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The effect of latency on optimal order execution policy

Chutian Ma, Giacinto Paolo Saggese, Paul Smith

TL;DR

The paper analyzes how order-latency (delay) impacts optimal limit-order execution under short-horizon price uncertainty modeled by a standard Brownian motion. It derives analytical expressions for fill and non-fill probabilities via hitting-time analysis, quantifies the risk of marketable limit orders due to latency, and uses an exponential-utility, risk-averse objective within a finite-horizon Markov decision process to obtain optimal repricing policies. The authors provide closed-form approximations, an MDP formulation, and Monte Carlo validation, showing how latency, spread, and risk tolerance shape the optimal trading strategy and the mix of limit versus market executions. The work offers a latency-aware framework for small- to medium-sized orders in limit-order books, with practical implications for traders seeking to balance speed, probability of fill, and execution costs.

Abstract

Market participants regularly send bid and ask quotes to exchange-operated limit order books. This creates an optimization challenge where their potential profit is determined by their quoted price and how often their orders are successfully executed. The expected profit from successful execution at a favorable limit price needs to be balanced against two key risks: (1) the possibility that orders will remain unfilled, which hinders the trading agenda and leads to greater price uncertainty, and (2) the danger that limit orders will be executed as market orders, particularly in the presence of order submission latency, which in turn results in higher transaction costs. In this paper, we consider a stochastic optimal control problem where a risk-averse trader attempts to maximize profit while balancing risk. The market is modeled using Brownian motion to represent the price uncertainty. We analyze the relationship between fill probability, limit price, and order submission latency. We derive closed-form approximations of these quantities that perform well in the practical regime of interest. Then, we utilize a mean-variance method where our total reward function features a risk-tolerance parameter to quantify the combined risk and profit.

The effect of latency on optimal order execution policy

TL;DR

The paper analyzes how order-latency (delay) impacts optimal limit-order execution under short-horizon price uncertainty modeled by a standard Brownian motion. It derives analytical expressions for fill and non-fill probabilities via hitting-time analysis, quantifies the risk of marketable limit orders due to latency, and uses an exponential-utility, risk-averse objective within a finite-horizon Markov decision process to obtain optimal repricing policies. The authors provide closed-form approximations, an MDP formulation, and Monte Carlo validation, showing how latency, spread, and risk tolerance shape the optimal trading strategy and the mix of limit versus market executions. The work offers a latency-aware framework for small- to medium-sized orders in limit-order books, with practical implications for traders seeking to balance speed, probability of fill, and execution costs.

Abstract

Market participants regularly send bid and ask quotes to exchange-operated limit order books. This creates an optimization challenge where their potential profit is determined by their quoted price and how often their orders are successfully executed. The expected profit from successful execution at a favorable limit price needs to be balanced against two key risks: (1) the possibility that orders will remain unfilled, which hinders the trading agenda and leads to greater price uncertainty, and (2) the danger that limit orders will be executed as market orders, particularly in the presence of order submission latency, which in turn results in higher transaction costs. In this paper, we consider a stochastic optimal control problem where a risk-averse trader attempts to maximize profit while balancing risk. The market is modeled using Brownian motion to represent the price uncertainty. We analyze the relationship between fill probability, limit price, and order submission latency. We derive closed-form approximations of these quantities that perform well in the practical regime of interest. Then, we utilize a mean-variance method where our total reward function features a risk-tolerance parameter to quantify the combined risk and profit.

Paper Structure

This paper contains 17 sections, 3 theorems, 64 equations, 15 figures, 2 tables.

Table of Contents

  1. Introduction
  2. The Brownian motion price model
  3. Formalism
  4. Single limit order execution: hitting times
  5. Risk of execution as a market order
  6. Close price given non-execution
  7. Expected close given a non-fill
  8. Basic order splitting
  9. Naive order splitting
  10. Reprice until filled
  11. Static repricing policy
  12. Expected execution price under indefinite static repricing
  13. Markov Decision Process formulation
  14. Optimal policy
  15. Discussion
  16. ...and 2 more sections

Key Result

Lemma 7

We compute $\int_{-\infty}^{y} e^{-\lambda c} p(c, h \leq y) dc$, which is used in eq exp utility II. The expression is equal to

Figures (15)

  • Figure 1: Probability of a market fill
  • Figure 2: Probability of a limit fill
  • Figure 3: Expected proportion of market fills among all fills
  • Figure 4: Expected close price given non-execution of a limit order
  • Figure 5: Close price pdf given non-execution of a limit order at various limit prices
  • ...and 10 more figures

Theorems & Definitions (13)

  • Remark 1
  • Remark 2
  • Remark 3: Effects of scaling
  • Remark 4: Effect of spread
  • Remark 5
  • Remark 6
  • Lemma 7
  • proof
  • Lemma 8
  • proof
  • ...and 3 more