Execution Times of Small Limit Orders: A Simpler Modeling Approach

Abstract

In an electronic stock market, an equity trader can submit two kinds of orders: a market order or a limit order. In a market order, the trade occurs at the best available price on the opposite side of the book. In a limit order, on the other hand, the trader specifies a price (lower limit in case of sell orders and higher limit in case of buy orders) beyond which they are not willing to transact. Limit orders supply liquidity to the market and aid in price discovery since they indicate the prices that traders are willing to pay at any point of time. One of the risks that a trader placing a limit order faces is the risk of delayed execution or non-execution. If the execution is delayed, then the trader also faces a “picking-off” risk, in the event of the arrival of new information. With these issues in the background, a trader placing a limit order at a certain price, given various economic variables such as recent price movements as well as characteristics of the company in question, is interested in the probability of execution of the order as a function of subsequent elapsed time. For example, if she places a small sell order at 0.5 percent above the last traded price for a given stock, what is the probability that the order will be executed in the next t minutes? With this motivation, this paper considers execution times of small limit orders in an electronic exchange, specifically the National Stock Exchange (NSE) of India. Order execution times have been studied in several other works, where they are modeled by reconstructing the history of the order book using high-frequency data. Here, for the first time, the much simpler approach of small hypothetical orders placed at certain prices at certain points of time has been used. Given that an order has been placed at a certain price, subsequent price movements determine the lower and upper bounds of the time to execution based on when (and if) the order price is first reached and when it is first crossed. Survival analysis with interval censoring is used to model the execution probability of an order as a function of time. Several Accelerated Failure Time models are built with historical trades and order book data for 50 stocks over 63 trading days. Additionally, choice of distributions, relative importance of covariates, and model reduction are discussed; and results qualitatively consistent with studies that did not use hypothetical orders are obtained. Interestingly, for the data, the differences between the above-mentioned bounds are not very large. Directly using them without interval censoring gives survival curves that bracket the correct curve obtained with interval censoring. The paper concludes that this approach, though data- and computation-wise much less intensive than traditional approaches, nevertheless yields useful insights on execution probabilities of small limit orders in electronic exchanges.