ABERGEL Frederic

The main objective of this thesis is to understand the various aspects of market impact. It consists of four chapters in which market impact is studied in different contexts and at different scales. The first chapter presents an empirical study of the market impact of limit orders in European equity markets. In the second chapter, we have extended the methodology presented for the equity markets to the options markets. This empirical study has shown that our definition of an options meta-order allows us to recover all the results highlighted in the equity markets. The third chapter focuses on market impact in the context of derivatives valuation. This chapter attempts to bring a microstructure component to the valuation of options by proposing a theory of market impact disturbances during the re-hedging process. In the fourth chapter, we explore a fairly simple model for metaorder relaxation. Metaorder relaxation is treated in this section as an informational process that is transmitted to the market. Thus, starting from the point of departure that at the end of the execution of a meta-order the information carried by it is maximal, we propose an interpretation of the relaxation phenomenon as being the result of the degradation of this information at the expense of the external noise of the market.

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.

This paper deals with a fundamental subject that has seldom been addressed in recent years, that of market impact in the options market. Our analysis is based on a proprietary database of metaorders-large orders that are split into smaller pieces before being sent to the market on one of the main Asian markets. In line with our previous work on the equity market [Said et al., 2018], we propose an algorithmic approach to identify metaorders, based on some implied volatility parameters, the at the money forward volatility and at the money forward skew. In both cases, we obtain results similar to the now well understood equity market: Square-root law, Fair Pricing Condition and Market Impact Dynamics.

The thesis deals with numerical schemes for Markovian decision problems (MDPs), partial differential equations (PDEs), backward stochastic differential equations (SRs), as well as reflected backward stochastic differential equations (SRDEs). The thesis is divided into three parts.The first part deals with numerical methods for solving MDPs, based on quantization and local or global regression. A market-making problem is proposed: it is solved theoretically by rewriting it as an MDP. and numerically by using the new algorithm. In a second step, a Markovian embedding method is proposed to reduce McKean-Vlasov type probabilities with partial information to MDPs. This method is implemented on three different McKean-Vlasov type problems with partial information, which are then numerically solved using numerical methods based on regression and quantization.In the second part, new algorithms are proposed to solve MDPs in high dimension. The latter are based on neural networks, which have proven in practice to be the best for learning high dimensional functions. The consistency of the proposed algorithms is proved, and they are tested on many stochastic control problems, which allows to illustrate their performances.In the third part, we focus on methods based on neural networks to solve PDEs, EDSRs and reflected EDSRs. The convergence of the proposed algorithms is proved and they are compared to other recent algorithms of the literature on some examples, which allows to illustrate their very good performances.

This thesis consists of two interrelated parts. In the first part, we empirically study the behavior of high-frequency traders on European financial markets. In the second part, we use the results obtained to build new multi-agent models. The main objective of these models is to provide regulators and trading platforms with innovative tools to implement microstructure relevant rules and to quantify the impact of various participants on market quality.In the first part, we perform two empirical studies on unique data provided by the French regulator. We have access to all orders and trades of CAC 40 assets, at the microsecond scale, with the identities of the actors involved. We begin by comparing the behavior of high-frequency traders to that of other players, particularly during periods of stress, in terms of liquidity provision and trading activity. We then deepen our analysis by focusing on liquidity consuming orders. We study their impact on the price formation process and their information content according to the different categories of flows: high-frequency traders, participants acting as clients and participants acting as principal.In the second part, we propose three multi-agent models. Using a Glosten-Milgrom approach, our first model constructs the entire order book (spread and volume available at each price) from the interactions between three types of agents: an informed agent, an uninformed agent and market makers. This model also allows us to develop a methodology for predicting the spread in case of a change in the price step and to quantify the value of the priority in the queue. In order to focus on an individual scale, we propose a second approach where the specific dynamics of the agents are modeled by nonlinear Hawkes-type processes that depend on the state of the order book. In this framework, we are able to compute several relevant microstructure indicators based on individual flows. In particular, it is possible to classify market makers according to their own contribution to volatility. Finally, we introduce a model where liquidity providers optimize their best bid and offer prices according to the profit they can generate and the inventory risk they face. We then theoretically and empirically highlight an important new relationship between inventory and volatility.

This paper deals with a fundamental subject that has seldom been addressed in recent years, that of market impact in the options market. Our analysis is based on a proprietary database of metaorders-large orders that are split into smaller pieces before being sent to the market on one of the main Asian markets. In line with our previous work on the equity market [Said et al., 2018], we propose an algorithmic approach to identify metaorders, based on some implied volatility parameters, the at the money forward volatility and at the money forward skew. In both cases, we obtain results similar to the now well understood equity market: Square-root law, Fair Pricing Condition and Market Impact Dynamics.

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.

This paper deals with a fundamental subject that has seldom been addressed in recent years, that of market impact in the options market. Our analysis is based on a proprietary database of metaorders-large orders that are split into smaller pieces before being sent to the market on one of the main Asian markets. In line with our previous work on the equity market [Said et al., 2018], we propose an algorithmic approach to identify metaorders, based on some implied volatility parameters, the at the money forward volatility and at the money forward skew. In both cases, we obtain results similar to the now well understood equity market: Square-root law, Fair Pricing Condition and Market Impact Dynamics.

In the first part, we study the existence and nonexistence of an inequality called the Uniform Hopf Inequality (UHI), for a linear equation of the form Lv = f with measurable bounded coefficients and under homogeneous Dirichlet conditions. The IHU is a variant of the maximum principle, it was applied in the regularity proof W1.p 0 for a singular semilinear problem: Lu = F(u) where the coefficients of L are in the space vmor (evanescent mean-swinging functions) and F(u) is singular in u = 0 F(0) = +∞. Moreover, if the coefficients are lipschitzian, we prove that the optimal regularity of the gradient of the solution u is bmor (bounded mean oscillation functions i.e. Grad u in bmor).In the second part, we are interested in the regularity of the elastic system (stationary equations of elastic waves) with a singular source function in the sense that it is only integrable with respect to the distance function at the domain edge. Via duality, we show, according to ~f , that the problem admits a so-called very weak solution whose gradient is not necessarily integrable on the whole domain but only locally. We also determine the vector functions ~f for which ~u has its gradient integrable on the whole working space.

This paper is devoted to the important yet little explored subject of the market impact of limit orders. Our analysis is based on a proprietary database of metaorders - large orders that are split into smaller pieces before being sent to the market. We first address the case of aggressive limit orders and then, that of passive limit orders. In both cases, we provide empirical evidence of a power law behaviour for the temporary market impact. The relaxation of the price following the end of the metaorder is also studied, and the long-term impact is shown to stabilize at a level of approximately two-thirds of the maximum impact. Finally, a fair pricing condition during the life cycle of the metaorders is empirically validated.

High-dimensional Hawkes processes with exponential kernels are used to describe limit order books in order-driven financial markets. The dependencies between orders of various types are carefully studied and modelled, based on a thorough empirical analysis. The observation of inhibition effects is particularly interesting, and leads us to the use of non-linear Hawkes processes. A specific attention is devoted to the calibration problem, in order to account for the high dimensionality of the problem and the very poor convexity properties of the MLE. Our analyses show a good agreement between the statistical properties of order book data and those of the model.

This thesis aims to analyze the systemic risk on commodity futures markets. Indeed, several research works highlight the importance of these futures in the determination of the physical price of commodities. Their incorporation into traditional finance as a diversifying asset has led to a similar price evolution to that of various financial assets since about 2004. The question that motivated this thesis was therefore to quantify this systemic risk (since it affects commodities, which are directly involved in the real economy), to see precisely the means of transmission (which markets affect which other markets) and finally to allow for an evaluation of the consequences, for example, based on scenarios (stress tests). It therefore makes it possible to develop market surveillance tools and could therefore contribute to the regulation of these markets.

This thesis consists of two parts that can be read independently. In the first part of the thesis, we study hedging and option pricing problems related to a risk measure. Our main approach is the use of an asymmetric risk function and an asymptotic framework in which we obtain optimal solutions through nonlinear partial differential equations (PDEs).In the first chapter, we focus on the valuation and hedging of European options. We consider the problem of optimizing the residual risk generated by a discrete-time hedge in the presence of an asymmetric risk criterion. Instead of analyzing the asymptotic behavior of the solution of the associated discrete problem, we study the asymmetric residual risk measure integrated in a Markovian framework. In this context, we show the existence of this asymptotic risk measure. We then describe an asymptotically optimal hedging strategy via the solution of a totally nonlinear PDE. The second chapter applies this hedging method to the problem of valuing the output of a power plant. Since the power plant generates maintenance costs whether it is on or off, we are interested in reducing the risk associated with the uncertain revenues of this power plant by hedging with futures contracts. In the second part of the thesis, we consider several control problems related to economics and finance.The third chapter is dedicated to the study of a class of McKean-Vlasov (MKV) type problem with common noise, called conditional polynomial MKV. We reduce this polynomial class by Markov folding to finite dimensional control problems.We compare three different probabilistic techniques for numerically solving the reduced problem: quantization, control randomization regression, and delayed regression. We provide many numerical examples, such as portfolio selection with uncertainty about an underlying trend.In the fourth chapter, we solve dynamic programming equations associated with financial valuations in the energy market. We consider that a calibrated model for the underlyings is not available and that a small sample obtained from historical data is accessible.Moreover, in this context, we assume that futures contracts are often governed by hidden factors modeled by Markov processes. We propose a non-intrusive method to solve these equations through empirical regression techniques using only the historical log price of observable futures contracts.

We begin this thesis report by evaluating the expected exposure, which represents one of the major components of XVA adjustments. Under the assumption of independence between exposure and financing and credit costs, we derive in Chapter 3 a new representation of expected exposure as the solution of an ordinary differential equation with respect to the time of default observation. For the one-dimensional case, we rely on arguments similar to those for Dupire's local volatility. And for the multidimensional case, we refer to the Co-aire formula. This representation allows us to explain the impact of volatility on the expected exposure: this time value involves the volatility of the underlyings as well as the first-order sensitivity of the price, evaluated on a finite set of points. Despite numerical limitations, this method is an accurate and fast approach for valuing unit XVA in dimension 1 and 2.The following chapters are dedicated to the risk aspects of correlations between exposure envelopes and XVA costs. We present a model of the general correlation risk through a multivariate stochastic diffusion, including both the underlying assets of the derivatives and the default intensities. In this framework, we present a new approach to valuation by asymptotic developments, such that the price of an XVA adjustment corresponds to the price of the zero-correlation adjustment, plus an explicit sum of corrective terms. Chapter 4 is devoted to the technical derivation and study of the numerical error in the context of the valuation of default contingent derivatives. The quality of the numerical approximations depends solely on the regularity of the credit intensity diffusion process, and is independent of the regularity of the payoff function. The valuation formulas for CVA and FVA are presented in Chapter 5. A generalization of the asymptotic developments for the bilateral default framework is addressed in Chapter 6.We conclude this dissertation by addressing a case of the specific correlation risk related to rating migration contracts. Beyond the valuation formulas, our contribution consists in presenting a robust approach for the construction and calibration of a rating transition model consistent with market implied default probabilities.

This thesis is composed of four chapters.The first chapter is a preliminary description of the Factset Ownership database. The first chapter is a preliminary description of the Factset Ownership database. We give a statistical description of the database and present some stylized facts characterizing the portfolio structure of financial institutions and investment funds, as well as the market capitalization of the companies listed in the database.The second chapter proposes a method for statistically evaluating the similarity between pairs of financial institutions' portfolios. The second chapter proposes a method to statistically evaluate the similarity between pairs of portfolios of financial institutions. Since a statistically significant pair leads to the creation of a similarity link between these two entities, we are able to project an originally bi-partite network (between financial institutions and firms) into a mono-partite network (between institutions only) in order to study the evolution of its structure over time. Indeed, from an economic point of view, it is suspected that similar investment motives constitute an important risk factor of financial contagion that can be at the origin of bankruptcies with significant systemic consequences.The third chapter focuses on the collective behavior of investment fund managers and, in particular, on the way in which the structure of the portfolio of these funds optimally takes into account, on average, transaction costs in the presence of weak investment constraints. This phenomenon, where in many situations the median or average of a group of people's estimates is surprisingly close to the true value, is known as the wisdom of the crowd.The fourth chapter is devoted to the simultaneous study of market data. We use over 6.7 billion trades from the Thomson-Reuters Tick History database, and portfolio data from the FactSet Ownership database. We study the tick-to-tick dynamics of the order book as well as the aggregate action, i.e. on a much larger time scale, of investment funds. In particular, we show that the long memory of the sign of market orders is much shorter in the presence of the action, absolute or directional, of investment funds. Conversely, we explain to what extent a stock characterized by a weak memory is subject to directional trading due to the action of investment funds.

This thesis is devoted to the theoretical and numerical study of two nonlinear problems in the McKean sense in finance. In the first part, we address the problem of calibrating a model with local and stochastic volatility to take into account the prices of European vanilla options observed on the market. This problem results in the study of a nonlinear stochastic differential equation (SDE) in the McKean sense because of the presence in the diffusion coefficient of a conditional expectation of the stochastic volatility factor with respect to the SDE solution. We obtain the existence of the process in the particular case where the stochastic volatility factor is a jump process with a finite number of states. We also obtain the weak convergence at order 1 of the time discretization of the nonlinear DHS in the McKean sense for general stochastic volatility factors. In the industry, the calibration is efficiently performed using a regularization of the conditional expectation by a Nadaraya-Watson type kernel estimator, as proposed by Guyon and Henry-Labordère in [JGPHL]. We also propose a half-time numerical scheme and study the associated particle system that we compare to the algorithm proposed by [JGPHL]. In the second part of the thesis, we focus on a problem of contract valuation with margin calls, a problem that appeared with the application of new regulations since the financial crisis of 2008. This problem can be modeled by an anticipatory stochastic differential equation (SDE) with dependence on the law of the solution in the generator. We show that this equation is well-posed and propose an approximation of its solution using standard linear SRDEs when the liquidation time of the option in case of default is small. Finally, we show that the computation of the solutions of these standard EDSRs can be improved using the multilevel Monte Carlo method introduced by Giles in [G].

The aim of this paper is to compare the performance of a theoretically optimal portfolio with that of a moving average-based strategy in the presence of parameter misspecification. The setting we consider is that of a stochastic asset price model where the trend follows an unobservable Ornstein–Uhlenbeck process. For both strategies, we provide the asymptotic expectation of the logarithmic return as a function of the model parameters. Then, numerical examples are given, showing that an investment strategy using a moving average crossover rule is more robust than the optimal strategy under parameter misspecification.

This thesis addresses different aspects of market microstructure modeling and market making problems, with a particular focus on the practitioner's point of view. The order book, at the heart of the financial market, is a complex high-dimensional queueing system. We aim to improve the knowledge of the LOB for the research community, propose new modeling ideas and develop applications for Market Makers. In particular, we thank the Automated Market Making team for providing the high-quality high-frequency database and a powerful computational grid, without which this research would not have been possible. Chapter 1 presents the motivation for this research and summarizes the main results of the different works. Chapter 2 focuses entirely on the LOB and aims at proposing a new model that better reproduces some stylized facts. Through this research, not only do we confirm the influence of historical order flow on the arrival of new ones, but a new model is also provided that replicates much better the dynamics of the LOB, in particular the volatility realized in high and low frequency. In Chapter 3, the objective is to study market making strategies in a more realistic context. This research contributes to two aspects: on the one hand the new proposed model is more realistic but still simple to apply for strategy design, on the other hand the practical Market Making strategy is much improved compared to a naive strategy and is promising for practical application. High frequency prediction with deep learning method is studied in Chapter 4. Numerous results of the 1-step and multi-step prediction found the non-linearity, stationarity and universality of the relationship between the microstructure indicators and the price change, as well as the limitation of this approach in practice.

This paper studies the question of filtering and maximizing terminal wealth from expected utility in a partially information stochastic volatility models. The special features is that the only information available to the investor is the one generated by the asset prices, and the unobservable processes will be modeled by a stochastic differential equations. Using the change of measure techniques, the partial observation context can be transformed into a full information context such that coefficients depend only on past history of observed prices (filter processes). Adapting the stochastic non-linear filtering, we show that under some assumptions on the model coefficients, the estimation of the filters depend on a priori models for the trend and the stochastic volatility. Moreover, these filters satisfy a stochastic partial differential equations named "Kushner-Stratonovich equations". Using the martingale duality approach in this partially observed incomplete model, we can characterize the value function and the optimal portfolio. The main result here is that the dual value function associated to the martingale approach can be expressed, via the dynamic programming approach, in terms of the solution to a semilinear partial differential equation which depends also on the filters estimate and the volatility. We illustrate our results with some examples of stochastic volatility models popular in the financial literature.

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We first review empirical evidence that asset prices have had episodes of large fluctuations and been inefficient for at least 200 years. We briefly review recent theoretical results as well as the neurological basis of trend following and finally argue that these asset price properties can be attributed to two fundamental mechanisms that have not changed for many centuries: an innate preference for trend following and the collective tendency to exploit as much as possible detectable price arbitrage, which leads to destabilizing feedback loops.

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.

The classical theory of derivatives valuation is based on the absence of transaction costs and infinite liquidity. However, these assumptions are no longer true in the real market, especially when the transaction is large and the assets illiquid. The first part of this thesis focuses on proposing a model that incorporates both the transaction cost and the impact on the price of the underlying asset. We start by deriving the continuous time asset dynamics as the limit of the discrete time dynamics. Under the constraint of a zero position on the asset at the beginning and at maturity, we obtain a quasi-linear equation for the price of the derivative, in the sense of viscosity. We offer the perfect hedging strategy when the equation admits a regular solution. As for the hedging of a covered European option under the gamma constraint, the dynamic program principle used previously is no longer valid. Following the techniques of the stochastic target and the partial differential equation, we show that the price of the over-replication has become a viscosity solution of a nonlinear equation of parabolic type. We also construct the ε-optimal strategy, and propose a numerical scheme.The second part of this thesis is devoted to studies on a new numerical scheme of EDSR, based on the branching process. We first approximate the Lipschitzian generator by a sequence of local polynomials, and then apply the Picard iteration. Each Picard iteration can be represented in terms of a branching process. We demonstrate the convergence of our scheme on the infinite time horizon. A concrete example is discussed at the end in order to illustrate the performance of our algorithm.

We study the influence of taking liquidity costs and market impact into account when hedging a contingent claim. In the continuous time setting and under the assumption of perfect replication, we derive a fully non-linear pricing partial differential equation, and characterize its parabolic nature according to the value of a numerical parameter interpreted as a relaxation coefficient for market impact. We also investigate the case of stochastic volatility models with pseudo-optimal strategies.

This paper is devoted to the important yet little explored subject of the market impact of limit orders. Our analysis is based on a proprietary database of metaorders - large orders that are split into smaller pieces before being sent to the market. We first address the case of aggressive limit orders and then, that of passive limit orders. In both cases, we provide empirical evidence of a power law behaviour for the temporary market impact. The relaxation of the price following the end of the metaorder is also studied, and the long-term impact is shown to stabilize at a level of approximately two-thirds of the maximum impact. Finally, a fair pricing condition during the life cycle of the metaorders is empirically validated.

This paper is devoted to the important yet little explored subject of the market impact of limit orders. Our analysis is based on a proprietary database of metaorders - large orders that are split into smaller pieces before being sent to the market. We first address the case of aggressive limit orders and then, that of passive limit orders. In both cases, we provide empirical evidence of a power law behaviour for the temporary market impact. The relaxation of the price following the end of the metaorder is also studied, and the long-term impact is shown to stabilize at a level of approximately two-thirds of the maximum impact. Finally, a fair pricing condition during the life cycle of the metaorders is empirically validated.

Consistently fitting vanilla option surfaces is an important issue in derivative modelling. In this paper, we consider three different models: local and stochastic volatility, local correlation, hybrid local volatility with stochastic rates, and address their exact, nonparametric calibration. This calibration process requires solving a nonlinear partial integro-differential equation. A modified alternating direction implicit algorithm is used, and its theoretical and numerical analysis is performed.

The quality of various Hawkes-process-based order book models are assessed using some objective criteria. We start with a precise empirical analysis of the dependencies between order arrivals of various types, then, models built from multivariate, possibly nonlinear, Hawkes processes with multiple exponential kernels are introduced. Models are evaluated based on the distribution of forward recurrence times and the signature plot. This approach allows us to discriminate between various Hawkes-process- based models, and provide a financial interpretation of the more successful ones in terms of their behaviour at various time scales, and the presence of inhibition as well as excitation effects.

The main objective of this thesis is to provide new theoretical results concerning the performance of investments based on stochastic models. To do so, we consider the optimal investment strategy in the framework of a risky asset model with constant volatility and a hidden Ornstein Uhlenbeck process. In the first chapter, we present the context and the objectives of this study. We present, also, the different methods used, as well as the main results obtained. In the second chapter, we focus on the feasibility of trend calibration. We answer this question with analytical results and numerical simulations. We close this chapter by also quantifing the impact of a calibration error on the trend estimate and exploit the results to detect its sign. In the third chapter, we assume that the agent is able to calibrate the trend well and we study the impact that the non-observability of the trend has on the performance of the optimal strategy. To do so, we consider the case of a logarithmic utility and an observed or unobserved trend. In each of the two cases, we explain the asymptotic limit of the expectation and the variance of the logarithmic return as a function of the signal-to-noise ratio and the speed of reversion to the mean of the trend. We conclude this study by showing that the asymptotic Sharpe ratio of the optimal strategy with partial observations cannot exceed 2/(3^1.5)∗100% of the asymptotic Sharpe ratio of the optimal strategy with complete information. The fourth chapter studies the robustness of the optimal strategy with calibration error and compares its performance to a technical analysis strategy. To do so, we characterize, analytically, the asymptotic expectation of the logarithmic return of each of these two strategies. We show, through our theoretical results and numerical simulations, that a technical analysis strategy is more robust than the poorly calibrated optimal strategy.

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The question of interest in this paper is the estimation of the trend of a financial asset, and the impact of its misspecification on investment strategies. The setting we consider is that of a stochastic asset price model where the trend follows an unobservable Ornstein-Uhlenbeck process. Motivated by the use of Kalman filtering as a forecasting tool, we address the problem of parameters estimation, and measure the effect of parameters mis-specification. Numerical examples illustrate the difficulty of trend forecasting in financial time series.

In the first part, we study the existence and nonexistence of an inequality called the Uniform Hopf Inequality (UHI), for a linear equation of the form Lv = f with measurable bounded coefficients and under homogeneous Dirichlet conditions. The IHU is a variant of the maximum principle, it was applied in the regularity proof W1.p 0 for a singular semilinear problem: Lu = F(u) where the coefficients of L are in the space vmor (evanescent mean-swinging functions) and F(u) is singular in u = 0 F(0) = +∞. Moreover, if the coefficients are lipschitzian, we prove that the optimal regularity of the gradient of the solution u is bmor (bounded mean oscillation functions i.e. Grad u in bmor).In the second part, we are interested in the regularity of the elastic system (stationary equations of elastic waves) with a singular source function in the sense that it is only integrable with respect to the distance function at the domain edge. Via duality, we show, according to ~f , that the problem admits a so-called very weak solution whose gradient is not necessarily integrable on the whole domain but only locally. We also determine the vector functions ~f for which ~u has its gradient integrable on the whole working space.

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Hawkes processes provide a natural framework for modelling dependencies between the intensities of point processes. In the context of order-driven financial markets, the relevance of such dependencies has been amply demonstrated from an empirical, as well as theoretical, standpoint. In this work, we build on previous empirical and numerical studies and introduce a mathematical model of limit order books based on Hawkes processes with exponential kernels. After proving a general stationarity result, we focus on the long-time behavior of the limit order book and the corresponding dynamics of the suitably rescaled price. A formula for the asymptotic (in time) volatility of the price dynamics induced by that of the order book is obtained, involving the average of functions of the various order book events under the stationary distribution.

Using unpublished high-frequency data, this thesis studies three types of clustering present in the foreign exchange market: the concentration of orders on certain prices, the concentration of transactions over time and the existence of groups of investors making the same decisions. We start by studying the statistical properties of the EBS order book for the EUR/USD and USD/JPY currency pairs and the impact of a reduction in tick size on its dynamics. A large proportion of limit orders are still placed on the old authorized prices, leading to the appearance of barrier prices, where the best limits appear most of the time. This congestion effect can be seen in the average shape of the book where peaks are present at full distances. We show that this concentration of prices is caused by manual traders who refuse to use the new price resolution. We then raise the question of the ability of Hawkes processes to capture market dynamics. We analyze the accuracy of such processes as the calibration interval is increased. Different kernels constructed from sums of exponentials are systematically compared. The FX market that never closes is particularly suitable for our purpose, as it avoids the complications due to the nightly closing of equity markets. We find that the modeling is valid according to the three statistical tests, if a two-exponential kernel is used to fit one hour, and two or three for a full day. Over longer periods the model is systematically rejected by the tests because of the non-stationarity of the endogenous process. The estimated self-excitation time scales are relatively short and the endogeneity factor is high but subcritical around 0.8. Most agent-based models implicitly assume that agents interact through asset prices and trading volumes. Some explicitly use a network of interaction between traders, on which rumors are propagated, while others use a network that represents groups making common decisions. Unlike other types of data, such networks, if they exist at all, are necessarily implicit, which makes their detection complicated. We study the transactions of customers of two liquidity providers over several years. Assuming that the links between agents are determined by the timing of their activity or inactivity, we show that interaction networks exist. Moreover, we find that the activity of some agents systematically leads to the activity of other agents, thus defining lead-lag relationships between agents. This implies that the flow of customers is predictable, which we verify using a sophisticated statistical learning method.

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In this paper, we establish a model for market making in options whose underlying is perfectly liquid. In our model framework, the stock price follows a generic stochastic volatility model under the real-world probability measure P. Market participants price options on this stock under a risk-neutral pricing measure Q, and they may misspecify the parameters controlling the dynamics of the volatility process. We consider that there is an agent who is willing to make markets in an option on the stock with the aim of maximizing his expected utility from terminal wealth at the maturity of this option. Since market impact is an important feature in the microscopic time scale and should be taken into account in high frequency trading, we study di erent forms of this function argued in the recent literature. Through the use of optimal stochastic control, we provide exact expressions of optimal bid and ask quotes of the market making strategy in the case where the agent is risk-neutral. Afterward, we suppose that the agent is risk-averse and wants to reduce the variance of the nal wealth. In addition, this agent tries not to accumulate a large inventory in order not to have a signi cant exposure to market risk. For this purpose, we perturb the utility function by a penalty on the variance of nal wealth and also on accumulated inventory. Using singular perturbation with respect to the penalty parameter, we provide analytic approximations of the optimal bid and ask quotes. In order to con rm our theoretical results, we perform Monte Carlo simulations of the optimal market making strategy in the case where the stock price process follows a Heston model. We show that the opti- mal strategy is more pro table than a zero-intelligence strategy. Besides, we highlight the e ects of the misspeci cation of the parameters on the performance of the strategy.

Document rendered obsolete by the release of the latest version of "Long time behaviour of a Hawkes process-based limit order book.

Hawkes processes provide a natural framework to model dependencies between the intensities of point processes. In the context of order-driven financial markets, the relevance of such dependencies has been amply demonstrated from an empirical, as well as theoretical, standpoint. In this work, we build on previous empirical and numerical studies and introduce a mathematical model of limit order books based on Hawkes processes with exponential kernels. After proving a general stationarity result, we focus on the long-time behaviour of the limit order book and the corresponding dynamics of the suitably rescaled price. A formula for the asymptotic (in time) volatility of the price dynamics induced by that of the order book is obtained, involving the average of functions of the various order book events under the stationary distribution.

This thesis explores theoretically and empirically the implications of the joint stock/option dynamics on various issues related to options trading. First, we study the joint dynamics between an option on a stock and an option on the market index. The CAPM model provides an adequate mathematical framework for this study because it allows to model the joint dynamics of a stock and its market index. Moving to option prices, we show that beta and idiosyncratic volatility, parameters of the model, allow us to characterize the relationship between the implied volatility surfaces of the stock and the index. We then turn to the estimation of the beta parameter under the risk-neutral probability using option prices. This measure, called implied beta, represents the information contained in the option prices about the realization of the beta parameter in the future.For this reason, we try to see, if implied beta has any predictive power of the future beta.By conducting an empirical study, we conclude that implied beta does not improve the predictive ability compared to the historical beta which is computed through the linear regression of the stock returns on the index returns. Better yet, we note that the oscillation of the implied beta around the future beta can lead to arbitrage opportunities, and we propose an arbitrage strategy that allows to monetize this gap. On the other hand, we show that the implied beta estimator could be used to hedge options on the stock using index instruments, this hedging concerns notably the volatility risk and also the delta risk. In the second part of our work, we are interested in the problem of market making on options. In this study, we assume that the model of the underlying's dynamics under the risk-neutral probability could be misspecified, which reflects a mismatch between the implied distribution of the underlying and its historical distribution.First, we consider the case of a risk-neutral market maker who aims to maximize the expectation of his future wealth. Using a stochastic optimal control approach, we determine the optimal call and put prices on the option and interpret the effect of price inefficiency on the optimal strategy. In a second step, we consider that the market maker is risk averse and therefore tries to reduce the uncertainty associated with his inventory. By solving an optimization problem based on a mean-variance criterion, we obtain analytical approximations of the optimal buying and selling prices. We also show the effects of inventory and price inefficiency on the optimal strategy. We then turn to the market making of options in a higher dimension. Thus, following the same reasoning, we present a framework for the market making of two options with different underlyings with the constraint of variance reduction related to the inventory risk held by the market maker. In the last part of our work, we study the joint dynamics between the implied volatility at the currency and the underlying, and we try to establish the link between these joint dynamics and the implied skew. We are interested in an indicator called "Skew Stickiness Ratio" which has been introduced in the recent literature. This indicator measures the sensitivity of the implied volatility of the currency to the movements of the underlying. We propose a method that allows us to estimate the value of this indicator under the risk-neutral probability without the need to admit assumptions on the dynamics of the underlying. [.].

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This book presents the works and research findings of physicists, economists, mathematicians, statisticians, and financial engineers who have undertaken data-driven modelling of market dynamics and other empirical studies in the field of Econophysics. During recent decades, the financial market landscape has changed dramatically with the deregulation of markets and the growing complexity of products. The ever-increasing speed and decreasing costs of computational power and networks have led to the emergence of huge databases. The availability of these data should permit the development of models that are better founded empirically, and econophysicists have accordingly been advocating that one should rely primarily on the empirical observations in order to construct models and validate them. The recent turmoil in financial markets and the 2008 crash appear to offer a strong rationale for new models and approaches. The Econophysics community accordingly has an important future role to play in market modelling. The Econophys-Kolkata VIII conference proceedings are devoted to the presentation of many such modelling efforts and address recent developments. A number of leading researchers from across the globe report on their recent work, comment on the latest issues, and review the contemporary literature.

Although it is widely assumed that the stock market is efficient, some empirical studies have already tried to address the issue of forecasting stock returns. As far as is known, it is hard to find a paper involving not only the forecasting statistics but also the forecasting profitability. This paper aims to provide an empirical evidence of the market inefficiency and to present some simple realistic strategies based on forecasting stocks returns. In order to achieve this study, some linear and non linear algorithms are used to prove the predictability of returns. Many regularization methods are introduced to enhance the linear regression model. In particular, the RIDGE method is used to address the colinearity problem and the LASSO method is used to perform variable selection. The different obtained results show that the stock market is inefficient and that profitable strategies can be computed based on forecasting returns. Empirical tests also show that simple forecasting methods perform almost as well as more complicated methods.

The goal of this thesis is to answer the real need to predict future stock price fluctuations. Indeed, the randomness governing these fluctuations constitutes for financial actors, such as market makers, one of the greatest sources of risk. Throughout this study, we highlight the possibility of reducing the uncertainty on future prices by using appropriate mathematical models. This study is made possible thanks to a large financial database and a powerful computational grid made available to us by the Automatic Market Making team of BNP Paribas. In this paper, we only present the results of the research concerning high frequency trading. The results concerning the low-frequency part are of less scientific interest to the academic world and are also confidential. In the first chapter, we present the context and the objectives of this study. We also present the different methods used, as well as the main results obtained. In chapter 2, we focus on the contribution of technological superiority in high frequency trading. For this purpose, we simulate an ultra-fast, omniscient, and aggressive trader, and then we calculate his total gain over 3 years. The gains obtained are very modest and reflect the limited contribution of technology in high frequency trading. In chapter 3, we study the predictability of prices based on order book indicators. Using conditional expectations, we present empirical evidence of statistical dependencies between prices and the different indicators. The importance of these dependencies results from the simplicity of the method, eliminating any risk of overlearning the data. We then focus on the combination of the different indicators by a linear regression and we analyze the different numerical and statistical problems related to this method. Finally, we conclude that prices are predictable for a time horizon of a few minutes and we question the market efficiency hypothesis.In chapter 4, we focus on the price formation mechanism based on the arrival of events in the order book. We classify the orders into twelve types whose statistical properties we analyze. We then study the dependencies between these different types of orders and propose an order book model in line with empirical observations. Finally, we use this model to predict prices and we support the hypothesis of the non-efficiency of markets, suggested in chapter 3.

We study the filtering problem and the maximization problem of expected utility from terminal wealth in a partial information context. The special features is that the only information available to the investor is the vector of sock prices. The mean rate of return processes are not directly observed and supposed to be driven by a process $\mu_{t}$ modeled by a stochastic differential equations. The main result in this paper is to show under which assumptions on the coefficients of the model, we can estimate the unobserved market price of risks. Using the innovation approach, we show that under globally Lipschitz conditions on the coefficients of $\mu_{t}$, the filters estimate of the risks satisfy a measure-valued Kushner-Stratonovich equations. On the other hand, using the pathwise density approach, we show that under a nondegenerate assumption and some regularity assumptions on the coefficients of $\mu_{t}$, the density of the conditional distribution of $\mu_{t}$ given the observation data, can be expressed in terms of the solution to a linear parabolic partial differential equation parameterized by the observation path. Also, we can obtain an explicit formulae for the optimal wealth, the optimal portfolio and the value function for the cases of logarithmic and power utility function.

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In this paper, we establish a model for market making in options whose underlying is perfectly liquid. In our model framework, the stock price follows a generic stochastic volatility model under the real-world probability measure P. Market participants price options on this stock under a risk-neutral pricing measure Q, and they may misspecify the parameters controlling the dynamics of the volatility process. We consider that there is an agent who is willing to make markets in an option on the stock with the aim of maximizing his expected utility from terminal wealth at the maturity of this option. Since market impact is an important feature in the microscopic time scale and should be taken into account in high frequency trading, we study di erent forms of this function argued in the recent literature. Through the use of optimal stochastic control, we provide exact expressions of optimal bid and ask quotes of the market making strategy in the case where the agent is risk-neutral. Afterward, we suppose that the agent is risk-averse and wants to reduce the variance of the nal wealth. In addition, this agent tries not to accumulate a large inventory in order not to have a signi cant exposure to market risk. For this purpose, we perturb the utility function by a penalty on the variance of nal wealth and also on accumulated inventory. Using singular perturbation with respect to the penalty parameter, we provide analytic approximations of the optimal bid and ask quotes. In order to con rm our theoretical results, we perform Monte Carlo simulations of the optimal market making strategy in the case where the stock price process follows a Heston model. We show that the opti- mal strategy is more pro table than a zero-intelligence strategy. Besides, we highlight the e ects of the misspeci cation of the parameters on the performance of the strategy.

We present in our work a continuous time Capital Asset Pricing Model where the volatilities of the market index and the stock are both stochastic. Using a singular perturbation technique, we provide approximations for the prices of european options on both the stock and the index. These approximations are functions of the model parameters. We show then that existing estimators of the parameter beta, proposed in the recent literature, are biased in our setting because they are all based on the assumption that the idiosyncratic volatility of the stock is constant. We provide then an unbiased estimator of the parameter beta using only implied volatility data. This estimator is a forward measure of the parameter beta in the sense that it represents the information contained in derivatives prices concerning the forward realization of this parameter, we test then its capacity of prediction of forward beta and we draw a conclusion concerning its predictive power.

Using a new high frequency quality data set we provide a precise empirical study of the interdealer spot market. We check that the main stylized facts of financial time series are valid for the FX market: fat-tailed distribution of returns, aggregational normality and volatility clustering. We report two standard microstructure phenomena: microstructure noise effects in the signature plot and the Epps effect. We find an unusual shape for the average book, the spread distribution being bimodal. We construct the order flow and analyse its main characteristics: volume, placement, arrival intensity and sign. Many quantities have been dramatically affected by the decrease of the tick size in March 2011. We argue that the coexistence of manual traders and algorithmic traders, who react differently to the new tick size, leads to a strong price clustering in all types of orders and affects the price formation.

The primary goal of this book is to present the research findings and conclusions of physicists, economists, mathematicians and financial engineers working in the field of "Econophysics" who have undertaken agent-based modelling, comparison with empirical studies and related investigations. Most standard economic models assume the existence of the representative agent, who is “perfectly rational” and applies the utility maximization principle when taking action. One reason for this is the desire to keep models mathematically tractable: no tools are available to economists for solving non-linear models of heterogeneous adaptive agents without explicit optimization. In contrast, multi-agent models, which originated from statistical physics considerations, allow us to go beyond the prototype theories of traditional economics involving the representative agent. This book is based on the Econophys-Kolkata VII Workshop, at which many such modelling efforts were presented. In the book, leading researchers in their fields report on their latest work, consider recent developments and review the contemporary literature.

In this paper, we revisit the "Smile Dynamics" problem. In a previous work, Bergomi built a class of linear stochastic volatility models in which he specified the joint dynamics between the underlying and its instantaneous forward variances. The author introduced a quantity, which he called the Skew Stickiness ratio, in order to relate two quantities of interest: the first quantity is the correlation between the increments of the at-the-money implied volatility of maturity T and the log-returns of the underlying, while the second quantity is the implied skew of the same maturity T. In our work, we continue the study of the Skew stickiness ratio both from theoretical and empirical point of view. First, we provide a method to estimate the SSR (skew stickiness ratio) from option prices, this measure is called the implied SSR as it is conducted under the risk-neutral pricing measure Q. Next to that, we recall how to measure the realized SSR under the real-world probability measure P and we point out empirically that there is a discrepancy between the implied SSR and the realized SSR. The empirical study shows also that the implied SSR, in the limit of short maturities, can take a value superior to 2 which is in discordance with the results obtained in linear stochastic volatility models. For this reason, we show that the positive quantity (SSR -2) is coherent with the presence of jumps in a stochastic volatility model.

We introduce a multivariate Hawkes process with constraints on its conditional density. It is a multivariate point process with conditional intensity similar to that of a multivariate Hawkes process but certain events are forbidden with respect to boundary conditions on a multidimensional constraint variable, whose evolution is driven by the point process. We study this process in the special case where the fertility function is exponential so that the process is entirely described by an underlying Markov chain, which includes the constraint variable. Some conditions on the parameters are established to ensure the ergodicity of the chain. Moreover, scaling limits are derived for the integrated point process. This study is primarily motivated by the stochastic modelling of a limit order book for high frequency financial data analysis.

This paper studies the question of filtering and maximizing terminal wealth from expected utility in a partially information stochastic volatility models. The special features is that the only information available to the investor is the one generated by the asset prices, and the unobservable processes will be modeled by a stochastic differential equations. Using the change of measure techniques, the partial observation context can be transformed into a full information context such that coefficients depend only on past history of observed prices (filter processes). Adapting the stochastic non-linear filtering, we show that under some assumptions on the model coefficients, the estimation of the filters depend on a priori models for the trend and the stochastic volatility. Moreover, these filters satisfy a stochastic partial differential equations named "Kushner-Stratonovich equations". Using the martingale duality approach in this partially observed incomplete model, we can characterize the value function and the optimal portfolio. The main result here is that the dual value function associated to the martingale approach can be expressed, via the dynamic programming approach, in terms of the solution to a semilinear partial differential equation which depends also on the filters estimate and the volatility. We illustrate our results with some examples of stochastic volatility models popular in the financial literature.

This thesis studies some aspects of stochastic modeling of order books. In the first part, we analyze a model in which the order arrival times are Poissonian independent. We show that the order book is stable (in the sense of Markov chains) and that it converges to its stationary distribution exponentially fast. We deduce that the price generated in this framework converges to a Brownian motion at large time scales. We illustrate the results numerically and compare them to market data, highlighting the successes of the model and its limitations. In a second part, we generalize the results to a framework where arrival times are governed by self- and mutually-existing processes, under assumptions about the memory of these processes. The last part is more applied and deals with the identification of a realistic multivariate model from the order flows. We detail two approaches: the first one by likelihood maximization and the second one from the covariance density, and succeed in having a remarkable agreement with the data. We apply the estimated model to two concrete algorithmic trading problems, namely the measurement of the execution probability and the cost of a limit order.

Recent regulatory changes, known as Reg NMS in the United States or MiFID in Europe, together with the effects of the financial crisis (mainly its impact on liquidity), induced major changes on market microstructure in two main aspects: • the fragmentation of the liquidity around several trading venues, with the appearance of newcomers in Europe like Chi-X, BATS Europe, or Turquoise, some of them being not regulated or “dark". • the rise of a new type of agents, the high frequency traders, liable for 40% to 70% of the transactions. These two effects are linked since the high frequency traders, being the main clients of the trading venues, have an implicit impact on the products offered by these venues. Combining a survey of recent academic findings and empirical evidences, this paper presents what we consider to be the key elements to understand the stakes of these changes, and also provides potential clues to mitigate some of them. A first section is dedicated to exposes the recent modifications in market microstructure. The second one explains the role of the price formation process and how, interacting with liquidity supply and demand, high frequency traders can reshape it. The next section discloses the various strategies used by these new market participants and their profitability. A final section discusses recent tools designed in order to assess and control the high frequency trading activity.

We present in our work a continuous time Capital Asset Pricing Model where the volatilities of the market index and the stock are both stochastic. Using a singular perturbation technique, we provide approximations for the prices of european options on both the stock and the index. These approximations are functions of the model parameters. We show then that existing estimators of the parameter beta, proposed in the recent literature, are biased in our setting because they are all based on the assumption that the idiosyncratic volatility of the stock is constant. We provide then an unbiased estimator of the parameter beta using only implied volatility data. This estimator is a forward measure of the parameter beta in the sense that it represents the information contained in derivatives prices concerning the forward realization of this parameter, we test then its capacity of prediction of forward beta and we draw a conclusion concerning its predictive power.

Lead/lag relationships are an important stylized fact at high frequency. Some assets follow the path of others with a small time lag. We provide indicators to measure this phenomenon using tick-by-tick data. Strongly asymmetric cross-correlation functions are empirically observed, especially in the future/stock case. We confirm the intuition that the most liquid assets (short intertrade duration, narrow bid/ask spread, small volatility, high turnover) tend to lead smaller stocks. However, the most correlated stocks are those with similar levels of liquidity. This lead/lag phenomenon is not constant throughout the day, it shows an intraday seasonality with changes of behaviour at very specific times such as the announcement of macroeconomic figures and the US market opening. These lead/lag relationships become more and more pronounced as we zoom on significant events. We reach 60% of accuracy when forecasting the next midquote variation of the lagger using only the past information of the leader, which is significantly better than using the information of the lagger only. However, a naive strategy based on market orders cannot make any profit of this effect because of the bid/ask spread.

The aim of this study is to quantify the low latency advantage of High Frequency Trading (HFT) and to compute, empirically, an optimal holding period of a HF trader. Critics claim that low latency leads to information asymmetry victimizing retail investors. However, objective studies measuring the gain due to this asymmetry are rare. In order to perform the study, new methods are introduced in this paper, in particular, the optimal strategy problem is formulated and ideas are given to compute it in a reasonable amount of time. A new measure, the weighted mean holding period, is introduced and an algorithm to compute it is suggested. Using the previous concepts, a large empirical study based on optimal omniscient strategy is presented and evidence of the low latency advantage limitation is provided. In particular, it is shown that the bid ask spread and the transaction costs lead to a trading frequency much lower than the information renewal frequency.

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The possibility that the collective dynamics of a set of stocks could lead to a speci c basket violating the e cient market hypothesis is investigated. Precisely, we show that it is systematically possible to form a basket with a non-trivial autocorrelation structure when the examined time scales are of the order of tens of seconds. Moreover, we show that this situation is persistent enough to allow some kind of forecasting.

In this note, I study further a new approach recently introduced for the hedging of derivatives in incomplete markets via non quadratic local risk minimization. A structure result is provided, which essentially shows the equivalence between non-quadratic risk minimization under the historical probability and quadratic local risk minimization under an equivalent, implicitly defined probability.

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In this note, we cast a Hawkes process-based order book model into a markovian setting and. using techniques from the theory of Markov chains and stochastic stability, show that the order book is stable and leads to a diffusive price limit at large time scales.

We introduce a multivariate Hawkes process with constraints on its conditional density. It is a multivariate point process with conditional intensity similar to that of a multivariate Hawkes process but certain events are forbidden with respect to boundary conditions on a multidimensional constraint variable, whose evolution is driven by the point process. We study this process in the special case where the fertility function is exponential so that the process is entirely described by an underlying Markov chain, which includes the constraint variable. Some conditions on the parameters are established to ensure the ergodicity of the chain. Moreover, scaling limits are derived for the integrated point process. This study is primarily motivated by the stochastic modelling of a limit order book for high frequency financial data analysis.

A limit order book provides information on available limit order prices and their volumes. Based on these quantities, we give an empirical result on the relationship between the bid-ask liquidity balance and trade sign and we show that liquidity balance on best bid/best ask is quite informative for predicting the future market order's direction. Moreover, we de ne price jump as a sell (buy) market order arrival which is executed at a price which is smaller (larger) than the best bid (best ask) price at the moment just after the precedent market order arrival. Features are then extracted related to limit order volumes, limit order price gaps, market order information and limit order event information. Logistic regression is applied to predict the price jump from the limit order book's feature. LASSO logistic regression is introduced to help us make variable selection from which we are capable to highlight the importance of di erent features in predicting the future price jump. In order to get rid of the intraday data seasonality, the analysis is based on two separated datasets: morning dataset and afternoon dataset. Based on an analysis on forty largest French stocks of CAC40, we nd that trade sign and market order size as well as the liquidity on the best bid (best ask) are consistently informative for predicting the incoming price jump.

This thesis explores theoretically and empirically some aspects of the formation and evolution of financial asset prices observed in high frequency. We begin by studying the joint dynamics of the option and its underlying. Since high-frequency data make the realized volatility process of the underlying observable, we investigate whether this information is used to price options. We find that the market does not exploit it. Stochastic volatility models are therefore to be considered as reduced-form models. Nevertheless, this study allows us to test the relevance of an empirical hedging measure that we call effective delta. It is the slope of the regression of the option price returns on those of the underlying. It provides a fairly satisfactory indicator of hedging that is independent of any modeling. For price dynamics, we turn in the following chapters to more explicit models of the market microstructure. One of the characteristics of market activity is its clustering. Hawkes processes, which are point processes with this characteristic, therefore provide an adequate mathematical framework for the study of this activity. The Markovian representation of these processes, as well as their affine character when the kernel is exponential, allow us to use the powerful analytical tools of the infinitesimal generator and Dynkin's formula to compute various quantities related to them, such as the moments or autocovariances of the number of events on a given interval. We start with a one-dimensional framework, simple enough to illuminate the approach, but rich enough to allow applications such as grouping order arrival times, predicting future market activity knowing past activity, or characterizing unusual, but nevertheless observed, forms of signature plot where the measured volatility decreases as the sampling frequency increases. Our calculations also allow us to make the calibration of Hawkes processes instantaneous by using the method of moments. The generalization to the multidimensional case then allows us to capture, with clustering, the mean reversion phenomenon that also characterizes the market activity observed at high frequency. General formulas for the signature plot are then obtained and allow us to link its shape to the relative importance of clustering or mean reversion. Our calculations also allow us to obtain the explicit form of the volatility associated with the diffusive limit, connecting the microscopic level dynamics to the volatility observed macroscopically, for example on a daily scale. Moreover, the modeling of buying and selling activities by Hawkes processes allows to compute the impact of a meta order on the asset price. We then find and explain the concave shape of this impact as well as its temporal relaxation. The analytical results obtained in the multidimensional case then provide the appropriate framework for the study of correlation. We then present general results on the Epps effect, as well as on the formation of the correlation and the lead lag.

Order splitting is a standard practice in trading : traders constantly scan the limit order book and choose to limit the size of their market orders to the quantity available at the best limit, thereby controlling the market impact of their orders. In this article, we focus on the other trades, multiple-limits trades that go through the best available price in the order book, or "trade-throughs". We provide various statistics on trade-throughs: frequency, volume, intraday distribution, market impact. and present a new method for the measurement of lead-lag parameters between assets, sectors or markets.

The aim of this article is to briefly review and make new studies of correlations and co-movements of stocks, so as to understand the "seasonalities" and market evolution. Using the intraday data of the CAC40, we begin by reasserting the findings of Allez and Bouchaud [New J. Phys. 13, 025010 (2011)]: the average correlation between stocks increases throughout the day. We then use multidimensional scaling (MDS) in generating maps and visualizing the dynamic evolution of the stock market during the day. We do not find any marked difference in the structure of the market during a day. Another aim is to use daily data for MDS studies, and visualize or detect specific sectors in a market and periods of crisis. We suggest that this type of visualization may be used in identifying potential pairs of stocks for "pairs trade".

We study the behaviour of very weak solutions of linear elliptic equations when the right-hand-side belongs to a weighted L1 space, the weight being a function of the distance to the boundary.

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We study the influence of taking liquidity costs and market impact into account when hedging a contingent claim, first in the discrete time setting, then in continuous time. In the latter case and in a complete market, we derive a fully non-linear pricing partial differential equation, and characterizes its parabolic nature according to the value of a numerical parameter naturally interpreted as a relaxation coefficient for market impact. We then investigate the more challenging case of stochastic volatility models, and prove the parabolicity of the pricing equation in a particular case.

We present a mathematical study of the order book as a multidimensional continuous- time Markov chain where the order flow is modeled by independent Poisson processes. Our aim is to bridge the gap between the microscopic description of price formation (agent- based modeling), and the Stochastic Differential Equations approach used classically to describe price evolution at macroscopic time scales. To do this, we rely on the theory of infinitesimal generators and Foster-Lyapunov stability criteria for Markov chains. We motivate our approach using an elementary example where the spread is kept constant ("perfect market making"). Then we compute the infinitesimal generator associated with the order book in a general setting, and link the price dynamics to the instantaneous state of the order book. In the last section, we prove that the order book is ergodic - in particular it has a stationary distribution - that it converges to its stationary state exponentially fast, and that the large-scale limit of the price process is a Brownian motion.

The objective of my thesis is to study mathematically a volatility breakout indicator widely used by practitioners in the trading room. The Bollinger Bands indicator belongs to the family of so-called technical analysis methods and is therefore based exclusively on the recent history of the price considered and a principle deduced from past market observations, independently of any mathematical model. My work consists in studying the performance of this indicator in a universe governed by stochastic differential equations (Black-Scholes) whose diffusion coefficient changes its value at an unknown and unobservable random time, for a practitioner wishing to maximize an objective function (for example, a certain expected utility of the portfolio value at a certain maturity). In the framework of the model, the Bollinger indicator can be interpreted as an estimator of the time of the next break. In the case of small volatilities, we show that the behavior of the density of the indicator depends on the volatility, which makes it possible to detect, for a large enough volatility ratio, the volatility regime in which the indicator's distribution is located. Also, in the case of high volatilities, we show by an approach via the Laplace transform, that the asymptotic behavior of the indicator's distribution tails depends on the volatility. This makes it possible to detect the change in the large volatilities. Then, we are interested in a comparative study between the Bollinger indicator and the classical estimator of the quadratic variation for the detection of change in volatility. Finally, we study the optimal portfolio management which is described by a non-standard stochastic problem in the sense that the admissible controls are constrained to be functionals of the observed prices. We solve this control problem by drawing on the work of Pham and Jiao to decompose the initial portfolio allocation problem into a post-breakdown management problem and a pre-breakdown problem, and each of these problems is solved by the dynamic programming method. Thus, a verification theorem is proved for this stochastic control problem.

In this note, we cast a Hawkes process-based order book model into a markovian setting and. using techniques from the theory of Markov chains and stochastic stability, show that the order book is stable and leads to a diffusive price limit at large time scales.

This thesis addresses two topics: (i) The use of a new numerical method for the valuation of options on a basket of assets, (ii) Liquidity risk, order book modeling and market microstructure. First topic: A greedy algorithm and its applications to solve partial differential equations. The typical example in finance is the valuation of an option on a basket of assets, which can be obtained by solving the Black-Scholes PDE having as dimension the number of assets considered. We propose to study an algorithm that has been proposed and studied recently in [ACKM06, BLM09] to solve high dimensional problems and try to circumvent the curse of dimension. The idea is to represent the solution as a sum of tensor products and to iteratively compute the terms of this sum using a gluttonous algorithm. The solution of PDEs in high dimension is strongly related to the representation of functions in high dimension. In Chapter 1, we describe different approaches to represent high-dimensional functions and introduce the high-dimensional problems in finance that are addressed in this thesis work. The method selected in this manuscript is a nonlinear approximation method called Proper Generalized Decomposition (PGD). Chapter 2 shows the application of this method for the approximation of the solution of a linear PDE (the Poisson problem) and for the approximation of an integrable square function by a sum of tensor products. A numerical study of the latter problem is presented in Chapter 3. The Poisson problem and the approximation of an integrable square function will be used as a basis in Chapter 4 to solve the Black-Scholes equation using the PGD approach. In numerical examples, we have obtained results up to dimension 10. In addition to approximating the solution of the Black-Scholes equation, we propose a variance reduction method of classical Monte Carlo methods for pricing financial options. Second topic: Liquidity risk, order book modeling, market microstructure. Liquidity risk and market microstructure have become very important topics in financial mathematics. The deregulation of financial markets and the competition between them to attract more investors is one of the possible reasons. In this work, we study how to use this information to optimally execute the sale or purchase of orders. Orders can only be placed in a price grid. At each moment, the number of pending buy (or sell) orders for each price is recorded. In [AFS10], Alfonsi, Fruth and Schied proposed a simple model of the order book. In this model, it is possible to explicitly find the optimal strategy to buy (or sell) a given quantity of shares before a maturity. The idea is to split the buy (or sell) order into other smaller orders in order to find the balance between the acquisition of new orders and their price. This thesis work focuses on an extension of the order book model introduced by Alfonsi, Fruth and Schied. Here, the originality is to allow the depth of the order book to depend on time, which is a new feature of the order book that has been illustrated by [JJ88, GM92, HH95, KW96]. In this framework, we solve the optimal execution problem for discrete and continuous strategies. This gives us, in particular, sufficient conditions to exclude price manipulation in the sense of Huberman and Stanzl [HS04] or Transaction-Triggered Price Manipulation (see Alfonsi, Schied and Slynko).

We study the influence of taking liquidity costs and market impact into account when hedging a contingent claim, first in the discrete time setting, then in continuous time. In the latter case and in a complete market, we derive a fully non-linear pricing partial differential equation, and characterizes its parabolic nature according to the value of a numerical parameter naturally interpreted as a relaxation coefficient for market impact. We then investigate the more challenging case of stochastic volatility models, and prove the parabolicity of the pricing equation in a particular case.

In this note, I study further a new approach recently introduced for the hedging of derivatives in incomplete markets via non quadratic local risk minimization. A structure result is provided, which essentially shows the equivalence between non-quadratic risk minimization under the historical probability and quadratic local risk minimization under an equivalent, implicitly defined probability.

The aim of this thesis is to deepen the academic knowledge on the joint variations of high-frequency financial assets by analyzing them from a novel perspective. We take advantage of a tick-by-tick price database to highlight new stylistic facts about high-frequency correlation, and also to test the empirical validity of multivariate models. In Chapter 1, we discuss why high-frequency correlation is of paramount importance to trading. Furthermore, we review the empirical and theoretical literature on correlation at small time scales. Then we describe the main characteristics of the dataset we use. Finally, we state the results obtained in this thesis. In chapter 2, we propose an extension of the subordination model to the multivariate case. It is based on the definition of a global event time that aggregates the financial activity of all the assets considered. We test the ability of our model to capture notable properties of the empirical multivariate distribution of returns and observe convincing similarities. In Chapter 3, we study high-frequency lead/lag relationships using a correlation function estimator fit to tick-by-tick data. We illustrate its superiority over the standard correlation estimator in detecting the lead/lag phenomenon. We draw a parallel between lead/lag and classical liquidity measures and reveal an arbitrage to determine the optimal pairs for lead/lag trading. Finally, we evaluate the performance of a lead/lag based indicator to forecast short-term price movements. In Chapter 4, we focus on the seasonal profile of intraday correlation. We estimate this profile over four stock universes and observe striking similarities. We attempt to incorporate this stylized fact into a tick-by-tick price model based on Hawkes processes. The model thus constructed captures the empirical correlation profile quite well, despite its difficulty to reach the absolute correlation level.

In this thesis, we are interested in hedging derivatives in incomplete markets. The chosen approach can be seen as an extension of M. Schweizer's work on local minimization of quadratic risk. Indeed, while remaining within the framework of asset modeling by semimartingales, our method consists in replacing the quadratic risk criterion by a more general risk criterion, in the form of a convex functional of the local cost. We first obtain existence, uniqueness and characterization results for optimal strategies in a frictionless market, in discrete and continuous time. Then we explain these strategies in the framework of diffusion models with and without jumps. We also extend our method to the case where liquidity is no longer infinite. Finally, we show through numerical simulations the effects of the choice of the risk functional on the constitution of the optimal portfolio.

Vanilla calibration is a major problem in finance. We try to solve it for three classes of models: local and stochastic volatility models, the so-called "local correlation" model and a hybrid model of local volatility with stochastic rates. From a mathematical point of view, the calibration equation is a particularly complex nonlinear and integro-differential equation. In a first part, we prove the existence of solutions for this equation, as well as for its adjoint (simpler to solve). These results are based on fixed point methods in Hölder spaces and require classical theorems related to parabolic partial differential equations, as well as some a priori estimates in short time. The second part deals with the application of these existence results to the three financial models mentioned above. We also present the numerical results obtained by solving the edp. The calibration by this method is quite satisfactory. Finally, we focus on the algorithm used for the numerical solution: a predictor-corrector ADI scheme, which is modified to take into account the nonlinear character of the equation. We also describe an instability phenomenon of the edp solution that we try to explain from a theoretical point of view thanks to the so-called "Hadamard instability".

This thesis gathers existence and uniqueness results for evolving and stationary free boundary problems from biology, oil extraction or fluid mechanics. One of the main features of these models is that they exclude any presence of surface tension. The first part deals with a class of free boundary problems including the so-called muskat and quasi-stationary stefan problems. In all these cases, we show by an analysis of the nonlinear and nonlocal terms, the existence of a unique classical local solution whose regularity is the same in time and space. In the second part, we study a class of stationary transport equations with a non-local and positive lower order term. This type of equation comes directly from the free boundary problem considered in part 3, but can also be studied as such. A pseudo-differential analysis gives the existence of a unique solution in the sobolev spaces. In part 3, we consider an incompressible three-dimensional fluid subject to gravity flowing along an inclined plane, in the absence of surface tension. In this context, we prove that there exists a unique solution in sobolev spaces with weights by using the results of the second part, and at the cost of a nash-moser theorem.