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arXiv2021

Càdlàg semimartingale strategies for optimal trade execution in stochastic order book models

We analyze an optimal trade execution problem in a financial market with stochastic liquidity. To this end we set up a limit order book model in continuous time. Both order book depth and resilience are allowed to evolve randomly in time. We allow for trading in both directions and for càdlàg semimartingales as execution strategies. We derive a quadratic BSDE that under appropriate assumptions characterizes minimal execution costs and identify conditions under which an optimal execution strategy exists. We also investigate qualitative aspects of optimal strategies such as, e.g., appearance of strategies with infinite variation or existence of block trades and discuss connections with the discrete-time formulation of the problem. Our findings are illustrated in several examples.

math.PR↗

arXiv2025

Path-dependent Kyle equilibrium model

We consider an auction type equilibrium model with an insider in line with the one originally introduced by Kyle in 1985 and then extended to the continuous time setting by Back in 1992. The novelty introduced with this paper is that we deal with a general price functional depending on the whole past of the aggregate demand, i.e. we work with path-dependency. By using the functional Itô calculus, we provide necessary and sufficient conditions for the existence of an equilibrium. Furthermore, we consider both the cases of a risk-neutral and a risk-averse insider.

q-fin.TR↗

arXiv2022

Evidence of Crowding on Russell 3000 Reconstitution Events

We develop a methodology which replicates in great accuracy the FTSE Russell indexes reconstitutions, including the quarterly rebalancings due to new initial public offerings (IPOs). While using only data available in the CRSP US Stock database for our index reconstruction, we demonstrate the accuracy of this methodology by comparing it to the original Russell US indexes for the time period between 1989 to 2019. A python package that generates the replicated indexes is also provided. As an application, we use our index reconstruction protocol to compute the permanent and temporary price impact on the Russell 3000 annual additions and deletions, and on the quarterly additions of new IPOs . We find that the index portfolios following the Russell 3000 index and rebalanced on an annual basis are overall more crowded than those following the index on a quarterly basis. This phenomenon implies that transaction costs of indexing strategies could be significantly reduced by buying new IPOs additions in proximity to quarterly rebalance dates.

q-fin.TR↗

arXiv2020

Hidden Markov Models Applied To Intraday Momentum Trading With Side Information

A Hidden Markov Model for intraday momentum trading is presented which specifies a latent momentum state responsible for generating the observed securities' noisy returns. Existing momentum trading models suffer from time-lagging caused by the delayed frequency response of digital filters. Time-lagging results in a momentum signal of the wrong sign, when the market changes trend direction. A key feature of this state space formulation, is no such lagging occurs, allowing for accurate shifts in signal sign at market change points. The number of latent states in the model is estimated using three techniques, cross validation, penalized likelihood criteria and simulation-based model selection for the marginal likelihood. All three techniques suggest either 2 or 3 hidden states. Model parameters are then found using Baum-Welch and Markov Chain Monte Carlo, whilst assuming a single (discretized) univariate Gaussian distribution for the emission matrix. Often a momentum trader will want to condition their trading signals on additional information. To reflect this, learning is also carried out in the presence of side information. Two sets of side information are considered, namely a ratio of realized volatilities and intraday seasonality. It is shown that splines can be used to capture statistically significant relationships from this information, allowing returns to be predicted. An Input Output Hidden Markov Model is used to incorporate these univariate predictive signals into the transition matrix, presenting a possible solution for dealing with the signal combination problem. Bayesian inference is then carried out to predict the securities $t+1$ return using the forward algorithm. Simple modifications to the current framework allow for a fully non-parametric model with asynchronous prediction.

q-fin.TR↗

arXiv2020

The Importance of Low Latency to Order Book Imbalance Trading Strategies

There is a pervasive assumption that low latency access to an exchange is a key factor in the profitability of many high-frequency trading strategies. This belief is evidenced by the "arms race" undertaken by certain financial firms to co-locate with exchange servers. To the best of our knowledge, our study is the first to validate and quantify this assumption in a continuous double auction market with a single exchange similar to the New York Stock Exchange. It is not feasible to conduct this exploration with historical data in which trader identity and location are not reported. Accordingly, we investigate the relationship between latency of access to order book information and profitability of trading strategies exploiting that information with an agent-based interactive discrete event simulation in which thousands of agents pursue archetypal trading strategies. We introduce experimental traders pursuing a low-latency order book imbalance (OBI) strategy in a controlled manner across thousands of simulated trading days, and analyze OBI trader profit while varying distance (latency) from the exchange. Our experiments support that latency is inversely related to profit for the OBI traders, but more interestingly show that latency rank, rather than absolute magnitude, is the key factor in allocating returns among agents pursuing a similar strategy.

q-fin.TR↗

arXiv2020

Liquidity Provider Returns in Geometric Mean Markets

Geometric mean market makers (G3Ms), such as Uniswap and Balancer, comprise a popular class of automated market makers (AMMs) defined by the following rule: the reserves of the AMM before and after each trade must have the same (weighted) geometric mean. This paper extends several results known for constant-weight G3Ms to the general case of G3Ms with time-varying and potentially stochastic weights. These results include the returns and no-arbitrage prices of liquidity pool (LP) shares that investors receive for supplying liquidity to G3Ms. Using these expressions, we show how to create G3Ms whose LP shares replicate the payoffs of financial derivatives. The resulting hedges are model-independent and exact for derivative contracts whose payoff functions satisfy an elasticity constraint. These strategies allow LP shares to replicate various trading strategies and financial contracts, including standard options. G3Ms are thus shown to be capable of recreating a variety of active trading strategies through passive positions in LP shares.

q-fin.MF↗

arXiv2020

Prior knowledge distillation based on financial time series

One of the major characteristics of financial time series is that they contain a large amount of non-stationary noise, which is challenging for deep neural networks. People normally use various features to address this problem. However, the performance of these features depends on the choice of hyper-parameters. In this paper, we propose to use neural networks to represent these indicators and train a large network constructed of smaller networks as feature layers to fine-tune the prior knowledge represented by the indicators. During back propagation, prior knowledge is transferred from human logic to machine logic via gradient descent. Prior knowledge is the deep belief of neural network and teaches the network to not be affected by non-stationary noise. Moreover, co-distillation is applied to distill the structure into a much smaller size to reduce redundant features and the risk of overfitting. In addition, the decisions of the smaller networks in terms of gradient descent are more robust and cautious than those of large networks. In numerical experiments, we find that our algorithm is faster and more accurate than traditional methods on real financial datasets. We also conduct experiments to verify and comprehend the method.

cs.LG↗

arXiv2021

Optimal Transport and Risk Aversion in Kyle's Model of Informed Trading

We establish connections between optimal transport theory and the dynamic version of the Kyle model, including new characterizations of informed trading profits via conjugate duality and Monge-Kantorovich duality. We use these connections to extend the model to multiple assets, general distributions, and risk-averse market makers. With risk-averse market makers, liquidity is lower, assets exhibit short-term reversals, and risk premia depend on market maker inventories, which are mean reverting. We illustrate the model by showing that implied volatilities predict stock returns when there is informed trading in stocks and options and market makers are risk averse.

q-fin.TR↗

arXiv2021

Learning a functional control for high-frequency finance

We use a deep neural network to generate controllers for optimal trading on high frequency data. For the first time, a neural network learns the mapping between the preferences of the trader, i.e. risk aversion parameters, and the optimal controls. An important challenge in learning this mapping is that in intraday trading, trader's actions influence price dynamics in closed loop via the market impact. The exploration--exploitation tradeoff generated by the efficient execution is addressed by tuning the trader's preferences to ensure long enough trajectories are produced during the learning phase. The issue of scarcity of financial data is solved by transfer learning: the neural network is first trained on trajectories generated thanks to a Monte-Carlo scheme, leading to a good initialization before training on historical trajectories. Moreover, to answer to genuine requests of financial regulators on the explainability of machine learning generated controls, we project the obtained "blackbox controls" on the space usually spanned by the closed-form solution of the stylized optimal trading problem, leading to a transparent structure. For more realistic loss functions that have no closed-form solution, we show that the average distance between the generated controls and their explainable version remains small. This opens the door to the acceptance of ML-generated controls by financial regulators.

math.OC↗

arXiv2020

Why Stake When You Can Borrow?

As smart contract platforms autonomously manage billions of dollars of capital, quantifying the portfolio risk that investors engender in these systems is increasingly important. Recent work illustrates that Proof of Stake (PoS) is vulnerable to financial attacks arising from on-chain lending and has worse capital efficiency than Proof of Work (PoW) \cite{fanti_pos_econ}. Numerous methods for improving capital efficiency have been proposed that allow stakers to create fungible derivative claims on their staked assets. In this paper, we construct a unifying model for studying the security risks of these proposals. This model combines birth-death Pólya processes and risk models adapted from the credit derivatives literature to assess token inequality and return profiles. We find that there is a sharp transition between 'safe' and 'unsafe' derivative usage. Surprisingly, we find that contrary to \cite{fanti2019compounding} there exist conditions where derivatives can \emph{reduce} concentration of wealth in these networks. This model also applies to Decentralized Finance (DeFi) protocols where staked assets are used as insurance. Our theoretical results are validated using agent-based simulation.

q-fin.GN↗

arXiv2020

A note on Almgren-Chriss optimal execution problem with geometric Brownian motion

We solve explicitly the Almgren-Chriss optimal liquidation problem where the stock price process follows a geometric Brownian motion. Our technique is to work in terms of cash and to use functional analysis tools. We show that this framework extends readily to the case of a stochastic drift for the price process and the liquidation of a portfolio.

q-fin.TR↗

arXiv2020

Deep Investing in Kyle's Single Period Model

The Kyle model describes how an equilibrium of order sizes and security prices naturally arises between a trader with insider information and the price providing market maker as they interact through a series of auctions. Ever since being introduced by Albert S. Kyle in 1985, the model has become important in the study of market microstructure models with asymmetric information. As it is well understood, it serves as an excellent opportunity to study how modern deep learning technology can be used to replicate and better understand equilibria that occur in certain market learning problems. We model the agents in Kyle's single period setting using deep neural networks. The networks are trained by interacting following the rules and objectives as defined by Kyle. We show how the right network architectures and training methods lead to the agents' behaviour converging to the theoretical equilibrium that is predicted by Kyle's model.

q-fin.CP↗

arXiv2020

Deep Learning modeling of Limit Order Book: a comparative perspective

The present work addresses theoretical and practical questions in the domain of Deep Learning for High Frequency Trading. State-of-the-art models such as Random models, Logistic Regressions, LSTMs, LSTMs equipped with an Attention mask, CNN-LSTMs and MLPs are reviewed and compared on the same tasks, feature space and dataset, and then clustered according to pairwise similarity and performance metrics. The underlying dimensions of the modeling techniques are hence investigated to understand whether these are intrinsic to the Limit Order Book's dynamics. We observe that the Multilayer Perceptron performs comparably to or better than state-of-the-art CNN-LSTM architectures indicating that dynamic spatial and temporal dimensions are a good approximation of the LOB's dynamics, but not necessarily the true underlying dimensions.

q-fin.TR↗

arXiv2020

Transaction Costs in Execution Trading

In the present work we develop a formalism to tackle the problem of optimal execution when trading market securities. More precisely, we introduce a utility function that balances market impact and timing risk, with this last being modelled as the very negative transaction costs incurred by our order execution. The framework is built upon existing theory on optimal trading strategies, but incorporates characteristics that enable distinctive execution strategies. The formalism is complemented by an analysis of various impact models and different distributional properties of market returns.

q-fin.TR↗

arXiv2020

Formally Verified Trades in Financial Markets

We introduce a formal framework for analyzing trades in financial markets. These days, all big exchanges use computer algorithms to match buy and sell requests and these algorithms must abide by certain regulatory guidelines. For example, market regulators enforce that a matching produced by exchanges should be fair, uniform and individual rational. To verify these properties of trades, we first formally define these notions in a theorem prover and then develop many important results about matching demand and supply. Finally, we use this framework to verify properties of two important classes of double sided auction mechanisms. All the definitions and results presented in this paper are completely formalized in the Coq proof assistant without adding any additional axioms to it.

cs.LO↗

arXiv2020

Insider Trading with Temporary Price Impact

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.

q-fin.TR↗

arXiv2020

Investment sizing with deep learning prediction uncertainties for high-frequency Eurodollar futures trading

In this work we show that prediction uncertainty estimates gleaned from deep learning models can be useful inputs for influencing the relative allocation of risk capital across trades. In this way, consideration of uncertainty is important because it permits the scaling of investment size across trade opportunities in a principled and data-driven way. We showcase this insight with a prediction model and find clear outperformance based on a Sharpe ratio metric, relative to trading strategies that either do not take uncertainty into account, or that utilize an alternative market-based statistic as a proxy for uncertainty. Of added novelty is our modelling of high-frequency data at the top level of the Eurodollar Futures limit order book for each trading day of 2018, whereby we predict interest rate curve changes on small time horizons. We are motivated to study the market for these popularly-traded interest rate derivatives since it is deep and liquid, and contributes to the efficient functioning of global finance -- though there is relatively little by way of its modelling contained in the academic literature. Hence, we verify the utility of prediction models and uncertainty estimates for trading applications in this complex and multi-dimensional asset price space.

q-fin.ST↗

arXiv2020

A central bank strategy for defending a currency peg

We consider a central bank strategy for maintaining a two-sided currency target zone, in which an exchange rate of two currencies is forced to stay between two thresholds. To keep the exchange rate from breaking the prescribed barriers, the central bank is generating permanent price impact and thereby accumulating inventory in the foreign currency. Historical examples of failed target zones illustrate that this inventory can become problematic, in particular when there is an adverse macroeconomic trend in the market. We model this situation through a continuous-time market impact model of Almgren--Chriss-type with drift, in which the exchange rate is a diffusion process controlled by the price impact of the central bank's intervention strategy. The objective of the central bank is to enforce the target zone through a strategy that minimizes the accumulated inventory. We formulate this objective as a stochastic control problem with random time horizon. It is solved by reduction to a singular boundary value problem that was solved by Lasry and Lions (1989). Finally, we provide numerical simulations of optimally controlled exchange rate processes and the corresponding evolution of the central bank inventory.

q-fin.TR↗