Quantitative Finance

This paper goes beyond the optimal trading Mean Field Game model introduced by Pierre Cardaliaguet and Charles-Albert Lehalle in [Cardaliaguet, P. and Lehalle, C.-A., Mean field game of controls and an application to trade crowding, Mathematics and Financial Economics (2018)]. It starts by extending it to portfolios of correlated instruments. This leads to several original contributions: first that hedging strategies naturally stem from optimal liquidation schemes on portfolios. Second we show the influence of trading flows on naive estimates of intraday volatility and correlations. Focussing on this important relation, we exhibit a closed form formula expressing standard estimates of correlations as a function of the underlying correlations and the initial imbalance of large orders, via the optimal flows of our mean field game between traders. To support our theoretical findings, we use a real dataset of 176 US stocks from January to December 2014 sampled every 5 minutes to analyze the influence of the daily flows on the observed correlations. Finally, we propose a toy model based approach to calibrate our MFG model on data.

We present a thorough empirical analysis of market impact on the Bitcoin/USD exchange market using a complete dataset that allows us to reconstruct more than one million metaorders. We empirically confirm the "square-root law'' for market impact, which holds on four decades in spite of the quasi-absence of statistical arbitrage and market marking strategies. We show that the square-root impact holds during the whole trajectory of a metaorder and not only for the final execution price. We also attempt to decompose the order flow into an "informed'' and "uninformed'' component, the latter leading to an almost complete long-term decay of impact. This study sheds light on the hypotheses and predictions of several market impact models recently proposed in the literature and promotes heterogeneous agent models as promising candidates to explain price impact on the Bitcoin market -- and, we believe, on other markets as well.

Traders are often faced with large block orders in markets with limited liquidity and varying volatility. Executing the entire order at once usually incurs a large trading cost because of this limited liquidity. In order to minimize this cost traders split up large orders over time. Varying volatility however implies that they now take on price risk, as the underlying assets' prices can move against the traders over the execution period. This execution problem therefore requires a careful balancing between trading slow to reduce liquidity cost and trading fast to reduce the volatility cost. R. Almgren solved this problem for a market with one asset and stochastic liquidity and volatility parameters, using a mean-variance framework. This leads to a nonlinear PDE that needs to be solved numerically. We propose a different approach using (quasi-)Monte Carlo which can handle any number of assets. Furthermore, our method can be run in real-time and allows the trader to change the parameters of the underlying stochastic processes on-the-fly.

It is well known that the distribution of returns from various financial instruments are leptokurtic, meaning that the distributions have "fatter tails" than a Normal distribution, and have skew toward zero. This paper presents a graceful micro-level explanation for such fat-tailed outcomes, using agents whose private valuations have Normally-distributed errors, but whose utility function includes a term for the percentage of others who also buy.

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 models by having market signals more properly expressed. For the opportunistic cost, we are able to go beyond the conventional Brownian-type motions. By working directly with the auto-covariances of the price dynamics, we remove the Markovian restriction associated with Brownians and thus allow potential memory effects in the price dynamics. In combination, the final execution model becomes a constrained quadratic programming problem in infinite-dimensional Hilbert spaces. Important linear constraints such as participation capping are all permissible. Uniqueness and existence of optimal solutions are established via the theory of positive compact operators in Hilbert spaces. Several typical numerical examples explain both the behavior and versatility of the model.

The analysis of the intraday dynamics of correlations among high-frequency returns is challenging due to the presence of asynchronous trading and market microstructure noise. Both effects may lead to significant data reduction and may severely underestimate correlations if traditional methods for low-frequency data are employed. We propose to model intraday log-prices through a multivariate local-level model with score-driven covariance matrices and to treat asynchronicity as a missing value problem. The main advantages of this approach are: (i) all available data are used when filtering correlations, (ii) market microstructure noise is taken into account, (iii) estimation is performed through standard maximum likelihood methods. Our empirical analysis, performed on 1-second NYSE data, shows that opening hours are dominated by idiosyncratic risk and that a market factor progressively emerges in the second part of the day. The method can be used as a nowcasting tool for high-frequency data, allowing to study the real-time response of covariances to macro-news announcements and to build intraday portfolios with very short optimization horizons.

We provide an economically sound micro-foundation to linear price impact models, by deriving them as the equilibrium of a suitable agent-based system. Our setup generalizes the well-known Kyle model, by dropping the assumption of a terminal time at which fundamental information is revealed so to describe a stationary market, while retaining agents' rationality and asymmetric information. We investigate the stationary equilibrium for arbitrary Gaussian noise trades and fundamental information, and show that the setup is compatible with universal price diffusion at small times, and non-universal mean-reversion at time scales at which fluctuations in fundamentals decay. Our model provides a testable relation between volatility of prices, magnitude of fluctuations in fundamentals and level of volume traded in the market.

Modern Algorithmic Trading ("Algo") allows institutional investors and traders to liquidate or establish big security positions in a fully automated or low-touch manner. Most existing academic or industrial Algos focus on how to "slice" a big parent order into smaller child orders over a given time horizon. Few models rigorously tackle the actual placement of these child orders. Instead, placement is mostly done with a combination of empirical signals and heuristic decision processes. A self-contained, realistic, and fully functional Child Order Placement (COP) model may never exist due to all the inherent complexities, e.g., fragmentation due to multiple venues, dynamics of limit order books, lit vs. dark liquidity, different trading sessions and rules. In this paper, we propose a reductionism COP model that focuses exclusively on the interplay between placing passive limit orders and sniping using aggressive takeout orders. The dynamic programming model assumes the form of a stochastic linear-quadratic regulator (LQR) and allows closed-form solutions under the backward Bellman equations. Explored in detail are model assumptions and general settings, the choice of state and control variables and the cost functions, and the derivation of the closed-form solutions.

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 Brownian motions. We then modify these bouncing GBMs to construct a discrete time stochastic process of trading times and trading prices, which is parameterized by a positive parameter δ . Under this model, it is shown that the inter-trading times are inverse Gaussian distributed, and the logarithmic returns between consecutive trading times follow a normal inverse Gaussian distribution. Our main results show that the logarithmic trading price process is a renewal reward process, and under a suitable scaling, this process converges to a standard Brownian motion as δ→0 . We also prove that the modified ask and bid processes approach the original bouncing GBMs as δ→0 . Finally, we derive a simple and effective prediction formula for trading prices, and illustrate the effectiveness of the prediction formula with an example using real stock price data.

We look at the effect of the tick size changes on the TOPIX 100 index names made by the Tokyo Stock Exchange on Jan-14-2014 and Jul-22-2104. The intended consequence of the change is price improvement and shorter time to execution. We look at security level metrics that include the spread, trading volume, number of trades and the size of trades to establish whether this goal is accomplished. An unintended effect might be the reduction in execution sizes, which would then mean that institutions with large orders would have greater difficulty in sourcing liquidity. We look at a sample of real orders to see if the execution costs have gone up across the orders since the implementation of this change. We study the mechanisms that affect how securities are traded on an exchange, before delving into the specifics of the TSE tick size events. Some of the topics we explore are: The Venue Menu and How to Increase Revenue; To Automate or Not to Automate; Microstructure under the Microscope; The Price of Connections to High (and Faraway) Places; Speed Thrills but Kills; Pick a Size for the Perfect Tick; TSE Tick Size Experiments, Then and Now; Sergey Bubka and the Regulators; Bird`s Eye View; Deep Dive; Possibilities for a Deeper Dive; Does Tick Size Matter? Tick Size Does Matter!