Quantitative Finance under rough volatility.

Summary This thesis aims at understanding several aspects of the roughness of volatility observed universally on financial assets. This is done in six steps. In the first part, we explain this property from the typical behaviors of agents in the market. More precisely, we build a microscopic price model based on Hawkes processes reproducing the important stylized facts of the market microstructure. By studying the long-run price behavior, we show the emergence of a rough version of the Heston model (called rough Heston model) with leverage. Using this original link between Hawkes processes and Heston models, we compute in the second part of this thesis the characteristic function of the log-price of the rough Heston model. This characteristic function is given in terms of a solution of a Riccati equation in the case of the classical Heston model. We show the validity of a similar formula in the case of the rough Heston model, where the Riccati equation is replaced by its fractional version. This formula allows us to overcome the technical difficulties due to the non-Markovian character of the model in order to value derivatives. In the third part, we address the issue of risk management of derivatives in the rough Heston model. We present hedging strategies using the underlying asset and the forward variance curve as instruments. This is done by specifying the infinite-dimensional Markovian structure of the model. Being able to value and hedge derivatives in the rough Heston model, we confront this model with the reality of financial markets in the fourth part. More precisely, we show that it reproduces the behavior of implied and historical volatility. We also show that it generates the Zumbach effect, which is a time-reversal asymmetry observed empirically on financial data. In the fifth part, we study the limiting behavior of the implied volatility at low maturity in the framework of a general stochastic volatility model (including the rough Bergomi model), by applying a density development of the asset price. While the approximation based on Hawkes processes has addressed several questions related to the rough Heston model, in Part 6 we consider a Markovian approximation applying to a more general class of rough volatility models. Using this approximation in the particular case of the rough Heston model, we obtain a numerical method for solving the fractional Riccati equations. Finally, we conclude this thesis by studying a problem not related to the rough volatility literature. We consider the case of a platform seeking the best make-take fee scheme to attract liquidity. Using the principal-agent framework, we describe the best contract to offer to the market maker as well as the optimal quotes displayed by the latter. We also show that this policy leads to better liquidity and lower transaction costs for investors.