Flexible Probabilistic Models for Capturing Extreme Risk in Cryptocurrency Returns

This study presents a comparative investigation of probabilistic mixture models for capturing extreme risk in cryptocurrency return distributions. Specifically, we evaluate the performance of the Gaussian Mixture Model (GMM), the Student-\((t)\) Mixture Model (TMM), and the Laplace Mixture Model (LMM) using standardized daily return series from ten leading cryptocurrencies over a five-year period. The primary objective is to assess each model's ability to accurately represent heavy tails and extreme return behavior, which are critical for reliable financial risk estimation. Model performance is evaluated using tail-sensitive risk measures, including Value-at-Risk (VaR) and Expected Shortfall (ES) at the 5% confidence level, alongside log-likelihood-based goodness-of-fit diagnostics. Empirical findings indicate that while the GMM effectively captures the central distribution, it systematically underestimates tail probabilities. The LMM provides moderate improvement in flexibility but does not consistently outperform alternative specifications in extreme regions. In contrast, the TMM demonstrates superior tail fit and delivers more accurate VaR and ES estimates across assets. These results underscore the importance of heavy-tailed mixture frameworks for robust risk modeling in high-volatility digital asset markets. JEL Classification: G17 , C13 , C58 , G32