SCOTTI Simone

Abstract We propose an extension of the $$\mathrm{\Gamma }$$ Γ -OU Barndorff-Nielsen and Shephard model taking into account jump clustering phenomena. We assume that the intensity process of the Hawkes driver coincides, up to a constant, with the variance process. By applying the theory of continuous-state branching processes with immigration, we prove existence and uniqueness of strong solutions of the SDE governing the asset price dynamics. We propose a measure change of self-exciting Esscher type in order to describe the relation between the risk-neutral and the historical dynamics, showing that the $$\mathrm{\Gamma }$$ Γ -OU Hawkes framework is stable under this probability change. By exploiting the affine features of the model we provide an explicit form for the Laplace transform of the asset log-return, for its quadratic variation and for the ergodic distribution of the variance process. We show that the proposed model exhibits a larger flexibility in comparison with the $$\mathrm{\Gamma }$$ Γ -OU model, in spite of the same number of parameters required. We calibrate the model on market vanilla option prices via characteristic function inversion techniques, we study the price sensitivities and propose an exact simulation scheme. The main financial achievement is that implied volatility of options written on VIX is upward shaped due to the self-exciting property of Hawkes processes, in contrast with the usual downward slope exhibited by the $$\mathrm{\Gamma }$$ Γ -OU Barndorff-Nielsen and Shephard model.

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We introduce a class of interest rate models, called the alpha-CIR model, which is a natural extension of the standard CIR model by adding a jump part driven by alpha-stable Levy processes with index alpha is an element of (1, 2]. We deduce an explicit expression for the bond price by using the fact that the model belongs to the family of CBI and affine processes, and analyze the bond price and bond yield behaviors. The alpha-CIR model allows us to describe in a unified and parsimonious way several recent observations on the sovereign bond market such as the persistency of low interest rates together with the presence of large jumps. Finally, we provide a thorough analysis of the jumps, and in particular the large jumps.Sino-French Research Program of Mathematics. NSFC of China [11671216]SCI(E)SSCIARTICLE3789-8132.

We introduce a class of interest rate models, called the α-CIR model, which gives a natural extension of the standard CIR model by adopting the α-stable Lévy process and preserving the branching property. This model allows to describe in a unified and parsimonious way several recent observations on the sovereign bond market such as the persistency of low interest rate together with the presence of large jumps at local extent. We emphasize on a general integral representation of the model by using random fields, with which we establish the link to the CBI processes and the affine models. Finally we analyze the jump behaviors and in particular the large jumps, and we provide numerical illustrations.

We use mixture of percentile functions to model credit spread evolution, which allows to obtain a flexible description of credit indices and their components at the same time. We show regularity results in order to extend mixture percentile to the dynamic case. We characterise the stochastic differential equation of the flow of cumulative distribution function and we link it with the ordered list of the components of the credit index. The main application is to introduce a functional version of Bollinger bands. The crossing of bands by the spread is associated with a trading signal. Finally, we show the richness of the signals produced by functional Bollinger bands compared with standard one with a pratical example.

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We introduce a class of interest rate models, called the α-CIR model, which gives a natural extension of the standard CIR model by adopting the α-stable Lévy process and preserving the branching property. This model allows to describe in a unified and parsimonious way several recent observations on the sovereign bond market such as the persistency of low interest rate together with the presence of large jumps at local extent. We emphasize on a general integral representation of the model by using random fields, with which we establish the link to the CBI processes and the affine models. Finally we analyze the jump behaviors and in particular the large jumps, and we provide numerical illustrations.

In this paper we introduce a jump-diffusion model of shot-noise type for stock prices, taking into account over and under-reaction of the market to incoming news. We work in a partial information setting, by supposing that standard investors do not have access to the market direction, the drift, (modeled via a random variable) after a jump. We focus on the expected (logarithmic) utility maximization problem by providing the optimal investment strategy in explicit form, both under full (i.e., from the insider point of view, aware of the right kind of market reaction at any time) and under partial information (i.e., from the standard investor viewpoint, who needs to infer the kind of market reaction from data). We test our results on market data relative to Enron and Ahold. The three main contributions of this paper are: the introduction of a new market model dealing with over and under-reaction to news, the explicit computation of the optimal filter dynamics using an original approach combining enlargement of filtrations with Innovation Theory and the application of the optimal portfolio allocation rule to market data.

We study the problem of an optimal exit strategy for an investment project which is unprofitable and for which the liquidation costs evolve stochastically. The firm has the option to keep the project going while waiting for a buyer, or liquidating the assets at immediate liquidity and termination costs. The liquidity and termination costs are governed by a mean-reverting stochastic process whereas the rate of arrival of buyers is governed by a regime-shifting Markov process. We formulate this problem as a multidimensional optimal stopping time problem with random maturity. We characterize the objective function as the unique viscosity solution of the associated system of variational Hamilton-Jacobi-Bellman inequalities. We derive explicit solutions and numerical examples in the case of power and logarithmic utility functions when the liquidity premium factor follows a mean-reverting CIR process.

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In this paper we present the detail computations involved in: Bernis, G. Carassus, L. Docq, G. and S. Scotti (2013), Optimal Credit Allocation under Regime Uncertainty with Sensitivity Analysis. First we propose a quick presentation of the methodology developed by Bouleau. Then, we apply this method to the problem of optimal credit allocation problem.

In this paper we present the detail computations involved in: Bernis, G. Carassus, L. Docq, G. and S. Scotti (2013), Optimal Credit Allocation under Regime Uncertainty with Sensitivity Analysis. First we propose a quick presentation of the methodology developed by Bouleau. Then, we apply this method to the problem of optimal credit allocation problem.

We study the problem of optimally liquidating a large portfolio position in a limit-order market. We allow for both limit and market orders and the optimal solution is a combination of both types of orders. Market orders deplete the order book, making future trades more expensive, whereas limit orders can be entered at more favorable prices but are not guaranteed to be filled. We model the bid-ask spread with resilience by a jump process, and the market-order arrival process as a controlled Poisson process. The objective is to minimize the execution cost of the strategy. We formulate the problem as a mixed stochastic continuous control and impulse problem for which the value function is shown to be the unique viscosity solution of the associated variational inequalities. We conclude with a calibration of the model on recent market data and a numerical implementation.

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Our objective is to study liquidity risk, in particular the so-called "bid-ask spread", as a by-product of market uncertainties. "Bid-ask spread", and more generally "limit order books" describe the existence of different sell and buy prices, which we explain by using different risk aversions of market participants. The risky asset follows a diffusion process governed by a Brownian motion which is uncertain. We use the error theory with Dirichlet forms to formalize the notion of uncertainty on the Brownian motion. This uncertainty generates noises on the trajectories of the underlying asset and we use these noises to expound the presence of bid-ask spreads. In addition, we prove that these noises also have direct impacts on the mid-price of the risky asset. We further enrich our studies with the resolution of an optimal liquidation problem under these liquidity uncertainties and market impacts. To complete our analysis, some numerical results will be provided.

This thesis is devoted to the study of applications of the theory of errors by Dirichlet forms. Our work is divided into three parts. The first one analyzes models governed by a stochastic differential equation. After a short technical chapter, an innovative model for order books is proposed. We consider that the bid-ask spread is not a defect, but rather an intrinsic property of the market. Uncertainty is carried by the Brownian motion that drives the asset. We show that the evolution of the spread can be evaluated using closed formulas and we study the impact of the uncertainty of the underlying on derivatives. We then introduce the PBS model for pricing European options. The innovative idea is to distinguish the market volatility from the parameter used by traders to hedge. We assume the former constant, while the latter becomes a subjective and erroneous estimate of the former. We prove that this model predicts a bid-ask spread and a volatility smile. The most interesting properties of this model are the existence of closed formulas for pricing, the impact of the underlying drift and an efficient calibration strategy. The second part focuses on models described by a partial differential equation. The linear and nonlinear cases are analyzed separately. In the first case we show interesting relations between error theory and wavelet theory. In the nonlinear case we study the sensitivity of the solutions using the error theory. Except in the case of an exact solution, there are two possible approaches: one can first discretize the PDE and study the sensitivity of the discretized problem, or demonstrate that the theoretical sensitivities verify PDEs. Both cases are studied, and we prove that the sharp and the bias are solutions of linear PDEs depending on the solution of the original PDE and we propose algorithms to numerically evaluate the sensitivities. Finally, the third part is dedicated to stochastic partial differential equations. Our analysis is divided into two chapters. First, we study the transmission of the uncertainty, present in the initial condition, to the solution of the SPDE. Then, we analyze the impact of a perturbation in the functional terms of the EDPS and in the coefficient of the associated Green's function. In both cases, we prove that the sharp and the bias are solutions of two linear EDPS depending on the solution of the original EDPS.

This thesis is devoted to the study of the applications of the error theory using Dirichlet forms. Our work is split into three parts. The first one deals with the models described by stochastic differential equations. After a short technical chapter, an innovative model for order books is proposed. We assume that the bid-ask spread is not an imperfection, but an intrinsic property of exchange markets instead. The uncertainty is carried by the Brownian motion guiding the asset. We find that spread evolutions can be evaluated using closed formulae and we estimate the impact of the underlying uncertainty on the related contingent claims. Afterwards, we deal with the PBS model, a new model to price European options. The seminal idea is to distinguish the market volatility with respect to the parameter used by traders for hedging. We assume the former constant, while the latter volatility being an erroneous subjective estimation of the former. We prove that this model anticipates a bid-ask spread and a smiled implied volatility curve. Major properties of this model are the existence of closed formulae for prices, the impact of the underlying drift and an efficient calibration strategy. The second part deals with the models described by partial differential equations. Linear and non-linear PDEs are examined separately. In the first case, we show some interesting relations between the error and wavelets theories. When non-linear PDEs are concerned, we study the sensitivity of the solution using error theory. Except when exact solution exists, two possible approaches are detailed: first, we analyze the sensitivity obtained by taking “derivatives” of the discrete governing equations. Then, we study the PDEs solved by the sensitivity of the theoretical solutions. In both cases, we show that sharp and bias solve linear PDE depending on the solution of the former PDE itself and we suggest algorithms to evaluate numerically the sensitivities. Finally, the third part is devoted to stochastic partial differential equations. Our analysis is split into two chapters. First, we study the transmission of an uncertainty, present on starting conditions, on the solution of SPDE. Then, we analyze the impact of a perturbation of the functional terms of SPDE and the coefficient of the related Green function. In both cases, we show that the sharp and bias verify linear SPDE depending on the solution of the former SPDE itself Cette thèse est consacrée à l'étude des applications de la théorie des erreurs par formes de Dirichlet.