Paper: arXiv 2406.13508

Authors: Oriol Zamora Font

Abstract

We derive a semi-analytical pricing formula for European VIX call options under the Heston-Hawkes stochastic volatility model introduced in arXiv:2210.15343. This arbitrage-free model incorporates the volatility clustering feature by adding an independent compound Hawkes process to the Heston volatility. Using the Markov property of the exponential Hawkes an explicit expression of $\text{VIX}^2$ is derived as a linear combination of the variance and the Hawkes intensity. We apply qualitative ODE theory to study the existence of some generalized Riccati ODEs. Thereafter, we compute the joint characteristic function of the variance and the Hawkes intensity exploiting the exponential affine structure of the model. Finally, the pricing formula is obtained by applying standard Fourier techniques.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 2.5/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents a rigorous theoretical derivation of VIX option pricing under a Heston-Hawkes stochastic volatility model using complex ODE analysis and Fourier techniques. However, it is purely theoretical and lacks empirical backtesting, real market data calibration, or numerical implementation results.

Research Flowchart

  flowchart TD
    A[Goal: Price European VIX Call Options under Heston-Hawkes Model] --> B[Model Formulation: Combine Heston Volatility with Compound Hawkes Process]
    B --> C[Derive VIX² Expression as Linear Combination of Variance & Intensity]
    C --> D[Solve Generalized Riccati ODEs via Qualitative ODE Theory]
    D --> E[Compute Joint Characteristic Function via Affine Structure]
    E --> F[Apply Fourier Transform Inversion Techniques]
    F --> G[Outcome: Semi-Analytical VIX Option Pricing Formula]

Cited References (24)

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and 16 more
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  2. VIX Option Pricing in a Jump-Diffusion Model (2008)
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  4. An analytical formula for VIX futures and its applications (2012)
  5. Pricing Vix Options with Stochastic Volatility and Random Jumps (2013)
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0 of 24 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.