  Since they were authorized by the U.S. Security and Exchange Commission in
1998, electronic exchanges have boomed, and by 2010 high frequency trading
accounted for over 70% of equity trades in the US. Such markets are thought to
increase liquidity because of the presence of market makers, who are willing to
trade as counterparties at any time, in exchange for a fee, the bid-ask spread.
In this paper, we propose an equilibrium model showing how such market makers
provide liquidity. The model relies on a codebook for client trades, the
implied alpha. After solving the individual clients optimization problems and
identifying their implied alphas, we frame the market maker stochastic
optimization problem as a stochastic control problem with an infinite
dimensional control variable. Assuming either identical time horizons for all
the clients, or a stochastic partial differential equation model for their
beliefs, we solve the market maker problem and derive tractable formulas for
the optimal strategy and the resulting limit-order book dynamics.
