Paper: arXiv 2311.08289

Authors: Alexandre Pannier

Abstract

We regard options on VIX and Realised Variance as solutions to path-dependent partial differential equations (PDEs) in a continuous stochastic volatility model. The modeling assumption specifies that the instantaneous variance is a $C^3$ function of a multidimensional Gaussian Volterra process; this includes a large class of models suggested for the purpose of VIX option pricing, either rough, or not, or mixed. We unveil the path-dependence of those volatility derivatives and, under a regularity hypothesis on the payoff function, we prove the well-posedness of the associated PDE. The latter is of heat type, because of the Gaussian assumption, and the terminal condition is also path-dependent. Furthermore, formulae for the greeks are provided, the implied volatility is shown to satisfy a quasi-linear path-dependent PDE and, in Markovian models, finite-dimensional pricing PDEs are obtained for VIX options.

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 provides a rigorous theoretical framework using functional Ito calculus and path-dependent PDEs for pricing volatility derivatives under Gaussian Volterra processes. While highly novel and mathematically sophisticated, it lacks empirical backtesting and implementation details with real market data.

Research Flowchart

  flowchart TD
    A[Goal: Price volatility derivatives via path-dependent PDEs] --> B[Input: Continuous stochastic volatility model driven by Gaussian Volterra process]
    B --> C[Method: Formulate functional heat-type PDE with path-dependent terminal conditions]
    C --> D[Analysis: Prove well-posedness under payoff regularity hypotheses]
    D --> E[Outcome 1: Exact pricing & Greeks formulas for VIX/Realised Variance options]
    D --> F[Outcome 2: Quasi-linear PDE for Implied Volatility & finite-D PDEs for Markovian cases]