### 统计代写|风险建模代写Financial risk modeling代考|Market Liquidity, Stock Characteristics and Order Cancellations

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## 统计代写|风险建模代写Financial risk modeling代考|The Case of Fleeting Orders

Estimates vary by the trading platform, sample period and sample stocks, but the range of limit order “cancellations” documented in the literature is generally between one-tenth and two-thirds of all order submissions for US-based equities. 1 While this is a significant proportion of total orders submitted, the focus of both theoretical and empirical finance has been on limit order “executions.” In studies that do model limit order cancellations, all cancellations are usually treated as homogenous or are characterized by a homogeneous index, which is the same for all limit orders. Such characterization often leads to misspecified distributions relative to the empirical properties exhibited by order cancellations.
Empirical observation of limit order termination shows that most of the nonmarketable limit orders submitted to the order book end up being cancelled without execution and the majority of cancelled orders get cancelled within a very short time. Figure $2.1$ presents the Weibull probability plot for the survival to cancellation probability $S(t)$ as a function of limit order duration $t$ for ask limit orders for one stock, Comcast Corporation (ticker: CMCSA), submitted via the INET ECN during regular trading hours on September 20,2006 . The limit order durations are marked on the logarithmic scale on the horizontal axis, while the monotonic double-negative logarithmic transformations $\sim \ln (\sim \ln (S(t)))$ of the order survival-to-cancellation function are reported on the vertical axis. The six equidistant dashed vertical lines on the graph mark the duration times $(0.01,0.1,1,10,100$ and 1000 seconds) since the limit order submission. ${ }^{2}$ Between 50 and 90 percent of limit orders in each order

aggressiveness strata (where aggressiveness is measured by the tick distance from the best same-side quote) are cancelled within two seconds after the order submission.

The four equidistant dashed horizontal lines are drawn at the levels corresponding to the survival rates 99 percent, 90 percent, 35 percent and $0.0001$ percent. Those values are chosen so that the incremental distances between the horizontal lines on the doublenegative logarithmic scale $\ln (\ln (0.90) / \ln (0.99)) \approx \ln (\ln (0.35) / \ln (0.90))$ $\approx \ln (\ln (0.0001) / \ln (0.35)) \approx \ln (10)$ are approximately equal to the incremental distances $\ln (10)$ between the consecutive vertical lines on the log-duration scale. Note that if order cancellations in any order aggressiveness category occurred according to the Weibull distribution, which is a popular choice of the survival function literature, then the survivalto-cancellation function $S(t)=\exp (A t \beta)$ in the transformed scales would be plotted as a straight line with slope $\sim \beta$. While the constancy of the Weibull parameter $0<\beta<1$ can be accepted if our focus is exclusively on relatively large times to cancellation, approaching the fleeting orders with the constant Weibull parameter assumption is clearly inappropriate.
While this is an illustration for one stock-day, our empirical analysis finds that this pattern is robust. Yet, apart from a recent paper by Hasbrouck and Saar (2007), these fleeting orders have not been the focus of any study. There must be reasons why traders or algorithms submit limit orders and subsequently cancel their orders within such a short period of time.

## 统计代写|风险建模代写Financial risk modeling代考|Literature review

A trader submitting an order to a limit order platform can choose between placing a marketable limit order (which gets immediate execution at the best prevailing price) or a limit order that enters the book and awaits execution. Once the order is in the book, it “risks” termination either by execution or by cancellation. Liu (2009) analyses the relation between limit order submission risk and monitoring costs borne by the trader. Limit order traders face two types of risk – first, they may be picked off due to expected price changes and second, they face the possibility of nonexecution. To mitigate these risks traders monitor the market and cancel or revise their orders as needed. But monitoring is costly, resulting in a trade-off between the cost of monitoring and the risks of limit order submission. The theoretical model predicts that if the stock is actively traded, limit order submission risks and order cancellations/revisions are positively related. Stocks with wide bid-ask spreads have lower rates of order cancellations and large capitalization stocks have lower costs of gathering information (and hence more intense monitoring of limit orders) and therefore more order revisions and cancellations. However, the empirical evidence from our study suggests that this is not the case. Apart from stock characteristics, order characteristics are also related to the rates of cancellation. Menkhoff and Schmeling (2005) separate limit orders into those that are aggressively priced – “screen orders ${ }^{\prime \prime}$ – and ordinary ones that wait in the book. They find that screen orders have a much lower cancellation rate than ordinary limit orders. In a study that focuses on time to cancellation of limit orders, Eisler et al. (2007) show that to correctly model the empirical properties of a limit order book and price formation therein, it is essential to specify a correct functional form for the cancellation process. They find that the transaction time,

the first passage time, the time to (first) fill and time to cancel are best described as asymptotically power-law distributed. In contrast, Challet and Stinchcombe (2003) show that for four highly liquid stocks on the Island ECN’s order book, ${ }^{4}$ the distribution for cancelled orders seems to have an algebraic decay with particular numerical values of the exponent. ${ }^{5}$ Rosu (2009) proposes a continuous time model of price formation and shows that to generate results that are close to those observed in the actual data, incorporating order cancellations is important.

While all the above studies tangentially focus on order cancellation, to our knowledge the first paper to actually make the distinction between order cancellations at longer durations and “fleeting order” cancellations is Hasbrouck and Saar (2002). In that version of their paper they use order data from the Island ECN and find that close to 28 percent of all visible submitted orders are cancelled within two seconds. They coin the term “fleeting orders” to describe these very short-lived orders. Although it is the first to describe fleeting orders, the 2002 study does not focus on these orders (in fact, these are eliminated from the sample in their subsequent analysis).

## 统计代写|风险建模代写Financial risk modeling代考|Data sample characteristics

Our sample comprises of the Nasdaq 100 stocks and the sample period is July-December 2006. We use the first three months (63 trading days) for estimation purposes and the next three for out-of-sample robustness checks. Our data comes from several sources. We use limit order placement and cancellation data from the INET ECN. The [email protected] data feed from INET provides anonymous histories for all limit orders – when the orders were placed, modified, cancelled and executed – for all stocks that are traded on that platform. We use CRSP data for our firm-specific covariates and TAQ data from the NYSE to build the “National Best Bid and Offer” (NBBO) series of spreads and depths for our sample stocks. Additionally, we use earnings announcements data from I/B/E/S to partition our sample days into anticipated news days versus nonnews days in order to check the robustness of our estimates.

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