A Random Walk in Illiquidity Modeling

Liquidity is a fundamental property of a well-functioning financial market and lack of liquidity is generally at the heart of many financial crises and disasters, such as the global financial crisis of 2007–2008 and the LTCM crisis of 1998. Liquidity refers more precisely to the efficiency or ease with which an asset or security can be sold in financial markets without affecting its market price and is always tied to the temporal granularity at which it is analyzed. Indeed, when modeling the price of an asset and its liquidity, we typically distinguish between at least three different time scales: highfrequency (fraction of seconds), intraday time scale (between seconds and hours) and low-frequency (from days to years for very large time scales). These three temporalities and their connections are the subject of the three parts constituting this thesis. The first part of the thesis hence deals with rough volatility as it arises from the modelling of market liquidity at high frequency and studies the connection of such models with longterm insurance/asset pricing in Chapter 1 and with portfolio insurance strategies in Chapter 2. The second part of the thesis then investigates the use of fractional processes for modelling the illiquidity process associated with low-frequency observations of asset prices; first with subdiffusions in Chapter 3 and then with a new fractional Hawkes process in Chapter 4. Finally, the third part of the thesis focuses on stochastic optimal control in finance, and more precisely on optimal liquidation problems as they consider an intermediate time scale. A new jump-dependent indirect price impact model is then proposed in Chapter 5, and a deep learning algorithm is developed in Chapter 6 for solving complex high-dimensional stochastic optimal control (and liquidation) problems without explicit solution.