Existence, uniqueness and positivity of solutions to the Guyon-Lekeufack path-dependent volatility model with general kernels

ArXiv ID: 2408.02477 “View on arXiv”

Authors: Unknown

Abstract

We show the existence and uniqueness of a continuous solution to a path-dependent volatility model introduced by Guyon and Lekeufack (2023) to model the price of an equity index and its spot volatility. The considered model for the trend and activity features can be written as a Stochastic Volterra Equation (SVE) with non-convolutional and non-bounded kernels as well as non-Lipschitz coefficients. We first prove the existence and uniqueness of a solution to the SVE under integrability and regularity assumptions on the two kernels and under a condition on the second kernel weighting the past squared returns which ensures that the activity feature is bounded from below by a positive constant. Then, assuming in addition that the kernel weighting the past returns is of exponential type and that an inequality relating the logarithmic derivatives of the two kernels with respect to their second variables is satisfied, we show the positivity of the volatility process which is obtained as a non-linear function of the SVE’s solution. We show numerically that the choice of an exponential kernel for the kernel weighting the past returns has little impact on the quality of model calibration compared to other choices and the inequality involving the logarithmic derivatives is satisfied by the calibrated kernels. These results extend those of Nutz and Valdevenito (2023).

Keywords: Stochastic Volterra Equation (SVE), Path-dependent volatility, Equity index, Existence and uniqueness, Equity

Complexity vs Empirical Score

• Math Complexity: 9.5/10
• Empirical Rigor: 3.0/10
• Quadrant: Lab Rats
• Why: The paper is heavily focused on theoretical proofs of existence, uniqueness, and positivity for a path-dependent volatility model using advanced stochastic calculus and Volterra equations, with no backtesting or implementation details provided in the excerpt.

  flowchart TD
    A["Research Goal<br>Prove existence, uniqueness, and positivity<br>for Guyon-Lekeufack Path-Dependent Volatility Model"] --> B["Methodology<br>Stochastic Volterra Equation (SVE) Framework"]

    B --> C{"Conditions Checked?"}
    
    C -- General Assumptions --> D["Main Theorem<br>Existence & Uniqueness<br>of Continuous Solution"]
    C -- Exponential Kernel &<br>Log-Derivative Inequality --> E["Positivity Theorem<br>Strictly Positive Volatility Process"]

    D --> F["Key Outcome 1<br>SVE Solution exists under<br>integrability & regularity constraints"]
    E --> G["Key Outcome 2<br>Volatility bounded below by<br>positive constant"]

    F --> H["Numerical Validation<br>Calibration to market data"]
    G --> H

    H --> I["Final Finding<br>Exponential kernels perform comparably<br>to other choices; inequality holds in practice"]