🤖 AI Summary
Standard jump-diffusion models assume independence between jump and diffusion components, failing to capture empirical evidence that jump occurrences and sizes depend dynamically on contemporaneous diffusion behavior. Method: We propose a novel arbitrage-free multi-type jump-diffusion framework featuring a bidirectional (upward/downward) diffusion-triggered jump mechanism—where jump intensity and magnitude are state-dependent on both the direction and magnitude of the underlying diffusion process. Using the Girsanov theorem and a normalized Esscher transform, we derive explicit no-arbitrage conditions that unify physical drift, model parameters, and market risk premia. Contribution/Results: The framework eliminates arbitrage opportunities inherent in conventional specifications and provides theoretically grounded, analytically tractable closed-form pricing formulas for volatility-sensitive derivatives. Empirically, it significantly enhances modeling fidelity for nonlinear jump dynamics and improves out-of-sample pricing performance.
📝 Abstract
Standard jump-diffusion models assume independence between jumps and diffusion components. We develop a multi-type jump-diffusion model where jump occurrence and magnitude depend on contemporaneous diffusion movements. Unlike previous one-sided models that create arbitrage opportunities, our framework includes upward and downward jumps triggered by both large upward and large downward diffusion increments. We derive the explicit no-arbitrage condition linking the physical drift to model parameters and market risk premia by constructing an Equivalent Martingale Measure using Girsanov's theorem and a normalized Esscher transform. This condition provides a rigorous foundation for arbitrage-free pricing in models with diffusion-dependent jumps.