Seminario Polynomial path-dependent volatility models

Relatore: Fabio Baschetti (Università di Verona)

Abstract:

Recent years have witnessed a growing interest in path-dependent volatility (PDV) models. This is particularly true after Guyon and Lekeufack (2023) showed that a simple functional form in terms of a trend feature (R_1) and an activity feature (R_2) explains approximately 85-90% of the variability in the levels of the major volatility indices. Not only does their multi-factor Markov PDV model have a large predictive power of volatility, but it is also shown to jointly fit S&P 500 (SPX) and VIX option market data. However, the computation of the VIX poses numerical challenges in their framework. To address these difficulties, we introduce a class of polynomial path-dependent volatility models. We establish well-posedness and provide a simple condition for positivity of the variance. Furthermore, we obtain a semi-closed-form expression for the forward variance process and derive an expansion of the smile at order two in small volatility of volatility. Empirically, our poly-PDV performs comparably to the original Guyon-Lekeufack volatility model both under P and Q, while enabling fast evaluation of the VIX via the moment formula. This is joint work with G. Gazzani and J. Guyon.