Optimal Order Routing in a Fragmented Market

In modern equity markets, participants have a choice of many exchanges at which to trade. Exchanges typically operate as electronic limit order books operating under the “price-time” priority rule. Taking into account the effect of investors’ order routing decisions across exchanges, we find that the equilibrium of this decentralized market exhibits a certain state space collapse property, whereby (a) the states at different exchanges are coupled in a fairly intuitive manner, (b) the behavior of the market is captured through a one-dimensional process that can be viewed as a weighted aggregate depth of the market at the best bid and offer across all exchanges, and (c) the behavior of the various exchanges is inferred through a set of simple mappings from that of the aggregated market depth process. This predicted dimension reduction is the result of high-frequency order routing decisions that essentially couple the dynamics across exchanges. We derive a characterization of the market equilibrium and the associated aggregated depth process. Analyzing a TAQ dataset for a sample of stocks over a one month period, we find strong support for the predicted state space collapse property.