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arXiv2012

Optimal execution and price manipulations in time-varying limit order books

This paper focuses on an extension of the Limit Order Book (LOB) model with general shape introduced by Alfonsi, Fruth and Schied. Here, the additional feature allows a time-varying LOB depth. We solve the optimal execution problem in this framework for both discrete and continuous time strategies. This gives in particular sufficient conditions to exclude Price Manipulations in the sense of Huberman and Stanzl or Transaction-Triggered Price Manipulations (see Alfonsi, Schied and Slynko). These conditions give interesting qualitative insights on how market makers may create or not price manipulations.

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arXiv2012

Optimal multifactor trading under proportional transaction costs

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. The displacement of the middle of the buffer from the costfree position is found to be proportional to the square of the width, and hence to the 2/3 power of the transaction cost; the proportionality constant depends on the expected short-term change in position.

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arXiv2012

Super-exponential bubbles in lab experiments: evidence for anchoring over-optimistic expectations on price

We analyze a controlled price formation experiment in the laboratory that shows evidence for bubbles. We calibrate two models that demonstrate with high statistical significance that these laboratory bubbles have a tendency to grow faster than exponential due to positive feedback. We show that the positive feedback operates by traders continuously upgrading their over-optimistic expectations of future returns based on past prices rather than on realized returns.

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arXiv2012

Optimal High Frequency Trading in a Pro-Rata Microstructure with Predictive Information

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 executed volumes are modelled by compound Poisson processes. Second, the high frequency trader faces the overtrading risk, which is the risk of brutal variations in her inventory. The consequences of this risk are investigated in the context of optimal liquidation. The optimal trading problem is studied by stochastic control and dynamic programming methods, which lead to a characterization of the value function in terms of an integro quasi-variational inequality. We then provide the associated numerical resolution procedure, and convergence of this computational scheme is proved. Next, we examine several situations where we can on one hand simplify the numerical procedure by reducing the number of state variables, and on the other hand focus on specific cases of practical interest. We examine both a market making problem and a best execution problem in the case where the mid-price process is a martingale. We also detail a high frequency trading strategy in the case where a (predictive) directional information on the mid-price is available. Each of the resulting strategies are illustrated by numerical tests.

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arXiv2013

Optimal starting times, stopping times and risk measures for algorithmic trading: Target Close and Implementation Shortfall

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, the p-variation, which reduces to the variance if the process is a brownian motion. We deduce the explicit formula for the TC and IS algorithms under a self-similar process. We show that there is an equivalence between selfsimilar models and a family of risk measures called p-variations: assuming a self-similar process and calibrating empirically the parameter p for the p-variation yields the same result as assuming a Brownian motion and using the p-variation as risk measure instead of the variance. We also show that p can be seen as a measure of the aggressiveness: p increases if and only if the TC algorithm starts later and executes faster. Finally, we show how the parameter p of the p-variation can be implied from the optimal starting time of TC, and that under this framework p can be viewed as a measure of the joint impact of market impact (i.e. liquidity) and volatility.

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arXiv2014

Price manipulation in a market impact model with dark pool

For a market impact model, price manipulation and related notions play a role that is similar to the role of arbitrage in a derivatives pricing model. Here, we give a systematic investigation into such regularity issues when orders can be executed both at a traditional exchange and in a dark pool. To this end, we focus on a class of dark-pool models whose market impact at the exchange is described by an Almgren--Chriss model. Conditions for the absence of price manipulation for all Almgren--Chriss models include the absence of temporary cross-venue impact, the presence of full permanent cross-venue impact, and the additional penalization of orders executed in the dark pool. When a particular Almgren--Chriss model has been fixed, we show by a number of examples that the regularity of the dark-pool model hinges in a subtle way on the interplay of all model parameters and on the liquidation time constraint. The paper can also be seen as a case study for the regularity of market impact models in general.

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arXiv2013

Point process bridges and weak convergence of insider trading models

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 when noise trades arrive. The construction also allows us to prove that Glosten-Milgrom equilibria converge weakly to Kyle-Back equilibrium, without the additional assumptions imposed in \textit{K. Back and S. Baruch, Econometrica, 72 (2004), pp. 433-465}, when the common intensity of the Poisson processes tends to infinity.

math.PR↗

arXiv2012

A Multi Period Equilibrium Pricing Model

In this paper, we propose an equilibrium pricing model in a dynamic multi-period stochastic framework with uncertain income streams. In an incomplete market, there exist two traded risky assets (e.g. stock/commodity and weather derivative) and a non-traded underlying (e.g. temperature). The risk preferences are of exponential (CARA) type with a stochastic coefficient of risk aversion. Both time consistent and time inconsistent trading strategies are considered. We obtain the equilibriums prices of a contingent claim written on the risky asset and non-traded underlying. By running numerical experiments we examine how the equilibriums prices vary in response to changes in model parameters.

math.OC↗

arXiv2012

A No-Arbitrage Model of Liquidity in Financial Markets involving Brownian Sheets

We consider a dynamic market model where buyers and sellers submit limit orders. If at a given moment in time, the buyer is unable to complete his entire order due to the shortage of sell orders at the required limit price, the unmatched part of the order is recorded in the order book. Subsequently these buy unmatched orders may be matched with new incoming sell orders. The resulting demand curve constitutes the sole input to our model. The clearing price is then mechanically calculated using the market clearing condition. We use a Brownian sheet to model the demand curve, and provide some theoretical assumptions under which such a model is justified. Our main result is the proof that if there exists a unique equivalent martingale measure for the clearing price, then under some mild assumptions there is no arbitrage. We use the Ito- Wentzell formula to obtain that result, and also to characterize the dynamics of the demand curve and of the clearing price in the equivalent measure. We find that the volatility of the clearing price is (up to a stochastic factor) inversely proportional to the sum of buy and sell order flow density (evaluated at the clearing price), which confirms the intuition that volatility is inversely proportional to volume. We also demonstrate that our approach is implementable. We use real order book data and simulate option prices under a particularly simple parameterization of our model. The no-arbitrage conditions we obtain are applicable to a wide class of models, in the same way that the Heath-Jarrow-Morton conditions apply to a wide class of interest rate models.

q-fin.CP↗

arXiv2012

High-frequency market-making with inventory constraints and directional bets

In this paper we extend the market-making models with inventory constraints of Avellaneda and Stoikov ("High-frequency trading in a limit-order book", Quantitative Finance Vol.8 No.3 2008) and Gueant, Lehalle and Fernandez-Tapia ("Dealing with inventory risk", Preprint 2011) to the case of a rather general class of mid-price processes, under either exponential or linear PNL utility functions, and we add an inventory-risk-aversion parameter that penalises the marker-maker if she finishes her day with a non-zero inventory. This general, non-martingale framework allows a market-maker to make directional bets on market trends whilst keeping under control her inventory risk. In order to achieve this, the marker-maker places non-symmetric limit orders that favour market orders to hit her bid (resp. ask) quotes if she expects that prices will go up (resp. down). With this inventory-risk-aversion parameter, the market-maker has not only direct control on her inventory risk but she also has indirect control on the moments of her PNL distribution. Therefore, this parameter can be seen as a fine-tuning of the marker-maker's risk-reward profile. In the case of a mean-reverting mid-price, we show numerically that the inventory-risk-aversion parameter gives the market-maker enough room to tailor her risk-reward profile, depending on her risk budgets in inventory and PNL distribution (especially variance, skewness, kurtosis and VaR). For example, when compared to the martingale benchmark, a market can choose to either increase her average PNL by more than 15% and carry a huge risk, on inventory and PNL, or either give up 5% of her benchmark PNL to increase her control on inventory and PNL, as well as increasing her Sharpe ratio by a factor bigger than 2.

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arXiv2012

A Utility Framework for Bounded-Loss Market Makers

We introduce a class of utility-based market makers that always accept orders at their risk-neutral prices. We derive necessary and sufficient conditions for such market makers to have bounded loss. We prove that hyperbolic absolute risk aversion utility market makers are equivalent to weighted pseudospherical scoring rule market makers. In particular, Hanson's logarithmic scoring rule market maker corresponds to a negative exponential utility market maker in our framework. We describe a third equivalent formulation based on maintaining a cost function that seems most natural for implementation purposes, and we illustrate how to translate among the three equivalent formulations. We examine the tradeoff between the market's liquidity and the market maker's worst-case loss. For a fixed bound on worst-case loss, some market makers exhibit greater liquidity near uniform prices and some exhibit greater liquidity near extreme prices, but no market maker can exhibit uniformly greater liquidity in all regimes. For a fixed minimum liquidity level, we give the lower bound of market maker's worst-case loss under some regularity conditions.

cs.GT↗

arXiv2012

Effective Trade Execution

This paper examines the role of algorithmic trading in modern financial markets. Additionally, order types, characteristics, and special features of algorithmic trading are described under the lens provided by the large development of high frequency trading technology. Special order types are examined together with an intuitive description of the implied dynamics of the order book conditional to special orders (iceberg and hidden). The chapter provides an analysis of the transaction costs associated with trading activity and examines the most common trading strategy employed in the market. It also examines optimal execution strategy with the description of the Efficient Trading Frontier. These concepts represent the tools needed to understand the most recent innovations in financial markets and the most recent advances in microstructures research.

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arXiv2012

Structural distortions in the Euro interbank market: The role of 'key players' during the recent market turmoil

We study the frictions in the patterns of trades in the Euro money market. We characterize the structure of lending relations during the period of recent financial turmoil. We use network-topology method on data from overnight transactions in the Electronic Market for Interbank Deposits (e-Mid) to investigate on two main issues. First, we characterize the division of roles between borrowers and lenders in long-run relations by providing evidence on network formation at a yearly frequency. Second, we identify the 'key players' in the marketplace and study their behaviour. Key players are 'locally-central banks' within a network that lend (or borrow) large volumes to (from) several counterparties, while borrowing (or lending) small volumes from (to) a small number of institutions. Our results are twofold. We show that the aggregate trading patterns in e-Mid are characterized by largely asymmetric relations. This implies a clear division of roles between lenders and borrowers. Second, the key players do not exploit their position of network leaders by imposing opportunistic pricing policies. We find that only a fraction of the networks composed by big players are characterized by interest rates that are statistically different from the average market rate throughout the turmoil period.

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arXiv2013

Large tick assets: implicit spread and optimal tick size

In this work, we provide a framework linking microstructural properties of an asset to the tick value of the exchange. In particular, we bring to light a quantity, referred to as implicit spread, playing the role of spread for large tick assets, for which the effective spread is almost always equal to one tick. The relevance of this new parameter is shown both empirically and theoretically. This implicit spread allows us to quantify the tick sizes of large tick assets and to define a notion of optimal tick size. Moreover, our results open the possibility of forecasting the behavior of relevant market quantities after a change in the tick value and to give a way to modify it in order to reach an optimal tick size.

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arXiv2012

Adaptive Execution: Exploration and Learning of Price Impact

We consider a model in which a trader aims to maximize expected risk-adjusted profit while trading a single security. In our model, each price change is a linear combination of observed factors, impact resulting from the trader's current and prior activity, and unpredictable random effects. The trader must learn coefficients of a price impact model while trading. We propose a new method for simultaneous execution and learning - the confidence-triggered regularized adaptive certainty equivalent (CTRACE) policy - and establish a poly-logarithmic finite-time expected regret bound. This bound implies that CTRACE is efficient in the sense that the (ε,δ)-convergence time is bounded by a polynomial function of 1/ε and log(1/δ) with high probability. In addition, we demonstrate via Monte Carlo simulation that CTRACE outperforms the certainty equivalent policy and a recently proposed reinforcement learning algorithm that is designed to explore efficiently in linear-quadratic control problems.

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arXiv2016

Investor's sentiment in multi-agent model of the continuous double auction

We introduce and treat rigorously a new multi-agent model of the continuous double auction or in other words the order book (OB). It is designed to explain collective behaviour of the market when new information affecting the market arrives. The novel feature of the model is two additional slow changing parameters, the so-called sentiment functions. These sentiment functions measure the conception of the fair price of two groups of investors, namely, bulls and bears. Our model specifies differential equations for the time evolution of sentiment functions and constitutes a nonlinear Markov process which exhibits long term correlations. We explain the intuition behind equations for sentiment functions and present numerical simulations which show that the behaviour of our model is similar to the behaviour of the real market. We also obtain a diffusion limit of the model, the Ornstein-Uhlenbeck type process with variable volatility. The volatility is proportional to the difference of opinions of bulls and bears about the fair price of a security. The paper is complimentary to our previous work where mathematical proofs are presented.

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arXiv2012

The Transition from Brownian Motion to Boom-and-Bust Dynamics in Financial and Economic Systems

Quasi-equilibrium models for aggregate variables are widely-used throughout finance and economics. The validity of such models depends crucially upon assuming that the systems' participants behave both independently and in a Markovian fashion. We present a simplified market model to demonstrate that herding effects between agents can cause a transition to boom-and-bust dynamics at realistic parameter values. The model can also be viewed as a novel stochastic particle system with switching and reinjection.

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arXiv2013

On statistical indistinguishability of the complete and incomplete markets

The possibility of statistical evaluation of the market completeness and incompleteness is investigated for continuous time diffusion stock market models. It is known that the market completeness is not a robust property: small random deviations of the coefficients convert a complete market model into a incomplete one. The paper shows that market incompleteness is also non-robust: small deviations can convert an incomplete model into a complete one. More precisely, it is shown that, for any incomplete market from a wide class of models, there exists a complete market model with arbitrarily close paths of the stock prices and the market parameters. This leads to a counterintuitive conclusion that the incomplete markets are indistinguishable from the complete markets in the terms of the market statistics.

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