A hypothetical risk-neutral agent who trades to maximize the expected profit of the next trade will approximately exhibit long-term optimal behavior as long as this agent uses the vector $p = \nabla V (t, x)$ as effective microstructure alphas, where V is the Bellman value function for a smooth relaxation of the problem. Effective microstructure alphas are the steepest-ascent direction of V , equal to the generalized momenta in a dual Hamiltonian formulation. This simple heuristics has wide-ranging practical implications; indeed, most utility-maximization problems that require implementation via discrete limit-order-book markets can be treated by our method.
We study the optimal trading policies for a wind energy producer who aims to sell the future production in the open forward, spot, intraday and adjustment markets, and who has access to imperfect dynamically updated forecasts of the future production. We construct a stochastic model for the forecast evolution and determine the optimal trading policies which are updated dynamically as new forecast information becomes available. Our results allow to quantify the expected future gain of the wind producer and to determine the economic value of the forecasts.
We study several optimal stopping problems that arise from trading a mean-reverting price spread over a finite horizon. Modeling the spread by the Ornstein-Uhlenbeck process, we analyze three different trading strategies: (i) the long-short strategy; (ii) the short-long strategy, and (iii) the chooser strategy, i.e. the trader can enter into the spread by taking either long or short position. In each of these cases, we solve an optimal double stopping problem to determine the optimal timing for starting and subsequently closing the position. We utilize the local time-space calculus of Peskir (…
Given two time series, A and B, sampled asynchronously at different times {t_A_i} and {t_B_j}, termed "ticks", how can one best estimate the correlation coefficient ρbetween changes in A and B? We derive a natural, minimum-variance estimator that does not use any interpolation or binning, then derive from it a fast (linear time) estimator that is demonstrably nearly as good. This "fast tickwise estimator" is compared in simulation to the usual method of interpolating changes to a regular grid. Even when the grid spacing is optimized for the particular parameters (not often possible in practice…
Optimal trading strategies for pairs trading have been studied by models that try to find either optimal shares of stocks by assuming no transaction costs or optimal timing of trading fixed numbers of shares of stocks with transaction costs. To find optimal strategies which determine optimally both trade times and number of shares in pairs trading process, we use a singular stochastic control approach to study an optimal pairs trading problem with proportional transaction costs. Assuming a cointegrated relationship for a pair of stock log-prices, we consider a portfolio optimization problem wh…
We study optimal trading in an Almgren-Chriss model with running and terminal inventory costs and general predictive signals about price changes. As a special case, this allows to treat optimal liquidation in "target zone models": asset prices with a reflecting boundary enforced by regulatory interventions. In this case, the optimal liquidation rate is the "theta" of a lookback option, leading to explicit formulas for Bachelier or Black-Scholes dynamics.
Optimal trading is a recent field of research which was initiated by Almgren, Chriss, Bertsimas and Lo in the late 90's. Its main application is slicing large trading orders, in the interest of minimizing trading costs and potential perturbations of price dynamics due to liquidity shocks. The initial optimization frameworks were based on mean-variance minimization for the trading costs. In the past 15 years, finer modelling of price dynamics, more realistic control variables and different cost functionals were developed. The inclusion of signals (i.e. short term predictors of price dynamics) i…
We summarize the fundamental issues at stake in algorithmic trading, and the progress made in this field over the last twenty years. We first present the key problems of algorithmic trading, describing the concepts of optimal execution, optimal placement, and price impact. We then discuss the most recent advances in algorithmic trading through the use of Machine Learning, discussing the use of Deep Learning, Reinforcement Learning, and Generative Adversarial Networks.
We consider the problem of optimal trading for a power producer in the context of intraday electricity markets. The aim is to minimize the imbalance cost induced by the random residual demand in electricity, i.e. the consumption from the clients minus the production from renewable energy. For a simple linear price impact model and a quadratic criterion, we explicitly obtain approximate optimal strategies in the intraday market and thermal power generation, and exhibit some remarkable properties of the trading rate. Furthermore, we study the case when there are jumps on the demand forecast and …
This paper studies four trading algorithms of a professional trader at a multilateral trading facility, observing a realistic two-sided limit order book whose dynamics are driven by the order book events. The identity of the trader can be either internalizing or regular, either a hedge fund or a brokery agency. The speed and cost of trading can be balanced by properly choosing active strategies on the displayed orders in the book and passive strategies on the hidden orders within the spread. We shall show that the price switching algorithms provide lower and upper bounds of the mixed trading a…
In this paper we propose a mathematical framework to address the uncertainty emergingwhen the designer of a trading algorithm uses a threshold on a signal as a control. We rely ona theorem by Benveniste and Priouret to deduce our Inventory Asymptotic Behaviour (IAB)Theorem giving the full distribution of the inventory at any point in time for a well formulatedtime continuous version of the trading algorithm.Since this is the first time a paper proposes to address the uncertainty linked to the use of athreshold on a signal for trading, we give some structural elements about the kind of signals …
We consider an agent who needs to buy (or sell) a relatively small amount of asset over some fixed short time interval. We work at the highest frequency meaning that we wish to find the optimal tactic to execute our quantity using limit orders, market orders and cancellations. To solve the agent's control problem, we build an order book model and optimize an expected utility function based on our price impact. We derive the equations satisfied by the optimal strategy and solve them numerically. Moreover, we show that our optimal tactic enables us to outperform significantly naive execution str…
We establish connections between optimal transport theory and the dynamic version of the Kyle model, including new characterizations of informed trading profits via conjugate duality and Monge-Kantorovich duality. We use these connections to extend the model to multiple assets, general distributions, and risk-averse market makers. With risk-averse market makers, liquidity is lower, assets exhibit short-term reversals, and risk premia depend on market maker inventories, which are mean reverting. We illustrate the model by showing that implied volatilities predict stock returns when there is inf…
We study the problem of optimal trading using general alpha predictors with linear costs and temporary impact. We do this within the framework of stochastic optimization with finite horizon using both limit and market orders. Consistently with other studies, we find that the presence of linear costs induces a no-trading zone when using market orders, and a corresponding market-making zone when using limit orders. We show that, when combining both market and limit orders, the problem is further divided into zones in which we trade more aggressively using market orders. Even though we do not sol…
In this paper, we generalize the Almgren-Chriss's market impact model to a more realistic and flexible framework and employ it to derive and analyze some aspects of optimal liquidation problem in a security market. We illustrate how a trader's liquidation strategy alters when multiple venues and extra information are brought into the security market and detected by the trader. This study gives some new insights into the relationship between liquidation strategy and market liquidity, and provides a multi-scale approach to the optimal liquidation problem with randomly varying volatility.
A great deal of academic and theoretical work has been dedicated to optimal liquidation of large orders these last twenty years. The optimal split of an order through time (`optimal trade scheduling') and space (`smart order routing') is of high interest \rred{to} practitioners because of the increasing complexity of the market micro structure because of the evolution recently of regulations and liquidity worldwide. This paper translates into quantitative terms these regulatory issues and, more broadly, current market design. It relates the recent advances in optimal trading, order-book simula…
We study price formation in intraday electricity markets in the presence of intermittent renewable generation. We consider the setting where a major producer may interact strategically with a large number of small producers. Using stochastic control theory we identify the optimal strategies of agents with market impact and exhibit the Nash equilibrium in closed form in the asymptotic framework of mean field games with a major player. This is a companion paper to [Féron, Tankov, and Tinsi, Price formation and optimal trading in intraday electricity markets, arXiv:2009.04786, 2020], where a simi…
In the present work we develop a formalism to tackle the problem of optimal execution when trading market securities. More precisely, we introduce a utility function that balances market impact and timing risk, with this last being modelled as the very negative transaction costs incurred by our order execution. The framework is built upon existing theory on optimal trading strategies, but incorporates characteristics that enable distinctive execution strategies. The formalism is complemented by an analysis of various impact models and different distributional properties of market returns.
We consider the optimal trade execution strategies for a large portfolio of single stocks proposed by Almgren (2003). This framework accounts for a nonlinear impact of trades on average market prices. The results of Almgren (2003) are based on the assumption that no shares of assets per unit of time are trade at the beginning of the period. We propose a general solution method that accomodates the case of a positive stock of assets in the initial period. Our findings are twofold. First of all, we show that the problem admits a solution with no trading in the opening period only if additional p…
Proportional transaction costs present difficult theoretical problems in trading algorithm design, on account of their lack of analytical tractability. The author derives a solution of DT-NT-DT form for an arbitrary model in which the the traded asset has diffusive dynamics described by one or more stochastic risk factors. The width of the NT zone is found to be, as expected, proportional to the cube root of the transaction cost. It is also proportional to the 2/3 power of the volatility of the target position, thereby causing a faster trading strategy to be buffered more than a slower one. Th…
This editorial article partially informs the algorithmic trading community about launching of the new journal "Algorithmic Trading and Controls" (ATC). ATC is an online open-access journal that publishes novel works on algorithmic trading and its control methodologies. In this inaugural article, we discuss nine major challenges that contemporary Algo trading faces. There is nothing superstitiously magical about the number "nine," but so is any other one. Several of these challenges are at the strategy level, including for example, trading of illiquid securities or optimal portfolio execution. …
Motivated by the industry practice of pairs trading, we study the optimal timing strategies for trading a mean-reverting price spread. An optimal double stopping problem is formulated to analyze the timing to start and subsequently liquidate the position subject to transaction costs. Modeling the price spread by an Ornstein-Uhlenbeck process, we apply a probabilistic methodology and rigorously derive the optimal price intervals for market entry and exit. As an extension, we incorporate a stop-loss constraint to limit the maximum loss. We show that the entry region is characterized by a bounded…
We reconsider the problem of optimal trading in the presence of linear and quadratic costs, for arbitrary linear costs but in the limit where quadratic costs are small. Using matched asymptotic expansion techniques, we find that the trading speed vanishes inside a band that is narrower than in the absence of quadratic costs, by an amount that scales as the one-third power of quadratic costs. Outside of the band, we find three regimes: a small boundary layer where the velocity vanishes linearly with the distance to the band, an intermediate region where the velocity behaves as a square-root of …
Signature-Based Optimal Execution for Statistical Arbitrage with Path-Dependent Trading Signals
Deep Blue Capital B.V.（深蓝资本有限公司）；Informatics Institute, University of Amsterdam（阿姆斯特丹大学信息学研究所）；Korteweg-de Vries Institute for Mathematics, University of Amsterdam（阿姆斯特丹大学科特韦格-德弗里斯数学研究所）
We model the impact costs of a strategy that trades a basket of correlated instruments, by extending to the multivariate case the linear propagator model previously used for single instruments. Our specification allows us to calibrate a cost model that is free of arbitrage and price manipulation. We illustrate our results using a pool of US stocks and show that neglecting cross-impact effects leads to an incorrect estimation of the liquidity and suboptimal execution strategies. We show in particular the importance of synchronizing the execution of correlated contracts.
We study optimal liquidation in the presence of linear temporary and transient price impact along with taking into account a general price predicting finite-variation signal. We formulate this problem as minimization of a cost-risk functional over a class of absolutely continuous and signal-adaptive strategies. The stochastic control problem is solved by following a probabilistic and convex analytic approach. We show that the optimal trading strategy is given by a system of four coupled forward-backward SDEs, which can be solved explicitly. Our results reveal how the induced transient price di…
Considering that a trader or a trading algorithm interacting with markets during continuous auctions can be modeled by an iterating procedure adjusting the price at which he posts orders at a given rhythm, this paper proposes a procedure minimizing his costs. We prove the a.s. convergence of the algorithm under assumptions on the cost function and give some practical criteria on model parameters to ensure that the conditions to use the algorithm are fulfilled (using notably the co-monotony principle). We illustrate our results with numerical experiments on both simulated data and using a finan…
The aim of this paper is to explain how parameters adjustments can be integrated in the design or the control of automates of trading. Typically, we are interested by the online estimation of the market impacts generated by robots or single orders, and how they/the controller should react in an optimal way to the informations generated by the observation of the realized impacts. This can be formulated as an optimal impulse control problem with unknown parameters, on which a prior is given. We explain how a mix of the classical Bayesian updating rule and of optimal control techniques allows one…
We revisit optimal execution of an active portfolio in the presence of slippage (aka linear, proportional, or absolute-value) costs. Market efficiency implies a close balance between active alphas and trading costs, so even small changes to trading optimization can make a big difference. It has been observed for some time that optimal trading involves a pattern of a no-trade zone with width $Δ$ increasing with slippage cost parameter $c$. In a setting of a reasonably stable (non-stochastic) forecast of future returns and a quadratic risk aversion, it is shown that $Δ\sim c^{1/2}$, which differ…
We study a single-period optimal transport problem on $\mathbb{R}^2$ with a covariance-type cost function $c(x,y) = (x_1-y_1)(x_2-y_2)$ and a backward martingale constraint. We show that a transport plan $γ$ is optimal if and only if there is a maximal monotone set $G$ that supports the $x$-marginal of $γ$ and such that $c(x,y) = \min_{z\in G}c(z,y)$ for every $(x,y)$ in the support of $γ$. We obtain sharp regularity conditions for the uniqueness of an optimal plan and for its representation in terms of a map. Our study is motivated by a variant of the classical Kyle model of insider trading f…
The theory of optimal trading under proportional transaction costs has been considered from a variety of perspectives. In this paper, we show that all the results can be interpreted using a universal law, illustrating the results in trading algorithm design.
Alpha signals for statistical arbitrage strategies are often driven by latent factors. This paper analyses how to optimally trade with latent factors that cause prices to jump and diffuse. Moreover, we account for the effect of the trader's actions on quoted prices and the prices they receive from trading. Under fairly general assumptions, we demonstrate how the trader can learn the posterior distribution over the latent states, and explicitly solve the latent optimal trading problem. We provide a verification theorem, and a methodology for calibrating the model by deriving a variation of the …
We study the optimal timing strategies for trading a mean-reverting price process with afinite deadline to enter and a separate finite deadline to exit the market. The price process is modeled by a diffusion with an affine drift that encapsulates a number of well-known models,including the Ornstein-Uhlenbeck (OU) model, Cox-Ingersoll-Ross (CIR) model, Jacobi model,and inhomogeneous geometric Brownian motion (IGBM) model.We analyze three types of trading strategies: (i) the long-short (long to open, short to close) strategy; (ii) the short-long(short to open, long to close) strategy, and (iii) …
We develop a dynamic trading strategy in the Linear Quadratic Regulator (LQR) framework. By including a price mean-reversion signal into the optimization program, in a trading environment where market impact is linear and stage costs are quadratic, we obtain an optimal trading curve that reacts opportunistically to price changes while retaining its ability to satisfy smooth or hard completion constraints. The optimal allocation is affine in the spot price and in the number of outstanding shares at any time, and it can be fully derived iteratively. It is also aggressive in the money, meaning th…
We propose a framework to study optimal trading policies in a one-tick pro-rata limit order book, as typically arises in short-term interest rate futures contracts. The high-frequency trader has the choice to trade via market orders or limit orders, which are represented respectively by impulse controls and regular controls. We model and discuss the consequences of the two main features of this particular microstructure: first, the limit orders sent by the high frequency trader are only partially executed, and therefore she has no control on the executed quantity. For this purpose, cumulative …
In this paper, we employ the Heston stochastic volatility model to describe the stock's volatility and apply the model to derive and analyze the optimal trading strategies for dealers in a security market. We also extend our study to option market making for options written on stocks in the presence of stochastic volatility. Mathematically, the problem is formulated as a stochastic optimal control problem and the controlled state process is the dealer's mark-to-market wealth. Dealers in the security market can optimally determine their ask and bid quotes on the underlying stocks or options con…
We develop a tractable equilibrium model for price formation in intraday electricity markets in the presence of intermittent renewable generation. Using stochastic control theory, we identify the optimal strategies of agents with market impact and exhibit the Nash equilibrium in closed form for a finite number of agents as well as in the asymptotic framework of mean field games. Our model reproduces the empirical features of intraday market prices, such as increasing price volatility at the approach of the delivery date and the correlation between price and renewable infeed forecasts, and rela…
Automated trading systems on developed and emerging capital markets are studied in this paper. The standard for developed market is automated trading system with 40-days simple moving average. We tested it for the index SIX Industrial for 1000 and 730 trading days of the slovak emerging capital market. The Buy and Hold trading system was 7.80 times more profitable than this etalon trading system for active trading. Taking of profitable standard trading system from a developed capital market does not lead to optimal results on the emerging capital markets. We then studied optimized standard tra…
Deep reinforcement learning for optimal trading with partial information
Scuola Normale Superiore, Pisa（比萨高等师范学院）；Department of Statistical Sciences, University of Toronto（多伦多大学统计科学系）；Oxford-Man Institute for Quantitative Finance, University of Oxford（牛津大学牛津-曼量化金融研究所）；Dipartimento di Matematica, University of Bologna（博洛尼亚大学数学系）
We study Nash equilibria for inventory-averse high-frequency traders (HFTs), who trade to exploit information about future price changes. For discrete trading rounds, the HFTs' optimal trading strategies and their equilibrium price impact are described by a system of nonlinear equations; explicit solutions obtain around the continuous-time limit. Unlike in the risk-neutral case, the optimal inventories become mean-reverting and vanish as the number of trading rounds becomes large. In contrast, the HFTs' risk-adjusted profits and the equilibrium price impact converge to their risk-neutral count…
Financial markets are often driven by latent factors which traders cannot observe. Here, we address an algorithmic trading problem with collections of heterogeneous agents who aim to perform optimal execution or statistical arbitrage, where all agents filter the latent states of the world, and their trading actions have permanent and temporary price impact. This leads to a large stochastic game with heterogeneous agents. We solve the stochastic game by investigating its mean-field game (MFG) limit, with sub-populations of heterogeneous agents, and, using a convex analysis approach, we show tha…
This paper studies the timing of trades under mean-reverting price dynamics subject to fixed transaction costs. We solve an optimal double stopping problem to determine the optimal times to enter and subsequently exit the market, when prices are driven by an exponential Ornstein-Uhlenbeck process. In addition, we analyze a related optimal switching problem that involves an infinite sequence of trades, and identify the conditions under which the double stopping and switching problems admit the same optimal entry and/or exit timing strategies. Among our results, we find that the investor general…
MOT: A Mixture of Actors Reinforcement Learning Method by Optimal Transport for Algorithmic Trading
School of Artificial Intelligence, University of Chinese Academy of Sciences（中国科学院大学人工智能学院）；Institute of Automation, Chinese Academy of Sciences（中国科学院自动化研究所）
We prove the existence of an equilibrium in a model with transaction costs and price impact where two agents are incentivized to trade towards a target. The two types of frictions -- price impact and transaction costs -- lead the agents to two distinct changes in their optimal investment approach: price impact causes agents to continuously trade in smaller amounts, while transaction costs cause the agents to cease trading before the end of the trading period. As the agents lose wealth because of transaction costs, the exchange makes a profit. We prove the existence of a strictly positive optim…
We seek a discussion about the most suitable feedback control structure for stock trading under the consideration of proportional transaction costs. Suitability refers to robustness and performance capability. Both are tested by considering different one-step ahead prediction qualities, including the ideal case, correct prediction of the direction of change in daily stock prices and the worst-case. Feedback control structures are partitioned into two general classes: stochastic model predictive control (SMPC) and genetic. For the former class three controllers are discussed, whereby it is dist…
As a fundamental problem in algorithmic trading, order execution aims at fulfilling a specific trading order, either liquidation or acquirement, for a given instrument. Towards effective execution strategy, recent years have witnessed the shift from the analytical view with model-based market assumptions to model-free perspective, i.e., reinforcement learning, due to its nature of sequential decision optimization. However, the noisy and yet imperfect market information that can be leveraged by the policy has made it quite challenging to build up sample efficient reinforcement learning methods …
Optimal execution, i.e., the determination of the most cost-effective way to trade volumes in continuous trading sessions, has been a topic of interest in the equity trading world for years. Electricity intraday trading slowly follows this trend but is far from being well-researched. The underlying problem is a very complex one. Energy traders, producers, and electricity wholesale companies receive various position updates from customer businesses, renewable energy production, or plant outages and need to trade these positions in intraday markets. They have a variety of options when it comes t…
We derive explicit recursive formulas for Target Close (TC) and Implementation Shortfall (IS) in the Almgren-Chriss framework. We explain how to compute the optimal starting and stopping times for IS and TC, respectively, given a minimum trading size. We also show how to add a minimum participation rate constraint (Percentage of Volume, PVol) for both TC and IS. We also study an alternative set of risk measures for the optimisation of algorithmic trading curves. We assume a self-similar process (e.g. Levy process, fractional Brownian motion or fractal process) and define a new risk measure, th…
We assume a continuous-time price impact model similar to Almgren-Chriss but with the added assumption that the price impact parameters are stochastic processes modeled as correlated scalar Markov diffusions. In this setting, we develop trading strategies for a trader who desires to liquidate his inventory but faces price impact as a result of his trading. For a fixed trading horizon, we perform coefficient expansion on the Hamilton-Jacobi-Bellman equation associated with the trader's value function. The coefficient expansion yields a sequence of partial differential equations that we solve to…
Optimal control models for limit order trading often assume that the underlying asset price is a Brownian motion since they deal with relatively short time scales. The resulting optimal bid and ask limit order prices tend to track the underlying price as one might expect. This is indeed the case with the model of Avellaneda and Stoikov (2008), which has been studied extensively. We consider here this model under the condition when the underlying price is mean reverting. Our main result is that when time is far from the terminal, the optimal price for bid and ask limit orders is constant, which…
We consider the Brownian market model and the problem of expected utility maximization of terminal wealth. We, specifically, examine the problem of maximizing the utility of terminal wealth under the presence of transaction costs of a fund/agent investing in futures markets. We offer some preliminary remarks about statistical arbitrage strategies and we set the framework for futures markets, and introduce concepts such as margin, gearing and slippage. The setting is of discrete time, and the price evolution of the futures prices is modelled as discrete random sequence involving Ito's sums. We …
We extend the framework of trading strategies of Gatheral [2010] from single stocks to a pair of stocks. Our trading strategy with the executions of two round-trip trades can be described by the trading rates of the paired stocks and the ratio of their trading periods. By minimizing the potential cost arising from cross-impacts, i.e., the price change of one stock due to the trades of another stock, we can find out an optimal strategy for executing a sequence of trades from different stocks. We further apply the model of the strategy to a specific case, where we quantify the cross-impacts of t…
