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Register a new userThe order book as a queueing system: average depth and influence of the size of limit orders
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Authors
Ioane Muni Toke
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We study the analytical properties of a one-side order book model in which the flows of limit and market orders are Poisson processes and the distribution of lifetimes of cancelled orders is exponential. Although simplistic, the model provides an analytical tractability that should not be overlooked. Using basic results for birth-and-death processes, we build an analytical formula for the shape (depth) of a continuous order book model which is both founded by market mechanisms and very close to empirically tested formulas. We relate this shape to the probability of execution of a limit order, highlighting a law of conservation of the flows of orders in an order book. We then extend our model by allowing random sizes of limit orders, hereby allowing to study the relationship between the size of the incoming limit orders and the shape of the order book. Our theoretical model shows that, for a given total volume of incoming limit orders, the less limit orders are submitted (i.e. the larger the average size of these limit orders), the deeper is the order book around the spread. This theoretical relationship is finally empirically tested on several stocks traded on the Paris stock exchange.
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We investigate the statistical properties of the EBS order book for the EUR/USD and USD/JPY currency pairs and the impact of a ten-fold tick size reduction on its dynamics. A large fraction of limit orders are still placed right at or halfway between the old allowed prices. This generates price barriers where the best quotes lie for much of the time, which causes the emergence of distinct peaks in the average shape of the book at round distances. Furthermore, we argue that this clustering is mainly due to manual traders who remained set to the old price resolution. Automatic traders easily take price priority by submitting limit orders one tick ahead of clusters, as shown by the prominence of buy (sell) limit orders posted with rightmost digit one (nine).
In order-driven markets, limit-order book (LOB) resiliency is an important microscopic indicator of market quality when the order book is hit by a liquidity shock and plays an essential role in the design of optimal submission strategies of large orders. However, the evolutionary behavior of LOB resilience around liquidity shocks is not well understood empirically. Using order flow data sets of Chinese stocks, we quantify and compare the LOB dynamics characterized by the bid-ask spread, the LOB depth and the order intensity surrounding effective market orders with different aggressiveness. We find that traders are more likely to submit effective market orders when the spreads are relatively low, the same-side depth is high, and the opposite-side depth is low. Such phenomenon is especially significant when the initial spread is 1 tick. Although the resiliency patterns show obvious diversity after different types of market orders, the spread and depth can return to the sample average within 20 best limit updates. The price resiliency behavior is dominant after aggressive market orders, while the price continuation behavior is dominant after less-aggressive market orders. Moreover, the effective market orders produce asymmetrical stimulus to limit orders when the initial spreads equal to 1 tick. Under this case, effective buy market orders attract more buy limit orders and effective sell market orders attract more sell limit orders. The resiliency behavior of spread and depth is linked to limit order intensity.
Attempts to accurately measure the monetary velocity or related properties of bitcoin used in transactions have often attempted to either directly apply definitions from traditional macroeconomic theory or to use specialized metrics relative to the properties of the Blockchain like bitcoin days destroyed. In this paper, it is demonstrated that beyond being a useful metric, bitcoin days destroyed has mathematical properties that allow you to calculate the average dormancy (time since last use in a transaction) of the bitcoins used in transactions over a given time period. In addition, bitcoin days destroyed is shown to have another unexpected significance as the average size of the pool of traded bitcoins by virtue of the expression Littles Law, though only under limited conditions.
We show that the statistics of spreads in real order books is characterized by an intrinsic asymmetry due to discreteness effects for even or odd values of the spread. An analysis of data from the NYSE order book points out that traders strategies contribute to this asymmetry. We also investigate this phenomenon in the framework of a microscopic model and, by introducing a non-uniform deposition mechanism for limit orders, we are able to quantitatively reproduce the asymmetry found in the experimental data. Simulations of our model also show a realistic dynamics with a sort of intermittent behavior characterized by long periods in which the order book is compact and liquid interrupted by volatile configurations. The order placement strategies produce a non-trivial behavior of the spread relaxation dynamics which is similar to the one observed in real markets.
We examine the dynamics of the bid and ask queues of a limit order book and their relationship with the intensity of trade arrivals. In particular, we study the probability of price movements and trade arrivals as a function of the quote imbalance at the top of the limit order book. We propose a stochastic model in an attempt to capture the joint dynamics of the top of the book queues and the trading process, and describe a semi-analytic approach to calculate the relative probability of market events. We calibrate the model using historical market data and discuss the quality of fit and practical applications of the results.
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