🤖 AI Summary
This paper investigates the well-posedness of solutions to the Guyon–Lekeufack path-dependent volatility model under general kernel functions. The underlying dynamics are formulated as stochastic Volterra equations with non-convolutional, unbounded kernels and non-Lipschitz coefficients—a setting that poses significant challenges for modeling trend and activity features. Methodologically, we integrate stochastic Volterra equation theory, path-dependent stochastic analysis, and kernel regularity analysis, complemented by numerical calibration and sensitivity validation. Our key contributions are threefold: (i) we establish, for the first time, global existence and uniqueness of solutions under non-convex, unbounded kernels and non-Lipschitz conditions; (ii) we introduce a novel positivity criterion based on logarithmic derivative inequalities, rigorously ensuring strict positivity of the volatility process; and (iii) we empirically verify that the calibrated kernel satisfies the proposed positivity condition, while demonstrating that exponential kernel choices exert negligible impact on calibration accuracy.
📝 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).