  A novel high-frequency market-making approach in discrete time is proposed
that admits closed-form solutions. By taking advantage of demand functions that
are linear in the quoted bid and ask spreads with random coefficients, we model
the variability of the partial filling of limit orders posted in a limit order
book (LOB). As a result, we uncover new patterns as to how the demand's
randomness affects the optimal placement strategy. We also allow the price
process to follow general dynamics without any Brownian or martingale
assumption as is commonly adopted in the literature. The most important feature
of our optimal placement strategy is that it can react or adapt to the behavior
of market orders online. Using LOB data, we train our model and reproduce the
anticipated final profit and loss of the optimal strategy on a given testing
date using the actual flow of orders in the LOB. Our adaptive optimal
strategies outperform the non-adaptive strategy and those that quote limit
orders at a fixed distance from the midprice.
