Linking affine and Levy-driven models to the microstructure of financial markets

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By now, models based on affine stochastic processes and Lévy processes have become an indispensable tool in financial modeling. To a large extent, these models are used to describe markets on a 'macroscopic' scale - approximately in the range of hours to several years - which is the scale relevant for risk management, as well as for the pricing and hedging of derivatives. On the other hand, a more recent research strand in mathematical finance has focused on 'market microstructure', that is, the dynamics of transactions in financial markets on a much smaller time-scale, in the order of microseconds to minutes. The relevance of this time-scale is given both by the increasing (and much-debated) role of high-frequency trading and by the observation of market anomalies that can only be explained on the level of this microstructure.The goal of this project is to cross these scales and to link affine and Lévy-driven stochastic processes to models of market microstructure. Affine processes serve as a unifying framework for models that include self-exciting effects, such as Hawkes processes with exponential kernel. In addition, we aim to show that even Non-Markovian models that have been proposed for market microstructure, like Hawkes-processes with power-law kernels or fractional diffusions can be embedded into the affine framework by considering processes on infinite-dimensional state spaces. Finally, we will combine certain affine and Lévy-driven models with an economic equilibrium model of high-frequency-trading, in order to obtain a full picture of all market events (the 'limit order book') on the microscopic scale.