Arbitrage-Free Pricing with Diffusion-Dependent Jumps

TL;DR

This paper introduces a new jump-diffusion model where jump behavior depends on current diffusion movements, ensuring arbitrage-free pricing by deriving explicit no-arbitrage conditions using advanced mathematical techniques.

Contribution

It develops a multi-type jump-diffusion framework with diffusion-dependent jumps and establishes explicit no-arbitrage conditions for arbitrage-free pricing.

Findings

Derived explicit no-arbitrage condition linking drift and model parameters.

Constructed an Equivalent Martingale Measure using Girsanov's theorem and Esscher transform.

Ensured the model's consistency with arbitrage-free market assumptions.

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.