Limit order books and trade informativeness

In the microstructure literature, information asymmetry is an important determinant of market liquidity. The classic setting is that uninformed dedicated liquidity suppliers charge price concessions when incoming market orders are likely to be informationally motivated. In limit order book (LOB) markets, however, this relationship is less clear, as market participants can switch roles, and freely choose to immediately demand or patiently supply liquidity by submitting either market or limit orders. We study the importance of information asymmetry in LOBs based on a recent sample of 30 German Deutscher Aktienindex (DAX) stocks. We find that Hasbrouck's (1991) measure of trade informativeness Granger causes book liquidity, in particular that required to fill large market orders. Picking-off risk due to public news-induced volatility is more important for top-of-the book liquidity supply. In our multivariate analysis, we control for volatility, trading volume, trading intensity and order imbalance to isolate the effect of trade informativeness on book liquidity.     

Liquidity,volume and price efficiency: The impact of order vs. quote driven trading

As equity trading becomes predominantly electronic, is there still value to a traditional, intermediated dealer system? We address this question by comparing the impact of the organization of trading on volume, liquidity, and price efficiency in a quote-driven dealer market and in an order-driven limit order book. Small order price impacts are higher and large order price impacts are lower in a dealer market. Prices are more efficient in the limit order book, except when the level of informed trading is high. Volume is higher in a limit order market, making this system most attractive for trading venues.     

Algorithmic trading in a microstructural limit order book model

We propose a microstructural modeling framework for studying optimal market-making policies in a FIFO (first in first out) limit order book (order book). In this context, the limit orders, market orders, and cancel orders arrivals in the order book are modeled as point processes with intensities that only depend on the state of the order book. These are high-dimensional models which are realistic from a micro-structure point of view and have been recently developed in the literature. In this context, we consider a market maker who stands ready to buy and sell stock on a regular and continuous basis at a publicly quoted price, and identifies the strategies that maximize their P&L penalized by their inventory. An extension of the methodology is proposed to solve market-making problems where the orders arrivals are modeled using Hawkes processes with exponential kernel.

We apply the theory of Markov Decision Processes and dynamic programming method to characterize analytically the solutions to our optimal market-making problem. The second part of the paper deals with the numerical aspect of the high-dimensional trading problem. We use a control randomization method combined with quantization method to compute the optimal strategies. Several computational tests are performed on simulated data to illustrate the efficiency of the computed optimal strategy. In particular, we simulated an order book with constant/ symmetric/ asymmetrical/ state dependent intensities, and compared the computed optimal strategy with naive strategies. Some codes are available on https://github.com/comeh.     

Can order exposure be mandated?

In this paper, we examine whether the hidden portion of limit orders represents depth that would be revealed if traders were not allowed to hide it, and the associated market quality implications. Specifically, we examine the decisions by the Toronto Stock Exchange to first abolish the use of hidden limit orders in 1996, and then reintroduce them in 2002. We find that quoted depth does not change following either decision, suggesting that the hidden portion of orders represents depth that would otherwise not be exposed. Using confidential order data for the period following the reintroduction of hidden limit orders, we find that total inside depth increases. For both events, volume does not change and the usage of the limit order book increases if hidden limit orders are allowed. This suggests that if traders are required to expose their orders they will not exit the market, but instead will switch to using market orders. We also find evidence to suggest that informed traders use hidden limit orders to minimize price impact if the probability of non-execution is small.     

Optimal order placement in limit order markets

To execute a trade, participants in electronic equity markets may choose to submit limit orders or market orders across various exchanges where a stock is traded. This decision is influenced by characteristics of the order flows and queue sizes in each limit order book, as well as the structure of transaction fees and rebates across exchanges. We propose a quantitative framework for studying this order placement problem by formulating it as a convex optimization problem. This formulation allows the study of how the optimal order placement decision depends on the interplay between the state of order books, the fee structure, order flow properties and the aversion to execution risk. In the case of a single exchange, we derive an explicit solution for the optimal split between limit and market orders. For the general case of order placement across multiple exchanges, we propose a stochastic algorithm that computes the optimal routing policy and study the sensitivity of the solution to various parameters. Our algorithm does not require an explicit statistical model of order flow but exploits data on recent order fills across exchanges in the numerical implementation of the algorithm to acquire this information through a supervised learning procedure.     

Approximating functionals of local martingales under lack of uniqueness of the Black–Scholes PDE solution

When the underlying stock price is a strict local martingale process under an equivalent local martingale measure, the Black–Scholes PDE associated with a European option may have multiple solutions. In this paper, we study an approximation for the smallest hedging price of such an European option. Our results show that a class of rebate barrier options can be used for this approximation. Among them, a specific rebate option is also provided with a continuous rebate function, which corresponds to the unique classical solution of the associated parabolic PDE. Such a construction makes existing numerical PDE techniques applicable for its computation. An asymptotic convergence rate is also studied when the knock-out barrier moves to infinity under suitable conditions.     

Market impact as anticipation of the order flow imbalance

In this paper, we assume that the permanent market impact of metaorders is linear and that the price is a martingale. Those two hypotheses enable us to derive the evolution of the price from the dynamics of the flow of market orders. For example, if the market order flow is assumed to follow a nearly unstable Hawkes process, we retrieve the apparent long memory of the flow together with a power law impact function, which is consistent with the celebrated square root law. We also link the long memory exponent of the sign of market orders with the impact function exponent. One of the originalities of our approach is that our results are derived without assuming that market participants are able to detect the beginning of metaorders.     

Liquidity distribution in the limit order book on the stock exchange of Thailand

The liquidity distribution, or the shape of the limit order book, influences trading behavior and choice of order submission by public liquidity suppliers. The present study seeks to discover whether liquidity providers are concerned about being picked off by informed traders, and whether they are less willing to supply liquidity at the market or demand higher price spreads. The results show that liquidity at the market is a small portion of total liquidity, and that firm size, minimum tick size, volatility, and trading volume play significant roles in determining the liquidity distribution within an order book.     

Liquidity,information, strategic trading in an electronic order book: New insights from the European carbon markets

The electronic limit order book (LOB hereafter) has rapidly become the primary way of trading European carbon assets over the 4 years of the EU ETS programme (2008–2012). In this first attempt of examining the informational content of an electronic order book, we evidence that order flow imbalances have a moderate capacity to predict short term price changes. However, we find that both LOB slope and immediacy costs help to forecast quote improvements and volatility in the next 30 min. Further, we explain why informed trading is highly influential and show that it consists in mixing order splitting strategies and posting fleeting orders once the asymmetric information is reduced (Rosu, 2009). Overall, the consolidated status of the order book mirrors a high level of market uncertainty and a low degree of informational efficiency. In this way, strategic trading can in itself explain some of order book properties, independently of the degree of traders’ sophistication and market competition.     

Order book characteristics and the volume–volatility relation: Empirical evidence from a limit order market

Using unique data, we address the issue of price formation in a limit order market. A standard volume–volatility relation is documented with the number of trades acting as the important component of volume. The main contribution of the paper is to identify strong evidence that volume, volatility, and the volume–volatility relation are negatively related to the order book slope. These results are robust to the inclusion of several liquidity measures. A significant empirical relationship between the order book slope and the coefficient of variation in earnings forecasts by financial analysts suggests that the slope is proxying for disagreement among investors. Hence, our results support models where investor heterogeneity intensifies the volume–volatility relation.     

基于异质信号的资产价格学习机制及均衡演化路径

本文从金融市场指令交易机制出发,将市场投资主体分为市场追随者和自我相信者两类,分析了两类投资者面对由异质分析资产价值信号形成的市场价格的学习机制,且对交易量变化以及均衡价格的达成路径进行了研究。研究表明,单期价格均衡下市场仍存在资产交易,且投资者随着均衡价格期数增加和价格精度增大逐渐撤离市场,较大的市场追随者比重将滞后混合策略市场的价格均衡期数。     

Dual pricing of multi-exercise options under volume constraints

In this paper, we study the pricing problem of multi-exercise options under volume constraints. The volume constraint is modelled by an adapted process with values in the positive integers, which describes the maximal number of rights to be exercised at a given time. We derive a representation of the marginal value of an additional nth right as a standard single stopping problem with a modified cash-flow process. This representation then leads to a dual pricing formula, which generalizes a result by Meinshausen and Hambly (Math. Finance 14:557–583, 2004) from the standard multi-exercise option (with at most one right per time step) to general constraints. We also state an explicit Monte Carlo algorithm for computing confidence intervals for the price of multi-exercise options under volume constraints and present numerical results for the pricing of a swing contract in an electricity market.     

Modelling the buy and sell intensity in a limit order book market

In this paper, we model the buy and sell arrival process in the limit order book market at the Australian Stock Exchange. Using a bivariate autoregressive intensity model we analyze the contemporaneous buy and sell intensity as a function of the state of the market. We find evidence that trading decisions are both information as well as liquidity driven. Confirming predictions from market microstructure theory traders submit market orders by inferring from the recent order flow and the book with respect to upper and lower tail expectations as well as trading directions. However, traders also tend to take liquidity when the liquidity supply is high. Moreover, we find evidence that traders pay more attention to recent order arrivals and the current state of the order book than to the past order flow.     

Second order multiscale stochastic volatility asymptotics: stochastic terminal layer analysis and calibration

Multiscale stochastic volatility models have been developed as an efficient way to capture the principal effects on derivative pricing and portfolio optimization of randomly varying volatility. The recent book by Fouque et al. (Multiscale Stochastic Volatility for Equity, Interest-Rate and Credit Derivatives, 2011) analyzes models in which the volatility of the underlying is driven by two diffusions – one fast mean-reverting and one slowly varying – and provides a first order approximation for European option prices and for the implied volatility surface, which is calibrated to market data. Here, we present the full second order asymptotics, which are considerably more complicated due to a terminal layer near the option expiration time. We find that to second order, the implied volatility approximation depends quadratically on log-moneyness, capturing the convexity of the implied volatility curve seen in data. We introduce a new probabilistic approach to the terminal layer analysis needed for the derivation of the second order singular perturbation term, and calibrate to S&P 500 options data.     

Implied stopping rules for American basket options from Markovian projection

This work addresses the problem of pricing American basket options in a multivariate setting, which includes among others, the Bachelier and Black–Scholes models. In high dimensions, nonlinear PDE methods for solving the problem become prohibitively costly due to the curse of dimensionality. Instead, this work proposes to use a stopping rule that depends on the dynamics of a low-dimensional Markovian projection of the given basket of assets. From a numerical analysis point of view, we split the given non-smooth high-dimensional problem into two subproblems, namely one dealing with a smooth high-dimensionality integration in the parameter space and the other dealing with a low-dimensional, non-smooth optimal stopping problem in the projected state space. Assuming that we know the density of the forward process and using the Laplace approximation, we first efficiently evaluate the diffusion coefficient corresponding to the low-dimensional Markovian projection of the basket. Then, we approximate the optimal early exercise boundary of the option by solving an HJB PDE in the projected, low-dimensional space. The resulting near-optimal early exercise boundary is used to produce an exercise strategy for the high-dimensional option, thereby providing a lower bound for the price of the American basket option. A corresponding upper bound is also provided. These bounds allow one to assess the accuracy of the proposed pricing method. Indeed, our approximate early exercise strategy provides a straightforward lower bound for the American basket option price. Following a duality argument due to Rogers, we derive a corresponding upper bound solving only the low-dimensional optimal control problem. Numerically, we show the feasibility of the method using baskets with dimensions up to 50. In these examples, the resulting option price relative errors are only of the order of few percent.     

Multiple-limit trades: empirical facts and application to lead–lag measures

Order splitting is a standard practice in trading: traders constantly scan the limit order book and choose to limit the size of their market orders to the quantity available at the best limit, thereby controlling the market impact of their orders. In this article, we focus on the other trades, multiple-limit trades that go through the best available price in the order book, or ‘trade-throughs’. We provide various statistics on trade-throughs: frequency, volume, intraday distribution, market impact, etc., and present a new method for the measurement of lead–lag parameters between assets, sectors or markets.     

Price limit performance: evidence from transactions data and the limit order book

In recent years, organized stock exchanges with daily price limits adopted wider limits as narrower limits were criticized for jeopardizing market efficiency. This study examines the impact of a wide price limit on price discovery processes, using data from the Kuala Lumpur Stock Exchange. Specifically, examined is the impact of daily price limits on (i) information asymmetry; (ii) arrival rates of informed traders; and (iii) order imbalance. Using both trade-to-trade transaction data and the limit order book, we compile evidence that price limits do not improve information asymmetry, delays the arrival of informed traders, and exacerbates order imbalance. These results suggest that price limits on individual securities do not improve price discovery processes but impose serious costs even when the limit band is as wide as 30%.     

Specialist participation and limit orders

We present a market microstructure model to examine specialist's strategic participation decisions in a security market where there are noise traders, limit order traders, an insider and a specialist. We argue that the specialist's participation rate depends on the depth of the limit book and its uncertainty. In particular, the specialist has incentives to trade against the market trend when the limit book depth is low and to trade with the market trend when the depth is high. Moreover, the specialist's participation rate is positively related to the limit book depth uncertainty and the asset price volatility, but is negative related to the average trading volume. We also discuss the specialist's participation strategies under the NYSE regulation that prohibits the specialist from trading with the market trend.     

Optimal consumption and investment for markets with random coefficients

We consider an optimal investment and consumption problem for a Black–Scholes financial market with stochastic coefficients driven by a diffusion process. We assume that an agent makes consumption and investment decisions based on CRRA utility functions. The dynamic programming approach leads to an investigation of the Hamilton–Jacobi–Bellman (HJB) equation which is a highly nonlinear partial differential equation (PDE) of the second order. By using the Feynman–Kac representation, we prove uniqueness and smoothness of the solution. Moreover, we study the optimal convergence rate of iterative numerical schemes for both the value function and the optimal portfolio. We show that in this case, the optimal convergence rate is super-geometric, i.e., more rapid than any geometric one. We apply our results to a stochastic volatility financial market.     

Competition,signaling and non-walking through the book: Effects on order choice

We investigate the effects of competition and signaling in a pure order driven market and examine the trading patterns of agents when walking through the book is not allowed. Our results suggest that the variables capturing the cost of a large market order are not informative for an impatient trader under this market mechanism. We also document that the competition effect is not present only at the top of the book but persistent beyond the best quotes. Moreover, it dominates the signaling effect for both a limit order and a market order trader. Finally, we show that institutional investors’ order submission strategies are characterized by only a few pieces of the limit order book information. This is consistent with informed traders placing orders based on their own private valuations rather than the state of the book.