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Equity Market Impact

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... Our third contribution is to present an empirical study on how dynamic portfolio allocations are affected by transient and permanent liquidity effects. We apply our method to solve a realistic cash-and-stock portfolio allocation problem, for which we adopt the power-law liquidity model of Almgren, Thum, Hauptmann, and Li (2005). We measure the certainty equivalent losses associated with ignoring liquidity issues, and illustrate the ability of our dynamic allocation to protect the investor's capital in illiquid markets. ...
... In reality, the shape of the limit order book differs by the characteristics of the portfolio assets. A power law of both liquidity cost and market impact for the U.S. stock markets has been found in Almgren et al. (2005). Obizhaeva and Wang (2013) assume a linear price shift and uses a negative exponential function to model the resilience of the limit order book. ...
... In this paper, we adopt the calibrated power law functions of Almgren et al. (2005) to analyze the impact of the market illiquidity on the dynamic portfolio selection problem. These power law functions are given by ...
Article
Full-text available
We present a simulation-and-regression method for solving dynamic portfolio optimization problems in the presence of general transaction costs, liquidity costs and market impact. This method extends the classical least squares Monte Carlo algorithm to incorporate switching costs, corresponding to transaction costs and transient liquidity costs, as well as multiple endogenous state variables, namely the portfolio value and the asset prices subject to permanent market impact. To handle endogenous state variables, we adapt a control randomization approach to portfolio optimization problems and further improve the numerical accuracy of this technique for the case of discrete controls. We validate our modified numerical method by solving a realistic cash-and-stock portfolio with a power-law liquidity model. We identify the certainty equivalent losses associated with ignoring liquidity effects, and illustrate how our dynamic optimization method protects the investor's capital under illiquid market conditions. Lastly, we analyze, under different liquidity conditions, the sensitivities of certainty equivalent returns and optimal allocations with respect to trading volume, stock price volatility, initial investment amount, risk aversion level and investment horizon. © 2018
... The literature devoted to the study of the market impact itself is quite vast, and it is more frequent to find an impact that varies as a power law of the size of the trade (see, e.g., [2]). Although our linear approach is clearly not the most realistic in terms of market microstructure, it has the advantage of avoiding arbitrage opportunities, as well as not being sensitive to the hedging frequency: in a nonlinear model, splitting an order in half would induce a different market response compared to executing a single trade, a situation that we want to avoid here, as we aim at deriving a time continuous formulation. ...
... Let us rewrite equation (2) as follows: ...
... There is also a clear connection between this work and the work by Soner, Touzi and Zhang [35] about dual formulation of second-order target problems, and also with the problem of optimal transport by controlled diffusions studied in [36]. Indeed observe that the elliptic operator F (γ ) in (2) is convex, and then applying the reasoning of [35] based on the Legendre transform representation of F , one can show formally that a solution to (2) will also be solution of the following variational problem: ...
Article
We consider a model of linear market impact, and address the problem of replicating a contingent claim in this framework. We derive a nonlinear Black–Scholes equation that provides an exact replication strategy. This equation is fully nonlinear and singular, but we show that it is well posed, and we prove existence of smooth solutions for a large class of final payoffs, both for constant and local volatility. To obtain regularity of the solutions, we develop an original method based on Legendre transforms. The close connections with the problem of hedging with gamma constraints [SIAM J. Control Optim. 39 (2000) 73–96, Math. Finance 17 (2007) 59–80, Ann. Inst. H. Poincaré Anal. Non Linéaire 22 (2005) 633–666], with the problem of hedging under liquidity costs [Finance Stoch. 14 (2010) 317–341] are discussed. The optimal strategy and associated diffusion are related with the second-order target problems of [Ann. Appl. Probab. 23 (2013) 308–347], and with the solutions of optimal transport problems by diffusions of [Ann. Probab. 41 (2013) 3201–3240]. We also derive a modified Black–Scholes formula valid for asymptotically small impact parameter, and finally provide numerical simulations as an illustration.
... Our third contribution is to present an empirical study on how dynamic portfolio allocations are affected by transient and permanent liquidity effects. We apply our method to solve a realistic cash-and-stock portfolio allocation problem, for which we adopt the power-law liquidity model of Almgren, Thum, Hauptmann, and Li (2005). We measure the certainty equivalent losses associated with ignoring liquidity issues, and illustrate the ability of our dynamic allocation to protect the investor's capital in illiquid markets. ...
... In reality, the shape of the limit order book differs by the characteristics of the portfolio assets. A power law of both liquidity cost and market impact for the U.S. stock markets has been found in Almgren et al. (2005). Obizhaeva and Wang (2013) assume a linear price shift and uses a negative exponential function to model the resilience of the limit order book. ...
... In this paper, we adopt the calibrated power law functions of Almgren et al. (2005) to analyze the impact of the market illiquidity on the dynamic portfolio selection problem. These power law functions are given by ...
... Three typical liquidity policies, namely cash, minimum weight and portfolio expected shortfall (ES) liquidity policies on a long equity portfolio are combined and used as constraints on the performance of the portfolio. We derive a power-law MSDC from the results of Almgren et al. (2005), correcting a formula presented by Finger (2011). ...
... In a recent study by Tian et al. (2013), the MSDC is approximated by exponentials, however, this requires high frequency price information in the order book to model a ladder MSDC. In order to determine a practical MSDC lacking this price information, a power-law MSDC is developed in Chapter 3 derived from the seminal work of Almgren et al. (2005) who proposed that impact is a 3 5 power law of block size. ...
... In this chapter, we quantify the potential cost of liquidity constraints on a long equity portfolio using the liquidity risk framework of Acerbi and Scandolo (2008) and a power-law MSDC derived in this thesis from the results of Almgren et al. (2005). We look at the relationship between the portfolio value and constraints the portfolio may be subject to in practice. ...
Thesis
Full-text available
In this thesis we quantify the potential cost of liquidity constraints on a long equity portfolio using the liquidity risk framework of Acerbi and Scandolo (2008). The model modifies the classical mark-to-market valuation model, and incorporates the impact of liquidity policies of portfolios on the liquidity adjustment valuation (LVA). Also, we suggest a quantitative indicator that scores market liquidity ranging from 0 to 1 (perfect liquidity) for a portfolio with possible liquidity constraints. The thesis consists of three major studies. In the first one, we compute LVA given the cash, minimum weight and portfolio expected shortfall (ES) liquidity policies on a long equity portfolio. Several numerical examples in the results demonstrate the importance associated the incorporation of the liquidity policy in the liquidity risk valuation. In the second study, we quantify the execution costs and the revenue risk when implementing trading strategies over multiple periods by employing the transaction costs measure of Garleanu and Pedersen (2013). The portfolio liquidity costs estimated from the model of Garleanu and Pedersen (2013) are compared with the costs estimated from the liquidity risk measure of Finger (2011). In the third study, we estimate the liquidity-adjusted portfolio ES for a long equity portfolio with the liquidity constraints. Portfolio pure market P&L scenarios are based on initial positions, and the liquidity adjustments are based on positions sold, which depend on the specified liquidity constraints. Portfolio pure market P&L scenarios and state-dependent liquidity adjustments are integrated to obtain liquidity-adjusted P&L scenarios. Then, we apply the liquidity score method (Meucci, 2012) on the liquidity-plus-market P&L distribution to quantify the market liquidity for the portfolio. The results show the importance of pricing liquidity risk with liquidity constraints. The liquidity costs can vary greatly on different liquidity policies, portfolio MtM values, market situation and time to liquidation.
... In this paper, we quantify the potential cost of liquidity constraints on a typical mutual fund and compute the value given the liquidity policy. Our contribution is to formulate and combine cash, minimum weight and portfolio expected shortfall (ES) liquidity policies on a typical mutual fund (long equity portfolio), and to quantify the portfolio liquidity risk in the presence of liquidity constraints using the liquidity risk framework of Acerbi & Scandolo (2008) and a power-law MSDC derived in this paper from the results of Almgren et al. (2005). We correct a formula presented by Finger (2011). ...
... In a recent study by Tian et al. (2013), the MSDC is approximated by exponentials, however, this requires high frequency price information in the order book to model a ladder MSDC. In order to determine a practical MSDC lacking this price information, we develop a power-law MSDC derived from the seminal work of Almgren et al. (2005) who proposed that impact is a 3 5 power law of block size. The power-law MSDC is a continuous approximation of the MSDC. ...
... The power-law MSDC has econometric parameters; the daily volatility and average daily volume to characterize the slope of the curve. Volatility, as Almgren et al. (2005) note, is empirically the most influential factor on the market impact cost. The flatter (steeper) the slope of the power-law demand curve, the more liquid (illiquid) the asset is. ...
Article
Full-text available
In the standard approach to fund valuation, it is often assumed that markets are perfectly liquid and hence assets have unique prices. In practice, however, as has been widely documented , this is not the case. Asset values are impacted by deterioration of market liquidity (market depth). However, the work by Acerbi and Scandolo (Quantitative Finance, 2008, 8(7), 681) highlighted, in addition, the key role that the liquidity policies of the fund can have on the fund value. Funds with the identical positions but differing liquidity policies have different values. In this paper, we describe and analyze, the relationship between the fund value and the liquidity policy (constraints that can potentially be called upon, and that are imposed externally on the fund manager). These constraints, for example, can be a requirement to be able to generate an amount of cash (due to possible adverse redemption), a minimum weight requirement per asset, or a market risk constraint such as a maximum expected shortfall (ES). We employ a power-law Marginal Supply-Demand Curve (MSDC) * Corresponding author: johara@essex.ac.uk 1 to model market depth, together with various types of liquidity policies. These liquidity restrictions affect, to a varying degree, the liquidation process of illiquid and high volatility assets. We find that as the liquidity policy becomes stricter with the portfolio ES constraint, the liquidity costs increase significantly.
... The non-linearity of temporary impact in the trading velocity has been addressed by Almgren (2003). Almgren et al. (2005) showed that empirical data fits a concave impact law with exponent 0.6. Obizhaeva and Kyle (2013) showed that the size exponent could be affected by heterogenities in the data and proposed an invariance principle to isolate a factor that captures the effect of market capitalization on impact, but still find a size exponent of 0.56 in a large dataset of portfolio trades. ...
... Integrating over time to compute the average price, it follows that the implementation shortfall also grows as a square root of the trade size. This result is in agreement with Torre (1997) and with phenomenological models including Lakonishok (1993) and(1995), Almgren et al. (2005), Bouchaud et al. (2009), Moro et al. (2009) andToth et al. (2011). However, the model is limited to a uniform participation rate, so it does not address the optimal execution problem. ...
... Price models (1) and (2) with price dynamics (3) or (4) provide us with a rich setting to exhibit a variety of models frequently used in the literature on portfolio liquidation and price impact modeling, e.g., see (Almgren andChriss, 2000/2001;Huberman and Stanzl, 2004;Almgren et al., 2005;Huberman and Stanzl, 2005;Moazeni et al., 2010Moazeni et al., , 2013Moazeni et al., , 2016Foucault et al., 2013;Amihud et al., 2013;Gueant, 2014). For instance, Bertsimas and Lo (1998), Almgren andChriss (2000/2001), and Almgren et al. (2005) consider the additive model (3) withL where the components of the l-vectorZ t are independent standard normals and Σ is an m × l volatility matrix of the asset returns. ...
... Price models (1) and (2) with price dynamics (3) or (4) provide us with a rich setting to exhibit a variety of models frequently used in the literature on portfolio liquidation and price impact modeling, e.g., see (Almgren andChriss, 2000/2001;Huberman and Stanzl, 2004;Almgren et al., 2005;Huberman and Stanzl, 2005;Moazeni et al., 2010Moazeni et al., , 2013Moazeni et al., , 2016Foucault et al., 2013;Amihud et al., 2013;Gueant, 2014). For instance, Bertsimas and Lo (1998), Almgren andChriss (2000/2001), and Almgren et al. (2005) consider the additive model (3) withL where the components of the l-vectorZ t are independent standard normals and Σ is an m × l volatility matrix of the asset returns. Hence, D t (P t−1 ) = P t−1 + ΣZ t . ...
Article
Arguments on the existence of dynamic arbitrage and price manipulation strategies are often invoked to guide modeling price impacts of large trades. We revisit the concept of dynamic arbitrage in illiquid markets in the presence of time-varying stochastic price impact functions and a broad class of market price dynamics. We first establish a sufficient condition under which searching in the space of F0-measurable admissible round-trip trades is enough to attain a no-dynamic arbitrage certificate. This result simplifies identifying price impact structures that rule out dynamic arbitrage and accredits the analysis in some existing literature, where its assessment is limited to the search in the set of F0-measurable round-trip trades. For time-varying stochastic linear price impact functions, we show that this condition is necessary and sufficient for the absence of dynamic arbitrage. The present quantitative analysis implies that a trader’s opinion on the existence of dynamic arbitrage opportunities for a price impact model depends on his belief about expected future price changes and expected future price impacts, which can be revised over time by the collection of new information. This motivates us to let the existence of such arbitrage opportunities not only depend on the investor’s belief on expected price movements but also depend on the trader’s risk attitude. We thus introduce the concept of risk-averse dynamic arbitrage using a general time-consistent dynamic risk measure and a risk-aversion threshold level. Similar sufficient conditions are studied under which searching in the space of static round-trip trades enables us to conclude on no-risk-averse dynamic arbitrage.
... The downside of such a delayed approach is that the execution of the entire trade order is postponed, which may lead to a loss in opportunities caused by (unfavorable) changes in market prices. The change in the price of a security is impacted by the size of the transaction and is often modelled as a concave monotonically increasing function of the trade size [1,44]. In that vein, Lillo et al. [26] and Gabaix et al. [14] model market impact costs as a concave power law function of the transaction size. ...
... In Barra's Market Impact Model Handbook [42], it is showed that the square-root formula fits transaction cost data remarkably well. An empirical study conducted by Almgren et al. [1] advocates to set the price change as proportional to a 3/5 power law of block size. ...
Article
We define a regularized variant of the Dual Dynamic Programming algorithm called REDDP (REgularized Dual Dynamic Programming) to solve nonlinear dynamic programming equations. We extend the algorithm to solve nonlinear stochastic dynamic programming equations. The corresponding algorithm, called SDDP-REG, can be seen as an extension of a regularization of the Stochastic Dual Dynamic Programming (SDDP) algorithm recently introduced which was studied for linear problems only and with less general prox-centers. We show the convergence of REDDP and SDDP-REG. We assess the performance of REDDP and SDDP-REG on portfolio models with direct transaction and market impact costs. In particular, we propose a risk-neutral portfolio selection model which can be cast as a multistage stochastic second-order cone program. The formulation is motivated by the impact of market impact costs on large portfolio rebalancing operations. Numerical simulations show that REDDP is much quicker than DDP on all problem instances considered (up to 184 times quicker than DDP) and that SDDP-REG is quicker on the instances of portfolio selection problems with market impact costs tested and much faster on the instance of risk-neutral multistage stochastic linear program implemented (8.2 times faster).
... 34 So, what should we use as τ i in (91)? The model of [Almgren et al, 2005] is reasonable for our purposes here. Let H i be the dollar amount traded for the stock labeled by i. ...
Preprint
We discuss - in what is intended to be a pedagogical fashion - generalized "mean-to-risk" ratios for portfolio optimization. The Sharpe ratio is only one example of such generalized "mean-to-risk" ratios. Another example is what we term the Fano ratio (which, unlike the Sharpe ratio, is independent of the time horizon). Thus, for long-only portfolios optimizing the Fano ratio generally results in a more diversified and less skewed portfolio (compared with optimizing the Sharpe ratio). We give an explicit algorithm for such optimization. We also discuss (Fano-ratio-inspired) long-short strategies that outperform those based on optimizing the Sharpe ratio in our backtests.
... Such kind of assumption is also made in the seminal paper [2], and reflects stylized facts on limit order books. The form (2.5) was suggested in several empirical studies, see [12], [17], [3], and used also in [8], [11]. ...
Preprint
This paper deals with numerical solutions to an impulse control problem arising from optimal portfolio liquidation with bid-ask spread and market price impact penalizing speedy execution trades. The corresponding dynamic programming (DP) equation is a quasi-variational inequality (QVI) with solvency constraint satisfied by the value function in the sense of constrained viscosity solutions. By taking advantage of the lag variable tracking the time interval between trades, we can provide an explicit backward numerical scheme for the time discretization of the DPQVI. The convergence of this discrete-time scheme is shown by viscosity solutions arguments. An optimal quantization method is used for computing the (conditional) expectations arising in this scheme. Numerical results are presented by examining the behaviour of optimal liquidation strategies, and comparative performance analysis with respect to some benchmark execution strategies. We also illustrate our optimal liquidation algorithm on real data, and observe various interesting patterns of order execution strategies. Finally, we provide some numerical tests of sensitivity with respect to the bid/ask spread and market impact parameters.
... However, empirical studies show that the price impacts are a non-linear concave function of trade sizes. For example, power-law models with an exponent γ ranging from 0.2 to 0.8 are uncovered from different datasets in terms of markets, periods and order properties 21,15,8,20,1,16,7,12,26 . Several theoretical explanations are proposed for the size dependence of price impact. ...
Preprint
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 findings may provide valuable insights for optimal trade execution.
... It is common knowledge that cost of market impact can cut down a large proportion of the trading strategy's profit. Therefore, we can use a simple permanent-temporary market impact model, inspired by [2]. ...
Preprint
Full-text available
In this article, we provide a flexible framework for optimal trading in an asset listed on different venues. We take into account the dependencies between the imbalance and spread of the venues, and allow for partial execution of limit orders at different limits as well as market orders. We present a Bayesian update of the model parameters to take into account possibly changing market conditions and propose extensions to include short/long trading signals, market impact or hidden liquidity. To solve the stochastic control problem of the trader we apply the finite difference method and also develop a deep reinforcement learning algorithm allowing to consider more complex settings.
... While the exchange providers make valid arguments that they provide different business propositions, there is a growing concern in the industry that the revenue model for these exchanges is now largely centered around providing market data and charging for exchange connectivity fees. 11 Because the venue quotes are protected, any serious operator needs to connect with the exchanges and leverage their direct market data feeds in their trading applications. With more exchanges, the more connections and fees these exchange can charge. ...
... During violent market routs, however, execution costs/slippage can be much higher for sell orders, and especially short-sales (see above). 78 The trading costs are based on the model of [Almgren et al, 2005] -see [Kakushadze and Serur, 2018] for details. 79 We assume no leverage in our backtests. ...
Preprint
We explain in a nontechnical fashion why dollar-neutral quant trading strategies, such as equities Statistical Arbitrage, suffered substantial losses (drawdowns) during the COVID-19 market selloff. We discuss: (i) why these strategies work during "normal" times; (ii) the market regimes when they work best; and (iii) their limitations and the reasons for why they "break" during extreme market events. An accompanying appendix (with a link to freely accessible source code) includes backtests for various strategies, which put flesh on and illustrate the discussion in the main text.
... The choice of the temporary impact function is more flexible. For instance, Almgren [Alm03] assumed that the magnitude of the temporary market impact is a power law function of the trading rate, which was estimated through a square-root law in [Alm03] and a 3/5 power law in [ATHL05] based on the available historical transaction data. To better capture the intertemporal nature of supply and demand in the market, Obizhaeva and Wang [OW13] proposed another kind of price impact that is persistent (or transient) with the impact of past trades on current prices decaying over time. ...
Thesis
Full-text available
Diese Dissertation analysiert BSDEs und PDEs mit singulären Endbedingungen, welche in Problemen der optimalen Portfolioliquidierung auftreten. In den vergangenen Jahren haben Portfolioliquidierungsprobleme in der Literatur zur Finanzmathematik große Aufmerksamkeit erhalten. Ihre wichtigste Eigenschaft ist die singuläre Endbedingung der durch die Liquidierungsbedingung induzierten Wertfunktion, welche eine singuläre Endbedingung der zugehörigen BSDE oder PDE impliziert. Diese Arbeit besteht aus drei Kapiteln. Das erste Kapitel analysiert ein Portfolioliquidierungsproblem für mehrere Wertpapiere mit sofortigem und anhaltendem Preiseinfluss und stochastischer Resilienz. Wir zeigen, dass die Wertfunktion durch eine mehrdimensionale BSRDE mit singulärer Endbedingung beschrieben werden kann. Wir weisen die Existenz einer Lösung dieser BSRDE nach und zeigen, dass diese durch eine Folge von Lösungen von BSRDEs mit endlicher und wachsender Endbedingung approximiert werden kann. Eine neue a priori-Abschätzung für die approximierenden BSRDEs wird für den Nachweis hergeleitet. Das zweite Kapitel betrachtet ein Portfolioliquidierungsproblem mit unbeschränkten Kostenkoeffizienten. Wir weisen die Existenz einer eindeutigen nichtnegativen Viskositätslösung der HJB-Gleichung nach. Das Existenzresultat basiert auf einem neuartigen Vergleichsprinzip für semi-stetige Viskositätssub-/-superlösungen für singuläre PDEs. Stetigkeit der Viskositätslösung ist hinreichend für das Verifikationsargument. Im dritten Kapitel untersuchen wir ein optimales Liquidierungsproblem unter Mehrdeutigkeit der Parameter des Preiseinflusses. In diesem Fall kann die Wertfunktion durch die Lösung einer semilinearen PDE mit superlinearem Gradienten beschrieben werden. Zuerst zeigen wir die Existenz einer Viskositätslösung indem wir unser Vergleichsprinzip für singuläre PDEs erweitern. Sodann weisen wir die Regularität mit einer asymptotischen Entwicklung der Lösung am Endzeitpunkt nach.
... h(S, t) dS (1) and the quantity flux is defined as ...
Preprint
Full-text available
We present a class of macroscopic models of the Limit Order Book to simulate the aggregate behaviour of market makers in response to trading flows. The resulting models are solved numerically and asymptotically, and a class of similarity solutions linked to order book formation and recovery is explored. The main result is that order book recovery from aggressive liquidity taking follows a simple t1/3t^{1/3} scaling law.
... 34 So, what should we use as τ i in (91)? The model of [Almgren et al, 2005] is reasonable for our purposes here. Let H i be the dollar amount traded for the stock labeled by i. ...
Article
We discuss - in what is intended to be a pedagogical fashion - generalized "mean-to-risk" ratios for portfolio optimization. The Sharpe ratio is only one example of such generalized "mean-to-risk" ratios. Another example is what we term the Fano ratio (which, unlike the Sharpe ratio, is independent of the time horizon). Thus, for long-only portfolios optimizing the Fano ratio generally results in a more diversified and less skewed portfolio (compared with optimizing the Sharpe ratio). We give an explicit algorithm for such optimization. We also discuss (Fano-ratio-inspired) long-short strategies that outperform those based on optimizing the Sharpe ratio in our backtests.
... However, empirical studies show that the price impacts are a non-linear concave function of trade sizes. For example, power-law models with an exponent γ ranging from 0.2 to 0.8 are uncovered from different datasets in terms of markets, periods and order properties 21,15,8,20,1,16,7,12,26 . Several theoretical explanations are proposed for the size dependence of price impact. ...
Article
Full-text available
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 findings may provide valuable insights for optimal trade execution.
... Such kind of assumption is also made in the seminal paper [2], and reflects stylized facts on limit order books. The form (2.5) was suggested in several empirical studies, see [13], [18], [3], and used also in [8], [11]. ...
... The price dynamics is the result of the interplay between the incoming order flow and the order book [4].Figure 1 is a schematic illustration of this process [11]. Note that we chose to represent quantities on the bid side of the book by non-positive numbers. ...
Article
Motivated by the desire to bridge the gap between the microscopic description of price formation (agent-based modeling) and the stochastic differential equations approach used classically to describe price evolution at macro-scopic time scales, we present a mathematical study of the order book as a mul-tidimensional continuous-time Markov chain and derive several mathematical results in the case of independent Poissonian arrival times. In particular, we show that the cancellation structure is an important factor ensuring the existence of a stationary distribution and the exponential convergence towards it. We also prove, by means of the functional central limit theorem (FCLT), that the rescaled-centered price process converges to a Brownian motion. We illustrate the analysis with numerical simulation and comparison against market data.
... Such kind of assumption is also made in the seminal paper [3], and reflects stylized facts on limit order books. The form (4.2.5) was suggested in several empirical studies, see [50], [60], [4], and used also in [28], [44]. ...
Article
We propose a quantitative approach to some high frequency trading problematics. We are interested in several aspects of this field, from minimizing indirect trading costs to market making, and more generally in profit maximization strategies over a finite time horizon. We build an original framework that reflects specificities of high frequency trading, and especially the distinction between passive and active trading, thanks to mixed stochastic control methods. We carefully model high fequency market phenomena, and for each of them we propose calibration methods that are compatible with practical constraints of algorithmic trading.
... A usual form (see e.g. [51], [71], [2]) of temporary price impact and transaction cost function f , suggested by empirical studies is ...
Article
We study the link between Backward SDEs and some stochastic optimal control problems and their application to mathematical finance. In the first part we focus on the BSDE representation of solution to impulse control and optimal switching. We first introduce the notion of constrained BSDEs with jumps and prove that it gives a representation of solutions to Markovian impulse control problems. We then bind these contrained BSDEs to BSDEs with oblique reflexion and optimal switching problems. In the secoond part, we study the time discretization of the previous BSDEs. We first state a discretization of constrained BSDE using the approximation given by the penalized BSDEs. We the provide a speed convergence for the natural scheme associated to BSDEs with oblique reflections. Finally, in the third part, we consider a liquidation problem under execution risk and cost. We characterize the associated value function as the minimal solution to the associated quasi-variational inequality.
... The square root impact of order flow on returns in the above specification reflects the concave impact of trades on returns commonly accepted in the literature (for example, Hasbrouck and Seppi (2001) and Almgren, Thum, Hauptmann, and Li (2005)). The use absolute order flow and sign (OFm t ) Rm t as dependent variable allows us to capture the heterogeneity among announcement types using the fixed effects am. ...
Research
Full-text available
We examine stock index and Treasury futures markets around releases of U.S. macroeconomic announcements. Seven out of 18 market-moving announcements show evidence of substantial informed trading before the official release time. Prices begin to move in the “correct” direction about 30 minutes before the release time. The pre-announcement price drift accounts on average for about half of the total price adjustment. These results imply that some traders have private information about macroeconomic fundamentals. The evidence points to leakage and proprietary data collection as the most likely sources of that private information.
... 9 Dutt and Harris (2005) evaluated ε from the transaction cost prediction models of ITG and Goldman, which focus on US markets. Instead, we evaluate ε as the market impact cost in the CSI 300 futures market according to the method of Almgren, Thum, Hauptmann and Li (2005). We choose 1-month high-frequency transaction data to estimate the market impact cost. ...
Article
The aim of this study was to find the optimal position limit for the Chinese stock index (CSI) 300 futures market. A low position limit helps to prevent price manipulations in the spot market, and thus keeps the magnitude of instantaneous price changes within the tolerance range of policymakers. However, setting a position limit that is too low may also have negative effects on market quality. We propose an artificial limit order market with heterogeneous interacting agents to examine the impact of different levels of position limits on market quality, measured as liquidity, return volatility, efficiency of information dissemination, and trading welfare. The simulation model is based on realistic trading mechanisms, investor structure, and order submission behavior observed in the CSI 300 futures market.
... [22] So first we consider the discrete-time model of an optimal execution problem with MI and then derive the continuous-time model as their limit. We consider the case when MI function is convex with respect to the execution volume of a trader, whereas some empirical studies tell us that MI function is concave ([3] etc.) But [7] pointed out that (much of this variation comes about because these studies actually measure different things. ...
Conference Paper
Full-text available
We study the optimal execution problem in the market model in consideration of market impact as a regular stochastic control problem. We focus on mathematical formulation and characterization as the viscosity solution of the corresponding non-linear partial differential equation (so-called HJB.) First we introduce the outline of the theory of an optimal portfolio management problem and liquidity problems which are both important in mathematical finance. Then we formulate our optimal execution problem as the discrete-time model and describe the value function with respect to a trader's optimization problem. By shortening the intervals of execution times, we derive the value function of the continuous-time model and then we study some properties of them. We show that the properties of the continuous-time value function vary by the strength of market impact. Moreover we introduce some examples of this model, which tell us that the forms of the optimal execution strategies entirely change according to the amount of the security holdings.
... These results are consistent with the findings of several empirical studies (for instance, [5] and [14] $)$ . And these also claim that to select an adequate execution policy is significant to control MI cost. ...
Article
Price impact is the adverse change of the asset price against trader's action. As a crucial part of the indirect trading cost, price impact has attracted increasing attention in both econometric and data science literature. In this paper, we draw upon both strands of the literature and develop a deep neural network enhanced recursive (DeRecv) model to estimate temporary and permanent price impact of an order or trade. The temporary price impact is calculated as the sum of the expected immediate impact at each time point after taking action in an ad hoc market condition. The permanent price impact is defined as a new permanent level at which the information of the incoming order is entirely absorbed by the market. Through the experimental evaluation based on data from 10 stocks at NASDAQ and Shanghai Stock Exchange, we show that the proposed DeRecv model is better than the reinforcement learning model and the traditional vector autoregressive model.
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We compare trade size to the prevailing market depth at the best level in the limit order book to detect and account for zero impact trades in an immediate price impact model. Our model also incorporates standard trade attributes (trade size, market capitalization and volatility) in a dynamic setting. The incorporation of market depth information reduces the mean absolute/squared forecast error of an immediate price impact prediction by about 60%. After controlling for trade attributes, market depth, price impact dynamics and intra-and inter- day periodicities (in order of relative importance) all improve the prediction of a trade’s price impact. We demonstrate the value of our model by showing that splitting a big order into a series of smaller trades results in a reduction of between 60% and 82% of the immediate price impact cost of the big order. We also find that our depth indicator helps with the prediction of order flow and permanent price impact.
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We define a regularized variant of the dual dynamic programming algorithm called DDP-REG to solve nonlinear dynamic programming equations. We extend the algorithm to solve nonlinear stochastic dynamic programming equations. The corresponding algorithm, called SDDP-REG, can be seen as an extension of a regularization of the stochastic dual dynamic programming (SDDP) algorithm recently introduced which was studied for linear problems only and with less general prox-centers. We show the convergence of DDP-REG and SDDP-REG. We assess the performance of DDP-REG and SDDP-REG on portfolio models with direct transaction and market impact costs. In particular, we propose a risk-neutral portfolio selection model which can be cast as a multistage stochastic second-order cone program. The formulation is motivated by the impact of market impact costs on large portfolio rebalancing operations. Numerical simulations show that DDP-REG is much quicker than DDP on all problem instances considered (up to 184 times quicker than DDP) and that SDDP-REG is quicker on the instances of portfolio selection problems with market impact costs tested and much faster on the instance of risk-neutral multistage stochastic linear program implemented (8.2 times faster).
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This work studies the constrained optimal execution problem with a random market depth in the limit order market. Motivated from the real trading activities, our execution model considers the execution bounds and allows the random market depth to be statistically correlated in different periods. Usually, it is difficult to achieve the analytical solution for this class of constrained dynamic decision problem. Thanks to the special structure of this model, by applying the proposed state separation theorem and dynamic programming, we successfully obtain the analytical execution policy. The revealed policy is of feedback nature. Examples are provided to illustrate our solution methods. Simulation results demonstrate the advantages of our model comparing with the classical execution policy.
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In this study, we introduce an explicit trading-volume process into the Almgren--Chriss model, which is a standard model for optimal execution. We propose a penalization method of deriving a verification theorem for an adaptive optimization problem. We also discuss the optimality of the volume-weighted average-price strategy of a risk-neutral trader. Moreover, we derive a second-order asymptotic expansion of the optimal strategy and verify its accuracy numerically.
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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.
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In this study, we introduce an explicit trading-volume process into the Almgren--Chriss model, which is a standard model for optimal execution. We propose a penalization method of deriving a verification theorem for an adaptive optimization problem. We also discuss the optimality of the volume-weighted average-price strategy of a risk-neutral trader. Moreover, we derive a second-order asymptotic expansion of the optimal strategy and verify its accuracy numerically.
Chapter
A thin market is a market with few buying or selling offers. The concept of market thinness, while general, is typically used in the context of financial markets. When the number of buying or selling offers is small, investors’ trading positions are large relative to market size. Trading then requires price concessions and thus exerts an impact on prices. A thin market is characterized by low trading volume, high volatility and high bid–ask spreads. This article discusses the modelling of thin markets, some typical phenomena of such markets, and their implications for market design.
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As a consequence of recent technological advances and the proliferation of algorithmic and high-frequency trading, the cost of trading in financial markets has irrevocably changed. One important change, known as price impact, relates to how trading affects prices. Price impact represents the largest cost associated with trading. Forecasting price impact is very important as it can provide estimates of trading profits after costs and also suggest optimal execution strategies. Although several models have recently been developed which may forecast the immediate price impact of individual trades, limited work has been done to compare their relative performance. We provide a comprehensive performance evaluation of these models and test for statistically significant outperformance amongst candidate models using out-of-sample forecasts. We find that normalizing price impact by its average value significantly enhances the performance of traditional non-normalized models as the normalization factor captures some of the dynamics of price impact. Copyright
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This work analyzes the relation between stock price changes and high volume trades in Brazil. Using a unique intra-day database, we evaluate 10 of the most liquid shares from 2001 to 2006. Unlike most international studies, which are based on data from funds or institutional investors, this article breaks new ground by working with publicly available information. Our results indicate a positive and significant relation between stock price changes and high volume trades. In line with existing literature, we show there are both temporary and partially permanent on stock prices after high volume trades. Our study also indicates the existence of asymmetry between purchases and sales.
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In this paper the problem of optimal trading in illiquid markets is addressed when the deviations from a given stochastic target function describing, for instance, external aggregate client flow are penalized. Using techniques of singular stochastic control, we extend the results of [F. Naujokat and N. Westray, Math. Financ. Econ., 4 (2011), pp. 299-335] to a two-sided limit order market with temporary market impact and resilience, where the bid ask spread is now also controlled. In addition to using market orders, the trader can also submit orders to a dark pool. We first show existence and uniqueness of an optimal control. In a second step, a suitable version of the stochastic maximum principle is derived which yields a characterization of the optimal trading strategy in terms of a nonstandard coupled forward-backward stochastic differential equation (FBSDE). We show that the optimal control can be characterized via buy, sell, and no-trade regions. The new feature is that we now get a nondegenerate no-trade region, which implies that market orders are used only when the spread is small. This allows us to describe precisely when it is optimal to cross the bid ask spread, which is a fundamental problem of algorithmic trading. We also show that the controlled system can be described in terms of a reflected BSDE. As an application, we solve the portfolio liquidation problem with passive orders.
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An optimization model for the execution of algorithmic orders at multiple trading venues is herein proposed and analyzed. The optimal trajectory consists of both market and limit orders, and takes advantage of any price or liquidity improvement in a particular market. The complexity of a multi-market environment poses a bi-level nonlinear optimization problem. The lower-level problem admits a unique solution thus enabling the second order conditions to be satisfied under a set of reasonable assumptions. The model is computationally affordable and solvable using standard software packages. The simulation results presented in the paper show the model’s effectiveness using real trade data. From the outset, great effort was made to ensure that this was a challenging practical problem which also had a direct real world application. To be able to estimate in realtime the probability of fill for tens of thousands of orders at multiple price levels in a liquidity fragmented market place and finally carry out an optimization procedure to find the most optimal order placement solution is a significant computational breakthrough.
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In this paper, we first propose a portfolio management model where the objective is to balance equity and liability. The asset price dynamics includes both permanent and temporary price impact, where the permanent impact is a linear function of the cumulative trading amount and the temporary impact is a kth (between 0 and 1) order power function of the instantaneous trading rate. We construct efficient frontiers to visualize the tradeoff between equity and liability and obtain analytical properties regarding the optimal trading strategies. In the second part, we further consider an optimal deleveraging problem with leverage constraints. It reduces to a non-convex polynomial optimization program with polynomial and box constraints. A Lagrangian method for solving the problem is presented and the quality of the solution is studied.
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