The Black–Scholes model laid the mathematical foundation for modern option pricing; however, its assumptions—stationary, independent, and Gaussian returns—are frequently violated in real markets, where long-memory volatility and sudden price jumps are well-documented. Two issues remain open: (1) Few option pricing models comprehensively incorporate long-memory and jump features. (2) The equivalence of the hedging, risk-neutral, and actuarial pricing methods, well-established under the standard Black–Scholes framework, has not been examined under jump–diffusion models. To address these gaps, we developed a sub-mixed fractional Brownian motion with Jumps (smfBm-J) model that jointly captures long memory, nonstationary increments, and jumps and derives a closed-form European call option pricing formula under the smfBm-J framework, highlighting the impact of model choice on valuation in incomplete markets.

This study examines whether financial leverage promotes sustainable agricultural productivity by accounting for the trade-off between profitability and default risk. We construct a risk-adjusted total factor productivity (TFP) measure by incorporating default risk as an undesirable output into a slack-based DEA framework (USBM), with risk estimated via contingent claims analysis (CCA). Using panel data on Chinese listed agricultural firms, we find a robust inverted-U relationship between leverage and TFP, indicating an optimal leverage range. Mechanism analysis reveals a dual-channel effect: leverage improves productivity through profitability and reduced financing constraints at low levels but increases default risk and undermines financial sustainability at high levels. Decomposition results show that leverage promotes efficiency catch-up but inhibits frontier technological progress, implying a trade-off between short-term efficiency gains and long-term sustainability. Substantial heterogeneity across subsectors and market structures further suggests that optimal leverage is context-dependent. This study contributes by developing a risk-adjusted productivity framework, identifying the nonlinear effects of leverage on sustainable TFP, and providing micro-level evidence from agriculture in a developing economy. The findings offer implications for capital structure optimization and sustainable agricultural finance.

The low-carbon transition has strengthened interactions between the cryptocurrency and carbon markets, while differences in cryptocurrency energy intensity may lead to heterogeneous risk spillover. We distinguish between energy-intensive and energy-efficient cryptocurrencies and use a multiscale quantile-on-quantile connectedness framework to examine spillovers in returns, volatility, skewness, and excess kurtosis. Three main findings emerge. First, connectedness is weak under normal market conditions but rises sharply when either market enters an extreme state. Return spillovers are concentrated mainly over short horizons, whereas volatility and higher-order risks show stronger connectedness over medium and long horizons. Second, net spillover roles vary across market states, investment horizons, and cryptocurrency types. Differences in energy intensity are particularly evident in volatility spillovers over long horizons and cross-state skewness transmission, while Ethereum shows a transitional pattern following its shift from proof-of-work to proof-of-stake. Third, the relative importance of direct and reverse connectedness varies across risk measures and horizons. These linkages are also sensitive to common external shocks, with major energy and monetary events producing distinct horizon-dependent responses. These findings highlight the importance of jointly considering energy intensity, higher-order risks, market states, and investment horizons in cross-market risk management.