di-versification, enhanced allocation stability, and greater resilience to volatility clustering and tail risk. This study contributes to the literature on robust portfolio optimization by integrating entropy as a formal objective within a scalarized multi-criteria framework. The proposed approach offers promising applications in sustainable investing, algorithmic asset allo-cation, and decentralized finance, especially under high-uncertainty market conditions.
Arthur Rodrigues Pereira de Carvalho, Felipe Quintino, Helton Saulo, Luan Carlos de Sena Monteiro Ozelim · 6 authors
This study introduces and evaluates stochastic models to describe Bitcoin price dynamics at different time scales, using daily data from January 2019 to December 2024 and intraday data from 20 January 2025. In the daily analysis, models based on are introduced to capture long memory, paired with both constant-volatility (CONST) and stochastic-volatility specifications via the Cox–Ingersoll–Ross (CIR) process. The novel family of models is based on Generalized Ornstein–Uhlenbeck processes with a fluctuating exponential trend (GOU-FE), which are modified to account for multiplicative fBm noise. Traditional Geometric Brownian Motion processes (GFBM) with either constant or stochastic volatilities are employed as benchmarks for comparative analysis, bringing the total number of evaluated models to four: GFBM-CONST, GFBM-CIR, GOUFE-CONST, and GOUFE-CIR models. Estimation by numerical optimization and evaluation through error metrics, information criteria (AIC, BIC, and EDC), and 95% Expected Shortfall (ES95) indicated better fit for the stochastic-volatility models (GOUFE-CIR and GFBM-CIR) and the lowest tail-risk for GOUFE-CIR, although residual analysis revealed heteroscedasticity and non-normality. For intraday data, Exponential, Weibull, and Generalized Gamma Autoregressive Conditional Duration (ACD) models, with adjustments for intraday patterns, were applied to model the time between transactions. Results showed that the ACD models effectively capture duration clustering, with the Generalized Gamma version exhibiting superior fit according to the Cox–Snell residual-based analysis and other metrics (AIC, BIC, and mean-squared error). Overall, this work advances the modeling of Bitcoin prices by rigorously applying and comparing stochastic frameworks across temporal scales, highlighting the critical roles of long memory, stochastic volatility, and intraday dynamics in understanding the behavior of this digital asset.
Η παρούσα μεταπτυχιακή διατριβή εξετάζει τη δυναμική της μεταβλητότητας στις αγορές κρυπτονομισμάτων και αναπτύσσει προβλεπτικά μοντέλα για την εις βάθος κατανόηση της συμπεριφοράς των τιμών σε ένα έντονα κερδοσκοπικό περιβάλλον. Με βάση δεδομένα από τέσσερα βασικά κρυπτονομίσματα, συγκεκριμένα, το Bitcoin (BTC), το Ethereum (ETH), το Litecoin (LITE) και το Dogecoin (DOGE) εφαρμόζονται προηγμένες οικονομετρικές μέθοδοι για την αποτύπωση κρίσιμων χαρακτηριστικών, όπως η συσσώρευση μεταβλητότητας, οι βαριές ουρές κατανομής και οι ασύμμετρες αντιδράσεις σε εξωγενείς διαταραχές. Το μεθοδολογικό πλαίσιο περιλαμβάνει μοντέλα GARCH(1,1) και EGARCH(1,1) με κατανομή GED, ARIMA - GARCH για τον από κοινού προσδιορισμό μέσης τιμής και διακύμανσης, GARCH-X με ενσωμάτωση μακροοικονομικών μεταβλητών, Στοχαστικά Μοντέλα Μεταβλητότητας Bayes (SV), καθώς και ARFIMA για τη διερεύνηση μακροχρόνιας μνήμης.Τα εμπειρικά αποτελέσματα δείχνουν ότι η μεταβλητότητα παραμένει ιδιαίτερα υψηλή σε όλα τα υπό εξέταση κρυπτονομίσματα, με εμφανή φαινόμενα μόχλευσης στο BTC, DOGE και LITE, ενώ το ETH εμφανίζει σχεδόν συμμετρικές αντιδράσεις. Η ενσωμάτωση μακροοικονομικών μεταβλητών αναδεικνύει τον καθοριστικό ρόλο του Δείκτη Τιμών Καταναλωτή (CPI), της προσφοράς χρήματος (Μ2) και των τιμών ενέργειας και τεχνολογίας, κυρίως στην περίπτωση του BTC, επιβεβαιώνοντας τη διασύνδεση των αγορών κρυπτονομισμάτων με τις διεθνείς χρηματοοικονομικές συνθήκες. Οι προβλέψεις ενός βήματος δείχνουν ότι BTC και LITE επιτυγχάνουν την υψηλότερη ακρίβεια, το ETH εμφανίζει μέτριο επίπεδο ακρίβειας, ενώ το DOGE παραμένει ιδιαίτερα απαιτητικό στη μοντελοποίηση, με τον έλεγχο VaR να καταδεικνύει συστηματική υπερεκτίμηση του καθοδικού κινδύνου. Τα Στοχαστικά Μοντέλα Μεταβλητότητας Bayes αποδεικνύονται πιο αποτελεσματικά από τα GARCH στην αποτύπωση αιφνίδιων μεταβολών και αλλαγών καθεστώτος, ενώ τα αποτελέσματα ARFIMA επιβεβαιώνουν περιορισμένη μακροχρόνια μνήμη για BTC και LITE, ήπια επιμονή για ETH και σχεδόν μηδενική για DOGE.Τα ευρήματα υπογραμμίζουν τη σημασία ευέλικτων, ασύμμετρων και βαριάς ουράς μοντέλων για ακριβή πρόβλεψη της μεταβλητότητας στις αγορές κρυπτονομισμάτων. Η μελέτη συμβάλλει στη διεθνή βιβλιογραφία μέσω ολοκληρωμένης συγκριτικής ανάλυσης μοντέλων μεταβλητότητας και προσφέρει πρακτικές κατευθύνσεις σε επενδυτές, διαχειριστές κινδύνου και φορείς χάραξης πολιτικής. Ειδικότερα, προτείνεται η αξιοποίηση προηγμένων μοντέλων για εκτίμηση κινδύνου, η ενσωμάτωση μακροοικονομικών δεικτών στο πλαίσιο πρόβλεψης, καθώς και η υιοθέτηση μετρικών ευαίσθητων στις ουρές κατανομής, όπως το Expected Shortfall, για τη διαχείριση ιδιαίτερα κερδοσκοπικών περιουσιακών στοιχείων.
Traditional portfolio optimization models, rooted in the mean–variance framework of Markowitz, rely heavily on variance as a risk measure. Although theoretically elegant, this approach becomes fragile in volatile and structurally unstable markets such as cryptocurrencies, where return distributions deviate significantly from normality, cor-relations are unstable, and concentration risk emerges. These limitations have motivated the search for alternative frameworks capable of capturing uncertainty in a more flexible and distribution-free manner. Entropy, originally introduced by Shannon as a measure of information, has gradually been recognized in the financial literature as a suitable proxy for diversification and systemic uncertainty. To address the shortcomings of variance-based models, this paper introduces the Weighted Shannon Entropy (WSE) model as a diversification-oriented alternative. By extending the classical Shannon entropy with asset-specific informational weights, the WSE framework provides additional flexibility for modeling heterogeneous asset char-acteristics, such as liquidity, informational value, or perceived reliability. Using the principle of maximum entropy and the method of Lagrange multipliers, we derive ex-ponential-form solutions for portfolio weights that naturally discourage concentration, ensure balanced allocations, and remain analytically tractable. The methodology is validated empirically on a portfolio of four leading cryptocurren-cies—Bitcoin (BTC), Ethereum (ETH), Solana (SOL), and Binance Coin (BNB)—using market data from January to March 2025. The results demonstrate that the entropy-based optimization framework produces well-diversified portfolios, robust to volatility and structural instability, and provides a distribution-free alternative to the classical mean–variance model. Beyond its empirical performance, the WSE formulation highlights the conceptual advantage of entropy in integrating return, risk, and diversification into a single unified framework. The paper contributes both theoretically and practically: it strengthens the mathematical foundation of entropy-based portfolio selection, extends its applicability to digital asset markets, and illustrates how weighting schemes can enrich the classical Shannon measure. Future research may extend this approach to multi-period optimization, gen-eralized entropies such as Tsallis and Kaniadakis, or integration with machine learning models for dynamic portfolio management.
ABSTRACT We propose a novel option pricing model that explicitly incorporates volatility‐of‐volatility (VOV) dynamics and its associated risk premium. Our framework integrates realized variance and realized quarticity to capture latent VOV dynamics, addressing key challenges in cryptocurrency option pricing. Using Fourier inversion methods, we derive a closed‐form pricing formula for European‐style options. Empirical analysis with high‐frequency cryptocurrency option data shows that our model improves pricing accuracy, reducing implied volatility errors by 8.55% compared to benchmark models. The model outperforms benchmarks across all moneyness levels, remains robust for both short‐ and long‐maturity contracts, and maintains accuracy under high volatility. This study contributes to the literature by introducing a tractable and empirically validated approach to cryptocurrency option pricing through explicit VOV modeling.
Xinran Huang, Linzhi Tan, Haozhe Su, Jeremy Eng‐Tuck Cheah
ABSTRACT One of the critical risks associated with cryptocurrency assets is the so‐called downside risk, or tail risk. Conditional Value‐at‐Risk (CVaR) is a measure of tail risks that is not normally considered in the construction of a cryptocurrency portfolio. In this paper, we propose a new approach to portfolio construction based on a deep learning CVaR utility function. This approach is designed to address the issue of tail risk. We evaluate the performance of this approach in comparison to other portfolio construction techniques, including the naïve, minimum variance and mean‐variance portfolios. Our findings indicate that the proposed approach outperforms traditional optimisation models.
This paper develops a novel modeling framework that integrates time-varying quantile-based spillover effects into a regime-switching realized volatility model. A dynamic spillover factor is constructed by identifying the most influential contributors to Bitcoin’s realized volatility across different quantile levels. This quantile-layered structure enables the model to capture heterogeneous spillover paths under varying market conditions at a macro level while also enhancing the sensitivity of volatility regime identification via its incorporation into a time-varying transition probability (TVTP) Markov-switching mechanism at a micro level. Empirical results based on the cryptocurrency market demonstrate the superior forecasting performance of the proposed TVTP-MS-HAR model relative to standard benchmark models. The model exhibits strong capability in identifying state-dependent spillovers and capturing nonlinear market dynamics. The findings further reveal an asymmetric dual-tail amplification and time-varying interconnectedness in the spillover effects, along with a pronounced asymmetry between market capitalization and systemic importance. Compared to decomposition-based approaches, the X-RV type of models—especially when combined with the proposed quantile-driven factor—offers improved robustness and predictive accuracy in the presence of extreme market behavior. This paper offers a coherent approach that bridges phenomenon identification, source localization, and predictive mechanism construction, contributing to both the academic understanding and practical risk assessment of cryptocurrency markets.
This article proposes a new nonparametric test for detecting short-lived locally explosive trends (drift bursts) in pure-jump processes. The new test is designed specifically to detect intraday flash crashes and gradual jumps in cryptocurrency prices recorded at a high frequency. Empirical analysis shows that drift bursts in bitcoin price occur, on average, every second day. Their economic importance is highlighted by showing that hedge funds holding cryptocurrency in their portfolios are exposed to a risk factor associated with the intensity of bitcoin crashes. On average, hedge funds do not profit from intraday bitcoin crashes and do not hedge against the associated risk.
Rüya Kaplan Yıldırım, Turgay Münyas, Gülden Kadooğlu Aydın
Constructing an effective asset allocation strategy requires building well-diversified portfolios that maintain robust performance beyond the sample data. The classical Markowitz portfolio optimisation, while widely used, is known to suffer from issues such as estimation errors and sensitivity to multicollinearity, which can significantly distort the allocation process and reduce performance reliability. In order to surmount the aforementioned challenges, the incorporation of Machine Learning echniques, specifically Ridge regression, into the portfolio creation process has been effected. This has resulted in the provision of a hybrid model that combines the strengths of Markowitz optimisation and Ridge regression. The integration of these approaches within the hybrid model serves to mitigate the prediction risks while maintaining the diversification benefits inherent to the Markowitz framework. The model was trained using an 80/20 split and cross-validation was employed to prevent overfitting. The findings indicate that this integrated approach attains the maximum Sharpe ratio, thereby significantly enhancing risk-adjusted returns and portfolio stability when applied to cryptoasset returns. The findings emphasise the merits of integrating classical optimisation methodologies with machine learning to develop more robust and adaptive asset allocation strategies. By analysing the impact of high-volatility cryptoassets on portfolio performance, it makes important contributions to both the literature and practical portfolio strategies for investors.
Hamdan Bukenya Ntare, John Weirstrass Muteba Mwamba, Franck Adékambi
There has been growing interest among investors to include cryptocurrencies in their portfolios because of their diversification potential. However, the diversification role of cryptocurrencies when added to South African bank equities is yet to be determined. This study rigorously evaluates asset co-movement and diversification benefits of integrating cryptocurrencies into South African bank equity portfolios. Using advanced financial engineering techniques, including multi-asset particle swarm optimizer (MA-PSO), random optimizer, and a static equal-weighted portfolio (EWP) model, this study analyzed the dynamic portfolio performance and diversification of cryptocurrencies in the 2017–2024 period. The portfolio performance of the three methods is also compared with the results from the traditional one-period mean–variance optimization (MVO) method. The findings underscore the superiority of dynamic models over static EWP in assessing the impact of cryptocurrency inclusion in bank equity portfolios. While pre-COVID-19 studies identified cryptocurrencies as effective hedges against market downturns, this protective role appears attenuated in the post-COVID-19 era. The dynamic MA-PSO model emerges as the optimal approach, delivering better-diversified portfolios. Consequently, South African portfolio managers must carefully evaluate investor risk tolerance before incorporating cryptocurrencies, with regulators imposing stringent guidelines to mitigate potential losses.
Purpose: This study aims to examines advanced portfolio management techniques using Long Short-Term Memory (LSTM) networks, the study was applied to investing in cryptocurrencies whose markets are characterized by high-frequency trading, and using behavioral finance models based on the concept of return-risk and deep learning based on the work of artificial neural networks (ANN) and long-term memory (LSTM) algorithms Design/Methodology/Approach: This study adopts quantitative approach. Moreover, A random portfolio consisting of 25 cryptocurrencies was selected based on the database of the website: https://finance.yahoo.com/crypto/ during the period 2021-2024 AD and programming the Python language. And an attempt to evaluate the performance of the models used in accurately predicting the optimal relative weights of the investment portfolio, which proved the relative effectiveness of deep learning models by estimating the values of the mean square error (MSE) at a level of 0.0218% to predict the optimal portfolio weights for 5 days based on training 80% and testing 20% of the study data. Findings: The second hypothesis of this study was accepted, which states the effectiveness of deep learning algorithms to predict the weights of optimal portfolios with a return estimated at 1.7239% and a risk of 1.1219% and a Sharpe index value estimated at 1.5365%, while the Markowitz return-risk model portfolio came with a return rate estimated at 31.15% and a risk of 39.05%. With no diversification of investment on all portfolio assets and a Sharpe index value of 0.7978%. Practical Implications: This study provides important insights that machine learning offers significant advantages in portfolio optimization, from improved forecasting of asset returns to dynamic rebalancing, better risk management, and automation. The ability to handle high-dimensional, non-linear, and non-stationary data makes ML an ideal tool for optimizing portfolios in complex and fast-moving markets; especially in cryptocurrency markets. However, challenges like data quality, overfitting, and interpretability must be addressed to ensure effective deployment of ML in real-world portfolio. Originality/Value: This study provides an original and timely contribution to understanding the use of deep learning for portfolio optimization represents a significant advancement over traditional financial models by offering several original and valuable benefits. These include the ability to capture complex non-linear relationships, dynamic rebalancing in response to real-time data, processing of unstructured data (like sentiment analysis), advanced risk management, and the integration of high-dimensional data. The combination of these capabilities enables more accurate, adaptive, and robust portfolio optimization, ultimately enhancing portfolio performance and reducing risk.
Traditional portfolio optimization techniques predominantly rely on the classical mean–variance framework introduced by Markowitz, which focuses on balancing expected returns against risk, typically measured by variance. However, in volatile and structur-ally unstable markets such as cryptocurrencies, this approach often fails to capture the full spectrum of uncertainty and diversification potential. This paper introduces an al-ternative methodology grounded in entropy, a fundamental concept in information theory that quantifies uncertainty and disorder. By incorporating entropy into the portfolio optimization process, we offer a more generalizable, distribution-free approach that enhances diversification and resilience.We develop and analyze three distinct en-tropy-based models: the maximum Shannon entropy model, the second-order entropy (Tsallis) model, and the maximum weighted Shannon entropy model. These formula-tions extend the traditional mean–variance approach by integrating nonlinear uncer-tainty measures, enabling a richer representation of investor preferences and asset in-terdependencies. Analytical solutions to the proposed models are derived using the method of Lagrange multipliers, ensuring mathematical rigor and interpretability.The proposed models are empirically validated using a portfolio composed of four leading cryptocurrencies—Bitcoin (BTC), Ethereum (ETH), Solana (SOL), and Binance Coin (BNB)—with market data from January to March 2025. The case studies demonstrate how entropy-based optimization leads to well-diversified portfolios, robust under market turbulence and heavy-tailed return distributions. Notably, the models facilitate dynamic adjustments in asset allocation in response to shifts in return–risk characteristics and entropy levels. This study contributes to the ongoing generalization of portfolio theory by positioning entropy as both a diversification enhancer and a structural risk measure. It provides theoretical insight, practical tools for asset allocation in high-volatility environments, and paves the way for future research in entropy-driven financial optimization frameworks.
Portfolio optimization is a fundamental problem in financial theory, aiming to balance risk and return in asset allocation. Traditional models, such as Mean–Variance optimization, are effective, but often fail to account for diversification adequately. This study introduces the Mean–Variance–Entropy (MVE) model, which integrates Tsallis entropy into the classic Mean–Variance framework to enhance portfolio diversification and risk management. Entropy, specifically second-order entropy, penalizes excessive concentration in the portfolio, encouraging a more balanced and diversified allocation of assets. The model is applied to a portfolio of five major cryptocurrencies: Bitcoin (BTC), Ethereum (ETH), Solana (SOL), Cardano (ADA), and Binance Coin (BNB). The performance of the MVE model is compared with that of the traditional Mean–Variance model, and results demonstrate that the entropy-enhanced model provides better diversification, although with a slightly lower Sharpe ratio. The findings suggest that while the entropy-adjusted model results in a slightly lower Sharpe ratio, it offers better diversification and a more resilient portfolio, especially in volatile markets. This study demonstrates the potential of incorporating entropy into portfolio optimization as a means to mitigate concentration risk and improve portfolio performance. The approach is particularly beneficial for markets such as cryptocurrency, where volatility and asset correlations fluctuate rapidly. This paper contributes to the growing body of literature on portfolio optimization by offering a more diversified, robust, and risk-adjusted approach to asset allocation
This paper investigates the volatility dynamics and underlying long memory features of four major cryptocurrencies-Bitcoin, Ethereum, Litecoin, and Ripple-which were selected due to their high liquidity, large trading volumes, and historical significance in the digital asset market. The long-range dependence exhibited in cryptocurrency markets is often overlooked. However, based on the strong evidence of persistent dependence in the return series, we adopt advanced volatility models that are capable of accommodating high volatility and heavy-tails, as well as the long memory properties of cryptocurrencies. Specifically, we employ long-memory extensions of the GAS (Long memory GAS) and GARCH (Fractionally Integrated Asymmetric Power ARCH) models, integrating heavy-tailed innovation distributions: the Generalized Hyperbolic Distribution (GHD) and Generalized Lambda Distribution (GLD). Standard GARCH and GAS models are included as benchmarks. The performance of the models are assessed using Value-at-Risk (VaR) estimation, backtesting (in-sample and out-of-sample) and volatility forecasting metrics. The results indicate that long memory models, particularly the FIAPARCH model, consistently outperforms the standard GAS and GARCH models in capturing tail risk and the volatility persistence. These findings emphasize the critical role of long memory in modeling the risk of cryptocurrencies, indicating that accounting for volatility persistence can significantly enhance the accuracy of risk estimates and strengthen risk management practices.
ABSTRACT This paper investigates the economic consequences for Bitcoin options' prices of a long memory in conditional volatility and conditional non‐normality of Bitcoin returns. The arbitrage‐free prices of Bitcoin options are determined by market consistent valuation and the conditional Esscher transform. Monte Carlo estimates for option prices from estimated models based on Bitcoin returns data are provided. Explanations for the results from an economic perspective are provided. Economic insights and implications of the results for the nature of cryptocurrencies, their risk evaluation, and the hedging of Bitcoin's derivatives are explored.
Whether financial assets movements exhibit correlation and memory has been an intriguing question for physicists. This study aims to investigate whether financial shocks exhibit non-Markovian behavior. In particular, it explores the presence of long-term memory and non-local fluctuations during financial crises. The non-Markovian behavior of volatility and return during the cryptocurrency crashes of 2017–2021 and 2021–2024 cycles are examined. The analysis shows that a scaling relation, which is valid for a singular Markovian process, breaks down in data sets spanning approximately 1 year and 3 years after the onset of the 2017 crash. A similar pattern was observed in the 2021 crash, although the analysis does not work for some data sets. In these time intervals, the crash process shows non-Markovian behavior with financial shocks demonstrating non-local fluctuations and evidence of long-term memory.