On the Guyon-Lekeufack Volatility Model

TL;DR

This paper proves the well-posedness and key properties of the Guyon-Lekeufack path-dependent volatility model, which combines past returns and volatility with exponential kernels, enhancing understanding of its mathematical robustness.

Contribution

It establishes the existence, uniqueness, and stability of solutions for the model's coupled stochastic differential equations, a key step in validating its theoretical foundation.

Findings

Model has a unique strong solution for all parameters.

Volatility process remains positive under certain conditions.

Associated price process maintains the martingale property.

Abstract

Guyon and Lekeufack recently proposed a path-dependent volatility model and documented its excellent performance in fitting market data and capturing stylized facts. The instantaneous volatility is modeled as a linear combination of two processes, one is an integral of weighted past price returns and the other is the square-root of an integral of weighted past squared volatility. Each of the weightings is built using two exponential kernels reflecting long and short memory. Mathematically, the model is a coupled system of four stochastic differential equations. Our main result is the wellposedness of this system: the model has a unique strong (non-explosive) solution for all parameter values. We also study the positivity of the resulting volatility process and the martingale property of the associated exponential price process.