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No-Dynamic-Arbitrage and Market Impact

Taylor & Francis
Quantitative Finance
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Abstract

Starting from a no-dynamic-arbitrage principle that imposes that trading costs should be non-negative on average and a simple model for the evolution of market prices, we demonstrate a relationship between the shape of the market impact function describing the average response of the market price to traded quantity and the function that describes the decay of market impact. In particular, we show that the widely assumed exponential decay of market impact is compatible only with linear market impact. We derive various inequalities relating the typical shape of the observed market impact function to the decay of market impact, noting that, empirically, these inequalities are typically close to being equalities.

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... We study operator learning in the context of linear propagator models for optimal order execution problems with transient price impactà la Bouchaud et al. (2004) andGatheral (2010). Transient price impact persists and decays over time according to some propagator kernel. ...
... We refer to the excellent monographs [CJP15,Gué16,BBDG18,Web23] for a comprehensive overview of this topic. This paper addresses an optimal liquidation problem in the context of linear propagator models proposed by [BGPW04,BFL09,Gat10]. Propagator models constitute a versatile class of transient price impact models, defined by a price impact kernel (propagator), which reliably captures in reduced form the interplay between price moves and current and past trades as empirically observed when executing market orders in limit order books. ...
... We consider a classical optimal order execution problem with price impact within the general framework of linear propagator models. This class of models was originally developed by [BGPW04,BFL09] in discrete time and formulated by [Gat10] in continuous time; see also [BBDG18,Chapter 13]. The associated optimal liquidation problem was explicitly solved only very recently in [AJN22], and we will also refer to [AJNV23, Section 3.2] for a simpler direct approach. ...
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We study operator learning in the context of linear propagator models for optimal order execution problems with transient price impact \`a la Bouchaud et al. (2004) and Gatheral (2010). Transient price impact persists and decays over time according to some propagator kernel. Specifically, we propose to use In-Context Operator Networks (ICON), a novel transformer-based neural network architecture introduced by Yang et al. (2023), which facilitates data-driven learning of operators by merging offline pre-training with an online few-shot prompting inference. First, we train ICON to learn the operator from various propagator models that maps the trading rate to the induced transient price impact. The inference step is then based on in-context prediction, where ICON is presented only with a few examples. We illustrate that ICON is capable of accurately inferring the underlying price impact model from the data prompts, even with propagator kernels not seen in the training data. In a second step, we employ the pre-trained ICON model provided with context as a surrogate operator in solving an optimal order execution problem via a neural network control policy, and demonstrate that the exact optimal execution strategies from Abi Jaber and Neuman (2022) for the models generating the context are correctly retrieved. Our introduced methodology is very general, offering a new approach to solving optimal stochastic control problems with unknown state dynamics, inferred data-efficiently from a limited number of examples by leveraging the few-shot and transfer learning capabilities of transformer networks.
... The first distinguishes between temporary trading costs, that only affect each trade separately, and permanent price impact that affects the current and all future trades in the same manner (cf., e.g., Bertsimas and Lo (1998); Almgren and Chriss (2001) and many more recent studies). The second takes into account the transient nature of price impact, which is caused by large trades but gradually wears off once these are completed, cf., e.g., Bouchaud et al. (2004Bouchaud et al. ( , 2006; Obizhaeva and Wang (2013); Gatheral (2010); Alfonsi et al. (2010); Predoiu et al. (2011); Gatheral and Schied (2013). ...
... This is a special case of the general framework for transient price impact studied in Gatheral (2010). 6 Put differently, each trade has a linear temporary price impact proportional to both trade size and speed, compare Guasoni and Weber (2015); Moreau et al. (2017) for more details. ...
... This is evidently satisfied in the most common specifications Λ = λI n or Λ = λΣ for a scalar λ > 0, for example. 8 Related models with transient price impact have been studied intensively in the optimal execution literature, compare, e.g., Obizhaeva and Wang (2013); Gatheral (2010); Alfonsi et al. (2010); Predoiu et al. (2011). the "transient" distortion of the actual price from its unaffected version thus has the following Ornstein-Uhlenbeck-type dynamics, (2.9) and the risky positions H evolve as ...
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We study portfolio selection in a model with both temporary and transient price impact introduced by Garleanu and Pedersen (2016). In the large-liquidity limit where both frictions are small, we derive explicit formulas for the asymptotically optimal trading rate and the corresponding minimal leading-order performance loss. We find that the losses are governed by the volatility of the frictionless target strategy, like in models with only temporary price impact. In contrast, the corresponding optimal portfolio not only tracks the frictionless optimizer, but also exploits the displacement of the market price from its unaffected level.
... They incorporated linear permanent and temporary market impact models, providing a more dynamic approach to the execution problem. Their work laid the groundwork for subsequent studies, including Ly Vath, Mnif, and Pham [35], Gatheral [29] and Obizhaeva and Wang [39], who examined different aspects of market impact and execution costs, proposing models that account for both temporary and permanent price impacts. Different extensions of these models have been introduced since then to include nonlinear impacts, reflecting more realistic trading conditions; see e.g. ...
... • Market impact: Drawing from the research of Bouchaud, Farmer, and Lillo [13] and Gatheral [29], we propose a price impact model in which the trading size influences the price Q in a concave manner 3 , meaning that v → V −1 (v) forms a continuous concave function from R + to R + . As empirical observations suggest that the price impact function often approximates a square-root function, we will employ a power-shaped density function • Hidden liquidity: To ensure clarity in our results, we will focus on two possible regimes i ∈ {1, 2} throughout the remainder of our study. ...
... By means of the uniqueness of the Doob-Meyer decomposition for semi-martingales (refer to Section III.3 in Protter [44]) with reference to (31), we establish that the finite variations in (29) and (30) are identical. Thus, we conclude that ...
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We study optimal liquidation strategies under partial information for a single asset within a finite time horizon. We propose a model tailored for high-frequency trading, capturing price formation driven solely by order flow through mutually stimulating marked Hawkes processes. The model assumes a limit order book framework, accounting for both permanent price impact and transient market impact. Importantly, we incorporate liquidity as a hidden Markov process, influencing the intensities of the point processes governing bid and ask prices. Within this setting, we formulate the optimal liquidation problem as an impulse control problem. We elucidate the dynamics of the hidden Markov chain's filter and determine the related normalized filtering equations. We then express the value function as the limit of a sequence of auxiliary continuous functions, defined recursively. This characterization enables the use of a dynamic programming principle for optimal stopping problems and the determination of an optimal strategy. It also facilitates the development of an implementable algorithm to approximate the original liquidation problem. We enrich our analysis with numerical results and visualizations of candidate optimal strategies.
... Accurate estimation of transactions' price impact is instrumental for designing profitable trading strategies. Propagator models serve as a central tool in describing these phenomena mathematically (see Bouchaud et al. [9], Gatheral [20]). These models express price moves in terms of the influence of past trades, and give a reliable reduced form view on the limit order book reaction for trades execution. ...
... The proof of Theorem 2.10 is given in Section 5. Examples where the Volterra price impact kernel G ⋆ is in fact a convolution kernel arise from empirical studies and are quite popular in the literature (see e.g., [9,20,35]). In the sequel, we incorporate this structural property of the price impact coefficient to enhance our estimation procedure. ...
... We present some typical examples for price impact convolution kernels, whose projection on a finite grid belongs to K ad . In [9,20] among others, the following kernel was introduced: [20]. The case where K(t) = e −ρt , for some constant ρ > 0, was proposed by Obizhaeva and Wang [35]. ...
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We consider an offline learning problem for an agent who first estimates an unknown price impact kernel from a static dataset, and then designs strategies to liquidate a risky asset while creating transient price impact. We propose a novel approach for a nonparametric estimation of the propagator from a dataset containing correlated price trajectories, trading signals and metaorders. We quantify the accuracy of the estimated propagator using a metric which depends explicitly on the dataset. We show that a trader who tries to minimise her execution costs by using a greedy strategy purely based on the estimated propagator will encounter suboptimality due to so-called spurious correlation between the trading strategy and the estimator and due to intrinsic uncertainty resulting from a biased cost functional. By adopting an offline reinforcement learning approach, we introduce a pessimistic loss functional taking the uncertainty of the estimated propagator into account, with an optimiser which eliminates the spurious correlation, and derive an asymptotically optimal bound on the execution costs even without precise information on the true propagator. Numerical experiments are included to demonstrate the effectiveness of the proposed propagator estimator and the pessimistic trading strategy.
... As mentioned above and in [22,14,3], we have that price impact can be temporary, transient and/or permanent. Market orders (MOs) that walk the LOB have a price impact because they obtain average split its sell order in different trade sizes, some of which could be very small at a particular time compared to the depth of the LOB. ...
... Moreover, once the volume sold is large enough for triggering a price impact, this temporary impact seems to be best reproduced by a power-law function with exponent below 1 and more precisely between 0.4 and 0.8. This is consistent with the literature and the so-called square-root law as investigated in [8,40,22,15]. Such behaviors of the temporary impact curves justify the use of an indirect price impact model based on a jump process, as we will introduce in the following Section 2.1. ...
... We fix a filtered probability space Ω, F, (F t ) 0≤t≤T , P where T is the deterministic trading horizon. As an extension of the price impact model of Gatheral in [22], we further assume that the execution price S ν t at time t follows ...
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We propose in this paper a new framework of optimal liquidation strategies for a trader seeking to liquidate his large inventory based on a jump-dependent price impact model with propagator. This new jump-dependent price impact model allows to best reproduce the empirical direct and indirect effects of market orders on the transaction price. More precisely, different choices of propagators are proposed and their implications in terms of temporary, permanent and transient impacts on the transaction price are discussed. For each choice of such kernels, we formulate the most relevant optimal liquidation problem faced by the trader, derive explicitly the related Hamilton-Jacobi-Bellman equation and solve it numerically. Moreover, we show how to extend our price impact model so to include the possibility for the trader to also use limit orders. We hence manage in this paper to propose an alternative more realistic and flexible description of the order book's dynamic and to make a bridge between high-frequency price models and optimal liquidation problems.
... Since the position is large, its liquidation affects asset prices in an unfavorable way, which creates additional execution costs. The temporal evolution of this price impact can be described by means of a kernel G, for which some empirical studies suggest a behavior of the form G(t) ∼ t −α for some α ∈ (0, 1); see, e.g., Gatheral (2010). Assumption (3) is reasonable in this context: it excludes the existence of price manipulation strategies that generate profit through their own price impact (Huberman and Stanzl, 2004;Gatheral, 2010). ...
... The temporal evolution of this price impact can be described by means of a kernel G, for which some empirical studies suggest a behavior of the form G(t) ∼ t −α for some α ∈ (0, 1); see, e.g., Gatheral (2010). Assumption (3) is reasonable in this context: it excludes the existence of price manipulation strategies that generate profit through their own price impact (Huberman and Stanzl, 2004;Gatheral, 2010). The term γ 2 T 0 ϕ(t) 2 dt can be interpreted as costs arising from 'slippage' or temporary price impact as in Almgren (2003). ...
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Motivated by the problem of optimal portfolio liquidation under transient price impact, we study the minimization of energy functionals with completely monotone displacement kernel under an integral constraint. The corresponding minimizers can be characterized by Fredholm integral equations of the second type with constant free term. Our main result states that minimizers are analytic and have a power series development in terms of even powers of the distance to the midpoint of the domain of definition and with nonnegative coefficients. We show moreover that our minimization problem is equivalent to the minimization of the energy functional under a nonnegativity constraint.
... Metaorders have started being recorded as such in a systematic way only recently, and mostly in proprietary databases that are not readily accessible to academic researchers in the field of market microstructure. The analyses presented in [Almgren et al., 2005] [ Moro et al., 2009] [Gatheral, 2010] [Toth et al., 2011] [ Bershova and Rakhlin, 2013] [Bacry et al., 2015] [Zarinelli et al., 2015] [Gomes and Waelbroeck, 2015] essentially cover all that is published about the market impact of large orders. ...
... It is clearly an important explanatory variable of the price discovery and is studied as such in several papers [Kyle, 1985] [Hautsch and Huang, 2012] [Farmer et al., 2004]. Temporary market impact is obviously the main source of trading costs, and models based on empirical measurements can be used in optimal trading schemes [Almgren et al., 2005] [Gatheral, 2010] [ Lehalle and Dang, 2010] [Almgren and Chriss, 2001] [Gatheral and Schied, 2013], or used by an investment firm in order to understand its trading costs [Bershova andRakhlin, 2013] [Brokmann et al., 2015] [ Mastromatteo et al., 2014]. One common conclusion to the studies is that the temporary market impact is mainly characterized by three components. ...
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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 behaviour 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.
... In each case Z is taken to be a predictable semimartingale with left limit process Z − and jumps ∆Z = Z − Z − . Models in this category include Ankirchner et al. (2016), Brown et al. (2010), Chen et al. (2014), Gatheral andSchied (2011), Schied (2013), Schied and Schöneborn (2009) Ting et al. (2007) in continuous time and Almgren and Chriss (2000), Bertsimas and Lo (1998) in discrete time. Other impact specifications can be found, for example, in Chen et al. (2015), Cheridito andSepin (2014), Forsyth (2011), Lorenz and Almgren (2011), Subramanian and Jarrow (2001) and Ting et al. (2007). ...
... More recent studies, excellently summarized in Gatheral (2010), consider an intermediate form of impact where the execution price is given by the formula ...
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We study optimal liquidation of a trading position (so-called block order or meta-order) in a market with a linear temporary price impact (Kyle, 1985). We endogenize the pressure to liquidate by introducing a downward drift in the unaffected asset price while simultaneously ruling out short sales. In this setting the liquidation time horizon becomes a stopping time determined endogenously, as part of the optimal strategy. We find that the optimal liquidation strategy is consistent with the square-root law which states that the average price impact per share is proportional to the square root of the size of the meta-order (Bershova and Rakhlin, 2013; Farmer et al., 2013; Donier et al., 2015; T\'oth et al., 2016). Mathematically, the Hamilton-Jacobi-Bellman equation of our optimization leads to a severely singular and numerically unstable ordinary differential equation initial value problem. We provide careful analysis of related singular mixed boundary value problems and devise a numerically stable computation strategy by re-introducing time dimension into an otherwise time-homogeneous task.
... From a practical perspective, an important question is to understand how well a trader's strategy performs within a limit order book and how the market reacts to this trader's strategy, noting that the placement of orders can have an adversarial effect (see, e.g. [16]). ...
... [47], as it can have a significant negative effect on profits from trading. It is suggested e.g. in [19,16,46] that the market impact is proportional to the square root of the order size P divided by the daily trading volume V , ...
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In this paper, we show how K-nearest neighbor (K-NN) resampling, an off-policy evaluation method proposed in \cite{giegrich2023k}, can be applied to simulate limit order book (LOB) markets and how it can be used to evaluate and calibrate trading strategies. Using historical LOB data, we demonstrate that our simulation method is capable of recreating realistic LOB dynamics and that synthetic trading within the simulation leads to a market impact in line with the corresponding literature. Compared to other statistical LOB simulation methods, our algorithm has theoretical convergence guarantees under general conditions, does not require optimization, is easy to implement and computationally efficient. Furthermore, we show that in a benchmark comparison our method outperforms a deep learning-based algorithm for several key statistics. In the context of a LOB with pro-rata type matching, we demonstrate how our algorithm can calibrate the size of limit orders for a liquidation strategy. Finally, we describe how K-NN resampling can be modified for choices of higher dimensional state spaces.
... Gatheral (2010) investigates a relationship between the shape of the market impact function and the decay of market impact. ...
... 9 shows the frequency histogram and the QQ-plot of the standardized daily logvolumes relative to the time period considered and it confirms that the observed total trading Kolmogorov-Smirnov test applied to the log-volume (upper panels) and log-price (lower panels) for SPX and NA indices, with the null hypothesis assuming normality of the time series. The y-axis range spans from 0.05 to 1.volume follows a GBM.Figure 4.10 shows the estimated Pearson correlation coefficients of price ...
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An empirical analysis, suggested by optimal Merton dynamics, reveals some unexpected features of asset volumes. These features are connected to traders' belief and risk aversion. This paper proposes a trading strategy model in the optimal Merton framework that is representative of the collective behavior of heterogeneous rational traders. This model allows for the estimation of the average risk aversion of traders acting on a specific risky asset, while revealing the existence of a price of risk closely related to market price of risk and volume rate. The empirical analysis, conducted on real data, confirms the validity of the proposed model.
... Consequently, it is only when the price level changes, the hidden liquidity is brought out into the market, and causes a reversion in market impact over time. The Propagator model [7] is a widely used model that explains the observed part of this phenomenon by dividing up the market impact into initial impacts and the reversion of these market impacts when estimating the total market impact over the current period. We too will use this framework presented by Gatheral et al for the purpose of our research. ...
... The second approach looks beyond the first-term lag. As described by the propagator model [7], we expect the price impact of trade to revert in a decaying fashion, as the time between the trade and price impact observation increases. This follows Equation 2, where the total market impact, M (t), over period t is the accumulation of all initial impacts made by trades across the period, f (ẋ s ), after the decay of these impacts have been taken into consideration following the decay kernel G(t − s), plus a noise term . ...
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In a financial exchange, market impact is a measure of the price change of an asset following a transaction. This is an important element of market microstructure, which determines the behaviour of the market following a trade. In this paper, we first provide a discussion on the market impact observed in the BTC/USD Futures market, then we present a novel multi-agent market simulation that can follow an underlying price series, whilst maintaining the ability to reproduce the market impact observed in the market in an explainable manner. This simulation of the financial exchange allows the model to interact realistically with market participants, helping its users better estimate market slippage as well as the knock-on consequences of their market actions. In turn, it allows various stakeholders such as industrial practitioners, governments and regulators to test their market hypotheses, without deploying capital or destabilising the system.
... In their work, temporary impact refers to "...temporary imbalances in supply in demand caused by our trading leading to temporary price movements away from equilibrium" and permanent impact "...means changes in the equilibrium price due to our trading which remain at least for the life of our liquidation". Theoretical approaches agree that the permanent price impact must be linear, Gatheral (2010) shows this condition is necessary to avoid dynamic arbitrage. He presents the principle of no-dynamic-arbitrage which states that the trading cost is non-negative for any round-trip trade strategy, i.e. ...
... That is, it is proportional to the square root of the volume of shares executed, as seen in Toth et al. (2016) and Almgren et al. (2005), or to the square root of the trade duration, as in Bershova and Rakhlin (2013). Moreover, different approaches using nonlinear temporary or permanent price impact have been studied in the literature, for example Gueant (2014) relates to the result in Gatheral (2010) and extends some theoretical results to the nonlinear setting, Barger and Lorig (2018) offers a theoretical approach, considering stochastic temporary and permanent price impacts. Alfonsi and Schied (2010) proposes a model where the impact depends on the theoretical shape of LOB given by a density (exponential), whereas our model does not rely on any shape and is thus capable of capture the actual book in simulations. ...
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This paper studies the optimal liquidation of stocks in the presence of temporary and permanent price impacts, and we focus in the case of cryptocurrencies. We start by presenting analytical solutions to the problem with linear temporary impact, and linear and quadratic permanent impact. Then, using data from the order book of the BNB cryptocurrency, we estimate the functional form of the temporary and permanent price impact in three different scenarios: underestimation, overestimation and average estimation, finding different functional forms for each scenario. Using finite differences and optimal policy iteration, we solve the problem numerically and observe interesting changes in the optimal liquidation policy when applying calibrated linear and power forms for the temporary and permanent price impacts. Then, with these optimal policies, we identify optimal liquidation trajectories and simulate the liquidation of initial inventories to compare the performance among the optimal strategies under different parametrizations and against a naive strategy. Finally, we characterize the optimal policies based on the functional form of the inventory and find that policies generating the highest revenue are those starting with a low trading rate and increasing it as time passes.
... Using a very careful statistical treatment, averaging over many metaorders eliminates part of this noise so that we can better identify universal properties of market impact, see for instance [ATHL05,BILL15,BR13,BBLB19]. When trading a metaorder, these empirical studies show that the price mechanically follows the metaorder, exhibiting a concave shape peaking at the end of the metaorder, followed by a convex relaxation, see [BR13,Gat10,Mor09]. During the execution of the metaorder, [ATHL05, Hop03, KO23, LFM03, Mor09] identified a square root dependence in the volume. ...
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While the market impact of aggressive orders has been extensively studied, the impact of passive orders, those executed through limit orders, remains less understood. The goal of this paper is to investigate passive market impact by developing a microstructure model connecting liquidity dynamics and price moves. A key innovation of our approach is to replace the traditional assumption of constant information content for each trade by a function that depends on the available volume in the limit order book. Within this framework, we explore scaling limits and analyze the market impact of passive metaorders. Additionally, we derive useful approximations for the shape of market impact curves, leading to closed-form formulas that can be easily applied in practice.
... The case of a limited availability of hedging products has not been dealt theoretically nor numerically so far in the literature to our knowledge. Most of the research currently focuses on developing some price impact features to model the liquidation of a position (see for example Gatheral (2010)) instead of imposing some depth limits. ...
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In Electricity markets, illiquidity, transaction costs and market price characteristics prevent managers to replicate exactly contracts. A residual risk is always present and the hedging strategy depends on a risk criterion chosen. We present an algorithm to hedge a position for a mean variance criterion taking into account the transaction cost and the small depth of the market. We show its effectiveness on a typical problem coming from the field of electricity markets.
... This model is sufficiently general as to capture a rich array of market impact models, including the ones used here and in [Chr24a] (as well as the original work on optimal execution in [AC97] and [AC01]). More broadly, the above model is general enough to describe as special cases some of the key models that have been studied empirically and theoretically including [Gat10a], [GS11], [GS13], [OW13] and [BFL09]. Also see [FGLM06], [ATHL05] and [ZTFL15] 4 . ...
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This is the third paper in a series concerning the game-theoretic aspects of position-building while in competition. The first paper set forth foundations and laid out the essential goal, which is to minimize implementation costs in light of how other traders are likely to trade. The majority of results in that paper center on the two traders in competition and equilibrium results are presented. The second paper, introduces computational methods based on Fourier Series which allows the introduction of a broad range of constraints into the optimal strategies derived. The current paper returns to the unconstrained case and provides a complete solution to finding equilibrium strategies in competition and handles completely arbitrary situations. As a result we present a detailed analysis of the value (or not) of trade centralization and we show that firms who naively centralize trades do not generally benefit and sometimes, in fact, lose. On the other hand, firms that strategically centralize their trades generally will be able to benefit.
... One of the most important 'what if' topics in finance is analyzing market impact, the change in asset prices caused by trading activity, which is crucial for understanding and navigating financial markets, e.g., developing optimal trading strategies and detecting manipulation. Most existing research in this area is relied heavily on strong assumptions and empirical formulas (Zarinelli et al., 2015;Almgren et al., 2005;Gatheral, 2010;Gatheral et al., 2012;2011), limited to costly and risky online experiments. In this section, we take the market impact as an example, showing how MarS can act as a reliable and powerful analysis platform and contribute to "what if" analysis. ...
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Generative models aim to simulate realistic effects of various actions across different contexts, from text generation to visual effects. Despite efforts to build real-world simulators, leveraging generative models for virtual worlds, like financial markets, remains underexplored. In financial markets, generative models can simulate market effects of various behaviors, enabling interaction with market scenes and players, and training strategies without financial risk. This simulation relies on the finest structured data in financial market like orders thus building the finest realistic simulation. We propose Large Market Model (LMM), an order-level generative foundation model, for financial market simulation, akin to language modeling in the digital world. Our financial Market Simulation engine (MarS), powered by LMM, addresses the need for realistic, interactive and controllable order generation. Key objectives of this paper include evaluating LMM's scaling law in financial markets, assessing MarS's realism, balancing controlled generation with market impact, and demonstrating MarS's potential applications. We showcase MarS as a forecast tool, detection system, analysis platform, and agent training environment. Our contributions include pioneering a generative model for financial markets, designing MarS to meet domain-specific needs, and demonstrating MarS-based applications' industry potential.
... There are strictly formal models, both linear and non-linear, models of limit and market order dynamics and models concerning market impact estimation. See [AC01], [AC01], [GS11], [GS13], [Gat10a], [OW13], [HH12], [ZTFL15] and [ATHL05] 1 . In this paper we will use linear models for both temporary and permanent market impact following [AC01] and define them next. ...
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This paper develops a mathematical framework for building a position in a stock over a fixed period of time while in competition with one or more other traders doing the same thing. We develop a game-theoretic framework that takes place in the space of trading strategies where action sets are trading strategies and traders try to devise best-response strategies to their adversaries. In this setup trading is guided by a desire to minimize the total cost of trading arising from a mixture of temporary and permanent market impact caused by the aggregate level of trading including the trader and the competition. We describe a notion of equilibrium strategies, show that they exist and provide closed-form solutions.
... In particular, should this hypothesis be verified, it would imply a violation of dynamic arbitrage in the sense of Gatheral, see Gatheral (2010) and Schneider and Lillo (2019). ...
... Monitoring and controlling price impact has been extensively researched in quantitative finance. It is well known that market impact is a function of volatility and order size as a percentage of daily volume under the square-root law, a basic form that has been both empirically and theoretically recognized (Tóth et al 2011;Gatheral 2010;Alfonsi et al 2012;Donier et al 2015). Although previous studies have validated the universality of the square-root function across different financial products and time periods, we are still interested to see its performance under the impact of abnormal market conditions, in particular during the recent pandemic-driven turbulence. ...
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During recent market turbulence, we have observed an anomalously high impact cost for stock market transactions. After conducting empirical analysis on actual execution data from 2020, we find that during such a market regime there appears to be a stronger crowding effect in the stock market, which means market participants have an increased propensity to trade the same stock on the same side at the same time, leading to larger transaction costs in general. Therefore, to address this problem, we extend our impact cost model beyond the typical factors such as order size, price volatility, trade volume and bid-offer spread. These factors alone are insufficient to fully capture the underlying pattern in this market regime as they would in normal times. We propose to add a size adjustment to the actual transaction size to account for this crowding effect and to achieve a more accurate estimate of the expected impact costs during such periods of elevated volatility. The adjustment is constructed using the Chicago Board Options Exchange Volatility Index, a well-known measure for expected volatility and "fear". With the enhanced model, we are able to provide a more robust estimation for transaction costs that significantly outperforms the original impact cost model in the face of market turmoil.
... The ratio of return volatility to the square root of the dollar trading volume has been widely adopted in the industry to capture the cross-sectional and time-series variations in trading costs. The ratio is similar in spirit to the illiquidity measure of Amihud (2002), and has been justified from various theoretical settings (e.g., Grinold and Kahn 1999;Gabaix, Gopikrishnan, Plerou, and Stanley 2006;Gatheral 2010). ...
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The classical Almgren-Chriss price impact model is generalized to incorporate contributions to the transacted price from a crowd of infinitesimal market makers, each identified by a characteristic time during which their inventory mean reverts. Upon completion of the execution, these market makers revert their capacities back to zero. The resulting price dynamics, in general, are neither Markovian nor semimartingale. Determining the optimal execution scheme for the liquidation problem thus becomes an infinite-dimensional stochastic control problem. Despite this complexity, the problem remains linear-quadratic, allowing its solution to be reduced to a system of operator Riccati equations that characterize the optimal value process and the associated optimal liquidation strategy. Remarkably, the operator Riccati differential equations can be explicitly solved in the observed relevant case where the price impact model reproduces the empirically detected power-law decay. A numerical implementation illustrates the theoretical findings.
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The developments in electronic markets have led to the diversification of trading activity, and traders need to manage the liquidity risk carefully. This problem is called the optimal execution problem and has become a significant issue among financial mathematicians, economists, and practitioners. This chapter aims to overview how the current financial market works and how one can analyze and build the algorithms for an “optimal execution strategy.” The first section gives a review of current financial markets, which leads to the basics for constructing a model of optimal execution from the viewpoints of market microstructure. In particular, I clarify the system of the “limit order book,” which includes an exposition about how traders place orders and influence the market. Also, this section presents the basic concepts of “large trader” and “market impact,” on top of which most execution models are built. The succeeding sections explain how one can incorporate market impact in modeling and formulate an execution problem through a fundamental model posed by Almgren and Chriss (J. Risk 3:5–39, 2000 [2]). I then describe an extensive model with a moderate change in market impact modeling, discussed in Ohnishi and Shimoshimizu (Quant. Financ. 20:1625–1644, 2020 [35]). These models embody the foundation of algorithms for optimal execution strategies.
Article
The curse of dimensionality significantly restricts the use of dynamic programming methods in solving complex problems. Consequently, researchers and practitioners often resort to approximate (suboptimal) control policies that strike a balance between ease of implementation and satisfactory performance. Information relaxation-based duality techniques generate both upper and lower bounds for the true values of stochastic dynamic programming problems, allowing us to evaluate the optimality of approximate policies through the dual gap. However, the literature still lacks guidance on handling cases where the gaps are excessively loose. In “Information Relaxation and a Duality-Driven Algorithm for Stochastic Dynamic Programs,” Chen, Ma, Liu, and Yu develop a novel DDP framework to obtain and tighten confidence interval estimates for the true value functions of SDP problems. Leveraging a new finding that the dual operation yields subsolutions, they establish convergence guarantees for DDP. Additionally, a regression-based Monte Carlo method is introduced, aimed at high-dimensional applications. Numerical examples demonstrate that DDP effectively improves dual gaps for various heuristics that are commonly used in the literature.
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We consider an optimal liquidation problem with instantaneous price impact and stochastic resilience for small instantaneous impact factors. Within our modelling framework, the optimal portfolio process converges to the solution of an optimal liquidation problem with general semimartingale controls when the instantaneous impact factor converges to zero. Our results provide a unified framework within which to embed the two most commonly used modelling frameworks in the liquidation literature and provide a foundation for the use of semimartingale liquidation strategies and the use of portfolio processes of unbounded variation. Our convergence results are based on novel convergence results for BSDEs with singular terminal conditions and novel representation results of BSDEs in terms of uniformly continuous functions of forward processes.
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The Handbook on Systemic Risk, written by experts in the field, provides researchers with an introduction to the multifaceted aspects of systemic risks facing the global financial markets. The Handbook explores the multidisciplinary approaches to analyzing this risk, the data requirements for further research, and the recommendations being made to avert financial crisis. The Handbook is designed to encourage new researchers to investigate a topic with immense societal implications as well as to provide, for those already actively involved within their own academic discipline, an introduction to the research being undertaken in other disciplines. Each chapter in the Handbook will provide researchers with a superior introduction to the field and with references to more advanced research articles. It is the hope of the editors that this Handbook will stimulate greater interdisciplinary academic research on the critically important topic of systemic risk in the global financial markets.
Chapter
The Handbook on Systemic Risk, written by experts in the field, provides researchers with an introduction to the multifaceted aspects of systemic risks facing the global financial markets. The Handbook explores the multidisciplinary approaches to analyzing this risk, the data requirements for further research, and the recommendations being made to avert financial crisis. The Handbook is designed to encourage new researchers to investigate a topic with immense societal implications as well as to provide, for those already actively involved within their own academic discipline, an introduction to the research being undertaken in other disciplines. Each chapter in the Handbook will provide researchers with a superior introduction to the field and with references to more advanced research articles. It is the hope of the editors that this Handbook will stimulate greater interdisciplinary academic research on the critically important topic of systemic risk in the global financial markets.
Chapter
The Handbook on Systemic Risk, written by experts in the field, provides researchers with an introduction to the multifaceted aspects of systemic risks facing the global financial markets. The Handbook explores the multidisciplinary approaches to analyzing this risk, the data requirements for further research, and the recommendations being made to avert financial crisis. The Handbook is designed to encourage new researchers to investigate a topic with immense societal implications as well as to provide, for those already actively involved within their own academic discipline, an introduction to the research being undertaken in other disciplines. Each chapter in the Handbook will provide researchers with a superior introduction to the field and with references to more advanced research articles. It is the hope of the editors that this Handbook will stimulate greater interdisciplinary academic research on the critically important topic of systemic risk in the global financial markets.
Article
For the “In This Issue” column: Trading costs play a central role in designing and implementing quantitative trading strategies. To quantify trading costs, optimal execution and trading algorithms rely on price impact models, such as the propagator model. Empirically, price impact is concave in trade sizes, leading to nonlinear models for which optimization problems are intractable and even qualitative properties, such as price manipulation, are poorly understood. This paper shows that, in the diffusion limit of small and frequent orders, the nonlinear model converges to a tractable linear model. In this high-frequency limit, a stochastic liquidity parameter approximates the original impact function’s nonlinearity. This allows us to derive simple formulas for optimal trading strategies and sharp conditions on market volumes to rule out price manipulation. A detailed empirical study using high-frequency limit-order data illustrates the practical performance of the theoretical results.
Article
We consider a mean-field control problem with càdlàg semimartingale strategies arising in portfolio liquidation models with transient market impact and self-exciting order flow. We show that the value function depends on the state process only through its law, and we show that it is of linear-quadratic form and that its coefficients satisfy a coupled system of nonstandard Riccati-type equations. The Riccati equations are obtained heuristically by passing to the continuous-time limit from a sequence of discrete-time models. A sophisticated transformation shows that the system can be brought into standard Riccati form, from which we deduce the existence of a global solution. Our analysis shows that the optimal strategy jumps only at the beginning and the end of the trading period. Funding: Financial support is through the National Natural Science Foundation of China [Grants 12101465 and 12101523], Hong Kong Research Grants Council (Early Career Scheme) [Grant 25215122], Hong Kong Polytechnic University [Internal Grant P0044694, Internal Grant P0045668, and Startup Grant P0035348], and the Hong Kong Research Centre for Quantitative Finance [Grant P0042708].
Article
We study the optimal order placement strategy with the presence of a liquidity cost. In this problem, a stock trader wishes to clear her large inventory by a predetermined time horizon T. A trader uses both limit and market orders, and a large market order faces an adverse price movement caused by the liquidity risk. First, we study a single period model where the trader places a limit order and/or a market order at the beginning. We show the behavior of optimal amount of market order, [Formula: see text], and optimal placement of limit order, [Formula: see text], under different market conditions. Next, we extend it to a multi-period model, where the trader makes sequential decisions of limit and market orders at multiple time points.
Chapter
The Handbook on Systemic Risk, written by experts in the field, provides researchers with an introduction to the multifaceted aspects of systemic risks facing the global financial markets. The Handbook explores the multidisciplinary approaches to analyzing this risk, the data requirements for further research, and the recommendations being made to avert financial crisis. The Handbook is designed to encourage new researchers to investigate a topic with immense societal implications as well as to provide, for those already actively involved within their own academic discipline, an introduction to the research being undertaken in other disciplines. Each chapter in the Handbook will provide researchers with a superior introduction to the field and with references to more advanced research articles. It is the hope of the editors that this Handbook will stimulate greater interdisciplinary academic research on the critically important topic of systemic risk in the global financial markets.
Chapter
The Handbook on Systemic Risk, written by experts in the field, provides researchers with an introduction to the multifaceted aspects of systemic risks facing the global financial markets. The Handbook explores the multidisciplinary approaches to analyzing this risk, the data requirements for further research, and the recommendations being made to avert financial crisis. The Handbook is designed to encourage new researchers to investigate a topic with immense societal implications as well as to provide, for those already actively involved within their own academic discipline, an introduction to the research being undertaken in other disciplines. Each chapter in the Handbook will provide researchers with a superior introduction to the field and with references to more advanced research articles. It is the hope of the editors that this Handbook will stimulate greater interdisciplinary academic research on the critically important topic of systemic risk in the global financial markets.
Chapter
The Handbook on Systemic Risk, written by experts in the field, provides researchers with an introduction to the multifaceted aspects of systemic risks facing the global financial markets. The Handbook explores the multidisciplinary approaches to analyzing this risk, the data requirements for further research, and the recommendations being made to avert financial crisis. The Handbook is designed to encourage new researchers to investigate a topic with immense societal implications as well as to provide, for those already actively involved within their own academic discipline, an introduction to the research being undertaken in other disciplines. Each chapter in the Handbook will provide researchers with a superior introduction to the field and with references to more advanced research articles. It is the hope of the editors that this Handbook will stimulate greater interdisciplinary academic research on the critically important topic of systemic risk in the global financial markets.
Article
We obtain the optimal execution strategy for two sequential trades in the presence of a transient price impact. We first present a novel and general solution method for the case of a single trade (a metaorder) that is executed as a sequence of sub-trades (child orders). We then analyze the case of two sequential metaorders, including the case where the size and direction of the second metaorder are uncertain at the time the first metaorder is initiated. We obtain the optimal execution strategy under two different cost functions. First, we minimize the total cost when each metaorder is benchmarked to the price at its initiation, the total separate costs approach widely used by practitioners. Although simple, we show that optimizing total separate costs can lead to a significant understatement of the real costs of trading whilst also adversely impacting order scheduling. We overcome these issues by introducing a new cost function that splits the second metaorder into two parts, one that is predictable when the first metaorder is initiated and a residual that is not. The predictable and residual parts of the second metaorder are benchmarked using the initiation prices of the first and second metaorders, respectively. We prove existence of an optimal execution trajectory for linear instantaneous price impact and positive definite decay, and derive the explicit form of the minimizer in the special case of exponentially decaying impact, however uniqueness in general remains unproven. Various numerical examples are included for illustration.
Article
This paper investigates the impact of anonymous trading on the agents' strategy in an optimal execution framework. It mainly explores the specificity of order attribution on the Toronto Stock Exchange, where brokers can choose to either trade with their own identity or under a generic anonymous code that is common to all the brokers. We formulate a stochastic differential game for the optimal execution problem of a population of N brokers and incorporate permanent and temporary price impacts for both the identity-revealed and anonymous trading processes. We then formulate the limiting mean-field game of controls with common noise and obtain a solution in closed-form via the probablistic approach for the Almgren-Chris price impact framework. Finally, we perform a sensitivity analysis to explore the impact of the model parameters on the optimal strategy.
Article
We propose a systematic algorithmic reverse‐stress testing methodology to create “worst case” scenarios for regulatory stress tests by accounting for losses that arise from distressed portfolio liquidations. First, we derive the optimal bank response for any given shock. Then, we introduce an algorithm which systematically generates scenarios that exploit the key vulnerabilities in banks' portfolio holdings and thus maximize contagion despite banks' optimal response to the shock. We apply our methodology to data of the 2016 European Banking Authority (EBA) stress test, and design worst case scenarios for the portfolio holdings of European banks at the time. Using spectral clustering techniques, we group 10,000 worst‐case scenarios into twelve geographically concentrated families. Our results show that even though there is a wide range of different scenarios within these 12 families, each cluster tends to affect the same banks. An “Anna Karenina” principle of stress testing emerges: Not all stressful scenarios are alike, but every stressful scenario stresses the same banks. These findings suggest that the precise specification of a scenario is not of primal importance as long as the most vulnerable banks are targeted and sufficiently stressed. Finally, our methodology can be used to uncover the weakest links in the financial system and thereby focus supervisory attention on these, thus building a bridge between macroprudential and microprudential stress tests.
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Using Trades and Quotes data from the Paris stock market, we show that the random walk nature of traded prices results from a very delicate interplay between two opposite tendencies: long-range correlated market orders that lead to super-diffusion (or persistence), and mean reverting limit orders that lead to sub-diffusion (or anti-persistence). We define and study a model where the price, at any instant, is the result of the impact of all past trades, mediated by a non constant `propagator' in time that describes the response of the market to a single trade. Within this model, the market is shown to be, in a precise sense, at a critical point, where the price is purely diffusive and the average response function almost constant. We find empirically, and discuss theoretically, a fluctuation-response relation. We also discuss the fraction of truly informed market orders, that correctly anticipate short term moves, and find that it is quite small.
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Recent empirical studies have demonstrated long-memory in the signs of orders to buy or sell in financial markets [2, 19]. We show how this can be caused by delays in market clearing. Under the common practice of order splitting, large orders are broken up into pieces and executed incrementally. If the size of such large orders is power law distributed, this gives rise to power law decaying autocorrelations in the signs of executed orders. More specifically, we show that if the cumulative distribution of large orders of volume v is proportional to v to the power -alpha and the size of executed orders is constant, the autocorrelation of order signs as a function of the lag tau is asymptotically proportional to tau to the power -(alpha - 1). This is a long-memory process when alpha < 2. With a few caveats, this gives a good match to the data. A version of the model also shows long-memory fluctuations in order execution rates, which may be relevant for explaining the long-memory of price diffusion rates.
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This paper investigates market manipulation trading strategies by large traders in a securities market. A large trader is defined as any investor whose trades change prices. A market manipulation trading strategy is one that generates positive real wealth with no risk. Market manipulation trading strategies are shown to exist under reasonable hypotheses on the equilibrium price process. Sufficient conditions for their nonexistence are also provided.
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We develop an approach to asset pricing in incomplete markets that bridges the gap between the two fundamental approaches in finance: model-based asset pricing and pricing by no arbitrage. We strengthen the absence of arbitrage assumption by precluding investment opportunities whose attractiveness to a benchmark investor exceeds a specified threshold. In our framework, the attractiveness of an investment opportunity is measured by the gain-loss ratio. We show that a restriction on the maximum gain-loss ratio is equivalent to a restriction on the ratio of the maximum to minimum values of the pricing kernel. By limiting the maximum gain-loss ratio, we can restrict the set of admissible pricing kernels, which in turn allows us to restrict the set of prices that can be assigned to assets. We illustrate our methodology by computing price bounds for call options in a Black-Scholes economy without intermediate trading. When we vary the maximum permitted gain-loss ratio, these bounds can range from the exact prices implied by a model-based pricing approach to the loose price bounds implied by the no-arbitrage approach.
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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, 2004, 4, 176, where the impact is instead 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 impact function is nearly lag independent. We provide new analysis of empirical data that confirm and make more precise our previous claims. We show that the power-law decay of the bare impact function comes both from an excess flow of limit order opposite to the market order flow, and to a systematic anti-correlation of the bid-ask motion between trades, two effects that create a 'liquidity molasses' which dampens market volatility.
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In this comment we discuss the problem of reconciling the linear efficiency of price returns with the long-memory of supply and demand. We present new evidence that shows that efficiency is maintained by a liquidity imbalance that co-moves with the imbalance of buyer vs. seller initiated transactions. For example, during a period where there is an excess of buyer initiated transactions, there is also more liquidity for buy orders than sell orders, so that buy orders generate smaller and less frequent price responses than sell orders. At the moment a buy order is placed the transaction sign imbalance tends to dominate, generating a price impact. However, the liquidity imbalance rapidly increases with time, so that after a small number of time steps it cancels all the inefficiency caused by the transaction sign imbalance, bounding the price impact. While the view presented by Bouchaud et al. of a fixed and temporary bare price impact is self-consistent and formally correct, we argue that viewing this in terms of a variable but permanent price impact provides a simpler and more natural view. This is in the spirit of the original conjecture of Lillo and Farmer, but generalized to allow for finite time lags in the build up of the liquidity imbalance after a transaction. We discuss the possible strategic motivations that give rise to the liquidity imbalance and offer an alternative hypothesis. We also present some results that call into question the statistical significance of large swings in expected price impact at long times.
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We investigate several statistical properties of the order book of three liquid stocks of the Paris Bourse. The results are to a large degree independent of the stock studied. The most interesting features concern (i) the statistics of incoming limit order prices, which follows a power-law around the current price with a diverging mean; and (ii) the humped shape of the average order book, which can be quantitatively reproduced using a `zero intelligence' numerical model, and qualitatively predicted using a simple approximation
Article
We introduce a new analysis of transaction costs that explicitly recognizes the importance of the timing of execution in assessing transaction costs. Time induces a risk/cost tradeoff. The price of immediacy results in higher costs for quickly executed orders while more gradual trading results in higher risk since the value of the asset can vary more over longer periods of time. We use a novel data set that allows a sequence of transactions to be associated with individual orders and measure and model the expected cost and risk associated with different order execution approaches. The model yields a risk/cost tradeoff that depends upon the state of the market and characteristics of the order. We show how to assess liquidation risk using the notion of liquidation value at risk (LVAR).
Article
Buying and selling stocks causes price changes, which are described by the price impact function. To explain the shape of this function, we study the Island ECN orderbook. In addition to transaction data, the orderbook contains information about potential supply and demand for a stock. The virtual price impact calculated from this information is four times stronger than the actual one and explains it only partially. However, we find a strong anticorrelation between price changes and order flow, which strongly reduces the virtual price impact and provides for an explanation of the empirical price impact function.
Article
This paper derives a static optimal execution strategy of a VWAP trade, in which the optimal execution strategy can be calculated by an iteration of a single variable optimization, rather than by a multivariable optimization. Analytical solutions are derived in some cases. We show that optimal execution times lag behind expected market trading volume distribution since price volatility tends to have a positive correlation with market trading volume. In a basket trade, execution error can be reduced by spreading out execution times according to the correlation of price movement. We confirm our theoretical results with actual trading data and simulations.
Article
It is known that the impact of transactions on stock price (market impact) is a concave function of the size of the order, but there exists little quantitative theory that suggests why this is so. I develop a quantitative theory for the market impact of hidden orders (orders that reflect the true intention of buying and selling) that matches the empirically measured result and that reproduces some of the non-trivial and universal properties of stock returns (returns are percent changes in stock price). The theory is based on a simple premise, that the stock market can be modeled in a mechanical way - as a device that translates order flow into an uncorrelated price stream. Given that order flow is highly autocorrelated, this premise requires that market impact (1) depends on past order flow and (2) is asymmetric for buying and selling. I derive the specific form for the dependence in (1) by assuming that current liquidity responds to information about all currently active hidden orders (liquidity is a measure of the price response to a transaction of a given size). This produces an equation that suggests market impact should scale logarithmically with total order size. Using data from the London Stock Exchange I empirically measure market impact and show that the result matches the theory. Also using empirical data, I qualitatively specify the asymmetry of (2). Putting all results together, I form a model for market impact that reproduces three universal properties of stock returns - that returns are uncorrelated, that returns are distributed with a power law tail, and that the magnitude of returns is highly autocorrelated (also known as clustered volatility).
Article
We investigate present some new statistical properties of order books. We analyse data from the Nasdaq and investigate (a) the statistics of incoming limit order prices, (b) the shape of the average order book, and (c) the typical life time of a limit order as a function of the distance from the best price. We also determine the `price impact' function using French and British stocks, and find a logarithmic, rather than a power-law, dependence of the price response on the volume. The weak time dependence of the response function shows that the impact is, surprisingly, quasi-permanent, and suggests that trading itself is interpreted by the market as new information.
Article
The supply/demand of a security in the market is an intertemporal, not a static, object and its dynamics is crucial in determining market participants' trading behavior. Previous studies on the optimal trading strategy to execute a given order focuses mostly on the static properties of the supply/demand. In this paper, we show that the dynamics of the supply/demand is of critical importance to the optimal execution strategy, especially when trading times are endogenously chosen. Using a limit-order-book market, we develop a simple framework to model the dynamics of supply/demand and its impact on execution cost. We show that the optimal execution strategy involves both discrete and continuous trades, not only continuous trades as previous work suggested. The cost savings from the optimal strategy over the simple continuous strategy can be substantial. We also show that the predictions about the optimal trading behavior can have interesting implications on the observed behavior of intraday volume, volatility and prices.
Article
In an environment where trading volume affects security prices and where prices are uncertain when trades are submitted, quasi-arbitrage is the availability of a series of trades that generate infinite expected profits with an infinite Sharpe ratio. We show that when the price impact of trades is permanent and time-independent, only linear price-impact functions rule out quasi-arbitrage and thus support viable market prices. When trades have also a temporary price impact, only the permanent price impact must be linear while the temporary one can be of a more general form. We also extend the analysis to a time-dependent framework. Copyright The Econometric Society 2004.
Article
In this article we revisit the classic problem of tatonnement in price formation from a microstructure point of view, reviewing a recent body of theoretical and empirical work explaining how fluctuations in supply and demand are slowly incorporated into prices. Because revealed market liquidity is extremely low, large orders to buy or sell can only be traded incrementally, over periods of time as long as months. As a result order flow is a highly persistent long-memory process. Maintaining compatibility with market efficiency has profound consequences on price formation, on the dynamics of liquidity, and on the nature of impact. We review a body of theory that makes detailed quantitative predictions about the volume and time dependence of market impact, the bid-ask spread, order book dynamics, and volatility. Comparisons to data yield some encouraging successes. This framework suggests a novel interpretation of financial information, in which agents are at best only weakly informed and all have a similar and extremely noisy impact on prices. Most of the processed information appears to come from supply and demand itself, rather than from external news. The ideas reviewed here are relevant to market microstructure regulation, agent-based models, cost-optimal execution strategies, and understanding market ecologies.
Article
We consider optimal execution strategies for block market orders placed in a limit order book (LOB). We build on the resilience model proposed by Obizhaeva and Wang (200516. Obizhaeva , A and Wang , J . 2005. Optimal trading strategy and supply/demand dynamics, Preprint Available online at: http://www.rhsmith.umd.edu/faculty/obizhaeva/OW060408.pdf (accessed 16 February 2009) [CrossRef]View all references) but allow for a general shape of the LOB defined via a given density function. Thus, we can allow for empirically observed LOB shapes and obtain a nonlinear price impact of market orders. We distinguish two possibilities for modelling the resilience of the LOB after a large market order: the exponential recovery of the number of limit orders, i.e. of the volume of the LOB, or the exponential recovery of the bid–ask spread. We consider both of these resilience modes and, in each case, derive explicit optimal execution strategies in discrete time. Applying our results to a block-shaped LOB, we obtain a new closed-form representation for the optimal strategy of a risk-neutral investor, which explicitly solves the recursive scheme given in Obizhaeva and Wang (200516. Obizhaeva , A and Wang , J . 2005. Optimal trading strategy and supply/demand dynamics, Preprint Available online at: http://www.rhsmith.umd.edu/faculty/obizhaeva/OW060408.pdf (accessed 16 February 2009) [CrossRef]View all references). We also provide some evidence for the robustness of optimal strategies with respect to the choice of the shape function and the resilience-type.
Market impact model handbook
Barra, 1997, Market impact model handbook,.