Endogenous Formation of Limit Order Books: Dynamics Between Trades. (arXiv:1605.09720v1 [q-fin.TR])

In this talk, we present a continuous time extension of the framework for modeling market microstructure, developed in our previous work. We use this extension to model the shape and dynamics of the Limit Order Book (LOB) between two consecutive trades. In this model, the LOB arises as an outcome of an equilibrium between multiple agents who have different beliefs about the future demand for the asset. These beliefs may change according to the information observed by the agents (e.g. represented by a relevant stochastic factor), implying a change in the shape of the LOB. This model is consistent with the empirical observation that most changes in the LOB are not due to trades. More importantly, it allows one to see how changing the relevant information signal affects the LOB. If the relevant signal is a function of the LOB itself, then, our approach allows one to model the "indirect" market impact (as opposed to the "direct" impact that a market order makes on the LOB, by eliminating certain limit orders instantaneously), showing how any change to the LOB causes further changes to it. On the mathematical side, we formulate the problem as a mixed control-stopping game, with a continuum of players. We manage to split the equilibrium problem into two parts, and represent one of them through a two-dimensional system of Reflected Backward Stochastic Differential Equations, and the other one with an infinite-dimensional fixed-point equation. We prove the existence of the solutions to both problems and show how they can be computed in a simple example.