Brokers and Informed Traders : Dealing With Toxic Flow and Extracting Trading Signals

We derive closed-form strategies for a broker who provides liquidity to an informed trader and to a noise trader over a finite-time and infinite-time trading horizon. The flow of the noise trader is uninformative and the broker trades with the noise trader at a profit, on average. On the other hand, the informed trader has privileged information about the trend in the price of the asset, so the broker trades with the informed trader at a loss, on average. These losses are payment for toxic flow from which the broker extracts the trend signal. The signal is one of the key ingredients in the broker's trading strategy to internalise (i.e., how much of the flow she keeps in her books), to externalise (i.e., how much she unwinds in a lit exchange), and to speculate in the lit market. The broker's dynamic strategy is a linear combination of four variables: the broker's inventory, the informed trader's inventory, the trend signal, and the uninformed trader's rate of trading with the broker. When the trading horizon is infinite (resp. finite), the coefficients of the four terms are constants (resp. deterministic functions of time). Finally, in the infinite horizon case, we show how the broker uses the flow of her clients to estimate the constant coefficients of the optimal strategy