CRC/TRR 388/1: "Microstructural foundations of rough volatility models" (SP B02)

Microstructure models of financial markets have been extensively studied in both the financial mathematics and the probability literature in the last two decades. While early work has primarily focused on the emergence of financial bubbles, the focus has recently shifted towards limit order book modelling and the microstructural foundations of stochastic volatility models. This project combines the PIs’ expertise on rough analysis and market microstructure to derive and analyze novel scaling limits for stochastic processes arising in microstructure models of financial markets with a particular focus on rough volatility models. This includes but is not limited to limit theorems for (quasi-)stationary Hawkes processes with self-exciting jump dynamics and Hawkes random measures with non-exponential decay kernels, Donsker-type theorems for fractional Brownian motion, a general convergence theory for rough stochastic integrals and rough stochastic differential equations, and the derivation of weak scaling limits based on the signature kernel and the corresponding maximum mean discrepancy distance as well as more generally from the point of view of regularity structures. This wide range of convergence results will allow us to derive high-frequency approximations for different classes of rough volatility models, including models of rough Heston-type, models of rough Bergomi-type, and a genuine new class of models based on rough stochastic integral processes. The overarching goal of the project is to improve our understanding of microstructural foundations of rough volatility models by allowing richer and more realistic microscopic dynamics, which result e.g. from the trading behaviour of heterogenous agents, and also by enhancing the available technical toolkit for the analysis of their scaling limits.