  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).
