Portfolio Execution with a Dark Pool Under Stochastic Volatility and Liquidity

Abstract

In this paper we study a portfolio execution problem in a discrete-time model in which orders can be submitted to a standard exchange and a dark pool. We model volatilities and correlations as stochastic processes and assume that trading at the standard exchange causes price impact. Orders sent to the dark pool do not affect prices. But they are not always filled. An efficient execution strategy has to find the right balance between favorable transaction prices and immediacy. Our main result gives a closed form expression for strategies that minimize a weighted average of expected implementation costs and exposure to market risk. As an example we simulate the optimal acquisition of two correlated assets and compare it to a program which trades the assets separately without taking into account correlation or cross price impact.