Date of Award

8-2018

Degree Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Statistics

Committee Chair

Jos´e E. Figueroa-L´opez

Committee Co-Chair

Raghu Pasupathy

Committee Member 1

Kiseop Lee

Committee Member 2

Michael Levine

Abstract

A Limit Order Book (LOB), a trading system used by most of the electronic financial trading exchanges across the world, collects all the buy and sell limit orders: orders that specify a desired price and quantity. Limit orders are executed by matching them to the market orders: orders to buy or sell certain quantities or shares at the best available prices out of the standing limit orders in the book. In an order-driven market using the LOB system, one of the stock trader’s major concerns is to clear his/her inventory, by using a combination of limit orders and market orders. In addition, most of the intraday traders have their own pre-determined time horizon for clearing their inventories. This time horizon may vary from a few milliseconds (for high frequency trades) to a few hours.

In this dissertation, we study the optimal placement problem, which consists of deciding whether to use a market or a limit buy order under a time-horizon constraint and determining an optimal price level to place the limit order for the latter case. While this dissertation focuses on the case when the investor is willing to place a buy order, this result, of course, can be applied to the case of sell orders.

We propose a trading strategy and resulting investor’s cost for a discrete LOB model and provide counterparts for a diffusive LOB where the average time-step between price changes and the tick-size are small enough so that the price process can be approximated well by a continuous time diffusive process. In the resulting investor’s cost, the initial LOB status and the order flow of orders in the book are taken into consideration.

We characterize the optimal limit order placement policy using both a Bachelier model and a Black-Scholes model as the diffusive approximating price process and analyze its behavior under different market conditions, including a small fee/rebate regime, which is plausible in practice. The Bachelier model and Black-Scholes model are wide-spread used in the mathematical finance field to describe the asset price movement.

One of our main results is that, for negative drift price processes, there exists a critical time t0 such that, for any time horizon longer than this critical time, there exists an optimal placement that is non-trivial in that it is different from one that is placed infinitesimally close to the best ask, such as the best bid and the second best bid. We investigate the assumptions regarding the behavior of a LOB which guarantees the existence of this non-trivial optimal placement.

The asymptotic behavior of t0 is investigated for both price process models and a simple method to approximate t0 is provided. Furthermore, the approximation of the optimal placement and the asymptotic of the optimal placement for the large time horizon are found in closed forms. Numerical and empirical analysis of the optimal placement using real LOB data validate the plausibility of our assumptions for the existence of the optimal placement strategy described above and show the performance of the proposed approach.

Finally, a different optimal placement problem is considered. While the trading strategy described above does not allow the cancellation and re-posting of the existing limit order, in the new approach we allow the investor to adjust the price level of a limit order before the investor’s time horizon. We investigate the behavior of optimal placement under this new strategy.

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