Entropy-Based Portfolio Optimization in Cryptocurrency Markets: A Comparative Study of Shannon, Tsallis, and Weighted Shannon Entropies

Traditional mean–variance portfolio optimization is ill-suited to cryptocurrency markets, where extreme volatility, fat-tailed distributions, and unstable correlations undermine variance as a risk measure. To overcome these limitations, this paper develops a unified entropy-based framework for portfolio diversification grounded in the Maximum Entropy Principle (MaxEnt). Within this formulation, Shannon entropy, Tsallis entropy, and Weighted Shannon Entropy (WSE) emerge as complementary specifications derived analytically via the method of Lagrange multipliers, ensuring mathematical tractability and interpretability. Empirical validation is conducted on a portfolio of four leading cryptocurrencies—Bitcoin (BTC), Ethereum (ETH), Solana (SOL), and Binance Coin (BNB)—using weekly return data from January to March 2025. Results reveal that Shannon entropy converges to near-uniform diversification, Tsallis entropy (q = 2) penalizes concentration more strongly and enhances robustness against tail risk, while WSE integrates asset-specific informational priorities, aligning allocations with investor preferences or market characteristics. Comparative analysis confirms that all three models yield allocations more resilient and structurally balanced than variance-driven portfolios, mitigating estimation risk and concentration effects. This study provides a coherent mathematical formulation of entropy-based portfolio optimization by embedding Shannon, Tsallis, and Weighted Shannon entropies within a common Maximum Entropy (MaxEnt) optimization framework. Beyond its immediate empirical scope, this work also opens several avenues for future research. First, entropy-based portfolio construction can be extended to dynamic multi-period settings with transaction costs and liquidity frictions, which are particularly relevant in cryptocurrency markets. Second, the framework may be generalized to incorporate alternative entropy measures such as Rényi or Kaniadakis entropy, enabling more refined sensitivity to tail risks and nonlinear dependencies. The proposed framework provides a flexible foundation for future extensions toward dynamic, multi-period portfolio optimization under uncertainty.