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arXiv2015

Calculating optimal limits for transacting credit card customers

We present a model of credit card profitability, assuming that the card-holder always pays the full outstanding balance. The motivation for the model is to calculate an optimal credit limit, which requires an expression for the expected outstanding balance. We derive its Laplace transform, assuming that purchases are made according to a marked point process and that there is a simplified balance control policy in place to prevent the credit limit being exceeded. We calculate optimal limits for a compound Poisson process example and show that the optimal limit scales with the distribution of the purchasing process and that the probability of exceeding the optimal limit remains constant. We establish a connection with the classic newsvendor model and use this to calculate bounds on the optimal limit for a more complicated balance control policy. Finally, we apply our model to real credit card purchase data.

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arXiv2015

Extension and calibration of a Hawkes-based optimal execution model

We provide some theoretical extensions and a calibration protocol for our former dynamic optimal execution model. The Hawkes parameters and the propagator are estimated independently on financial data from stocks of the CAC40. Interestingly, the propagator exhibits a smoothly decaying form with one or two dominant time scales, but only so after a few seconds that the market needs to adjust after a large trade. Motivated by our estimation results, we derive the optimal execution strategy for a multi-exponential Hawkes kernel and backtest it on the data for round trips. We find that the strategy is profitable on average when trading at the midprice, which is in accordance with violated martingale conditions. However, in most cases, these profits vanish when we take bid-ask costs into account.

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arXiv2015

Analysis of Ornstein-Uhlenbeck process stopped at maximum drawdown and application to trading strategies with trailing stops

We propose a strategy for automated trading, outline theoretical justification of the profitability of this strategy and overview the hypothetical results in application to currency pairs trading. The proposed methodology relies on the assumption that processes reflecting the dynamics of currency exchange rates are in a certain sense similar to the class of Ornstein-Uhlenbeck processes and exhibits the mean reverting property. In order to describe the quantitative characteristics of the projected return of the strategy, we derive the explicit expression for the running maximum of the Ornstein-Uhlenbeck process stopped at maximum drawdown and look at the correspondence between derived characteristics and the observed ones.

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arXiv2015

The time scales of the aggregate learning and sorting in market entry games with large number of players

We consider the dynamics of player's strategies in repeated market games, where the selection of strategies is determined by a learning model. Prior theoretical analysis and experimental data show that after large number of plays the average number of agents who decide to enter, per round of the game, approaches the market capacity and, after a longer wait, agents are being sorted into two groups: the agents in one group rarely enter the market, and in the other, the agents enter almost all the time. In this paper we obtain estimates of the characteristic times it takes for both patterns to emerge in the repeated plays of the game. The estimates are given in terms of the parameters of the game, assuming that the number of agents is large, the number of rounds of the game per unit of time is large, and the characteristic change of the propensity per game is small. Our approach is based on the analysis of the partial differential equation for the function $f(t,q)$ that describes the distribution of agents according to their level of propensity to enter the market, $q,$ at time $t.$

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arXiv2015

Optimum Liquidation Problem Associated with the Poisson Cluster Process

In this research, we develop a trading strategy for the discrete-time optimal liquidation problem of large order trading with different market microstructures in an illiquid market. In this framework, the flow of orders can be viewed as a point process with stochastic intensity. We model the price impact as a linear function of a self-exciting dynamic process. We formulate the liquidation problem as a discrete-time Markov Decision Processes, where the state process is a Piecewise Deterministic Markov Process (PDMP). The numerical results indicate that an optimal trading strategy is dependent on characteristics of the market microstructure. When no orders above certain value come the optimal solution takes offers in the lower levels of the limit order book in order to prevent not filling of orders and facing final inventory costs.

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arXiv2015

How to predict the consequences of a tick value change? Evidence from the Tokyo Stock Exchange pilot program

The tick value is a crucial component of market design and is often considered the most suitable tool to mitigate the effects of high frequency trading. The goal of this paper is to demonstrate that the approach introduced in Dayri and Rosenbaum (2015) allows for an ex ante assessment of the consequences of a tick value change on the microstructure of an asset. To that purpose, we analyze the pilot program on tick value modifications started in 2014 by the Tokyo Stock Exchange in light of this methodology. We focus on forecasting the future cost of market and limit orders after a tick value change and show that our predictions are very accurate. Furthermore, for each asset involved in the pilot program, we are able to define (ex ante) an optimal tick value. This enables us to classify the stocks according to the relevance of their tick value, before and after its modification.

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arXiv2015

Volatility Harvesting: Extracting Return from Randomness

Studying Binomial and Gaussian return dynamics in discrete time, we show how excess volatility can be traded to create growth. We test our results on real world data to confirm the observed model phenomena while also highlighting implicit risks.

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arXiv2015

Financial Knudsen number: breakdown of continuous price dynamics and asymmetric buy and sell structures confirmed by high precision order book information

We generalise the description of the dynamics of the order book of financial markets in terms of a Brownian particle embedded in a fluid of incoming, exiting and annihilating particles by presenting a model of the velocity on each side (buy and sell) independently. The improved model builds on the time-averaged number of particles in the inner layer and its change per unit time, where the inner layer is revealed by the correlations between price velocity and change in the number of particles (limit orders). This allows us to introduce the Knudsen number of the financial Brownian particle motion and its asymmetric version (on the buy and sell sides). Not being considered previously, the asymmetric Knudsen numbers are crucial in finance in order to detect asymmetric price changes. The Knudsen numbers allows us to characterise the conditions for the market dynamics to be correctly described by a continuous stochastic process. Not questioned until now for large liquid markets such as the USD/JPY and EUR/USD exchange rates, we show that there are regimes when the Knudsen numbers are so high that discrete particle effects dominate, such as during market stresses and crashes. We document the presence of imbalances of particles depletion rates on the buy and sell sides that are associated with high Knudsen numbers and violent directional price changes. This indicator can detect the direction of the price motion at the early stage while the usual volatility risk measure is blind to the price direction.

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arXiv2017

Liquidity Effects of Trading Frequency

In this article, we present a discrete time modeling framework, in which the shape and dynamics of a Limit Order Book (LOB) arise endogenously from an equilibrium between multiple market participants (agents). We use the proposed modeling framework to analyze the effects of trading frequency on market liquidity in a very general setting. In particular, we demonstrate the dual effect of high trading frequency. On the one hand, the higher frequency increases market efficiency, if the agents choose to provide liquidity in equilibrium. On the other hand, it also makes markets more fragile, in the sense that the agents choose to provide liquidity in equilibrium only if they are market-neutral (i.e., their beliefs satisfy certain martingale property). Even a very small deviation from market-neutrality may cause the agents to stop providing liquidity, if the trading frequency is sufficiently high, which represents an endogenous liquidity crisis (aka flash crash) in the market. This framework enables us to provide more insight into how such a liquidity crisis unfolds, connecting it to the so-called adverse selection effect.

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arXiv2016

Identifying collusion groups using spectral clustering

In an illiquid stock, traders can collude and place orders on a predetermined price and quantity at a fixed schedule. This is usually done to manipulate the price of the stock or to create artificial liquidity in the stock, which may mislead genuine investors. Here, the problem is to identify such group of colluding traders. We modeled the problem instance as a graph, where each trader corresponds to a vertex of the graph and trade corresponds to edges of the graph. Further, we assign weights on edges depending on total volume, total number of trades, maximum change in the price and commonality between two vertices. Spectral clustering algorithms are used on the constructed graph to identify colluding group(s). We have compared our results with simulated data to show the effectiveness of spectral clustering to detecting colluding groups. Moreover, we also have used parameters of real data to test the effectiveness of our algorithm.

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arXiv2015

Market Making with Model Uncertainty

Pari-mutuel markets are trading platforms through which the common market maker simultaneously clears multiple contingent claims markets. This market has several distinctive properties that began attracting the attention of the financial industry in the 2000s. For example, the platform aggregates liquidity from the individual contingent claims market into the common pool while shielding the market maker from potential financial loss. The contribution of this paper is two-fold. First, we provide a new economic interpretation of the market-clearing strategy of a pari-mutuel market that is well known in the literature. The pari-mutuel auctioneer is shown to be equivalent to the market maker with extreme ambiguity aversion for the future contingent event. Second, based on this theoretical understanding, we present a new market-clearing algorithm called the Knightian Pari-mutuel Mechanism (KPM). The KPM retains many interesting properties of pari-mutuel markets while explicitly controlling for the market maker's ambiguity aversion. In addition, the KPM is computationally efficient in that it is solvable in polynomial time.

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arXiv2015

Quadratic Hawkes processes for financial prices

We introduce and establish the main properties of QHawkes ("Quadratic" Hawkes) models. QHawkes models generalize the Hawkes price models introduced in E. Bacry et al. (2014), by allowing all feedback effects in the jump intensity that are linear and quadratic in past returns. A non-parametric fit on NYSE stock data shows that the off-diagonal component of the quadratic kernel indeed has a structure that standard Hawkes models fail to reproduce. Our model exhibits two main properties, that we believe are crucial in the modelling and the understanding of the volatility process: first, the model is time-reversal asymmetric, similar to financial markets whose time evolution has a preferred direction. Second, it generates a multiplicative, fat-tailed volatility process, that we characterize in detail in the case of exponentially decaying kernels, and which is linked to Pearson diffusions in the continuous limit. Several other interesting properties of QHawkes processes are discussed, in particular the fact that they can generate long memory without necessarily be at the critical point. Finally, we provide numerical simulations of our calibrated QHawkes model, which is indeed seen to reproduce, with only a small amount of quadratic non-linearity, the correct magnitude of fat-tails and time reversal asymmetry seen in empirical time series.

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arXiv2015

Volume Weighted Average Price Optimal Execution

We study the problem of optimal execution of a trading order under Volume Weighted Average Price (VWAP) benchmark, from the point of view of a risk-averse broker. The problem consists in minimizing mean-variance of the slippage, with quadratic transaction costs. We devise multiple ways to solve it, in particular we study how to incorporate the information coming from the market during the schedule. Most related works in the literature eschew the issue of imperfect knowledge of the total market volume. We instead incorporate it in our model. We validate our method with extensive simulation of order execution on real NYSE market data. Our proposed solution, using a simple model for market volumes, reduces by 10% the VWAP deviation RMSE of the standard "static" solution (and can simultaneously reduce transaction costs).

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arXiv2016

Price response in correlated financial markets: empirical results

Previous studies of the stock price response to individual trades focused on single stocks. We empirically investigate the price response of one stock to the trades of other stocks. How large is the impact of one stock on others and vice versa? -- This impact of trades on the price change across stocks appears to be transient instead of permanent. Performing different averages, we distinguish active and passive responses. The two average responses show different characteristic dependences on the time lag. The passive response exhibits a shorter response period with sizeable volatilities, and the active response a longer period. We also study the response for a given stock with respect to different sectors and to the whole market. Furthermore, we compare the self-response with the various cross-responses. The correlation of the trade signs is a short-memory process for a pair of stocks, but it turns into a long-memory process when averaged over different pairs of stocks.

q-fin.ST↗

arXiv2015

Performance analysis of the optimal strategy under partial information

The question addressed in this paper is the performance of the optimal strategy, and the impact of partial information. The setting we consider is that of a stochastic asset price model where the trend follows an unobservable Ornstein-Uhlenbeck process. We focus on the optimal strategy with a logarithmic utility function under full or partial information. For both cases, we provide the asymptotic expectation and variance of the logarithmic return as functions of the signal-to-noise ratio and of the trend mean reversion speed. Finally, we compare the asymptotic Sharpe ratios of these strategies in order to quantify the loss of performance due to partial information.

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arXiv2015

On the Efficient Market Hypothesis of Stock Market Indexes: The Role of Non-synchronous Trading and Portfolio Effects

In this article, the long-term behavior of the stock market index of the New York Stock Exchange is studied, for the period 1950 to 2013. Specifically, the CRSP Value-Weighted and CRSP Equal-Weighted index are analyzed in terms of market efficiency, using the standard ratio variance test, considering over 1600 one week rolling windows. For the equally weighted index, the null hypothesis of random walk is rejected in the whole period, while for the weighted market value index, the null hypothesis start to be accepted from the 1990s. In order to explain this difference, we raised the hypothesis that this behavior can be explained by the joint action of portfolios and non-synchronous trading effects. To check the feasibility of the above assumption, we performed a simulation of both effects, on two- and six-asset portfolios. The results showed that it is possible to explain the empirical difference between the two index, almost entirely by the joint effects of portfolio and non-synchronous trading.

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arXiv2015

Optimal Portfolio Liquidation and Dynamic Mean-variance Criterion

In this paper, we consider the optimal portfolio liquidation problem under the dynamic mean-variance criterion and derive time-consistent solutions in three important models. We give adapted optimal strategies under a reconsidered mean-variance subject at any point in time. We get explicit trading strategies in the basic model and when random pricing signals are incorporated. When we consider stochastic liquidity and volatility, we construct a generalized HJB equation under general assumptions for the parameters. We obtain an explicit solution in stochastic volatility model with a given structure supported by empirical studies.

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arXiv2015

Learning Unfair Trading: a Market Manipulation Analysis From the Reinforcement Learning Perspective

Market manipulation is a strategy used by traders to alter the price of financial securities. One type of manipulation is based on the process of buying or selling assets by using several trading strategies, among them spoofing is a popular strategy and is considered illegal by market regulators. Some promising tools have been developed to detect manipulation, but cases can still be found in the markets. In this paper we model spoofing and pinging trading, two strategies that differ in the legal background but share the same elemental concept of market manipulation. We use a reinforcement learning framework within the full and partial observability of Markov decision processes and analyse the underlying behaviour of the manipulators by finding the causes of what encourages the traders to perform fraudulent activities. This reveals procedures to counter the problem that may be helpful to market regulators as our model predicts the activity of spoofers.

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