Optimal trade execution in an order book model with stochastic liquidity parameters
We analyze an optimal trade execution problem in a financial market with stochastic liquidity. To this end we set up a limit order book model in which both order book depth and resilience evolve randomly in time. Trading is allowed in both directions. In discrete time, we discuss an explicit recursion that, under certain structural assumptions, characterizes minimal execution costs and observe some qualitative differences with related models. In continuous time, due to the stochastic dynamics of the order book depth and resilience, optimal execution strategies are typically of infinite variation, and the first thing to be discussed it how to extend the state dynamics and the cost functional to allow for general semimartingale strategies. We then derive a quadratic BSDE that under appropriate assumptions characterizes minimal execution costs, identify conditions under which an optimal execution strategy exists and, finally, illustrate our findings in several examples. This is a joint work with Julia Ackermann and Thomas Kruse.