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 distortion provides together with the predictive signal an additional predictor about future price changes. As a consequence, the optimal signal-adaptive trading rate trades off exploiting the predictive signal against incurring the transient displacement of the execution price from its unaffected level. This answers an open question from Lehalle and Neuman [29] as we show how to derive the unique optimal signal-adaptive liquidation strategy when price impact is not only temporary but also transient.
We model an informed agent with information about the future value of an asset trying to maximize profits when subjected to a transaction cost as well as a market maker tasked with setting fair transaction prices. In a single auction model, equilibrium is characterized by the unique root of a particular polynomial. Analysis of this polynomial with small levels of risk-aversion and transaction costs reveal a dimensionless parameter which captures several orders of asymptotic accuracy of the equilibrium behaviour. In a continuous time analogue of the single auction model, incorporation of a tran…
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 op…
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…
We define the concept of good trade execution and we construct explicit adapted good trade execution strategies in the framework of linear temporary market impact. Good trade execution strategies are dynamic, in the sense that they react to the actual realisation of the traded asset price path over the trading period; this is paramount in volatile regimes, where price trajectories can considerably deviate from their expected value. Remarkably however, the implementation of our strategies does not require the full specification of an SDE evolution for the traded asset price, making them robust …
A deterministic trading strategy can be regarded as a signal processing element that uses external information and past prices as inputs and incorporates them into future prices. This paper uses a market maker based method of price formation to study the price dynamics induced by several commonly used financial trading strategies, showing how they amplify noise, induce structure in prices, and cause phenomena such as excess and clustered volatility.
This note continues investigation of randomness-type properties emerging in idealized financial markets with continuous price processes. It is shown, without making any probabilistic assumptions, that the strong variation exponent of non-constant price processes has to be 2, as in the case of continuous martingales.
We formulate and solve a multi-player stochastic differential game between financial agents who seek to cost-efficiently liquidate their position in a risky asset in the presence of jointly aggregated transient price impact, along with taking into account a common general price predicting signal. The unique Nash-equilibrium strategies reveal how each agent's liquidation policy adjusts the predictive trading signal to the aggregated transient price impact induced by all other agents. This unfolds a quantitative relation between trading signals and the order flow in crowded markets. We also form…
This paper presents an equilibrium model of dynamic trading, learning, and pricing by strategic investors with trading targets and price impact. Since trading targets are private, rebalancers and liquidity providers filter the child order flow over time to estimate the latent underlying parent trading demand imbalance and its expected impact on subsequent price pressure dynamics. We prove existence of the equilibrium and solve for equilibrium trading strategies and prices in terms of the solution to a system of coupled ODEs. We show that trading strategies are combinations of trading towards i…
Traders in a stock market exchange stock shares and form a stock trading network. Trades at different positions of the stock trading network may contain different information. We construct stock trading networks based on the limit order book data and classify traders into $k$ classes using the $k$-shell decomposition method. We investigate the influences of trading behaviors on the price impact by comparing a closed national market (A-shares) with an international market (B-shares), individuals and institutions, partially filled and filled trades, buyer-initiated and seller-initiated trades, a…
We develop a single-period model for a large economic agent who trades with market makers at their utility indifference prices. A key role is played by a pair of conjugate saddle functions associated with the description of Pareto optimal allocations in terms of the utility function of a representative market maker.
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…
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 imb…
We use standard physics techniques to model trading and price formation in a market under the assumption that order arrival and cancellations are Poisson random processes. This model makes testable predictions for the most basic properties of a market, such as the diffusion rate of prices, which is the standard measure of financial risk, and the spread and price impact functions, which are the main determinants of transaction cost. Guided by dimensional analysis, simulation, and mean field theory, we find scaling relations in terms of order flow rates. We show that even under completely random…
The purpose of this paper is to showcase trading strategies that give solutions to three difficult and intriguing problems in business finance, economics and statistics. The paper discusses trading strategies for both commodities and stocks but the main focus is on stock market trading at the New York Stock Exchange. Problem 1: Buy Low and Sell High. The buy low and sell high problem can be summarized like this: suppose the price of a commodity or stock fluctuates indefinitely, is there any explicit strategy for a trader to "ride the price waves" by buying low and selling high to eventually wi…
Motivated by how transaction amount constrain trading volume and price volatility in stock market, we, in this paper, study the relation between volume and price if amount of transaction is given. We find that accumulative trading volume gradually emerges a kurtosis near the price mean value over a trading price range when it takes a longer trading time, regardless of actual price fluctuation path, time series, or total transaction volume in the time interval. To explain the volume-price behavior, we, in terms of physics, propose a transaction energy hypothesis, derive a time-independent trans…
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…
The asymmetric price impact between the institutional purchases and sales of 32 liquid stocks in Chinese stock markets in year 2003 is carefully studied. We analyze the price impact in both drawup and drawdown trends with consecutive positive and negative daily price changes, and test the dependence of the price impact asymmetry on the market condition. For most of the stocks institutional sales have a larger price impact than institutional purchases, and larger impact of institutional purchases only exists in few stocks with primarily increasing tendencies. We further study the mean return of…
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…
We develop a theoretical trading conditioning model subject to price volatility and return information in terms of market psychological behavior, based on analytical transaction volume-price probability wave distributions in which we use transaction volume probability to describe price volatility uncertainty and intensity. Applying the model to high frequent data test in China stock market, we have main findings as follows: 1) there is, in general, significant positive correlation between the rate of mean return and that of change in trading conditioning intensity; 2) it lacks significance in …
Technical trading represents a class of investment strategies for Financial Markets based on the analysis of trends and recurrent patterns of price time series. According standard economical theories these strategies should not be used because they cannot be profitable. On the contrary it is well-known that technical traders exist and operate on different time scales. In this paper we investigate if technical trading produces detectable signals in price time series and if some kind of memory effect is introduced in the price dynamics. In particular we focus on a specific figure called supports…
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…
In this paper we use fuzzy systems theory to convert the technical trading rules commonly used by stock practitioners into excess demand functions which are then used to drive the price dynamics. The technical trading rules are recorded in natural languages where fuzzy words and vague expressions abound. In Part I of this paper, we will show the details of how to transform the technical trading heuristics into nonlinear dynamic equations. First, we define fuzzy sets to represent the fuzzy terms in the technical trading rules; second, we translate each technical trading heuristic into a group o…
The dynamics of a stock market with heterogeneous agents is discussed in the framework of a recently proposed spin model for the emergence of bubbles and crashes. We relate the log returns of stock prices to magnetization in the model and find that it is closely related to trading volume as observed in real markets. The cumulative distribution of log returns exhibits scaling with exponents steeper than 2 and scaling is observed in the distribution of transition times between bull and bear markets.
In this paper we propose a deep recurrent architecture for the probabilistic modelling of high-frequency market prices, important for the risk management of automated trading systems. Our proposed architecture incorporates probabilistic mixture models into deep recurrent neural networks. The resulting deep mixture models simultaneously address several practical challenges important in the development of automated high-frequency trading strategies that were previously neglected in the literature: 1) probabilistic forecasting of the price movements; 2) single objective prediction of both the dir…
We develop from basic economic principles a continuous-time model for a large investor who trades with a finite number of market makers at their utility indifference prices. In this model, the market makers compete with their quotes for the investor's orders and trade among themselves to attain Pareto optimal allocations. We first consider the case of simple strategies and then, in analogy to the construction of stochastic integrals, investigate the transition to general continuous dynamics. As a result, we show that the model's evolution can be described by a nonlinear stochastic differential…
We develop a fundamentally different stochastic dynamic programming model of trading costs. Built on a strong theoretical foundation, our model provides insights to market participants by splitting the overall move of the security price during the duration of an order into the Market Impact (price move caused by their actions) and Market Timing (price move caused by everyone else) components. We derive formulations of this model under different laws of motion of the security prices, starting with a simple benchmark scenario and extending this to include multiple sources of uncertainty, liquidi…
We consider a limit order book, where buyers and sellers register to trade a security at specific prices. The largest price buyers on the book are willing to offer is called the market bid price, and the smallest price sellers on the book are willing to accept is called the market ask price. Market ask price is always greater than market bid price, and these prices move upwards and downwards due to new arrivals, market trades, and cancellations. We model these two price processes as "bouncing geometric Brownian motions (GBMs)", which are defined as exponentials of two mutually reflected Browni…
Studying the micro-trading behaviors before stock price jumps is an important problem for financial regulations and investment decisions. In this study, we provide a new framework to study pre-jump trading behaviors based on multivariate time series analysis. Different from the existing literature, our methodology takes into account the temporal information embedded in the trading-related attributes and can better evaluate and compare the abnormality levels of different attributes. Moreover, it can explore the joint informativeness of the attributes as well as select a subset of highly informa…
Few assets in financial history have been as notoriously volatile as cryptocurrencies. While the long term outlook for this asset class remains unclear, we are successful in making short term price predictions for several major crypto assets. Using historical data from July 2015 to November 2019, we develop a large number of technical indicators to capture patterns in the cryptocurrency market. We then test various classification methods to forecast short-term future price movements based on these indicators. On both PPV and NPV metrics, our classifiers do well in identifying up and down marke…
We propose a mathematical procedure for finding informed traders in ultra-high frequency trading. We wrote it as Vector ARMA and found condition of its stationarity. For the price exposure complied with ARMA(1,2) we proved that underlying asset price difference can be derived as ARMA(1,1) process. For validation of the model, we test an influence of informed traders in EUR/USD, GBP/USD, USD/RUB pairs and futures, in gold and futures prices, in Russian Trade System share index (RTS) and futures trading. We found some evidence of such influence in gold and currency pair USD/RUB pricing, in RTS i…
High-speed computerized trading, often called "high-frequency trading" (HFT), has increased dramatically in financial markets over the last decade. In the US and Europe, it now accounts for nearly one-half of all trades. Although evidence suggests that HFT contributes to the efficiency of markets, there are concerns it also adds to market instability, especially during times of stress. Currently, it is unclear how or why HFT produces these outcomes. In this paper, I use data from NASDAQ to show that HFT synchronizes prices in financial markets, making the values of related securities change co…
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…
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…
We study the statistical properties of the recurrence intervals $τ$ between successive trading volumes exceeding a certain threshold $q$. The recurrence interval analysis is carried out for the 20 liquid Chinese stocks covering a period from January 2000 to May 2009, and two Chinese indices from January 2003 to April 2009. Similar to the recurrence interval distribution of the price returns, the tail of the recurrence interval distribution of the trading volumes follows a power-law scaling, and the results are verified by the goodness-of-fit tests using the Kolmogorov-Smirnov (KS) statistic, t…
An investor trades a safe and several risky assets with linear price impact to maximize expected utility from terminal wealth. In the limit for small impact costs, we explicitly determine the optimal policy and welfare, in a general Markovian setting allowing for stochastic market, cost, and preference parameters. These results shed light on the general structure of the problem at hand, and also unveil close connections to optimal execution problems and to other market frictions such as proportional and fixed transaction costs.
We empirically study the trading activity in the electronic on-book segment and in the dealership off-book segment of the London Stock Exchange, investigating separately the trading of active market members and of other market participants which are non-members. We find that (i) the volume distribution of off-book transactions has a significantly fatter tail than the one of on-book transactions, (ii) groups of members and non-members can be classified in categories according to their trading profile (iii) there is a strong anticorrelation between the daily inventory variation of a market membe…
We consider an optimal trading problem over a finite period of time during which an investor has access to both a standard exchange and a dark pool. We take the exchange to be an order-driven market and propose a continuous-time setup for the best bid price and the market spread, both modelled by Lévy processes. Effects on the best bid price arising from the arrival of limit buy orders at more favourable prices, the incoming market sell orders potentially walking the book, and deriving from the cancellations of limit sell orders at the best ask price are incorporated in the proposed price dyna…
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…
We construct an equilibrium for the continuous time Kyle's model with stochastic liquidity, a general distribution of the fundamental price, and correlated stock and volatility dynamics. For distributions with positive support, our equilibrium allows us to study the impact of the stochastic volatility of noise trading on the volatility of the asset. In particular, when the fundamental price is log-normally distributed, informed trading forces the log-return up to maturity to be Gaussian for any choice of noise-trading volatility even though the price process itself comes with stochastic volati…
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…
In this paper we extend the analysis of an agent-based model for adaptive trading, called asynchronous stochastic price pump (ASPP) introduced by Perepelitsa and Timofeyev (2019), to the model with heterogeneous distribution of psychological parameters of speculative optimism and pessimism across the population of traders. We show that the new model has a range of qualitatively different dynamics when the correlation between those factors ranges from low negative to large positive values. A statistical parameter estimation suggests a heterogeneous ASPP with negative correlation as a model of p…
We make several improvements to the mean-variance framework for optimal pre-trade algorithmic execution, by working with volume measures and generic price dynamics. Volume measures are the continuum analogies for discrete volume profiles commonly implemented in the execution industry. Execution then becomes an absolutely continuous measure over such a measure space, and its Radon-Nikodym derivative is commonly known as the Participation of Volume (PoV) function. The four impact cost components are all consistently built upon the PoV function. Some novel efforts are made for these linear impact…
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…
Kyle (1985) builds a pioneering and influential model, in which an insider with long-lived private information submits an optimal order in each period given the market maker's pricing rule. An inconsistency exists to some extent in the sense that the ``constant pricing rule " actually assumes an adaptive expected price with pricing rule given before insider making the decision, and the ``market efficiency" condition, however, assumes a rational expected price and implies that the pricing rule can be influenced by insider's strategy. We loosen the ``constant pricing rule " assumption by taking …
Crowded trades by similarly trading peers influence the dynamics of asset prices, possibly creating systemic risk. We propose a market clustering measure using granular trading data. For each stock the clustering measure captures the degree of trading overlap among any two investors in that stock. We investigate the effect of crowded trades on stock price stability and show that market clustering has a causal effect on the properties of the tails of the stock return distribution, particularly the positive tail, even after controlling for commonly considered risk drivers. Reduced investor pool …
This paper analyzes the dynamic incentives for technology adoption under a transferable permits system, which allows for strategic trading on the permit market. Initially, firms can invest both in low-emitting production technologies and trade permits. In the model, technology adoption and allowance price are generated endogenously and are inter-dependent. It is shown that the non-cooperative permit trading game possesses a pure-strategy Nash equilibrium, where the allowance value reflects the level of uncovered pollution (demand), the level of unused allowances (supply), and the technological…
This note explores the consequences of nonlinear price impact functions on price dynamics within the chartist-fundamentalist framework. Price impact functions may be nonlinear with respect to trading volume. As indicated by recent empirical studies, a given transaction may cause a large (small) price change if market depth is low (high). Simulations reveal that such a relationship may create endogenous complex price fluctuations even if the trading behavior of chartists and fundamentalists is linear.
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…
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.
Price movement prediction has always been one of the traders' concerns in financial market trading. In order to increase their profit, they can analyze the historical data and predict the price movement. The large size of the data and complex relations between them lead us to use algorithmic trading and artificial intelligence. This paper aims to offer an approach using Twin-Delayed DDPG (TD3) and the daily close price in order to achieve a trading strategy in the stock and cryptocurrency markets. Unlike previous studies using a discrete action space reinforcement learning algorithm, the TD3 i…
During a speculative episode the price of an item jumps from an initial level p_1 to a peak level p_2 before more or less returning to level p_1. The ratio p_2/p_1 is referred to as the amplitude A of the peak. This paper shows that for a given market the peak amplitude is a linear function of the logarithm of the price at the beginning of the speculative episode; with p_1 expressed in 1999 euros the relationship takes the form: $ A=a\ln p_1 +b $; the values of the parameter a turn out to be relatively independent of the market considered: $ a \simeq 0.5 $, the values of the parameter b are mo…
We study trade-based manipulation of stock prices from the perspective of complex trading networks constructed by using detailed information of trades. A stock trading network consists of nodes and directed links, where every trader is a node and a link is formed from one trader to the other if the former sells shares to the latter. Specifically, three abnormal network motifs are investigated, which are found to be formed by a few traders, implying potential intention of price manipulation. We further investigate the dynamics of volatility, trading volume, average trade size and turnover aroun…
