Optimal Liquidation of Child Limit Orders

The present paper studies the optimal placement problem of a child order. In practice, short-term momentum indicators inferred from order book data play important roles in order placement decisions. In the present work, we first propose to explicitly model the short-term momentum indicator and then formulate the order placement problem as an optimal multiple stopping problem. In contrast to the common formulation in the existing literature, we allow zero refracting period between consecutive stopping times to take into account the common practice of submitting multiple orders at the same time. This optimal multiple stopping problem will be explored over both infinite and finite horizons. It is shown that the optimal placement of a child order and the optimal replacement of outstanding limit orders by market ones are determined by first passage times of the short-term momentum indicator across a sequence of time-dependent boundaries. The aggressiveness of the optimal order replacement strategy is also examined via several numerical examples. In particular, our work illustrates that the optimal order replacement strategy is more aggressive when the bid–ask spread is smaller, when the impact from the momentum indicator is larger, or when the remaining time becomes shorter. All of these decision-making behaviors predicted by our model are natural and agree with empirical studies in the existing literature.