  We develop a new market-making model, from the ground up, which is tailored
towards high-frequency trading under a limit order book (LOB), based on the
well-known classification of order types in market microstructure. Our flexible
framework allows arbitrary order volume, price jump, and bid-ask spread
distributions as well as the use of market orders. It also honors the
consistency of price movements upon arrivals of different order types. For
example, it is apparent that prices should never go down on buy market orders.
In addition, it respects the price-time priority of LOB. In contrast to the
approach of regular control on diffusion as in the classical Avellaneda and
Stoikov [1] market-making framework, we exploit the techniques of optimal
switching and impulse control on marked point processes, which have proven to
be very effective in modeling the order-book features. The
Hamilton-Jacobi-Bellman quasi-variational inequality (HJBQVI) associated with
the control problem can be solved numerically via finite-difference method. We
illustrate our optimal trading strategy with a full numerical analysis,
calibrated to the order-book statistics of a popular Exchanged-Traded Fund
(ETF). Our simulation shows that the profit of market-making can be severely
overstated under LOBs with inconsistent price movements.
