Narrow Results

We construct a model for liquidity risk and price impacts in a limit order book setting with depth, resilience and tightness. We derive a wealth equation and a characterization of illiquidity costs. We show that we can separate liquidity costs due to depth and resilience from those related to tightness, and obtain a reduced model in which proportional costs due to the bid-ask spread is removed. From this, we obtain conditions under which the model is arbitrage free. By considering the standard utility maximization problem, this also allows us to obtain a stochastic discount factor and an asset pricing formula which is consistent with empirical findings (e.g., Brennan and Subrahmanyam (1996); Amihud and Mendelson (1986)). Furthermore, we show that in limiting cases for some parameters of the model, we derive many existing liquidity models present in the arbitrage pricing literature, including Çetin et al. (2004) and Rogers and Singh (2010). This offers a classification of different types of liquidity costs in terms of the depth and resilience of prices.

We investigate whether the bid/ask queue imbalance in a limit order book (LOB) provides significant predictive power for the direction of the next mid-price movement. For each of 10 liquid stocks on Nasdaq, we fit logistic regressions between the queue imbalance and the direction of the subsequent mid-price movement, and we find a strongly statistically significant relationship in each case. Compared to a simple null model, we find that our logistic regression fits provide a considerable improvement in both binary and probabilistic classification of mid-price movements for large-tick stocks and a moderate improvement in both binary and probabilistic classification of mid-price movements for small-tick stocks. We also perform local logistic regression fits on the same data, and find that this semi-parametric approach slightly outperforms our logistic regression fits, at the expense of being more computationally intensive to implement.

One popular approach to model the limit order books dynamics of the best bid and ask at level-1 is to use the reduced-form diffusion approximations. It is well known that the biggest contributing factor to the price movement is the imbalance of the best bid and ask. We investigate the data of the level-1 limit order books of a basket of stocks and study the numerical evidence of drift, correlation, volatility and their dependence on the imbalance. Based on the numerical discoveries, we develop a non-parametric discrete model for the dynamics of the best bid and ask, which can be approximated by a reduced-form model that incorporates the empirical data of correlation and volatilities with analytical tractability that can fit the empirical data of the probability of price movement.

We propose a limit order book (LOB) model with dynamics that account for both the impact of the most recent order and volume imbalance. To model these effects jointly we introduce a discrete Markov chain model. We then find the policy for optimal order choice and control. The optimal policy derived uses limit orders, cancellations and market orders. It looks to avoid non-execution and adverse selection risk simultaneously. Using ultra high-frequency data from the NASDAQ stock exchange we compare our policy with other submission strategies that use a subset of all available order types and show that ours significantly outperforms.

We investigate the behavior of limit order books (LOBs) on the meso-scale motivated by order execution scheduling algorithms. To do so, we carry out empirical analysis of the order flows from market and limit order submissions, aggregated from tick-by-tick data via volume-based bucketing, as well as various LOB depth and shape metrics. We document a nonlinear relationship between trade imbalance and price change, which however can be converted into a linear link by considering a weighted average of market and limit order flows. We also document a hockey-stick dependence between trade imbalance and one-sided limit order flows, highlighting numerous asymmetric effects between the active and passive sides of the LOB. To address the phenomenological features of price formation, we construct regression models to identify the most significant predictors, confirming the predictive power of limit order flows. Another finding is that the deeper LOB shape, rather than just the book imbalance, is more relevant on this timescale. The empirical results are based on analysis of six large-tick assets from Nasdaq.

Market making is one of the most important aspects of algorithmic trading, and it has been studied quite extensively from a theoretical point of view. The practical implementation of so-called “optimal strategies” however suffers from the failure of most order-book models to faithfully reproduce the behavior of real market participants.

This paper is two-fold. First, some important statistical properties of order-driven markets are identified, advocating against the use of purely Markovian order-book models. Then, market making strategies are designed and their performances are compared, based on simulation as well as backtesting. We find that incorporating some simple non-Markovian features in the limit order book greatly improves the performances of market making strategies in a realistic context.