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Optimal High Frequency Trading with limit and market orders
Fabien Guilbaud, Huyen Pham
TL;DR
The paper addresses how a market maker can optimize limit and market orders while controlling inventory in a stochastic limit order book. It models spread dynamics and order execution through a mixed regular/impulse control framework, derives tractable reductions and calibration procedures, and reports computational results including inventory-risk effects and improved information ratios. The analysis excludes pro-rata priority markets and notes that simulated performance may overstate real-world results.
Problem
The paper studies how to maximize transaction profit over a finite horizon while controlling inventory under uncertain limit-order execution and costly market orders.
Method
The paper models the spread as a finite-state Markov chain driven by a Poisson clock and formulates limit orders as continuous controls and market orders as discrete impulse controls.
Results
The optimal strategy significantly improves the information ratio relative to a benchmark, while increasing inventory penalization reduces inventory risk through more frequent market unwinding.
Takeaways & Limitations
The framework supports market-making policies that trade off execution speed, quote profitability, market-order costs, and inventory risk, with calibration procedures for spread and execution dynamics.
Takeaways & Limitations
Simulated backtest performance may be overestimated relative to real-world performance and requires completion with a real-data backtest.
Abstract
from arXiv · showhide
We propose a framework for studying optimal market making policies in a limit order book (LOB). The bid-ask spread of the LOB is modelled by a Markov chain with finite values, multiple of the tick size, and subordinated by the Poisson process of the tick-time clock. We consider a small agent who continuously submits limit buy/sell orders and submits market orders at discrete dates. The objective of the market maker is to maximize her expected utility from revenue over a short term horizon by a tradeoff between limit and market orders, while controlling her inventory position. This is formulated as a mixed regime switching regular/ impulse control problem that we characterize in terms of quasi-variational system by dynamic programming methods. In the case of a mean-variance criterion with martingale reference price or when the asset price follows a Levy process and with exponential utility criterion, the dynamic programming system can be reduced to a system of simple equations involving only the inventory and spread variables. Calibration procedures are derived for estimating the transition matrix and intensity parameters for the spread and for Cox processes modelling the execution of limit orders. Several computational tests are performed both on simulated and real data, and illustrate the impact and profit when considering execution priority in limit orders and market orders
1 Introduction
The paper develops an optimal market-making framework for order-driven markets, where market makers balance limit-order execution and market-order costs while controlling inventory and several forms of risk.
- Market making: Market makers provide liquidity by posting simultaneous buy and sell limit orders, earning the difference between ask and bid prices.Their inventory changes as they intermediate between investors trading at different times.
- Risks: Inventory, adverse selection, and execution risk are the three principal risks affecting market-making strategies.Uncertain limit-order execution gives market makers only partial control over inventory.
- Model: The framework lets a market maker choose between continuously updated limit orders and market orders available at discrete dates.Limit orders execute uncertainly, whereas market orders execute immediately but incur costs; quotes may be placed at the best price or improved by one tick.
- Model: The bid-ask spread is represented by a finite-state Markov chain on tick-size multiples, while the stock mid-price follows a general Markov process.The spread chain is subordinated to a Poisson tick-time clock.
- Scope: The analysis focuses on the main equity-market priority mechanism and excludes limit order books operating under pro-rata priority.The paper therefore does not cover some futures-market mechanisms.
- Solution approach: The resulting optimization is a mixed regular/impulse control problem characterized by a Hamilton-Jacobi-Bellman quasi-variational inequality.The paper also describes calibration methods and reductions of the state variables for selected utility criteria.
2 A market-making model
The model combines a discrete Markov spread process with limit-order execution and costly market-order impulses, while accounting for inventory and execution priority. It also specifies calibration procedures for spread and execution intensities and reports empirical patterns from SOGN.PA data.
- Spread and price dynamics: The spread takes finite tick-multiple values and evolves as a time-changed Markov chain driven by a Poisson tick-time clock.Its calendar-time intensity matrix satisfies r_ij(t) = λ(t)ρ_ij for i ≠ j.
- Limit-order strategies: The agent can quote at the best price or improve it by one tick, trading off higher execution priority against less favorable quote prices.When the spread is one tick, an improved limit quote becomes a market order.
- Limit-order execution: Limit orders execute through independent Cox processes whose intensities depend on the quote and spread, encoding price and execution priority.Improved quotes receive higher execution intensity than orders at the current best quote.
- Market-order strategies: Market orders are modeled as impulse controls with immediate execution, inventory reduction, and a cost determined by order size, mid-price, spread, and fixed fee.The small-agent assumption makes total market orders execute immediately at the best bid or ask.
3 Optimal limit/market order strategies
The paper formulates market making with limit and market orders as a mixed regular/impulse control problem and characterizes its value through a quasi-variational inequality. Dynamic programming yields a viscosity-solution system whose operators account for price, spread, and order-induced cash and inventory changes.
- Control problem: The state consists of cash, inventory, mid-price, and finite-valued spread, while limit-order controls select quotes and submitted quantities.The associated nonlocal transitions change cash and inventory when limit orders execute.
- Control problem: The market maker maximizes finite-horizon transaction profit while controlling inventory and liquidating it at the terminal date.The control includes both limit-order and market-order strategies, with a terminal inventory constraint handled through liquidation.
- Control problem: The liquidation function values immediate market-order liquidation as cash plus inventory marked at the mid-price minus spread liquidation cost.It is defined as L(x, y, p, s) = x + yp − |y|s.
- Dynamic programming equation: The QVI combines the mid-price generator, spread Markov-chain generator, limit-order jump operator, and market-order impulse operator.The limit-order operator captures instantaneous cash and inventory jumps, while market orders are represented through impulses.
- Dynamic programming equation: The value function is the unique viscosity solution of the QVI system under suitable growth conditions on utility and inventory-penalty functions.The terminal condition evaluates utility at the liquidation value.
1. Mean criterion with penalty on inventory
Under a martingale stock-price assumption with inventory penalization, the dynamic programming solution reduces to one-dimensional equations. The resulting optimal strategies depend on inventory and spread rather than price or the martingale model.
- Mean criterion with penalty on inventory: The martingale stock-price assumption represents the market maker having no information about future price direction, while inventory penalization keeps holdings near zero.The setup typically starts from zero stock endowment.
- Reduced value function: The value function reduces to vi(t, x, y, p) = x + yp + φi(t, y).The residual function depends on time and inventory for each spread state.
- Numerical solution: The reduced one-dimensional IDEs can be solved numerically by finite differences over time and the inventory grid.The scheme discretizes the time derivative of φ and the grid space in inventory.
- Strategy dependence: The optimal market-making strategies are price independent and depend only on inventory and spread.The IDEs also imply independence from the particular martingale model for the stock price.
2. Exponential utility criterion
Under exponential utility and a Bachelier price model, the dynamic programming solution again reduces to one-dimensional integro-differential equations. More generally, the reduction holds for Lévy price processes but retains dependence on price-model characteristics such as volatility.
- Model assumptions: The risk-averse criterion uses exponential utility U(x) = −exp(−ηx) with η > 0 and assumes a Bachelier price process dPt = bdt + σdWt.The arithmetic Brownian-motion model may theoretically become negative, but is considered similar to geometric Brownian motion over short horizons.
- Reduced system: The dynamic programming solution reduces to a form involving functions φi that solve a system of one-dimensional integro-differential equations.The reduced system is supplemented by a terminal condition.
- Lévy extension: The same reduced form holds more generally when the mid-price follows a Lévy process.The reduction uses a generator property in which the price operator acts through a function ψ depending on inventory and Lévy characteristics.
- Strategy dependence: Optimal strategies remain independent of price levels and depend on inventory and spread, but they depend on the stock-price model, typically through volatility.This contrasts with the martingale mean-criterion case, where the policies do not depend on the martingale price model.
4 Computational results
The computational study evaluates the optimal strategy against alternative market-making policies using simulated backtests, examining performance, inventory control, execution balance, policy shape, and inventory-penalty effects.
- Shape of the optimal policy: The optimal policy depends on time, inventory, and spread, dividing decisions between limit-order making and market-order taking.The policy is represented in two zones, with the market-making zone characterized by price regimes.
- Shape of the optimal policy: Near the terminal date, the optimal policy becomes strongly time-dependent and unwinds inventory more aggressively to satisfy the terminal constraint.Near the beginning of the horizon, the policy is mainly time invariant under constant tick-time intensity.
- Benchmarked empirical performance analysis: The backtest compares the optimal strategy with WoMO, constant, and random strategies using simulated paths and empirical means and standard deviations.The simulations use an Euler scheme and evaluate quantities including terminal wealth and execution counts.
- Benchmarked empirical performance analysis: The optimal strategy significantly improves the information ratio relative to the benchmark, with the empirical terminal-wealth distribution providing confirmation.The reported comparisons include performance gains against constant and WoMO strategies.
- Benchmarked empirical performance analysis: 0.056 euros per trade is reported as surplus profit, mainly attributed to revenue from making the spread because execution counts are comparable with the constant strategy.This surplus is roughly twice the typical 0.03-euro clearing fee per execution cited for multilateral trading facilities.
- Benchmarked empirical performance analysis: The optimal strategy keeps maximum absolute inventory close to zero while allowing higher limit-order volume because market orders can immediately unwind positions.Its maximum absolute inventory is higher than WoMO but remains lower than the levels reached by constant and random strategies.
- Efficient frontier: The highest net information ratio occurs near γ ≃ 0.8, while increasing inventory penalization expands the market-trading zone and reduces inventory risk.The annualized net information ratio is reported as 47 by extrapolation, but the simulated backtest requires completion with real-data testing.