Paper: arXiv 2512.15071

Authors: Hamza Virk, Yihren Wu, Majnu John

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.

Complexity vs Empirical Score

  • Math Complexity: 7.5/10
  • Empirical Rigor: 2.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents a rigorous theoretical framework for arbitrage-free pricing with diffusion-dependent jumps using Girsanov’s theorem and Esscher transforms. While the mathematical derivation and model formulation are highly detailed and clear, the provided text lacks empirical backtesting, real market data implementation, and reproducible code.

Research Flowchart

  flowchart TD
    A["Research Goal: Arbitrage-Free Pricing<br>with Diffusion-Dependent Jumps"] --> B["Inputs: Multi-Type Jump-Diffusion Framework<br>(Upward & Downward Jumps/Increments)"]
    B --> C["Methodology: Construct Equivalent Martingale Measure<br>(Girsanov Theorem + Normalized Esscher Transform)"]
    C --> D["Derive Explicit No-Arbitrage Condition<br>(Link Physical Drift, Parameters, Risk Premia)"]
    D --> E["Outcomes: Rigorous No-Arbitrage Model<br>Eliminating One-Sided Arbitrage Opportunities"]

Cited References (10)

  1. A Jump-Diffusion Process for Asset Price with Non-Independent Jumps (2022) — link
  2. Stock market forecasting using hidden Markov model: a new approach (2005) — link
  3. Martingales and stochastic integrals in the theory of continuous trading (1981) — link
  4. The Pricing of Options and Corporate Liabilities (1973) — link
  5. Option pricing by Esscher transforms. (1995)
  6. Option pricing when underlying stock returns are discontinuous (1976) — link
  7. On Transforming a Certain Class of Stochastic Processes by Absolutely Continuous Substitution of Measures (1960) — link
  8. On the probability function in the collective theory of risk (1932) — link
and 2 more
  1. Derivative Pricing: By establishing the EMM Q , derivative securities can be priced using the principle of risk-neutral valuation, i.e., Price t = E Q [ e − r ( T − t ) Payoff T |F t ]
  2. Model Consistency: Ensuring that any specific parameterization of the model under the physical measure P is internally consistent and does not admit trivial arbitrage strategies

0 of 10 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.