Optimal Execution with Stochastic and Deterministic Delay
Abstract
We show how traders use aggressive immediate execution limit orders (IELOs) to liquidate a position when there are random delays in all the steps of a trade, i.e., there is latency in the marketplace and latency is random. We frame our model as a delayed impulse control problem in which the trader controls the times and the price limit of the IELOs she sends to the exchange. Our paper is the first to study an optimal liquidation problem that accounts for: random delays, price impact, and transaction costs. We introduce a new type of impulse control problem with stochastic (or deterministic) delay, not previously studied in the literature. The value functions are characterised as the solution to a coupled system of a Hamilton-Jacobi-Bellman quasi-variational inequality (HJBQVI) and a partial differential equation. We use a Feynman-Kac representation to reduce the system to a HJBQVI, for which we prove existence and uniqueness in a viscosity sense. We employ foreign exchange high-frequency trade data to implement the random-latency-optimal strategy and to compare it with four benchmarks: executing the entire order at once, optimal execution with deterministic latency, optimal execution with zero latency, and time-weighted average price. For example, in the EUR/USD currency pair, we show that the random-latency-optimal strategy and the deterministic-latency optimal strategy outperform the benchmarks between 3 USD per million EUR traded and 110 USD per million EUR traded, this is between one and 37 times the value of the transaction fees paid by liquidity takers. Joint work with and Leandro Sanchez Betancourt.
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