Front-running is trading stock by a broker who has inside knowledge a future client transaction is about to affect its price. It is called front because the broker trades before clients. It is illegal, and the profits of the broker come from losses of clients.
What if the broker trade after the client’s order? If the broker has the information that the client trade contains no information. The broker can make use of this information and earn profits by behind-running.
After a trade, there are permanent effects and temporary effects. The temporary effects should decay with time, and the price moves back to the equilibrium price. The broker can bet the mean reversion of temporary trading impact.
Let us start with a simple market-making model. There are two kinds of traders in the market, informed traders and uninformed traders. A market maker has no private information but uses Bayesian Learning to infer information from order flow. When the market maker receives single side orders, she will adjust her quotes to express the new information and worry about her inventory risk. Let us assume we are a broker, and we help our client, a big buy-side firm, to sell 1M share SPY. The big buy-side mutual fund usually has no private information, according to our historical experience. We trade SPY with the market maker. Market maker suspects this one-side order flow contains information and reduce her ask according to order flow information and inventory risk. After finishing trading clients’ orders, we can purchase stocks from market makers using our own account (double trading) and bet the price would return to its equilibrium price. After a long time, the market maker finds order flow balanced and no news, she realized the previous large trading was uninformed liquid trading. Then the market maker makes her quotes back to the equilibrium prices. Because the big buy-side firm has no private information, the permanent impact is zero. We then sell our stock and make a profit by this mean reversion effect of trading temporary impact.
In conclusion, “knowing some trades are uninformative” is itself informative. A broker can make use of this information and bet the mean reversion of temporary trading impact. This strategy makes the market more efficient by making the price return to equilibrium faster, which justifies the broker’s profit.