Damiano Brigo (Imperial College London)
Optimal Execution Under Different Dynamics, Criteria and Solutions Class

We solve a version of the optimal trade execution problem when the mid asset price follows a displaced diffusion. Optimal strategies in the adapted class under various risk criteria, namely value-at-risk, expected shortfall and a new criterion called “squared asset expectation” (SAE), related to a version of the cost variance measure, are derived and compared. It is well known that displaced diffusions (DD) exhibit dynamics which are in-between arithmetic Brownian motions (ABM) and geometric Brownian motions (GBM) depending of the choice of the shift parameter. Furthermore, DD allows for changes in the support of the mid asset price distribution, allowing one to include a minimum permitted value for the mid price, either positive or negative. We study the dependence of the optimal solution on the choice of the risk aversion criterion. Optimal solutions across criteria and asset dynamics are comparable although differences are not negligible for high levels of risk aversion and low market impact assets. This is illustrated with numerical examples. If time permits we will illustrate the discrepancy between optimal solutions in the static/deterministic class and optimal solutions in the larger adapted class in a few benchmark models.

Related Papers

Static vs Adapted Optimal Execution Strategies in Two Benchmark Trading Models

Optimal Execution Comparison Across Risks and Dynamics, with Solutions for Displaced Diffusions


Jose Figueroa-Lopez (Washington University in St. Louis)
Optimal Placement of a Small Order in a Diffusive Limit Order Book

We study the optimal placement problem of a stock trader who wishes to clear his/her inventory by a predetermined time horizon t, by using a limit order or a market order. For a diffusive market, we characterize the optimal limit order placement policy and analyze its behavior under different market conditions. In particular, we show that, in the presence of a negative drift, there exists a critical time t0 > 0 such that, for any time horizon t > t0, there exists an optimal placement, which, contrary to earlier work, is different from one that is placed”infinitesimally” close to the best ask, such as the best bid and second best bid. We also propose a simple method to approximate the critical time t0 and the optimal order placement.

This is a joint work with Hyoeun Lee and Raghu Pasupathy from Purdue.

Related Paper

Optimal Placement of a Small Order in a Diffusive Limit Order Book


Johannes Muhle-Karbe (Carnagie Mellon University)
Equilibrium Price Impact

We derive the equilibrium price at which the orders of a large trader are absorbed by a liquidity provider, who can gradually transfer these positions to a market of end users at a cost.

Joint work in progress with Peter Bank and Ibrahim Ekren.




Marvin Mueller (ETH Zurich)
Limit Order Books: Tractable SPDE Models

Complexity of nowadays electronic financial markets with high trading frequencies demands for new classes of high dimensional models. Macroscopic descriptions for the dynamics of buy and sell side of the limit order book lead to a system of stochastic partial differential equations, which can be shown to be very tractable. We discuss specifications of these models which admit explicit representations and can be easily calibrated to market data.

Following empirical observations, we use the order flow imbalance as a predictor for the next price move. On that way, the limit order book model induces a model for the dynamics of the mid-price process.


Sasha Stoikov (Cornell University)
The Micro-Price

I define the micro-price to be the limit of a sequence of expected mid-prices and provide conditions for this limit to exist. The micro-price is a martingale by construction and can be considered to be the ‘fair’ price of an asset, conditional on the information in the order book. The micro-price may be expressed as an adjustment to the mid-price that takes into account the bid-ask spread and the imbalance. The micro-price can be estimated using high frequency data. I show empirically that it is a better predictor of short term prices than the mid-price or the weighted mid-price.

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The Micro-Price: A High Frequency Estimator of Future Prices




Eyal Neuman (Imperial College London)
Incoporating Signals Into Optimal Trading

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) in optimal trading is a recent development and it is also the subject of this work.
We incorporate a Markovian signal in the optimal trading framework which was initially proposed by Gatheral, Schied, and Slynko and provide results on the existence and uniqueness of an optimal trading strategy. Moreover, we derive an explicit singular optimal strategy for the special case of an Ornstein-Uhlenbeck signal and an exponentially decaying transient market impact. The combination of a mean-reverting signal along with a market impact decay is of special interest, since they affect the short term price variations in opposite directions.

Later, we show that in the asymptotic limit were the transient market impact becomes instantaneous, the optimal strategy becomes continuous. This result is compatible with the optimal trading framework which was proposed by Cartea and Jaimungal.

This is a joint  work with Charles-Albert Lehalle.

Related Paper

Incorporating Signals Into Optimal Trading



Khalil Dayiri (Bloomberg)
Practical Considerations For Trading In Dark Pools

We go over some previously proposed models for trading in dark pool and explore their strengths and weaknesses from a practical perspective. We also showcase stylized facts relating to trade arrivals and price mouvements on different dark pools and make the argument that simpler optimization methods coupled with statistical estimators are preferred in a sell side trading environment and lead to more interesting options, trading styles and of course products.


David Fellah (J.P. Morgan)
Active Learning in Trading Algorithms 

In this presentation, we discuss some of the practical considerations in applying reinforcement learning to different trading problems. We first outline order book dynamics and market feedback mechanisms required for off-line simulation, highlighting some of the differences we observe around the globe in equities.  We then go on to describe some of the newer methods in reinforcement learning for high noise to signal ratio, and further apply them to various trading problems such as liquidation, pairs, and simple market-making problems.  Finally, we highlight some of our own experiences in applying the method in limit order book placement algorithms.



Pamela Saliba (Autorité des Marchés Financiers)
The Behaviour of High-Frequency Traders Under Different Market Stress Scenario

There is a big controversy about the consequences of High-Frequency Traders (HFTs) activity on market quality. This empirical study uses a unique data set provided by the French regulator “Autorité des Marchés Financiers” and gives some evidence concerning the practices of these members under market stress. On the one hand, we investigate the changes in HFTs behaviour related to the level of market stress at a daily scale. On the other hand, we analyse the intraday reactions of these specific market participants to various announcements. We find that in the absence of significant news, whatever the market conditions, HFTs are the main contributors to liquidity with a participation of 80% in the market depth at the three best limits. They constitute 60% of the traded amounts, with an aggressive/passive ratio equal to 53%. We identify a change of regime in the presence of scheduled news that goes beyond the expected reaction to volatility variations. This is notably characterised by an additional decrease by 15% of HFTs share in market depth. Moreover, we focus on two specific events: the European Central Bank announcements of the 3rd of December 2015 that occurred within a trading day and the Brexit that occurred prior to the market opening. Based on these two particular cases, we get that in extreme situations, when non-HFTs have time to adjust their trading tactics, they act as liquidity providers in place of HFTs.

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The Behaviour of High-Frequency Traders Under Different Market Stress Scenario


Carla Ysusi (Financial Conduct Authority)
Are High-Frequency Traders Anticipating the Order Flow? Cross-Venue Evidence from the UK Market

There have been allegations that high-frequency traders prey on other participants making profits taking no or minimal risk in the process by predicting with near certainty where orders will be routed. We investigate whether there is evidence of this happening systematically in the UK equity market. We examine whether high frequency traders exploit their milliseconds latency advantages to anticipate orders arriving in quick succession at different trading venues. We also analyze whether they can anticipate the order flow over longer time frames.

We do not find evidence that the first behavior is occurring systematically in UK markets. We find patterns consistent with the later but we cannot say whether this is due to them reacting faster to information.



Fabrizio Lillo (University of Bologna)
A Score-Driven Conditional Correlation Model for Noisy and Asynchronous Data: an Application to High-Frequency Covariance Dynamics

We propose a new multivariate conditional correlation model able to deal with data featuring both observational noise and asynchronicity. When modelling high-frequency multivariate financial time-series, the presence of both problems and the requirement for positive-definite estimates makes the estimation and forecast of the intraday dynamics of conditional covariance matrices particularly difficult. Our approach tackles all these challenging tasks within a new Gaussian state-space model with score-driven time-varying parameters that can be estimated using standard maximum likelihood methods. Similarly to DCC models, large dimensionality is handled by separating the estimation of correlations from individual volatilities. As an interesting outcome of this approach, intra-day patterns are recovered without the need of any time or cross-sectional averaging, allowing, for instance, to estimate the real-time response of the market covariances to macro-news announcements.

This is a joint work with G. Buccheri, G. Bormetti and F. Corsi.

Related Paper

A Score-Driven Conditional Correlation Model for Noisy and Asynchronous Data: An Application to High-frequency Covariance Dynamics


Iacopo Mastromatteo (Capital Fund Management)
Insights From Cross-Impact: What Really Is a Financial Instrument?

Recent results in the field of market microstructure have provided some statistical evidence on how the prices of financial instruments react to the order flow of correlated products. In this talk I will illustrate the implications of these findings in the field of price formation and optimal order execution, and speculate on how this can change the idea of what does a ‘single’ financial product mean. I will finally present some applications of these findings in the field of optimal order execution.