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

TL;DR

This paper proves the existence, uniqueness, and positivity of solutions to a complex path-dependent volatility model with general kernels, extending previous theoretical results and providing numerical insights into kernel choices.

Contribution

It establishes new mathematical results for a non-Lipschitz, non-convolutional Volterra equation model with general kernels, including positivity conditions and numerical calibration analysis.

Findings

Existence and uniqueness of solutions under broad conditions.

Positivity of the volatility process under specific kernel assumptions.

Minimal impact of exponential kernels on model calibration quality.

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).