Talk by Eduardo Abi Jaber, Assistant Professor Ecole Polytechnique
Title: Volatility dynamics and memory: from the Quintic model to Signatures
Abstract: We introduce the Quintic Ornstein-Uhlenbeck model, designed for the joint calibration of SPX and VIX options. The model is also capable of aligning with the term structure of the skew-stickiness ratio. At the same time, it remains mathematically tractable, enabling fast pricing of SPX options via Fourier techniques and VIX options via simple integration against a Gaussian density.
We then turn to a broader class of stochastic volatility models in which volatility evolves as a (possibly infinite) linear combination of the time-extended signature of Brownian motion. We show that signature volatility models can be seen as a kind of universal machine for modeling path-dependent volatility. Their structure is rich enough to encompass a wide range of existing models, including the Bergomi model and its path-dependent extensions. Crucially, they retain enough algebraic structure to admit Fourier-based pricing and hedging. Finally, we provide a characterization of the martingale property of the price process in terms of the signature truncation order.
Based on joint works with Paul Gassiat, Louis-Amand Gérard, Yuxing Huang, Camille Illand, Shaun Li, Xuyang Lin & Dimitri Sotnikov.