Narrow Results

We have examined the order book characteristics and market impact on the Korean stock index futures market (KOSPI 200 index futures). The distribution of order volumes generally follows power-law distribution. The estimated exponents are 1.9 for market order, 2.5 for limit order, and 2.1 for cancel order. This result is different from the case of stocks where the exponent of market order is larger than that of limit order. The order likelihood is distinctively high in every 50's of order volume, which implies the behavioral characteristics of human preference on round-up numbers. The distributions of bid–ask spread and the best quotes volume provide the evidence of the liquidity of KOSPI 200 index futures market. We have obtained the concave relationship between market impact and transaction volume as well. Finally, the market response behavior is observed regarding various transaction sizes. The size of market response is estimated to be proportional to the size of transaction. Also, the larger the transaction size is, the longer it takes to recover the stability from the impact triggered by transaction.

With an alternative choice of risk criterion, we solve the HJB equation explicitly to find a closed-form solution for the optimal trade execution strategy in the Almgren–Chriss framework assuming the underlying unaffected stock price process is geometric Brownian motion.

We consider an optimal trading problem over a finite period of time during which an investor has access to both a standard exchange and a dark pool. We take the exchange to be an order-driven market and propose a continuous-time setup for the best bid price and the market spread, both modeled by Lévy processes. Effects on the best bid price arising from the arrival of limit buy orders at more favorable prices, the incoming market sell orders potentially walking the book, and deriving from the cancellations of limit sell orders at the best ask price are incorporated in the proposed price dynamics. A permanent impact that occurs when ‘lit’ pool trades cannot be avoided is built in, and an instantaneous impact that models the slippage, to which all lit exchange trades are subject, is also considered. We assume that the trading price in the dark pool is the mid-price and that no fees are due for posting orders. We allow for partial trade executions in the dark pool, and we find the optimal trading strategy in both venues. Since the mid-price is taken from the exchange, the dynamics of the limit order book also affects the optimal allocation of shares in the dark pool. We propose a general objective function and we show that, subject to suitable technical conditions, the value function can be characterized by the unique continuous viscosity solution to the associated partial integro-differential equation. We present two explicit examples of the price and the spread models, derive the associated optimal trading strategy numerically. We discuss the various degrees of the agent's risk aversion and further show that roundtrips are not necessarily beneficial.

We consider an optimal liquidation problem with infinite horizon in the Almgren–Chriss framework, where the unaffected asset price follows a Lévy process. The temporary price impact is described by a general function that satisfies some reasonable conditions. We consider a market agent with constant absolute risk aversion, who wants to maximize the expected utility of the cash received from the sale of the agent’s assets, and show that this problem can be reduced to a deterministic optimization problem that we are able to solve explicitly. In order to compare our results with exponential Lévy models, which provide a very good statistical fit with observed asset price data for short time horizons, we derive the (linear) Lévy process approximation of such models. In particular we derive expressions for the Lévy process approximation of the exponential variance–gamma Lévy process, and study properties of the corresponding optimal liquidation strategy. We then provide a comparison of the liquidation trajectories for reasonable parameters between the Lévy process model and the classical Almgren–Chriss model. In particular, we obtain an explicit expression for the connection between the temporary impact function for the Lévy model and the temporary impact function for the Brownian motion model (the classical Almgren–Chriss model), for which the optimal liquidation trajectories for the two models coincide.

We study the relation between the trading behavior of agents and volatility in toy markets of adaptive inductively rational agents. We show that excess volatility, in such simplified markets, arises as a consequence of (i) the neglect of market impact implicit in price taking behavior and of (ii) excessive reactivity of agents. These issues are dealt with in detail in the simple case without public information. We also derive, for the general case, the critical learning rate above which trading behavior leads to turbulent dynamics of the market.

Single period risks acceptable to the market at zero cost are modeled by a convex set of random variables leading to bid and ask prices that are trade size dependent. The theory of nonlinear expectations is employed to construct dynamically consistent sequences of bid and ask unit size prices that are size and trade date contingent. We then study the optimal design of spot and forward trading to minimize execution costs. Finally, we illustrate the construction of a two period execution cost frontier trading a decrease in execution costs for additional exposure to price risk. Most structured products already have prices that depend on the direction of the trade. Additionally markets already exist for large block trades with their own price structure that takes account of the time allowed for its execution. This paper outlines arbitrage free mechanisms for generating such price structures that could lead to automating quotation systems for such markets. Additionally, we describe the tradeoffs implicit in seeking to commit to trades now as opposed to assuming the price risk implicit in delaying commitments. Such an explicit analysis could lead to the development of optimal execution algorithms that economize on the level of price risk absorbed into the execution strategies.

We solve a version of the optimal trade execution problem when the mid asset price follows a displaced diffusion (DD). Optimal strategies in the adapted class under various risk criteria, namely value-at-risk (VaR), expected shortfall (ES) and a new criterion called squared asset expectation (SAE), related to a version of the cost variance measure, are derived and compared. It is well known that DDs exhibit dynamics that are in-between arithmetic Brownian motions (ABM) and geometric Brownian motions (GBM) depending of the choice of the shift parameter. Furthermore, DD allows for changes in the support of the mid asset price distribution, allowing one to include a minimum permitted value for the mid price, either positive or negative. We study the dependence of the optimal solution on the choice of the risk aversion criterion. Optimal solutions across criteria and asset dynamics are comparable although differences are not negligible for high levels of risk aversion and low market impact assets. This is illustrated with numerical examples.

We make an extensive empirical study of the market impact of large orders (metaorders) executed in the US equity market between 2007 and 2009. We show that the square root market impact formula, which is widely used in the industry and supported by previous published research, provides a good fit only across about two orders of magnitude in order size. A logarithmic functional form fits the data better, providing a good fit across almost five orders of magnitude. We introduce the concept of an “impact surface” to model the impact as a function of both the duration and the participation rate of the metaorder, finding again a logarithmic dependence. We show that during the execution the price trajectory deviates from the market impact, a clear indication of non-VWAP executions. Surprisingly, we find that sometimes the price starts reverting well before the end of the execution. Finally we show that, although on average the impact relaxes to approximately $2∕3$ of the peak impact, the precise asymptotic value of the price depends on the participation rate and on the duration of the metaorder. We present evidence that this might be due to a herding phenomenon among metaorders.

Using a proprietary dataset of meta-orders and prediction signals, and assuming a quasi-linear impact model, we deconvolve market impact from past correlated trades and a predictable return component to elicit the temporal dependence of the market impact of a single daily meta-order, over a 10-day horizon in various equity markets. We find that the impact of single meta-orders is to a first approximation universal and slowly decays to zero (or to a small value), possibly as a power-law. We show that autocorrelated order-flows and trade information contents fully accounts for the apparent plateau observed in the raw data. We discuss the possible bias introduced by the quasi-linear assumption.

We present a thorough empirical analysis of market impact on the Bitcoin/USD exchange market using a complete dataset that allows us to reconstruct more than one million metaorders. We empirically confirm the “square-root law” for market impact, which holds on four decades in spite of the quasi-absence of statistical arbitrage and market marking strategies. We show that the square-root impact holds during the whole trajectory of a metaorder and not only for the final execution price. We also attempt to decompose the order flow into an “informed” and “uninformed” component, the latter leading to an almost complete long-term decay of impact. This study sheds light on the hypotheses and predictions of several market impact models recently proposed in the literature and promotes heterogeneous agent models as promising candidates to explain price impact on the Bitcoin market — and, we believe, on other markets as well.

This study is the first to decipher market impact at all time scales on the same database, from a trade-by-trade scale to a daily one. Moreover, the very concentrated nature of the database (400,000 metaorders issued by investors, electronically traded, during one year—2010—, on European market all regulated by the same directive—MiFID—) ensures investors did not change their habits during the study.

At the intraday scale, we confirm a square root temporary impact in the daily participation, and we shed light on a duration factor in $1∕T^{\gamma }$ with $\gamma \simeq 0.25$. Including this factor in the fits reinforces the square root shape of impact. We observe a power-law for the transient impact with an exponent between $0.5$ (for long metaorders) and $0.8$ (for shorter ones). Moreover, we show that the market does not anticipate the size of the metaorders. The intraday decay seems to exhibit two regimes (though hard to identify precisely): a “slow” regime right after the execution of the metaorder followed by a faster one. At the daily time scale, we show price moves after a metaorder can be split between realizations of expected returns that have triggered the investing decision and an idiosynchratic impact that slowly decays to zero.

Moreover, we propose a class of toy models based on Hawkes processes (the Hawkes Impact Model, HIM) to illustrate our reasoning. We show how the Impulsive-HIM model, despite its simplicity, embeds appealing features like transience and decay of impact. The latter is parametrized by a parameter C having a macroscopic interpretation: the ratio of contrarian reaction (i.e., impact decay) and of the “herding” reaction (i.e., impact amplification).

We consider intraday hedging of an option position, for a large trader who experiences temporary and permanent market impact. We formulate the general model including overnight risk, and solve explicitly in two cases which we believe are representative. The first case is an option with approximately constant gamma: the optimal hedge trades smoothly towards the classical Black–Scholes delta, with trading intensity proportional to instantaneous mishedge and inversely proportional to illiquidity. The second case is an arbitrary non-linear option structure but with no permanent impact: the optimal hedge trades toward a value offset from the Black–Scholes delta. We estimate the effects produced on the public markets if a large collection of traders all hedge similar positions. We construct a stable hedge strategy with discrete time steps.

We consider the optimal execution of a book of options when market impact is a driver of the option price. We aim at minimizing the mean-variance risk criterion for a given market impact function. First, we develop a framework to justify the choice of our market impact function. Our model is inspired from Leland’s option replication with transaction costs where the market impact is directly part of the implied volatility function. The option price is then expressed through a Black– Scholes-like PDE with a modified implied volatility directly dependent on the market impact. We set up a stochastic control framework and solve an Hamilton–Jacobi–Bellman equation using finite differences methods. The expected cost problem suggests that the optimal execution strategy is characterized by a convex increasing trading speed, in contrast to the equity case where the optimal execution strategy results in a rather constant trading speed. However, in such mean valuation framework, the underlying spot price does not seem to affect the agent’s decision. By taking the agent risk aversion into account through a mean-variance approach, the strategy becomes more sensitive to the underlying price evolution, urging the agent to trade faster at the beginning of the strategy.

The aim of this paper is to explain how parameters adjustments can be integrated in the design or the control of automates of trading. Typically, we are interested in the online estimation of the market impacts generated by robots or single orders, and how they/the controller should react in an optimal way to the information generated by the observation of the realized impacts. This can be formulated as an optimal impulse control problem with unknown parameters, on which a prior is given. We explain how a mix of the classical Bayesian updating rule and of optimal control techniques allows one to derive the dynamic programming equation satisfied by the corresponding value function, from which the optimal policy can be inferred. We provide an example of convergent finite difference scheme and consider typical examples of applications.

We leverage Kepler Cheuvreux client order database over the period October 2014 — October 2016 (349,442 trades corresponding to a EUR92.3bn turnover) to estimate new models of market impact. We find a multiplicative relationship between the market impact and the explanatory factors (the volatility, the trading period participation rate and the trading duration). Furthermore, the relationship between the participation rate and the duration on one side and the market impact on the other is concave, with the effect of the participation rate on the market impact scaling as a square root.

We introduce a new indicator of resiliency, which measures the ability of the order book to resist the aggressive order flow in a given period. This indicator shows a positive correlation with the residuals of our standard model of market impact, clearly demonstrating that the more resilient the stock, the more resistant it is to market impact. Thus, we are able to calibrate an enhanced model of market impact using our indicator of resiliency, which improves the explanatory power of the model compared to standard approaches. Our resiliency indicator thereby exposes the relationship between the market impact at the meta-order scale and the market impact at the elementary trade scale.

This paper complements the inspiring work on dimensional analysis and market microstructure by Kyle and Obizhaeva (2017). Following closely these authors, our main result shows, by a similar argument as usually applied in physics, the following remarkable fact. If the market impact of a meta-order only depends on four well-defined and financially meaningful variables, and some obvious scaling relations as well as the assumption of leverage neutrality are satisfied, then there is only one possible form of this dependence. In particular, the market impact is proportional to the square-root of the size of the meta-order.

This theorem can be regarded as a special case of a more general result of Kyle and Obizhaeva. These authors consider five variables which might have an influence on the size of the market impact. In this case, one finds a richer variety of possible functional relations which we precisely characterize. We also discuss the analogies to classical arguments from physics, such as the period of a pendulum.

This paper is devoted to the important yet little explored subject of the market impact of limit orders. Our analysis is based on a proprietary database of metaorders — large orders that are split into smaller pieces before being sent to the market. We first address the case of aggressive limit orders and then that of passive limit orders. In both cases, we provide empirical evidence of a power law behavior for the temporary market impact. The relaxation of the price following the end of the metaorder is also studied, and the long-term impact is shown to stabilize at a level of approximately two-thirds of the maximum impact. Finally, a fair pricing condition during the life cycle of the metaorders is empirically validated.

We present an empirical study of price reversion after the executed metaorders. We use a dataset with more than 8 million metaorders executed by institutional investors in the US equity market. We show that relaxation takes place as soon as the metaorder ends: while at the end of the same day, it is on average $\approx 2∕3$ of the peak impact, the decay continues for the next few days, following a power-law function at short-time scales, and converges to a non-zero asymptotic value at long-time scales ($\sim 50$ days) equal to $\approx 1∕2$ of the impact at the end of the first day, that is $\approx 1∕3$ of peak impact. Due to a significant, multiday correlation of the sign of executed metaorders, a careful deconvolution of the observed impact must be performed to extract the estimate of the impact decay of isolated metaorders.

We study optimal liquidation in “target zone models” — asset prices with a reflecting boundary enforced by regulatory interventions. This can be treated as a special case of an Almgren–Chriss model with running and terminal inventory costs and general predictive signals about price changes. The optimal liquidation rate in target-zone models can in turn be characterized as the “theta” of a lookback option, leading to explicit formulas for Bachelier or Black–Scholes dynamics.

We consider a market impact game for $n$ risk-averse agents that are competing in a market model with linear transient price impact and additional transaction costs. For both finite and infinite time horizons, the agents aim to minimize a mean-variance functional of their costs or to maximize the expected exponential utility of their revenues. We give explicit representations for corresponding Nash equilibria and prove uniqueness in the case of mean-variance optimization. A qualitative analysis of these Nash equilibria is conducted by means of numerical analysis.