This study proposes a versatile model for the dynamics of the best bid and ask prices using an extended Hawkes process. The model incorporates the zero intensities of the spread-narrowing processes at the minimum bid-ask spread, spread-dependent intensities, possible negative excitement, and nonnegative intensities. We apply the model to high-frequency best bid and ask price data from US stock markets. The empirical findings demonstrate a spread-narrowing tendency, excitations of the intensities caused by previous events, the impact of flash crashes, characteristic trends in fast trading over time, and the different features of market participants in the various exchanges.
This study examine the theoretical and empirical perspectives of the symmetric Hawkes model of the price tick structure. Combined with the maximum likelihood estimation, the model provides a proper method of volatility estimation specialized in ultra-high-frequency analysis. Empirical studies based on the model using the ultra-high-frequency data of stocks in the S\&P 500 are performed. The performance of the volatility measure, intraday estimation, and the dynamics of the parameters are discussed. A new approach of diffusion analogy to the symmetric Hawkes model is proposed with the distribut…
Price changes are induced by aggressive market orders in stock market. We introduce a bivariate marked Hawkes process to model aggressive market order arrivals at the microstructural level. The order arrival intensity is marked by an exogenous part and two endogenous processes reflecting the self-excitation and cross-excitation respectively. We calibrate the model for an SSE stock. We find that the exponential kernel with a smooth cut-off (i.e. the subtraction of two exponentials) produces much better calibration than the monotonous exponential kernel (i.e. the sum of two exponentials). The ex…
A point process model for order flows in limit order books is proposed, in which the conditional intensity is the product of a Hawkes component and a state-dependent factor. In the LOB context, state observations may include the observed imbalance or the observed spread. Full technical details for the computationally-efficient estimation of such a process are provided, using either direct likelihood maximization or EM-type estimation. Applications include models for bid and ask market orders, or for upwards and downwards price movements. Empirical results on multiple stocks traded in Euronext …
It has been suggested that marked point processes might be good candidates for the modelling of financial high-frequency data. A special class of point processes, Hawkes processes, has been the subject of various investigations in the financial community. In this paper, we propose to enhance a basic zero-intelligence order book simulator with arrival times of limit and market orders following mutually (asymmetrically) exciting Hawkes processes. Modelling is based on empirical observations on time intervals between orders that we verify on several markets (equity, bond futures, index futures). …
We provide some theoretical extensions and a calibration protocol for our former dynamic optimal execution model. The Hawkes parameters and the propagator are estimated independently on financial data from stocks of the CAC40. Interestingly, the propagator exhibits a smoothly decaying form with one or two dominant time scales, but only so after a few seconds that the market needs to adjust after a large trade. Motivated by our estimation results, we derive the optimal execution strategy for a multi-exponential Hawkes kernel and backtest it on the data for round trips. We find that the strategy…
High-frequency market making is a liquidity-providing trading strategy that simultaneously generates many bids and asks for a security at ultra-low latency while maintaining a relatively neutral position. The strategy makes a profit from the bid-ask spread for every buy and sell transaction, against the risk of adverse selection, uncertain execution and inventory risk. We design realistic simulations of limit order markets and develop a high-frequency market making strategy in which agents process order book information to post the optimal price, order type and execution time. By introducing t…
Bid-ask spread is taken as an important measure of the financial market liquidity. In this article, we study the dynamics of the spread return and the spread volatility of four liquid stocks in the Chinese stock market, including the memory effect and the multifractal nature. By investigating the autocorrelation function and the Detrended Fluctuation Analysis (DFA), we find that the spread return is lack of long-range memory, while the spread volatility is long-range time correlated. Moreover, by applying the Multifractal Detrended Fluctuation Analysis (MF-DFA), the spread return is observed t…
We study statistical aspects of state-dependent Hawkes processes, which are an extension of Hawkes processes where a self- and cross-exciting counting process and a state process are fully coupled, interacting with each other. The excitation kernel of the counting process depends on the state process that, reciprocally, switches state when there is an event in the counting process. We first establish the existence and uniqueness of state-dependent Hawkes processes and explain how they can be simulated. Then we develop maximum likelihood estimation methodology for parametric specifications of t…
We introduce and establish the main properties of QHawkes ("Quadratic" Hawkes) models. QHawkes models generalize the Hawkes price models introduced in E. Bacry et al. (2014), by allowing all feedback effects in the jump intensity that are linear and quadratic in past returns. A non-parametric fit on NYSE stock data shows that the off-diagonal component of the quadratic kernel indeed has a structure that standard Hawkes models fail to reproduce. Our model exhibits two main properties, that we believe are crucial in the modelling and the understanding of the volatility process: first, the model …
We develop a theory of bid and ask price dynamics where the two prices form due to interaction of buy and sell orders. In this model the two prices are represented by eigenvalues of a 2x2 price operator corresponding to "bid" and "ask" eigenstates. Matrix elements of price operator fluctuate in time which results in phase jitter for eigenstates. We show that the theory reflects very important characteristics of bid and ask dynamics and order density in the order book. Calibration examples are provided for stocks at various time scales. Lastly, this model allows to quantify and measure risk ass…
High frequency financial data is burdened by a level of randomness that is unavoidable and obfuscates the task of modelling. This idea is reflected in the intraday evolution of limit orders book data for many financial assets and suggests several justifications for the use of stochastic models. For instance, the arbitrary distribution of inter arrival times and the subsequent dependence structure between consecutive book events. This has lead to the development of many stochastic models for the dynamics of limit order books. In this paper we look to examine the adaptability of one family of su…
Motivated by the literature on investment flows and optimal trading, we examine intraday predictability in the cross-section of stock returns. We find a striking pattern of return continuation at half-hour intervals that are exact multiples of a trading day, and this effect lasts for at least 40 trading days. Volume, order imbalance, volatility, and bid-ask spreads exhibit similar patterns, but do not explain the return patterns. We also show that short-term return reversal is driven by temporary liquidity imbalances lasting less than an hour and bid-ask bounce. Timing trades can reduce execut…
This study presents an agent-based computational cross-market model for Chinese equity market structure, which includes both stocks and CSI 300 index futures. In this model, we design several stocks and one index futures to simulate this structure. This model allows heterogeneous investors to make investment decisions with restrictions including wealth, market trading mechanism, and risk management. Investors' demands and order submissions are endogenously determined. Our model successfully reproduces several key features of the Chinese financial markets including spot-futures basis distributi…
In this paper, we study the dynamics of absolute return, trading volume and bid-ask spread after the trading halts using high-frequency data from the Shanghai Stock Exchange. We deal with all three types of trading halts, namely intraday halts, one-day halts and inter-day halts, of 203 stocks in Shanghai Stock Exchange from August 2009 to August 2011. We find that absolute return, trading volume, and in case of bid-ask spread around intraday halts share the same pattern with a sharp peak and a power law relaxation after that. While for different types of trading halts, the peaks' height and th…
In the domain of the so called Econophysics some attempts already have been made for applying the theory of Thermodynamics and Statistical Mechanics to economics and financial markets. In this paper a similar approach is made from a different perspective, trying to model the limit order book and price formation process of a given stock by the Grand-Canonical Gibbs Ensemble for the bid and ask processes. As a consequence we can define in a meaningful way expressions for the temperatures of the ensembles of bid orders and of ask orders, which are a function of maximum bid, minimum ask and closur…
Financial markets show a number of non-stationarities, ranging from volatility fluctuations over ever changing technical and regulatory market conditions to seasonalities. On the other hand, financial markets show various stylized facts which are remarkably stable. It is thus an intriguing question to find out how these stylized facts emerge. As a first example, we here investigate how the bid-ask-spread between best sell and best buy offer for stocks develops during the trading day. For rescaled and properly smoothed data we observe collapsing curves for many different NASDAQ stocks, with a s…
Based on the order flow data of a stock and its warrant, the immediate price impacts of market orders are estimated by two competitive models, the power-law model (PL model) and the logarithmic model (LG model). We find that the PL model is overwhelmingly superior to the LG model, regarding the robustness of the estimated parameters and the accuracy of out-of-sample forecasting. We also find that the price impacts of ask and bid orders are consistent with each other for filled trades, since significant positive correlations are observed between the model parameters of both types of orders. Our…
Periodic Double Auctions (PDAs) are commonly used in the real world for trading, e.g. in stock markets to determine stock opening prices, and energy markets to trade energy in order to balance net demand in smart grids, involving trillions of dollars in the process. A bidder, participating in such PDAs, has to plan for bids in the current auction as well as for the future auctions, which highlights the necessity of good bidding strategies. In this paper, we perform an equilibrium analysis of single unit single-shot double auctions with a certain clearing price and payment rule, which we refer …
Using ultra-high-frequency data extracted from the order flows of 23 stocks traded on the Shenzhen Stock Exchange, we study the empirical regularities of order placement in the opening call auction, cool period and continuous auction. The distributions of relative logarithmic prices against reference prices in the three time periods are qualitatively the same with quantitative discrepancies. The order placement behavior is asymmetric between buyers and sellers and between the inside-the-book orders and outside-the-book orders. In addition, the conditional distributions of relative prices in th…
Intraday Limit Order Price Change Transition Dynamics Across Market Capitalizations Through Markov Analysis
National Institute of Technology Sikkim（锡金国立技术学院）；Chennai Mathematical Institute（马德拉斯数学研究所）；University G. d’Annunzio of Chieti-Pescara（基耶蒂-佩斯卡拉G. d'安努齐奥大学）
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 nonparametric discrete model for the dynamics of the best bid and ask, which can be approximated by…
An ability to postpone one's execution without penalty provides an important strategic advantage in high-frequency trading. To elucidate competition between traders one has to formulate to a quantitative theory of formation of the execution price from market expectations and quotes. This theory was provided in 2005 by Foucault, Kadan and Kandel. I derive asymptotic distribution of the bids/offers as a function of the ratio of patient and impatient traders using the dynamic version of the Foucault, Kadan and Kandel dynamic Limit Order Book (LOB) model. Dynamic version of the LOB model allows st…
Lead/lag relationships are an important stylized fact at high frequency. Some assets follow the path of others with a small time lag. We provide indicators to measure this phenomenon using tick-by-tick data. Strongly asymmetric cross-correlation functions are empirically observed, especially in the future/stock case. We confirm the intuition that the most liquid assets (short intertrade duration, narrow bid/ask spread, small volatility, high turnover) tend to lead smaller stocks. However, the most correlated stocks are those with similar levels of liquidity. This lead/lag phenomenon is not con…
Exploiting a precise reproduction of a stock exchange, the robustness of the Continuous Double Auction (CDA) mechanism, evaluated by means of the waiting time distributions, has been proved versus 36 different set ups made by varying both the operators' behaviour and the market micro structure. The obtained results demonstrate that the CDA remains able to clear strongly different order flows, though the Milan stock exchange seemed to be a little more efficient than the NYSE under the allocative point of view, witnessing the intrinsic complexity of the stock market. The simulation has been buil…
A consistency criterion for price impact functions in limit order markets is proposed that prohibits chain arbitrage exploitation. Both the bid-ask spread and the feedback of sequential market orders of the same kind onto both sides of the order book are essential to ensure consistency at the smallest time scale. All the stocks investigated in Paris Stock Exchange have consistent price impact functions.
Indian Stock Market Prediction using Augmented Financial Intelligence ML
University College London（伦敦大学学院）；National Institute of Technology Patna（巴特那国立理工学院）；SRM University AP（SRM大学安得拉邦校区）；Amirkabir University of Technology, Tehran Polytechnic（阿米尔卡比尔理工大学（德黑兰理工学院））；IIT Kanpur（印度理工学院坎普尔分校）
Market liquidity plays a vital role in the field of market micro-structure, because it is the vigor of the financial market. This paper uses a variable called convexity to measure the potential liquidity provided by order-book. Based on the high-frequency data of each stock included in the SSE (Shanghai Stock Exchange) 50 Index for the year 2011, we report several statistical properties of convexity and analyze the association between convexity and some other important variables (bid/ask-depth, spread, volatility, return.)
Rough volatility is a well-established statistical stylised fact of financial assets. This property has lead to the design and analysis of various new rough stochastic volatility models. However, most of these developments have been carried out in the mono-asset case. In this work, we show that some specific multivariate rough volatility models arise naturally from microstructural properties of the joint dynamics of asset prices. To do so, we use Hawkes processes to build microscopic models that reproduce accurately high frequency cross-asset interactions and investigate their long term scalin…
In this paper, we employ the Heston stochastic volatility model to describe the stock's volatility and apply the model to derive and analyze the optimal trading strategies for dealers in a security market. We also extend our study to option market making for options written on stocks in the presence of stochastic volatility. Mathematically, the problem is formulated as a stochastic optimal control problem and the controlled state process is the dealer's mark-to-market wealth. Dealers in the security market can optimally determine their ask and bid quotes on the underlying stocks or options con…
We study the dynamics of order flows around large intraday price changes using ultra-high-frequency data from the Shenzhen Stock Exchange. We find a significant reversal of price for both intraday price decreases and increases with a permanent price impact. The volatility, the volume of different types of orders, the bid-ask spread, and the volume imbalance increase before the extreme events and decay slowly as a power law, which forms a well-established peak. The volume of buy market orders increases faster and the corresponding peak appears earlier than for sell market orders around positive…
We observe the effects of the three different events that cause spread changes in the order book, namely trades, deletions and placement of limit orders. By looking at the frequencies of the relative amounts of price changing events, we discover that deletions of orders open the bid-ask spread of a stock more often than trades do. We see that once the amount of spread changes due to deletions exceeds the amount of the ones due to trades, other observables in the order book change as well. We then look at how these spread changing events affect the prices of stocks, by means of the price respon…
Motivated by empirical data, we develop a statistical description of the queue dynamics for large tick assets based on a two-dimensional Fokker-Planck (diffusion) equation, that explicitly includes state dependence, i.e. the fact that the drift and diffusion depends on the volume present on both sides of the spread. "Jump" events, corresponding to sudden changes of the best limit price, must also be included as birth-death terms in the Fokker-Planck equation. All quantities involved in the equation can be calibrated using high-frequency data on best quotes. One of our central finding is the th…
We study the optimal placement problem of a stock trader who wishes to clear his/her inventory by a predetermined time horizon t, by using a limit order or a market order. For a diffusive market, we characterize the optimal limit order placement policy and analyze its behavior under different market conditions. In particular, we show that, in the presence of a negative drift, there exists a critical time t0>0 such that, for any time horizon t>t0, there exists an optimal placement, which, contrary to earlier work, is different from one that is placed "infinitesimally" close to the best ask, suc…
Although behavioral economics has demonstrated that there are many situations where rational choice is a poor empirical model, it has so far failed to provide quantitative models of economic problems such as price formation. We make a step in this direction by developing empirical models that capture behavioral regularities in trading order placement and cancellation using data from the London Stock Exchange. For order placement we show that the probability of placing an order at a given price is well approximated by a Student distribution with less than two degrees of freedom, centered on the…
A limit order book provides information on available limit order prices and their volumes. Based on these quantities, we give an empirical result on the relationship between the bid-ask liquidity balance and trade sign and we show that liquidity balance on best bid/best ask is quite informative for predicting the future market order's direction. Moreover, we define price jump as a sell (buy) market order arrival which is executed at a price which is smaller (larger) than the best bid (best ask) price at the moment just after the precedent market order arrival. Features are then extracted relat…
R. Cont and A. de Larrard (SIAM J. Finan. Math, 2013) introduced a tractable stochastic model for the dynamics of a limit order book, computing various quantities of interest such as the probability of a price increase or the diffusion limit of the price process. As suggested by empirical observations, we extend their framework to 1) arbitrary distributions for book events inter-arrival times (possibly non-exponential) and 2) both the nature of a new book event and its corresponding inter-arrival time depend on the nature of the previous book event. We do so by resorting to Markov renewal proc…
Standard models in economics stress the role of intelligent agents who maximize utility. However, there may be situations where, for some purposes, constraints imposed by market institutions dominate intelligent agent behavior. We use data from the London Stock Exchange to test a simple model in which zero intelligence agents place orders to trade at random. The model treats the statistical mechanics of order placement, price formation, and the accumulation of revealed supply and demand within the context of the continuous double auction, and yields simple laws relating order arrival rates to …
While the long-ranged correlation of market orders and their impact on prices has been relatively well studied in the literature, the corresponding studies of limit orders and cancellations are scarce. We provide here an empirical study of the cross-correlation between all these different events, and their respective impact on future price changes. We define and extract from the data the "bare" impact these events would have, if they were to happen in isolation. For large tick stocks, we show that a model where the bare impact of all events is permanent and non-fluctuating is in good agreement…
We study the dynamics of the limit order book of liquid stocks after experiencing large intra-day price changes. In the data we find large variations in several microscopical measures, e.g., the volatility the bid-ask spread, the bid-ask imbalance, the number of queuing limit orders, the activity (number and volume) of limit orders placed and canceled, etc. The relaxation of the quantities is generally very slow that can be described by a power law of exponent $\approx0.4$. We introduce a numerical model in order to understand the empirical results better. We find that with a zero intelligence…
We consider a limit order book, where buyers and sellers register to trade a security at specific prices. The largest price buyers on the book are willing to offer is called the market bid price, and the smallest price sellers on the book are willing to accept is called the market ask price. Market ask price is always greater than market bid price, and these prices move upwards and downwards due to new arrivals, market trades, and cancellations. We model these two price processes as "bouncing geometric Brownian motions (GBMs)", which are defined as exponentials of two mutually reflected Browni…
We develop a theory which applies to any market dynamics that satisfy a fair market assumption on the nullity of the average profit of simple market making strategies. We show that for any such fair market, there exists a martingale fair price which corresponds to the average liquidation value (at the ask or the bid) of an infinitesimal quantity of stock. We show that this fair price is a natural reference price to compute the ex post gain of limit orders. Using only the fair market assumption, we link the spread to the impact of market orders on the fair price. We use our definition of the fa…
Market makers continuously set bid and ask quotes for the stocks they have under consideration. Hence they face a complex optimization problem in which their return, based on the bid-ask spread they quote and the frequency at which they indeed provide liquidity, is challenged by the price risk they bear due to their inventory. In this paper, we consider a stochastic control problem similar to the one introduced by Ho and Stoll and formalized mathematically by Avellaneda and Stoikov. The market is modeled using a reference price $S_t$ following a Brownian motion with standard deviation $σ$, arr…
Understanding the statistical properties of recurrence intervals of extreme events is crucial to risk assessment and management of complex systems. The probability distributions and correlations of recurrence intervals for many systems have been extensively investigated. However, the impacts of microscopic rules of a complex system on the macroscopic properties of its recurrence intervals are less studied. In this Letter, we adopt an order-driven stock market model to address this issue for stock returns. We find that the distributions of the scaled recurrence intervals of simulated returns ha…
Large tick assets, i.e. assets where one tick movement is a significant fraction of the price and bid-ask spread is almost always equal to one tick, display a dynamics in which price changes and spread are strongly coupled. We introduce a Markov-switching modeling approach for price change, where the latent Markov process is the transition between spreads. We then use a finite Markov mixture of logit regressions on past squared returns to describe the dependence of the probability of price changes. The model can thus be seen as a Double Chain Markov Model. We show that the model describes the …
This article provides a simple explanation of the asymptotic concavity of the price impact of a meta-order via the microstructural properties of the market. This explanation is made more precise by a model in which the local relationship between the order flow and the fundamental price (i.e. the local price impact) is linear, with a constant slope, which makes the model dynamically consistent. Nevertheless, the expected impact on midprice from a large sequence of co-directional trades is nonlinear and asymptotically concave. The main practical conclusion of the proposed explanation is that, th…
Stock prices are observed to be random walks in time despite a strong, long term memory in the signs of trades (buys or sells). Lillo and Farmer have recently suggested that these correlations are compensated by opposite long ranged fluctuations in liquidity, with an otherwise permanent market impact, challenging the scenario proposed in Quantitative Finance 4, 176 (2004), where the impact is *transient*, with a power-law decay in time. The exponent of this decay is precisely tuned to a critical value, ensuring simultaneously that prices are diffusive on long time scales and that the response …
We propose a limit order book (LOB) model with dynamics that account for both the impact of the most recent order and the shape of the LOB. We present an empirical analysis showing that the type of the last order significantly alters the submission rate of immediate future orders, even after accounting for the state of the LOB. To model these effects jointly we introduce a discrete Markov chain model. Then on these improved LOB dynamics, we find the policy for optimal order choice and placement in the share purchasing problem by framing it as a Markov decision process. The optimal policy deriv…
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 …
