Optimal High Frequency Trading with Limit and Market Orders

Abstract

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 at best bid/ask quotes, and may also set limit orders at best bid (resp. ask) plus (resp. minus) a tick for getting the execution order priority, which is a crucial issue in high frequency trading. By trading with limit orders, the agent faces an execution risk since her orders are executed only when they meet counterpart market orders, which are modelled by Cox processes with intensities depending on the spread and on her limit prices. By holding non-zero positions on the risky asset, the agent is also subject to the inventory risk related to price volatility. Then the agent can also choose to trade with market orders, and therefore get immediate execution, but at a least favorable price because she has to cross the bid-ask spread.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.

Risk Metrics and Fine Tuning of High‐frequency Trading Strategies

“…For example, there are algorithms that are designed to find the best execution prices for investors who wish to minimize the price impact of large buy or sell orders, e.g., Bertsimas and Lo (1998), Almgren (2003), Kharroubi and Pham (2010), and Bayraktar and Ludkovski (2012). Moreover, there are HF strategies that specialize in arbitraging across different trading venues, and that seek to profit from market making and from short-term deviations in stock prices, e.g., Cartea and Jaimungal (2013), Guilbaud and Pham (2013), Cartea, Jaimungal and Ricci (2014), and Guéant, Lehalle and Fernandez Tapia (2011).…”

Optimal hedging through limit orders

“…A pretty recent literature investigates optimal trading strategies with limit orders or with the combined use of market and limit orders [1,5,12,13,15,18,19] and option market making [7,26] , but to our knowledge, a hedging problem with limit orders has not been studied so far. Although the above-cited models adopt several simplifying assumptions, they result in stochastic control problems, which are rarely analytically tractable.…”

Short‐Term Stock Price Prediction Based on Limit Order Book Dynamics

“…These traders arrive randomly, choose whether they want to execute immediately, determine how large their order sizes should be, and even cancel their orders at any time to exit the market strategically. Despite its complexity, models of order book dynamics are of great interest not only because they can be applied to optimize trade execution strategies (Alfonsi et al, 2010;Obizhaeva and Wang, 2006;Predoiu et al, 2011;Guilbaud and Pham, 2013;Goettler et al, 2005Goettler et al, , 2009), but also because these models provide insight into the relationship between supply and demand (Farmer et al, 2004;Foucault et al, 2005). Empirical studies indicate that the formation of short-term price behavior depends on the evolution of limit order books (Parlour, 1998;Harris and Panchapagesan, 2005;Rosu, 2009).…”