Portfolio optimization is a cornerstone of modern financial decision-making, tradition-ally based on the meanâvariance model introduced by Markowitz. However, this framework relies on restrictive assumptionsâsuch as normally distributed returns and symmetric risk preferencesâthat 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 MeanâDeviationâEntropy (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 consideredâShannon entropy, Tsallis entropy, and cumulative residual SharmaâTanejaâMittal entropy (CR-STME)âto 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 meanâvariance and meanâMAD 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 CoxâIngersollâRoss (CIR) process. The novel family of models is based on Generalized OrnsteinâUhlenbeck 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 CoxâSnell 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.
Traditional portfolio optimization models, rooted in the meanâvariance 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-ciesâBitcoin (BTC), Ethereum (ETH), Solana (SOL), and Binance Coin (BNB)âusing 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 meanâvariance 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.
Xinran Huang, Linzhi Tan, Haozhe Su, Jeremy EngâTuck Cheah
ABSTRACT One of the critical risks associated with cryptocurrency assets is the soâcalled downside risk, or tail risk. Conditional ValueâatâRisk (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 naĂŻve, minimum variance and meanâvariance 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 Bitcoinâs 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 modelsâespecially when combined with the proposed quantile-driven factorâoffers 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.
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.
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 2017â2024 period. The portfolio performance of the three methods is also compared with the results from the traditional one-period meanâvariance 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 meanâvariance 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 meanâvariance 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 cryptocurrenciesâBitcoin (BTC), Ethereum (ETH), Solana (SOL), and Binance Coin (BNB)âwith 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 returnârisk 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 MeanâVariance optimization, are effective, but often fail to account for diversification adequately. This study introduces the MeanâVarianceâEntropy (MVE) model, which integrates Tsallis entropy into the classic MeanâVariance 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 MeanâVariance 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.
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 2017â2021 and 2021â2024 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.
This study introduces a novel hybrid stochastic modeling framework for simulating Ethereum price dynamics by integrating Poisson and Gaussian processes. The model captures both abrupt price jumps, modeled using a Poisson process, and continuous price variations, represented by a Gaussian process. Our analysis reveals that significant price fluctuations occur approximately every 4.33 days, with an average daily return of 0.0041 and an annualized volatility of 0.8631, underscoring the extreme volatility inherent in Ethereumâs market behavior. By combining these processes, the model effectively encapsulates the intrinsic price patterns of Ethereum, including persistent oscillations and sudden surges. Simulations of future price trajectories demonstrate the modelâs efficacy in replicating real world Ethereum price dynamics, offering valuable insights for traders and analysts in devising risk management strategies and making informed decisions in highly volatile cryptocurrency markets. The findings highlight the importance of hybrid models in addressing the unique challenges of modeling Ethereumâs price behavior.
Traditional portfolio optimization techniques predominantly rely on the classical meanâvariance 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 meanâvariance 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 cryptocurrenciesâBitcoin (BTC), Ethereum (ETH), Solana (SOL), and Binance Coin (BNB)âwith 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 returnârisk 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.
Cryptocurrencies have rapidly emerged as a significant financial asset class, influencing global monetary systems and financial markets. However, their extreme volatility, speculative nature, and evolving regulatory landscape pose challenges to investors, policymakers, and financial analysts. This study presents an in-depth quantitative analysis of cryptocurrency volatility and risk assessment, focusing on Bitcoin (BTC-USD) and its correlation with traditional financial assets, including the EUR/USD exchange rate and S&P 500 index. Our research employs Generalized Autoregressive Conditional Heteroskedasticity (GARCH) modeling to measure the dynamic volatility patterns of Bitcoin, revealing the assetâs substantial fluctuations over time and its sensitivity to market shocks. Additionally, we utilize Monte Carlo simulations to forecast potential future price movements of Bitcoin, highlighting risk scenarios and the probability distribution of price trajectories over a one-year period. The Value-at-Risk (VaR) model is implemented to estimate potential losses within a given confidence interval, providing a robust measure of downside risk. Furthermore, the study examines the integration of cryptocurrency markets with traditional financial instruments by analyzing cross-asset correlations and volatility spillover effects. The findings suggest that while Bitcoin remains a highly volatile asset, its correlation with the broader financial system is increasing, indicating a potential shift towards mainstream financial adoption. The results contribute to the ongoing debate on whether cryptocurrencies serve primarily as speculative instruments or as viable components of diversified investment portfolios. These insights are valuable for institutional investors, risk managers, and policymakers in designing more effective risk mitigation strategies for cryptocurrency investments.
ABSTRACT This paper investigates the current landscape of option trading platforms for cryptocurrencies, encompassing both centralized and decentralized exchanges. Option contracts in cryptocurrency markets offer functionalities akin to traditional markets, providing investors with tools to mitigate risks, particularly those arising from price volatility, while also allowing them to capitalize on future volatility trends. The paper discusses these applications of option contracts in the context of decentralized finance (DeFi), emphasizing their utility in managing market uncertainties. Despite a recent surge in the trading volume of options contracts on cryptocurrencies, decentralized platforms account for less than 1 % of this total volume. Hence, this paper takes a closer look by examining the design choices of these platforms to understand the challenges hindering their growth and adoption. It identifies technical, financial, and adoptionârelated challenges that decentralized exchanges face and provides commentary on existing platform responses. Subsequently, the paper analyzes the impact of absent options markets on the inefficiencies of automated market maker liquidity. It examines historical onâchain data for 14 ERC20 token pairs on Ethereum. The analysis shows 1143 instances in which deeper liquidity levels, as high as more, could have been achieved by establishing an options market.
This study aims to develop a dynamic portfolio trading system for high-risk profiles of cryptocurrencies in two phases: 1) portfolio selection and 2) portfolio construction. In the first phase, we propose a novel algorithmic trading model applying a Convolutional Neural Network (CNN) using a 2-D convolution layer with eight kernels of 3Ă3 sizes based on the prediction of selected technical indicators to predict buy/sell trading signals. To effectively increase the accuracy of the CNN model, first, the H-step ahead predictions of the selected technical indicators based on Long-short-term-memory (LSTM) along with the indicators themselves have been used to construct input matrices of the CNN model. A new price labeling approach was proposed to determine buying or selling points using the zigzag indicator (ZZ) in our CNN model. Assets with buy signals have been selected to construct the proposed portfolio. In the second phase, we propose a novel robust approach based on Holt-Winters-Multiplicative (HWM) to determine the realized crypto portfolio weights robustly by considering the seasonal effects. The experimental results show that our developed system outperforms the competing models for 30 cryptocurrencies with a high-risk profile in the two phases.
Since the introduction of Modern Portfolio Theory (MPT) in 1952, its practical applications, associated challenges, and computational efficiency on large and high-frequency datasets, particularly datasets not used to develop and optimize the model, have drawn extensive research interest. This study has examined the performance of various portfolio models that have explored the concept of MPT on U.S. stock and cryptocurrency markets, i.e., discrete Markowitz portfolio selection (DMPS), the optimal dynamic portfolio (ODP), the binary unconstrained ODP (BUODP) with a quantum annealing solver, and the 1/N naive diversification (ND). Their performance is then compared to the indices that measure the performance of corresponding market exchange-traded funds (ETFs) for stock markets. Our findings show that the DMPS and ODP perform better than other models, delivering better returns in a shorter period. Both run significantly faster than the BUODP (with quantum annealing) with computation time of approximately 0.5 seconds for the S&P 400, 500, and 600 markets whereas BUODP takes 30 seconds; they also mitigate risk and outperform ETFs and ND model for the out-of-sample test with diversified portfolios combining NASDAQ stocks and cryptocurrencies. Furthermore, we have analyzed the impact of data with different frequency intervals, i.e., weekly, daily, hourly, and one-minute, on portfolio performance of the stock markets. The results suggest that data collection frequencies do not make differences in portfolio selections and weights. This study contributes to the advancement of portfolio theory, providing insights and practical values, especially in addressing computational efficiency for high-frequency and large-scale datasets and saving computational costs.
The cryptocurrency market poses a huge challenge to portfolio optimization due to its high volatility and complex market dynamics. To address these issues, this paper uses reinforcement learning (RL) algorithms for dynamic portfolio optimization, aiming to improve the return and risk control capabilities of the portfolio through intelligent decision-making. This paper adopts a strategy based on deep reinforcement learning. By interacting with the cryptocurrency market, the agent can continuously optimize asset allocation, maximize investment returns while controlling volatility. The experimental results show that compared with traditional strategies, the reinforcement learning model has obvious advantages in key indicators such as cumulative return rate, annualized volatility, maximum drawdown and Sharpe ratio. Specifically, the cumulative return rate of the reinforcement learning model reaches 85.12%, the annualized volatility is 45.76%, and the maximum drawdown is controlled at -22.34%, showing strong income acquisition and risk management capabilities. In addition, the dynamic adjustment of asset allocation has optimized the weights of various cryptocurrencies, effectively dispersed risks, and improved the overall performance of the investment portfolio.
The paper presents two series representations of a L{\'e}vy process for the Generalized Tempered Stable (GTS) distribution: a series representation generated by the inverse tail integral and a short noise representation. Both series representations are used to simulate the daily returns of Bitcoin and Ethereum. The Q-Q plot analysis shows smooth linear patterns, indicating strong agreement between the empirical and theoretical GTS distributions.