Econometric Models of Limit-Order Executions

Motivation(s)

The limit order is one of the most important tool for trading equity securities: it accounts for 45% of total NYSE orders. Its effect on the market, market orders, and market makers have been thoroughly studied in the past decade. However, there have been few studies on the limit-order execution times. The existing models impose strong assumptions such as knowing the entire limit-order book or restricting the market to purely limit-orders.

Proposed Solution(s)

The authors propose an econometric model of limit-order execution times. Since limit-order execution times are nonnegative, random, and temporally ordered, they can be interpreted as failure times, which makes them amenable to survival analysis (e.g. Kaplan-Meier estimator). The authors estimate the conditional distribution of limit-order execution times as a function of economic variables such as the limit price, order size, current market conditions, and censored observations i.e. limit orders that expire or are canceled before they are executed. The incorporation of censored observations is one reason why survival analysis have less estimator bias than the existing models that ignore it.

Evaluation(s)

The authors analyzed the limit orders of the top 100 S&P 500 stocks between 1994 and 1995 from ITG dataset. Every action is timestamped and recorded, but there were few retail orders since the clients are almost exclusively institutional investors and other brokers/dealers.

They assume a parametric generalized gamma distribution for survival analysis and estimate separate models for time-to-first-fill and time-to-completion for both buy and sell limit orders. Four separate models are needed because a limit order can require multiple fills, and market conditions can affect execution times differently for buy limit orders and sell limit orders. To simplify the analysis, the models are conditioned on the explanatory variables. However, the resulting distributions closely match the ITG data and passes several diagnostic goodness-of-fit tests. The parameter estimates show that execution times can be quite sensitive to certain explanatory variables such as market depth, the spread between the limit price and the quote midpoint, and market volatility.

The authors evaluate their econometric model against hypothetical limit-order executions. The execution time of a limit order can be modeled as a first-passage time (FPT) i.e. the first time the transaction price reaches or crosses the limit price. This model assumes the stock price adheres to a geometric Brownian motion with drift. The empirical analysis demonstrates that the FPT model vastly underestimates limit-order execution times because

  • it disallows explicit roles for price discreteness,

  • it does not accommodate the impact of limit-order size,

  • it cannot accommodate time priority,

  • it makes no distinction between time-to-first-fill, time-to-completion, and time-to-censoring, and

  • it cannot easily incorporate the effects of explanatory variables such as price volatility, spreads, and market conditions.

Future Direction(s)

  • To what extend does the accuracy of limit-order execution times affect dynamic order-submission strategies?

  • How would the proposed model handle a stream of delayed data?

Question(s)

  • How did the authors decide on these explanatory variables?

Analysis

The behavior of limit-order execution times can be quantified by the proposed econometric model based on survival analysis and actual limit-order data. Even though the generalized gamma model with an accelerated failure time specification fits the data nicely, it should be noted that execution times are sensitive to some explanatory variables (e.g. limit price) but not to others (e.g. limit shares).

While statistical significance is a reasonable way to measure the accuracy of a model, the end goal is to improve the return on investment. The paper did not illustrate how the extra accuracy would increase returns. Furthermore, a live limit-order book costs money, so it is unclear whether the extra expense is worthwhile.

Notes

Limit-order Data

  • The NYSE order-handling rules dictate that time priority only applies to the first order at a given price; subsequent orders have equal time priority.

  • An order’s execution involves several partial fills before it is completed, but partial fills do not change the time priority.

  • Day-order and good-till-canceled order respectively comprise 82% and 17% of the market limit orders.

References

LMZ02

Andrew W Lo, A Craig MacKinlay, and June Zhang. Econometric models of limit-order executions. Journal of Financial Economics, 65(1):31–71, 2002.