Quantitative Finance

We propose an analytically tractable class of models for the dynamics of a limit order book, described through a stochastic partial differential equation (SPDE) with multiplicative noise for the order book centered at the mid-price, along with stochastic dynamics for the mid-price which is consistent with the order flow dynamics. We provide conditions under which the model admits a finite dimensional realization driven by a (low-dimensional) Markov process, leading to efficient estimation and computation methods. We study two examples of parsimonious models in this class: a two-factor model and a model with mean-reverting order book depth. For each model we analyze in detail the role of different parameters, the dynamics of the price, order book depth, volume and order imbalance, provide an intuitive financial interpretation of the variables involved and show how the model reproduces statistical properties of price changes, market depth and order flow in limit order markets.

A general framework is suggested to describe human decision making in a certain class of experiments performed in a trading laboratory. We are in particular interested in discerning between two different moods, or states of the investors, corresponding to investors using fundamental investment strategies, technical analysis investment strategies respectively. Our framework accounts for two opposite situations already encountered in experimental setups: i) the rational expectations case, and ii) the case of pure speculation. We consider new experimental conditions which allow both elements to be present in the decision making process of the traders, thereby creating a dilemma in terms of investment strategy. Our theoretical framework allows us to predict the outcome of this type of trading experiments, depending on such variables as the number of people trading, the liquidity of the market, the amount of information used in technical analysis strategies, as well as the dividends attributed to an asset. We find that it is possible to give a qualitative prediction of trading behavior depending on a ratio that quantifies the fluctuations in the model.

We introduce a new matching design for financial transactions in an electronic market. In this mechanism, called ad-hoc electronic auction design (AHEAD), market participants can trade between themselves at a fixed price and trigger an auction when they are no longer satisfied with this fixed price. In this context, we prove that a Nash equilibrium is obtained between market participants. Furthermore, we are able to assess quantitatively the relevance of ad-hoc auctions and to compare them with periodic auctions and continuous limit order books. We show that from the investors' viewpoint, the microstructure of the asset is usually significantly improved when using AHEAD.

In this article, we consider the optimal execution problem associated to accelerated share repurchase contracts. When firms want to repurchase their own shares, they often enter such a contract with a bank. The bank buys the shares for the firm and is paid the average market price over the execution period, the length of the period being decided upon by the bank during the buying process. Mathematically, the problem is new and related to both option pricing (Asian and Bermudan options) and optimal execution. We provide a model, along with associated numerical methods, to determine the optimal stopping time and the optimal buying strategy of the bank.

In this article, we provide a flexible framework for optimal trading in an asset listed on different venues. We take into account the dependencies between the imbalance and spread of the venues, and allow for partial execution of limit orders at different limits as well as market orders. We present a Bayesian update of the model parameters to take into account possibly changing market conditions and propose extensions to include short/long trading signals, market impact or hidden liquidity. To solve the stochastic control problem of the trader we apply the finite difference method and also develop a deep reinforcement learning algorithm allowing to consider more complex settings.

For more than a decade Vytelingum's Adaptive-Aggressive (AA) algorithm has been recognized as the best-performing automated auction-market trading-agent strategy currently known in the AI/Agents literature; in this paper, we demonstrate that it is in fact routinely outperformed by another algorithm when exhaustively tested across a sufficiently wide range of market scenarios. The novel step taken here is to use large-scale compute facilities to brute-force exhaustively evaluate AA in a variety of market environments based on those used for testing it in the original publications. Our results show that even in these simple environments AA is consistently out-performed by IBM's GDX algorithm, first published in 2002. We summarize here results from more than one million market simulation experiments, orders of magnitude more testing than was reported in the original publications that first introduced AA. A 2019 ICAART paper by Cliff claimed that AA's failings were revealed by testing it in more realistic experiments, with conditions closer to those found in real financial markets, but here we demonstrate that even in the simple experiment conditions that were used in the original AA papers, exhaustive testing shows AA to be outperformed by GDX. We close this paper with a discussion of the methodological implications of our work: any results from previous papers where any one trading algorithm is claimed to be superior to others on the basis of only a few thousand trials are probably best treated with some suspicion now. The rise of cloud computing means that the compute-power necessary to subject trading algorithms to millions of trials over a wide range of conditions is readily available at reasonable cost: we should make use of this; exhaustive testing such as is shown here should be the norm in future evaluations and comparisons of new trading algorithms.

Adversarial samples have drawn a lot of attention from the Machine Learning community in the past few years. An adverse sample is an artificial data point coming from an imperceptible modification of a sample point aiming at misleading. Surprisingly, in financial research, little has been done in relation to this topic from a concrete trading point of view. We show that those adversarial samples can be implemented in a trading environment and have a negative impact on certain market participants. This could have far reaching implications for financial markets either from a trading or a regulatory point of view.

We revisit the "epsilon-intelligence" model of Toth et al.(2011), that was proposed as a minimal framework to understand the square-root dependence of the impact of meta-orders on volume in financial markets. The basic idea is that most of the daily liquidity is "latent" and furthermore vanishes linearly around the current price, as a consequence of the diffusion of the price itself. However, the numerical implementation of Toth et al. was criticised as being unrealistic, in particular because all the "intelligence" was conferred to market orders, while limit orders were passive and random. In this work, we study various alternative specifications of the model, for example allowing limit orders to react to the order flow, or changing the execution protocols. By and large, our study lends strong support to the idea that the square-root impact law is a very generic and robust property that requires very few ingredients to be valid. We also show that the transition from super-diffusion to sub-diffusion reported in Toth et al. is in fact a cross-over, but that the original model can be slightly altered in order to give rise to a genuine phase transition, which is of interest on its own. We finally propose a general theoretical framework to understand how a non-linear impact may appear even in the limit where the bias in the order flow is vanishingly small.

We present the use of the fitted Q iteration in algorithmic trading. We show that the fitted Q iteration helps alleviate the dimension problem that the basic Q-learning algorithm faces in application to trading. Furthermore, we introduce a procedure including model fitting and data simulation to enrich training data as the lack of data is often a problem in realistic application. We experiment our method on both simulated environment that permits arbitrage opportunity and real-world environment by using prices of 450 stocks. In the former environment, the method performs well, implying that our method works in theory. To perform well in the real-world environment, the agents trained might require more training (iteration) and more meaningful variables with predictive value.

We propose a microstructural modeling framework for studying optimal market making policies in a FIFO (first in first out) limit order book (LOB). In this context, the limit orders, market orders, and cancel orders arrivals in the LOB are modeled as Cox point processes with intensities that only depend on the state of the LOB. These are high-dimensional models which are realistic from a micro-structure point of view and have been recently developed in the literature. In this context, we consider a market maker who stands ready to buy and sell stock on a regular and continuous basis at a publicly quoted price, and identifies the strategies that maximize her P\&L penalized by her inventory. We apply the theory of Markov Decision Processes and dynamic programming method to characterize analytically the solutions to our optimal market making problem. The second part of the paper deals with the numerical aspect of the high-dimensional trading problem. We use a control randomization method combined with quantization method to compute the optimal strategies. Several computational tests are performed on simulated data to illustrate the efficiency of the computed optimal strategy. In particular, we simulated an order book with constant/ symmet-ric/ asymmetrical/ state dependent intensities, and compared the computed optimal strategy with naive strategies. Some codes are available on this https URL.