<p>This study addresses the complex challenge of pricing carbon options in an incomplete market environment characterized by asymmetric returns and discontinuous price trajectories. We propose a financial asset pricing framework based on geometric skew Brownian-jump motion (GSBJM), where the skew parameter captures distributional asymmetry in asset prices while jump components accommodate market shocks and regulatory announcements. To resolve the pricing measure selection problem inherent in incomplete markets, we employ the minimal entropy pricing measure, which minimizes the information divergence from the real-world probability measure and ensures optimal consistency with observed market dynamics. A closed-form pricing formula for European-style carbon options is derived under the GSBJM specification. Empirical analysis utilizing one-year futures data from the European Energy Exchange demonstrates that the GSBJM model outperforms conventional geometric Brownian motion, jump diffusion and geometric skew Brownian motion models, providing superior fit to market data and more accurate pricing performance. These results confirm the practical value of the proposed framework for pricing carbon derivatives under realistic market conditions featuring skewness, jumps, and incompleteness.</p>