C\`adl\`ag semimartingale strategies for optimal trade execution in stochastic order book models

TL;DR

This paper develops a continuous-time limit order book model with stochastic liquidity, deriving a quadratic BSDE to characterize optimal trade execution strategies that account for random order book dynamics and allow for complex trading behaviors.

Contribution

It introduces a novel stochastic model for limit order books with dynamic depth and resilience, providing a framework to determine optimal execution strategies using quadratic BSDEs.

Findings

Optimal strategies characterized by quadratic BSDEs

Existence conditions for optimal execution strategies

Insights into strategies with infinite variation and block trades

Abstract

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 continuous time. Both order book depth and resilience are allowed to evolve randomly in time. We allow for trading in both directions and for c\`adl\`ag semimartingales as execution strategies. We derive a quadratic BSDE that under appropriate assumptions characterizes minimal execution costs and identify conditions under which an optimal execution strategy exists. We also investigate qualitative aspects of optimal strategies such as, e.g., appearance of strategies with infinite variation or existence of block trades and discuss connections with the discrete-time formulation of the problem. Our findings are illustrated in several examples.