TANKOV Peter

We develop a model for the industry dynamics in the electricity market, based on mean-field games of optimal stopping. In our model, there are two types of agents: the renewable producers and the conventional producers. The renewable producers choose the optimal moment to build new renewable plants, and the conventional producers choose the optimal moment to exit the market. The agents interact through the market price, determined by matching the aggregate supply of the two types of producers with an exogenous demand function. Using a relaxed formulation of optimal stopping mean-field games, we prove the existence of a Nash equilibrium and the uniqueness of the equilibrium price process. An empirical example, inspired by the UK electricity market is presented. The example shows that while renewable subsidies clearly lead to higher renewable penetration, this may entail a cost to the consumer in terms of higher peakload prices. In order to avoid rising prices, the renewable subsidies must be combined with mechanisms ensuring that sufficient conventional capacity remains in place to meet the energy demand during peak periods.

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Recent empirical analyses have revealed the existence of the Zumbach effect. This discovery led to the development of the quadratic Hawkes process, adapted to reproduce this effect. Since this model does not relate to the price formation process, we extended it to the order book with a generalized quadratic Hawkes process (GQ-Hawkes). Using market data, we show that there is a Zumbach-like effect that decreases future liquidity. Microfounding the Zumbach effect, it is responsible for a potential destabilization of financial markets. Moreover, the exact calibration of a QM-Hawkes process tells us that markets are at the edge of criticality. This empirical evidence has therefore prompted us to analyze an order book model constructed with a Zumbach-type coupling. We therefore introduced the Santa Fe quadratic model and proved numerically that there is a phase transition between a stable market and an unstable market subject to liquidity crises. Thanks to a finite size analysis we were able to determine the critical exponents of this transition, belonging to a new universality class. Not being analytically solvable, this led us to introduce simpler models to describe liquidity crises. Putting aside the microstructure of the order book, we obtain a class of spread models where we have computed the critical parameters of their transitions. Even if these exponents are not those of the Santa Fe quadratic transition, these models open new horizons to explore the spread dynamics. One of them has a nonlinear coupling that reveals a metastable state. This elegant alternative scenario does not need critical parameters to obtain an unstable market, even if the empirical evidence is not in its favor. Finally, we looked at order book dynamics from another angle: reaction-diffusion. We modeled a liquidity that reveals itself in the order book with a certain frequency. Solving this model in equilibrium reveals that there is a stability condition on the parameters beyond which the order book empties completely, corresponding to a liquidity crisis. By calibrating it on market data, we were able to qualitatively analyze the distance to this unstable region.

In this thesis, we are mainly interested in three research topics, relatively independent, but nevertheless related through the thread of interactions and incentives, as highlighted in the introduction constituting the first chapter.In the first part, we present extensions of contract theory, allowing in particular to consider a multitude of players in principal-agent models, with drift and volatility control, in the presence of moral hazard. In particular, Chapter 2 presents a continuous-time optimal incentive problem within a hierarchy, inspired by the one-period model of Sung (2015) and enlightening in two respects: on the one hand, it presents a framework where volatility control occurs in a perfectly natural way, and, on the other hand, it highlights the importance of considering continuous-time models. In this sense, this example motivates the comprehensive and general study of hierarchical models carried out in the third chapter, which goes hand in hand with the recent theory of second-order stochastic differential equations (2EDSR). Finally, in Chapter 4, we propose an extension of the principal-agent model developed by Aïd, Possamaï, and Touzi (2019) to a continuum of agents, whose performances are in particular impacted by a common hazard. In particular, these studies guide us towards a generalization of the so-called revealing contracts, initially proposed by Cvitanić, Possamaï and Touzi (2018) in a single-agent model.In the second part, we present two applications of principal-agent problems to the energy domain. The first one, developed in Chapter 5, uses the formalism and theoretical results introduced in the previous chapter to improve electricity demand response programs, already considered by Aïd, Possamaï and Touzi (2019). Indeed, by taking into account the infinite number of consumers that a producer has to supply with electricity, it is possible to use this additional information to build the optimal incentives, in particular to better manage the residual risk implied by weather hazards. In a second step, chapter 6 proposes, through a principal-agent model with adverse selection, an insurance that could prevent some forms of precariousness, in particular fuel precariousness.Finally, we end this thesis by studying in the last part a second field of application, that of epidemiology, and more precisely the control of the diffusion of a contagious disease within a population. In chapter 7, we first consider the point of view of individuals, through a mean-field game: each individual can choose his rate of interaction with others, reconciling on the one hand his need for social interactions and on the other hand his fear of being contaminated in turn, and of contributing to the wider diffusion of the disease. We prove the existence of a Nash equilibrium between individuals, and exhibit it numerically. In the last chapter, we take the point of view of the government, wishing to incite the population, now represented as a whole, to decrease its interactions in order to contain the epidemic. We show that the implementation of sanctions in case of non-compliance with containment can be effective, but that, for a total control of the epidemic, it is necessary to develop a conscientious screening policy, accompanied by a scrupulous isolation of the individuals tested positive.

This thesis is organized in three parts. The first part examines the relationship between microscopic and macroscopic market dynamics by focusing on the properties of volatility. In the second part, we focus on the stochastic optimal control of point processes. Finally, in the third part, we study two market design problems. We start this thesis by studying the links between the no-arbitrage principle and the volatility irregularity. Using a scaling method, we show that we can effectively connect these two notions by analyzing the market impact of metaorders. More precisely, we model the market order flow using linear Hawkes processes. We then show that the no-arbitrage principle and the existence of a non-trivial market impact imply that volatility is rough and more precisely that it follows a rough Heston model. We then examine a class of microscopic models where the order flow is a quadratic Hawkes process. The objective is to extend the rough Heston model to continuous models allowing to reproduce the Zumbach effect. Finally, we use one of these models, the quadratic rough Heston model, for the joint calibration of the SPX and VIX volatility slicks. Motivated by the intensive use of point processes in the first part, we are interested in the stochastic control of point processes in the second part. Our objective is to provide theoretical results for applications in finance. We start by considering the case of Hawkes process control. We prove the existence of a solution and then propose a method to apply this control in practice. We then examine the scaling limits of stochastic control problems in the context of population dynamics models. More precisely, we consider a sequence of models of discrete population dynamics which converge to a model for a continuous population. For each model we consider a control problem. We prove that the sequence of optimal controls associated to the discrete models converges to the optimal control associated to the continuous model. This result is based on the continuity, with respect to different parameters, of the solution of a backward-looking schostatic differential equation.In the last part we consider two market design problems. First, we examine the question of the organization of a liquid derivatives market. Focusing on an options market, we propose a two-step method that can be easily applied in practice. The first step is to select the options that will be listed on the market. For this purpose, we use a quantization algorithm that allows us to select the options most in demand by investors. We then propose a pricing incentive method to encourage market makers to offer attractive prices. We formalize this problem as a principal-agent problem that we solve explicitly. Finally, we find the optimal duration of an auction for markets organized in sequential auctions, the case of zero duration corresponding to the case of a continuous double auction. We use a model where the market takers are in competition and we consider that the optimal duration is the one corresponding to the most efficient price discovery process. After proving the existence of a Nash equilibrium for the competition between market takers, we apply our results on market data. For most assets, the optimal duration is between 2 and 10 minutes.

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In recent years, environmental concerns resulted in an increase in the use of renewable resources such as wind energy. However, high penetration of the wind power is a challenge due to the intermittency of this resource. In this context, the wind energy forecasting has become a major issue. In particular, for the end users of wind energy forecasts, a critical but often neglected issue is the economic value of the forecast. In this work, we investigate the economic value of forecasting from 30 min to 3 h ahead, also known as nowcasting. Nowcasting is mainly used to inform trading decisions in the intraday market. Two sources of uncertainty affecting wind farm revenues are investigated, namely forecasting errors and price variations. The impact of these uncertainties is assessed for six wind farms and several balancing strategies using market data. Results are compared with the baseline case of no nowcasting and with the idealized case of perfect nowcast. The three settings show significant differences while the impact of the choice of a specific balancing strategy appears minor.

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We develop an open-source Python software integrating flexibility needs from Variable Renewable Energies (VREs) in the development of regional energy mixes. It provides a flexible and extensible tool to researchers/engineers, and for education/outreach. It aims at evaluating and optimizing energy deployment strategies with high shares of VRE. assessing the impact of new technologies and of climate variability. conducting sensitivity studies. Specifically, to limit the algorithm’s complexity, we avoid solving a full-mix cost-minimization problem by taking the mean and variance of the renewable production-demand ratio as proxies to balance services. Second, observations of VRE technologies being typically too short or nonexistent, the hourly demand and production are estimated from climate time-series and fitted to available observations. We illustrate e4clim’s potential with an optimal recommissioning-study of the 2015 Italian PV-wind mix testing different climate-data sources and strategies and assessing the impact of climate variability and the robustness of the results.

In this thesis, we study several financial mathematics problems related to the valuation of derivatives. Through different asymptotic approaches, we develop methods to compute accurate approximations of the price of certain types of options in cases where no explicit formula exists.In the first chapter, we focus on the valuation of options whose payoff depends on the trajectory of the underlying by Monte Carlo methods, when the underlying is modeled by an affine process with stochastic volatility. We prove a principle of large trajectory deviations in long time, which we use to compute, using Varadhan's lemma, an asymptotically optimal change of measure, allowing to significantly reduce the variance of the Monte-Carlo estimator of option prices.The second chapter considers the valuation by Monte-Carlo methods of options depending on multiple underlyings, such as basket options, in Wishart's stochastic volatility model, which generalizes the Heston model. Following the same approach as in the previous chapter, we prove that the process vérifie a principle of large deviations in long time, which we use to significantly reduce the variance of the Monte Carlo estimator of option prices, through an asymptotically optimal change of measure. In parallel, we use the principle of large deviations to characterize the long-time behavior of the Black-Scholes implied volatility of basket options.In the third chapter, we study the valuation of realized variance options, when the spot volatility is modeled by a constant volatility diffusion process. We use recent asymptotic results on the densities of hypo-elliptic diffusions to compute an expansion of the realized variance density, which we integrate to obtain the expansion of the option price, and then their Black-Scholes implied volatility.The final chapter is devoted to the valuation of interest rate derivatives in the Lévy model of the Libor market, which generalizes the classical Libor market model (log-normal) by adding jumps. By writing the former as a perturbation of the latter and using the Feynman-Kac representation, we explicitly compute the asymptotic expansion of the price of interest rate derivatives, in particular, caplets and swaptions.

This thesis aims at understanding several aspects of the roughness of volatility observed universally on financial assets. This is done in six steps. In the first part, we explain this property from the typical behaviors of agents in the market. More precisely, we build a microscopic price model based on Hawkes processes reproducing the important stylized facts of the market microstructure. By studying the long-run price behavior, we show the emergence of a rough version of the Heston model (called rough Heston model) with leverage. Using this original link between Hawkes processes and Heston models, we compute in the second part of this thesis the characteristic function of the log-price of the rough Heston model. This characteristic function is given in terms of a solution of a Riccati equation in the case of the classical Heston model. We show the validity of a similar formula in the case of the rough Heston model, where the Riccati equation is replaced by its fractional version. This formula allows us to overcome the technical difficulties due to the non-Markovian character of the model in order to value derivatives. In the third part, we address the issue of risk management of derivatives in the rough Heston model. We present hedging strategies using the underlying asset and the forward variance curve as instruments. This is done by specifying the infinite-dimensional Markovian structure of the model. Being able to value and hedge derivatives in the rough Heston model, we confront this model with the reality of financial markets in the fourth part. More precisely, we show that it reproduces the behavior of implied and historical volatility. We also show that it generates the Zumbach effect, which is a time-reversal asymmetry observed empirically on financial data. In the fifth part, we study the limiting behavior of the implied volatility at low maturity in the framework of a general stochastic volatility model (including the rough Bergomi model), by applying a density development of the asset price. While the approximation based on Hawkes processes has addressed several questions related to the rough Heston model, in Part 6 we consider a Markovian approximation applying to a more general class of rough volatility models. Using this approximation in the particular case of the rough Heston model, we obtain a numerical method for solving the fractional Riccati equations. Finally, we conclude this thesis by studying a problem not related to the rough volatility literature. We consider the case of a platform seeking the best make-take fee scheme to attract liquidity. Using the principal-agent framework, we describe the best contract to offer to the market maker as well as the optimal quotes displayed by the latter. We also show that this policy leads to better liquidity and lower transaction costs for investors.

The increase in the share of intermittent renewable energies in the energy mix is generating problems related to the predictability of electricity production. In particular, on a seasonal basis, transmission system operators are forced to project the availability of generation resources and forecast demand. This allows them to guarantee supply for the next winter or summer. Nevertheless, current projections are mainly based on historical data (climatology) of temperatures (consumption), winds (wind production), or solar radiation (photovoltaic production). The thesis presents four works: three in the framework of seasonal forecasting, and one study on the realism of surface wind as modeled by the European Center's weather forecasting model.If wind energy forecasting at short time scales ranging from minutes to days as well as wind tendency at climatic scales have been widely studied, wind production forecasting at intermediate time scales ranging from a fortnight to the season has received little attention. The predictability of mid-latitude weather at these distant horizons is indeed still an open question. However, several studies have shown that seasonal numerical forecasting models are able to provide information on the variability of large-scale atmospheric circulation through the prediction of large-scale circulation oscillations, such as ENSO in the Pacific, or the NAO in the North Atlantic. It has also been shown that these oscillations have a strong impact on precipitation, temperature, and surface wind. Building the relationship between these indicators of large-scale atmospheric circulation and surface wind in France can therefore take into account the interannual variability of surface wind, which is not capable by definition climatology. This is the idea developed in the 3 studies concerning seasonal forecasting. In order to forecast the wind resource and production on a seasonal scale, two probabilistic models are developed. One is parametric, based on the prediction of the seasonal distribution of surface wind at different locations in France. The other is non-parametric, based on the estimation of the probability density of daily surface wind conditional on the state of the atmosphere. The third study proposes to reconstruct the joint probability of French national consumption and production, thus allowing to measure the risk of imbalance between supply and demand.

This thesis deals with the theory of stochastic equations in finite dimension. In the first part, we derive necessary and sufficient geometric conditions on the coefficients of a stochastic differential equation for the existence of a solution constrained to remain in a closed domain, under weak regularity conditions on the coefficients.In the second part, we address existence and uniqueness problems of stochastic Volterra equations of convolutional type. These equations are in general non-Markovian. We establish their correspondence with infinite-dimensional equations which allows us to approximate them by finite-dimensional Markovian stochastic differential equations. Finally, we illustrate our results by an application in mathematical finance, namely the modeling of rough volatility. In particular, we propose a stochastic volatility model that provides a good compromise between flexibility and tractability.

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We consider the problem of a wind producer who has access to the spot and intraday electricity markets and has the possibility of partially storing the produced energy using a battery storage facility. The aim of the producer is to maximize the expected gain of selling in the market the energy produced during a 24-hour period. We propose and calibrate statistical models for the power production and the intraday electricity price, and compute the optimal strategy of the producer via dynamic programming.

We prove a large deviations principle for the class of multidimensional affine stochastic volatility models considered in (Gourieroux, C. and Sufana, R., J. Bus. Econ. Stat., 28(3), 2010), where the volatility matrix is modelled by a Wishart process. This class extends the very popular Heston model to the multivariate setting, thus allowing to model the joint behaviour of a basket of stocks or several interest rates. We then use the large deviation principle to obtain an asymptotic approximation for the implied volatility of basket options and to develop an asymptotically optimal importance sampling algorithm, to reduce the number of simulations when using Monte-Carlo methods to price derivatives.

This thesis focuses on valuation and financial strategies for hedging risks in energy markets. These markets present particularities that distinguish them from standard financial markets, notably illiquidity and incompleteness. Illiquidity is reflected in high transaction costs and constraints on volumes traded. Incompleteness is the inability to perfectly replicate derivatives. We focus on different aspects of market incompleteness. The first part deals with valuation in Lévy models. We obtain an approximate formula for the indifference price and we measure the minimum premium to be brought over the Black-Scholes model. The second part concerns the valuation of spread options in the presence of stochastic correlation. Spread options deal with the price difference between two underlying assets -- for example gas and electricity -- and are widely used in the energy markets. We propose an efficient numerical procedure to calculate the price of these options. Finally, the third part deals with the valuation of a product with an exogenous risk for which forecasts exist. We propose an optimal dynamic strategy in the presence of volume risk, and apply it to the valuation of wind farms. In addition, a section is devoted to asymptotic optimal strategies in the presence of transaction costs.

This thesis deals with dependence problems between stochastic processes in continuous time. In a first part, new copulas are established to model the dependence between two Brownian movements and to control the distribution of their difference. It is shown that the class of admissible copulas for Brownians contains asymmetric copulas. With these copulas, the survival function of the difference of the two Brownians is higher in its positive part than with a Gaussian dependence. The results are applied to the joint modeling of electricity prices and other energy commodities. In a second part, we consider a discretely observed stochastic process defined by the sum of a continuous semi-martingale and a compound Poisson process with mean reversion. An estimation procedure for the mean-reverting parameter is proposed when the mean-reverting parameter is large in a high frequency finite horizon statistical framework. In a third part, we consider a doubly stochastic Poisson process whose stochastic intensity is a function of a continuous semi-martingale. To estimate this function, a local polynomial estimator is used and a window selection method is proposed leading to an oracle inequality. A test is proposed to determine if the intensity function belongs to a certain parametric family. With these results, the dependence between the intensity of electricity price peaks and exogenous factors such as wind generation is modeled.

We consider the problem of a wind producer who has access to the spot and intraday electricity markets and has the possibility of partially storing the produced energy using a battery storage facility. The aim of the producer is to maximize the expected gain of selling in the market the energy produced during a 24-hour period. We propose and calibrate statistical models for the power production and the intraday electricity price, and compute the optimal strategy of the producer via dynamic programming.

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An avenue for modelling part of the long-term variability of the wind energy resource from knowledge of the large-scale state of the atmosphere is investigated. The timescales considered are monthly to seasonal, and the focus is on France and its vicinity. On such timescales, one may obtain information on likely surface winds from the large-scale state of the atmosphere, determining for instance the most likely paths for storms impinging on Europe. In a first part, we reconstruct surface wind distributions on monthly and seasonal timescales from the knowledge of the large-scale state of the atmosphere , which is summarized using a principal components analysis. We then apply a multi-polynomial regression to model surface wind speed distributions in the parametric context of the Weibull distribution. Several methods are tested for the reconstruction of the parameters of the Weibull distribution , and some of them show good performance. This proves that there is a significant potential for information in the relation between the synoptic circulation and the surface wind speed. In the second part of the paper, the knowledge obtained on the relationship between the large-scale situation of the atmosphere and surface wind speeds is used in an attempt to forecast wind speeds distributions on a monthly horizon. The forecast results are promising but they also indicate that the Numerical Weather Prediction seasonal forecasts on which they are based, are not yet mature enough to provide reliable information for timescales exceeding one month.

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We build and evaluate a probabilistic model designed for forecasting the distribution of the daily mean wind speed at the seasonal timescale in France. On such long-term timescales, the variability of the surface wind speed is strongly influenced by the atmosphere large-scale situation. Our aim is to predict the daily mean wind speed distribution at a specific location using the information on the atmosphere large-scale situation, summarized by an index. To this end, we estimate, over 20 years of daily data, the conditional probability density function of the wind speed given the index. We next use the ECMWF seasonal forecast ensemble to predict the atmosphere large-scale situation and the index at the seasonal timescale. We show that the model is sharper than the climatology at the monthly horizon, even if it displays a strong loss of precision after 15 days. Using a statistical postprocessing method to recalibrate the ensemble forecast leads to further improvement of our probabilistic forecast, which then remains sharper than the climatology at the seasonal horizon.

In this paper we consider the pricing of options on interest rates such as caplets and swaptions in the Levy Libor model developed by Eberlein and Ozkan (Financ. Stochast. 9:327-348 (2005) [9]). This model is an extension to Levy driving processes of the classical log-normal Libor market model (LMM) driven by a Brownian motion. Option pricing is significantly less tractable in this model than in the LMM due to the appearance of stochastic terms in the jump part of the driving process when performing the measure changes which are standard in pricing of interest rate derivatives. To obtain explicit approximation for option prices, we propose to treat a given Levy Libor model as a suitable perturbation of the log-normal LMM. The method is inspired by recent works by Cerný, Denkl, and Kallsen (Preprint (2013) [6]) and Menasse and Tankov (Preprint (2015) [14]). The approximate option prices in the Levy Libor model are given as the corresponding LMM prices plus correction terms which depend on the characteristics of the underlying Levy process and some additional terms obtained from the LMM model.

In this paper we consider the pricing of options on interest rates such as caplets and swaptions in the Lévy Libor model developed by Eberlein and Özkan (Financ. Stochast. 9:327-348 (2005) [9]). This model is an extension to Lévy driving processes of the classical log-normal Libor market model (LMM) driven by a Brownian motion. Option pricing is significantly less tractable in this model than in the LMM due to the appearance of stochastic terms in the jump part of the driving process when performing the measure changes which are standard in pricing of interest rate derivatives. To obtain explicit approximation for option prices, we propose to treat a given Lévy Libor model as a suitable perturbation of the log-normal LMM. The method is inspired by recent works by Černý, Denkl, and Kallsen (Preprint (2013) [6]) and Ménassé and Tankov (Preprint (2015) [14]). The approximate option prices in the Lévy Libor model are given as the corresponding LMM prices plus correction terms which depend on the characteristics of the underlying Lévy process and some additional terms obtained from the LMM model.

We introduce a new functional measure of tail dependence for weakly dependent (asymptotically independent) random vectors, termed weak tail dependence function. The new measure is defined at the level of copulas and we compute it for several copula families such as the Gaussian copula, copulas of a class of Gaussian mixture models, certain Archimedean copulas and extreme value copulas. The new measure allows to quantify the tail behavior of certain functionals of weakly dependent random vectors at the log scale.Comment: Replaced with revised versio.

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We study the optimal trading policies for a wind energy producer who aims to sell the future production in the open forward, spot, intraday and adjustment markets , and who has access to imperfect dynamically updated forecasts of the future production. We construct a stochastic model for the forecast evolution and determine the optimal trading policies which are updated dynamically as new forecast information becomes available. Our results allow to quantify the expected future gain of the wind producer and to determine the economic value of the forecasts.

In the paper, we characterize the asymptotic behavior of the implied volatility of a basket call option at large and small strikes in a variety of settings with increasing generality. First, we obtain an asymptotic formula with an error bound for the left wing of the implied volatility, under the assumption that the dynamics of asset prices are described by the multidimensional Black-Scholes model. Next, we find the leading term of asymptotics of the implied volatility in the case where the asset prices follow the multidimensional Black-Scholes model with time change by an independent increasing stochastic process. Finally, we deal with a general situation in which the dependence between the assets is described by a given copula function. In this setting, we obtain a model-free tail-wing formula that links the implied volatility to a special characteristic of the copula called the weak lower tail dependence function.

We review and extend the now considerable literature on Levy copulas. First, we focus on Monte Carlo methods and present a new robust algorithm for the simulation of multidimensional Levy processes with dependence given by a Levy copula. Next, we review statistical estimation techniques in a parametric and a non-parametric setting. Finally, we discuss the interplay between Levy copulas and multivariate regular variation and briefly review the applications of Levy copulas in risk management. In particular, we provide a new easy-to-use sufficient condition for multivariate regular variation of Levy measures in terms of their Levy copulas.

We study stochastic delay differential equations (SDDE) where the coefficients depend on the moving averages of the state process. As a first contribution, we provide sufficient conditions under which the solution of the SDDE and a linear path functional of it admit a finite-dimensional Markovian representation. As a second contribution, we show how approximate finite-dimensional Markovian representations may be constructed when these conditions are not satisfied, and provide an estimate of the error corresponding to these approximations. These results are applied to optimal control and optimal stopping problems for stochastic systems with delay. © 2014, Springer Science+Business Media New York.

We consider the hedging error of a derivative due to discrete trading in the presence of a drift in the dynamics of the underlying asset. We suppose that the trader wishes to find rebalancing times for the hedging portfolio which enable him to keep the discretization error small while taking advantage of market trends. Assuming that the portfolio is readjusted at high frequency, we introduce an asymptotic framework in order to derive optimal discretization strategies. More precisely, we formulate the optimization problem in terms of an asymptotic expectation-error criterion. In this setting, the optimal rebalancing times are given by the hitting times of two barriers whose values can be obtained by solving a linear-quadratic optimal control problem. In specific contexts such as in the Black-Scholes model, explicit expressions for the optimal rebalancing times can be derived.

The financial markets have expanded considerably over the last three decades and have seen the emergence of a variety of derivative products. The most widely used of these derivatives are American options.

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We analyse the behaviour of the implied volatility smile for options close to expiry in the exponential L\'evy class of asset price models with jumps. We introduce a new renormalisation of the strike variable with the property that the implied volatility converges to a non-constant limiting shape, which is a function of both the diffusion component of the process and the jump activity (Blumenthal-Getoor) index of the jump component. Our limiting implied volatility formula relates the jump activity of the underlying asset price process to the short end of the implied volatility surface and sheds new light on the difference between finite and infinite variation jumps from the viewpoint of option prices: in the latter, the wings of the limiting smile are determined by the jump activity indices of the positive and negative jumps, whereas in the former, the wings have a constant model-independent slope. This result gives a theoretical justification for the preference of the infinite variation L\'evy models over the finite variation ones in the calibration based on short-maturity option prices.

In the first part, I am interested in a portfolio insurance problem for a manager of an investment fund, who wants to structure a financial product for investors with a capital guarantee. If the value of the product is below a fixed threshold, the investor will be reimbursed up to this threshold by the fund's insurer. In exchange, the insurer imposes a constraint on the risk that the manager can tolerate, measured with a risk measure. I give the solution to this problem and prove that the choice of the risk measure is a crucial point for the existence of an optimal portfolio. I apply my results for the entropy, spectral and G-divergence risk measures. Next, I focus on the quadratic hedging problem. The market is described by a three-dimensional Markov process with jumps. The first variable models the hedging instrument that is tradable on the market, the second one a financial asset that disturbs the dynamics of the hedging instrument and that is not tradable, such as a volatility factor. The third one represents a risk source that affects the option to be hedged and that is also not tradable. I prove that the value function of the problem is characterized by the unique solution of a system of three integro-differential equations, one of which is semilinear and does not depend on the option to be hedged, and the other two are linear. This allows me to characterize the optimal strategy. This result is demonstrated if the process is non-degenerate (strictly elliptic Brownian component) and if it is pure jump. I conclude with an application in the electricity market.

This paper summarizes my contributions to the study of discretization of processes with jumps, and to the modeling of liquidity risk and jump risk in financial markets. Chapter 2 gathers more theoretical results in the field of discretization of stochastic processes with jumps, with in particular a study of the discretization error of hedging strategies, and new simulation schemes of stochastic differential equations directed by Lévy processes. Chapter 3 presents and studies via stochastic control an investment and consumption optimization problem in illiquid financial markets. Chapter 4 contains more applied work on modeling jump risk in portfolio insurance strategies, derivatives, and electricity markets.

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This thesis deals with the modelling of stock prices by the exponentials of Lévy processes. In the first part we develop a non-parametric method allowing to calibrate exponential Lévy models, that is, to reconstruct such models from the prices of market-quoted options. We study the stability and convergence properties of this calibration method, describe its numerical implementation and give examples of its use. Our approach is first to reformulate the calibration problem as that of finding a risk-neutral exponential Lévy model that reproduces the option prices with the best possible precision and has the smallest relative entropy with respect to a given prior process, and then to solve this problem via the regularization methodology, used in the theory of ill-posed inverse problems. Applying this calibration method to the empirical data sets of index options allows us to study some properties of Lévy measures, implied by market prices. The second part of this thesis proposes a method allowing to characterize the dependence structures among the components of a multidimensional Lévy process and to construct multidimensional exponential Lévy models. This is done by introducing the notion of Lévy copula, which can be seen as an analog for Lévy processes of the notion of copula, used in statistics to model dependence between real-valued random variables. We give examples of parametric families of Lévy copulas and develop a method for simulating multidimensional Lévy processes with dependence given by a Lévy copula.