Optimal Execution for High Frequency Trading

In high-frequency trading, large buy (or sell) orders may cause other traders to raise (or lower) their offered price. This additional implementation cost due to short-term liquidity demand is often called market impact.

Practitioners commonly view this as a significant cost to track, avoid, or minimize. This can be done by slowly trading the asset, which usually reduces market impact by minimally affecting demand and supply over time. However, this exposes the trader to stock price fluctuations and risks falling behind a trading schedule. Hence, many trading algorithms seek to strategically trade off costs and risks in trade execution.

We analyze a continuous-time stochastic model for optimal execution using both market and limit orders. Our objective is to investigate how to optimally allocate aggressive (market) and passive (limit) orders over time to achieve trading goals. The aggressive nature of market orders tends to result in a higher market impact, whereas limit orders have less market impact but are less likely to fill. We model this fill uncertainty using an affine function of the trading rate in limit orders; this leads to an additional diffusion term that is correlated with the stock price process.

The trader’s objective is to maximize the expectation of the compensated P&L by choosing trading rates for market and limit orders. This amounts to solving a constrained stochastic control problem. This leads us to study the associated nonlinear Hamilton-Jacobi-Bellman (HJB) partial differential equation (PDE) problem. An analytic solution to the stochastic control problem is derived in the paper.

We explore the special cases of constant and linear uncertainty to examine the properties of the solutions and analyze the corresponding explicit optimal strategies.