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 transaction costs allows the informed agent's optimal trading strategy to be obtained in feedback form. Linear equilibrium is characterized by the unique solution to a system of two ordinary differential equations, of which one is forward in time and one is backward. When transaction costs are in effect, the price set by the market maker in equilibrium is not fully revealing of the informed agent's private signal, leaving an information gap at the end of the trading interval. When considering vanishing transaction costs, the equilibrium trading strategy and pricing rules converge to their frictionless counterparts.
In this paper, we present a multi-period trading model in the style of Kyle (1985)'s inside trading model, by assuming that there are at least two insiders in the market with long-lived private information, under the requirement that each insider publicly discloses his stock trades after the fact. Based on this model, we study the influences of "public disclosure" and "competition among insiders" on the trading behaviors of insiders. We find that the "competition among insiders" leads to higher effective price and lower insiders' profits, and the "public disclosure" makes each insider play a m…
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 …
Insider trading is one of the numerous white collar crimes that can contribute to the instability of the economy. Traditionally, the detection of illegal insider trades has been a human-driven process. In this paper, we collect the insider tradings made available by the US Securities and Exchange Commissions (SEC) through the EDGAR system, with the aim of initiating an automated large-scale and data-driven approach to the problem of identifying illegal insider tradings. The goal of the study is the identification of interesting patterns, which can be indicators of potential anomalies. We use t…
In this paper, we present a multi-period trading model by assuming that traders face not only asymmetric information but also heterogenous prior beliefs, under the requirement that the insider publicly disclose his stock trades after the fact. We show that there is an equilibrium in which the irrational insider camouflages his trades with a noise component so that his private information is revealed slowly and linearly whenever he is overconfident or underconfident. We also investigate the relationship between the heterogeneous beliefs and the trade intensity in the presence of trade disclosur…
The continuous-time version of Kyle's (1985) model is studied, in which market makers are not fiduciaries. They have some market power which they utilize to set the price to their advantage, resulting in positive expected profits. This has several implications for the equilibrium, the most important being that by setting a modest fee conditional of the order flow, the market maker is able to obtain a profit of the order of magnitude, and even better than, a perfectly informed insider. Our model also indicates why speculative prices are more volatile than predicted by fundamentals.
We construct explicitly a bridge process whose distribution, in its own filtration, is the same as the difference of two independent Poisson processes with the same intensity and its time 1 value satisfies a specific constraint. This construction allows us to show the existence of Glosten-Milgrom equilibrium and its associated optimal trading strategy for the insider. In the equilibrium the insider employs a mixed strategy to randomly submit two types of orders: one type trades in the same direction as noise trades while the other cancels some of the noise trades by submitting opposite orders …
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…
We consider a one-period Kyle (1985) framework where the insider can be subject to a penalty if she trades. We establish existence and uniqueness of equilibrium for virtually any penalty function when noise is uniform. In equilibrium, the demand of the insider and the price functions are in general non-linear and remain analytically tractable because the expected price function is linear. We use this result to investigate the trade off between price efficiency and 'fairness': we consider a regulator that wants to minimise post-trade standard deviation for a given level of uninformed traders' l…
We propose a static equilibrium model for limit order book where profit-maximizing investors receive an information signal regarding the liquidation value of the asset and execute via a competitive dealer with random initial inventory, who trades against a competitive limit order book populated by liquidity suppliers. We show that an equilibrium exists for bounded signal distributions, obtain closed form solutions for Bernoulli-type signals and propose a straightforward iterative algorithm to compute the equilibrium order book for the general case. We obtain the exact analytic asymptotics for …
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…
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 …
This paper investigates the equilibrium interactions between trading targets and private information in a multi-period Kyle (1985) market. There are two investors who each follow dynamic trading strategies: A strategic portfolio rebalancer who engages in order splitting to reach a cumulative trading target and an unconstrained strategic insider who trades on long-lived information. We consider cases in which the constrained rebalancer is partially informed as well as the special case in which the rebalancer is ex ante uninformed. We derive a linear Bayesian Nash equilibrium, describe an algori…
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…
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 utilize a fundamentally different model of trading costs to look at the effect of the opening of the Hong Kong Shanghai Connect that links the stock exchanges in the two cities, arguably the biggest event in international business and finance since Christopher Columbus set sail for India. We design a novel methodology that compensates for the lack of data on trading costs in China. We estimate trading costs across similar positions on the dual listed set of securities in Hong Kong and China, hoping to provide useful pieces of information to help scale 'The Great Wall of Chinese Securities T…
Market impact has become a subject of increasing concern among academics and industry experts. We put forward a price impact model which considers the heteroscedasticity of price in the time dimension and dependency between permanent impact and temporary impact. We discuss and derive the extremum of the expectation of permanent impact and realized impact by constructing several special trading trajectories. Given our use of a large trade and quote tick records of 17,213,238,343 compiled from the Chinese stock market, the model assessment ultimately suggest that our model is better than Almgren…
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…
When prices reflect all available information, they oscillate around an equilibrium level. This oscillation is the result of the temporary market impact caused by waves of buyers and sellers. This price behavior can be approximated through an Ornstein-Uhlenbeck (O-U) process. Market makers provide liquidity in an attempt to monetize this oscillation. They enter a long position when a security is priced below its estimated equilibrium level, and they enter a short position when a security is priced above its estimated equilibrium level. They hold that position until one of three outcomes occur:…
Executing a basket of co-integrated assets is an important task facing investors. Here, we show how to do this accounting for the informational advantage gained from assets within and outside the basket, as well as for the permanent price impact of market orders (MOs) from all market participants, and the temporary impact that the agent's MOs have on prices. The execution problem is posed as an optimal stochastic control problem and we demonstrate that, under some mild conditions, the value function admits a closed-form solution, and prove a verification theorem. Furthermore, we use data of fi…
We develop a theory for the market impact of large trading orders, which we call metaorders because they are typically split into small pieces and executed incrementally. Market impact is empirically observed to be a concave function of metaorder size, i.e., the impact per share of large metaorders is smaller than that of small metaorders. We formulate a stylized model of an algorithmic execution service and derive a fair pricing condition, which says that the average transaction price of the metaorder is equal to the price after trading is completed. We show that at equilibrium the distributi…
A large body of empirical literature has shown that market impact of financial prices is transient. However, from a theoretical standpoint, the origin of this temporary nature is still unclear. We show that an implied transient impact arises from the Nash equilibrium between a directional trader and one arbitrageur in a market impact game with fixed and permanent impact. The implied impact is the one that can be empirically inferred from the directional trader's trading profile and price reaction to order flow. Specifically, we propose two approaches to derive the functional form of the decay …
