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.