Stochastic Volatility Models

COGARCH models are continuous time versions of the well‐known GARCH models of financial returns. The first aim of this paper is to show how the method of prediction‐based estimating functions can be applied to draw statistical inference from observations of a COGARCH(1,1) model if the higher‐order structure of the process is clarified. A second aim of the paper is to provide recursive expressions for the joint moments of any fixed order of the process. Asymptotic results are given, and a simulation study shows that the method of prediction‐based estimating function outperforms the other available estimation methods.

We show the advantage of using Google search engine trends to forecast the volatility of the shortterm (weekly) exchange rate between the Mexican peso and United States dollar. We perform a comparison of models in the literature that have used Google Trends to examine explanatory variables. Some of the models are based on time series, whereas others are based on the similarity function, which captures the cognitive form of human reasoning. For example, an investor who needs to know the value that a variable will take in the future will take into account relevant, known, and available information, and weigh it to calculate the forecast. We conclude that taking into account the Google Trends variable helps explains partially the behaviour of volatility; and it is necessary to incorporate more aggregation levels. Moreover, to the best of our knowledge, literature on the subject of using Google Trends to explain relevant economic variables is relatively scarce.

In this study, the volatility of deposit banks’ credit stock in Turkey is investigated by using weekly data from June 2000 through June 2007. To determine the high and low volatility states, two state switching autoregressive conditional heteroscedasticity (SWARCH) models are estimated. The economic and political events that resulted in high volatility in credit stock volume are also analyzed. Results show that volatility of credit stock volume in Turkey is sensitive to domestic government bonds’ interest rate and foreign portfolio investments. The volatility of credit stock volume is also affected by political events.

We show the advantage of using Google search engine trends to forecast the volatility of the shortterm (weekly) exchange rate between the Mexican peso and United States dollar. We perform a comparison of models in the literature that have used Google Trends to examine explanatory variables. Some of the models are based on time series, whereas others are based on the similarity function, which captures the cognitive form of human reasoning. For example, an investor who needs to know the value that a variable will take in the future will take into account relevant, known, and available information, and weigh it to calculate the forecast. We conclude that taking into account the Google Trends variable helps explains partially the behaviour of volatility; and it is necessary to incorporate more aggregation levels. Moreover, to the best of our knowledge, literature on the subject of using Google Trends to explain relevant economic variables is relatively scarce.

In this presentation, we address the topic of valuing European options using the Heston (1993) stochastic volatility model via Monte Carlo simulation. To do this, we present the theoretical characteristics of the model, as well as the valuation formula found in Heston’s seminal paper. The Monte Carlo simulation in this context refers to generating artificial trajectories for the underlying asset price and its variance over time within a system of correlated stochastic differential equations (SDEs). These trajectories can then be used to obtain prices for European or other types of options. We propose the discretization of the system using an Euler scheme. We will execute a Monte Carlo exercise with variations in the model's parameters and observe the corresponding effect on the distribution of the underlying asset and its variance.

Keywords: Quantitative Finance, Financial Engineering, Derivatives Pricing, European Option Valuation, Stochastic Volatility Models, Heston Model, Monte Carlo Simulation Methods, Numerical Solutions of Stochastic, Differential Equations (SDEs), Euler Discretization Scheme, Milstein, Discretization Scheme, Variance Reduction Techniques (Antithetic Variates),  Risk-Neutral Valuation

Alignment with UN Sustainable Development Goals (SDGs)
SDG 8: Decent Work and Economic Growth By developing more sophisticated models for pricing derivatives and managing financial risk (moving beyond basic models like Black-Scholes to incorporate stochastic volatility), this work contributes to the stability and efficiency of financial markets. Robust financial systems with accurate risk assessment tools are fundamental prerequisites for sustained and inclusive economic growth (Target 8.10: strengthen the capacity of domestic financial institutions).

SDG 9: Industry, Innovation, and Infrastructure This presentation represents applied research within the financial services industry. It utilizes advanced mathematical modeling and computational techniques (Monte Carlo simulation with variance reduction) to solve practical valuation problems. This directly supports the enhancement of scientific research and the upgrading of technological and analytical capabilities within the financial infrastructure (Target 9.5).

This paper aims to investigate the direct relationship between inflation and inflation uncertainty by employing a dynamic method for the monthly country-region-place United States data for the time period 1976-2007. While the bulk of previous studies has employed GARCH models in investigating the link between inflation and inflation uncertainty, in this study Stochastic Volatility in Mean models are used to capture the shocks to inflation uncertainty within a dynamic framework. These models allow researchers to assess the dynamic effects of innovations in inflation as well as inflation volatility on inflation and inflation volatility over time, by incorporating the unobserved volatility as an explanatory variable in the mean (inflation) equation. Empirical findings suggest that innovations in inflation volatility increases inflation. This evidence is robust across various definitions of inflation and different sub-periods.

This paper investigates the effect of inflation uncertainty innovations on inflation over time by considering the monthly United States data for the time period 1976-2006. In order to investigate the effect of inflation uncertainty innovation on inflation, a Stochastic Volatility in Mean model (SVM) has been employed. SVM models are generally used to capture the innovation to inflation uncertainty, which cannot be achieved in the framework of popular deterministic ARCH type of models. Empirical evidence provided here suggests that innovations in inflation volatility increases inflation persistently. This evidence is robust across various definitions of inflation and different sub-periods.

We propose a new model for electricity pricing based on the price cap principle. The particularity of the model is that the asset price is an exponential functional of a jump L\'evy process. This model can capture both mean reversion and jumps which are observed in electricity market. It is shown that the value of an European option of this asset is the unique viscosity solution of a partial integro-differential equation (PIDE). A numerical approximation of this solution by the finite differences method is provided. The consistency, stability and convergence results of the scheme are given. Numerical simulations are performed under a smooth initial condition.

In this paper, a study of a stochastic volatility model for asset pricing is described. Originally presented by J. Da Fonseca, M. Grasselli and C. Tebaldi, the Wishart volatility model identifies the volatility of the asset as the trace of a Wishart process. Contrary to a classic multifactor Heston model, this model allows to add degrees of freedom with regard to the stochastic correlation. Thanks to its flexibility, this model enables a better fit of market data than the Heston model. Besides, the Wishart volatility model keeps a clear interpretation of its parameters and conserves an efficient tractability. Firstly, we recall the Wishart volatility model and we present a Monte Carlo simulation method in sight of the evaluation of complex options. Regarding stochastic volatility models, implied volatility surfaces of vanilla options have to be obtained for a short time. The aim of this article is to provide an accurate approximation method to deal with asymptotic smiles and to appl...

In Monte Carlo path simulations, which are used extensively in computational fi-nance, one is interested in the expected value of a quantity which is a functional of the solution to a stochastic differential equation [M.B. Giles, Multilevel Monte Carlo Path Simulation: Operations Research, 56(3) (2008) 607-617] where we have a scalar function with a uniform Lipschitz bound. Normally, we discretise the stochas-tic differential equation numerically. The simplest estimate for this expected value is the mean of the payoff (the value of an option at the terminal period) values from N independent path simulations. The multilevel Monte Carlo path simula-tion method recently introduced by Giles exploits strong convergence properties to improve the computational complexity by combining simulations with different lev-els of resolution. This new method improves on the computational complexity of the standard Monte Carlo approach by considering Monte Carlo simulations with a geometric sequence ...

The Black-Scholes Model is a cornerstone of financial economics, revolutionizing options pricing and modern finance. Developed by Fischer Black and Myron Scholes in 1973, it provides a mathematical framework for valuing options and derivatives. This research thoroughly examines the Black-Scholes Model, encompassing its historical context, theoretical foundations, practical applications, limitations, and critically reviews the existing literature on the proposed exact as well as the numerical solutions to the Black-Scholes model and recent advancements. By analyzing its multifaceted aspects, this paper aims to deepen understanding and shed light on its significance in contemporary financial markets.

The objective of this research work was to determine the price of American put option: - When the underlying price does not experience jumps, i.e., the price is assumed to follow the Black-Scholes model and - When the underlying price experiences jumps, i.e., the price is assumed to follow the Merton Jump Diffusion model. An American option is more difficult to price than a European option because the American option has the possibility of early exercise at every instant of its lifetime. It’s never optimal to exercise early an American call option with non-dividend stock. Since we were interested in an early exercise problem with a non-dividend stock option, we opted for an American put option. A brief introduction to options is given in chapter one. This chapter also includes brief description of concepts, such as arbitrage and risk free rate. The chapter finally ends with the introductory summary of the American options. The second chapter introduces the American option and stopping time. We discuss the Black-Scholes model in section 2.1. This section also mentions about the boundary conditions and free boundary problem for American put, which lead to a linear complementarity problem (LCP) under certain assumptions. In section 2.2, we discuss how the solutions of the discrete LCPs are approximated using the operator splitting method. We briefly describes various methods of discretization applied to the Black-Scholes Greeks and boundary conditions in section 2.3. The application of implicit Crank-Nicolson finite difference method on Black-Scholes model is described in section 2.4. The final section of the chapter describes the Thomas algorithm using implicit Crank-Nicolson scheme to price an American put option. We have implemented this method using Matlab to compute the American put price in Black Scholes model. The chapter 3 deals with the pricing model for the American put option when the price of the underlying follows the jumps. In our thesis, we have considered the jump diffusion model proposed by Merton. The section 3.1 describes various integrals associated with the jump process. The section 3.2 introduces the Itô-Doeblin formula for a jump process. We discuss the application of Itô lemma to a jump diffusion process in section 3.3. The section 3.4 mentions the derivation of partial integro-differential equation for a jump diffusion process of an American put option. The section 3.5 describes the discretization approach to a partial integro-differential equation for jump diffusion process. The last section of this chapter describes the generalized implicit-explicit finite difference method to price an American put option under the Merton jump diffusion model. We have also implemented this method using Matlab to compute the American put price with jumps.

In this paper, we introduce a unifying approach to option pricing under continuous-time stochastic volatility models with jumps. For European style options, a new semi-closed pricing formula is derived using the generalized complex Fourier transform of the corresponding partial integro-differential equation. This approach is successfully applied to models with different volatility diffusion and jump processes. We also discuss how to price options with different payoff functions in a similar way. In particular, we focus on a log-normal and a log-uniform jump diffusion stochastic volatility model, originally introduced by Bates and Yan and Hanson, respectively. The comparison of existing and newly proposed option pricing formulas with respect to time efficiency and precision is discussed. We also derive a representation of an option price under a new approximative fractional jump diffusion model that differs from the aforementioned models, especially for the out-of-the money contracts.

In this paper we perform robustness and sensitivity analysis of several continuous-time stochastic volatility (SV) models with respect to the process of market calibration. The analyses should validate the hypothesis on importance of the jump part in the underlying model dynamics. Also an impact of the long memory parameter is measured for the approximative fractional SV model. For the first time, the robustness of calibrated models is measured using bootstrapping methods on market data and Monte-Carlo filtering techniques. In contrast to several other sensitivity analysis approaches for SV models, the newly proposed methodology does not require independence of calibrated parameters-an assumption that is typically not satisfied in practice. Empirical study is performed on data sets of Apple Inc. equity options traded in April and May 2015.

In this paper we perform robustness and sensitivity analysis of several continuous-time stochastic volatility (SV) models with respect to the process of market calibration. The analyses should validate the hypothesis on importance of the jump part in the underlying model dynamics. Also an impact of the long memory parameter is measured for the approximative fractional SV model. For the first time, the robustness of calibrated models is measured using bootstrapping methods on market data and Monte-Carlo filtering techniques. In contrast to several other sensitivity analysis approaches for SV models, the newly proposed methodology does not require independence of calibrated parameters-an assumption that is typically not satisfied in practice. Empirical study is performed on data sets of Apple Inc. equity options traded in April and May 2015.

In this paper, we introduce a unifying approach to option pricing under continuous‐time stochastic volatility models with jumps. For European style options, a new semi‐closed pricing formula is derived using the generalized complex Fourier transform of the corresponding partial integro‐differential equation. This approach is successfully applied to models with different volatility diffusion and jump processes. We also discuss how to price options with different payoff functions in a similar way.In particular, we focus on a log‐normal and a log‐uniform jump diffusion stochastic volatility model, originally introduced by Bates and Yan and Hanson, respectively. The comparison of existing and newly proposed option pricing formulas with respect to time efficiency and precision is discussed. We also derive a representation of an option price under a new approximative fractional jump diffusion model that differs from the aforementioned models, especially for the out‐of‐the money contracts....

In this paper we present a new model for pricing and hedging a portfolio of derivatives that takes into account the effect of an extreme movement in the underlying. We make no assumptions about the timing of this 'crash' or the probability distribution of its size, except that we put an upper bound on the latter. The pricing and hedging follow from the assumption that the worst scenario actually happens i.e. the size and time of the crash are such as to give the option its worst value. The optimal static hedge follows from the desire to make the best of this worst value. There are many applications for this crash modelling, we shall focus on using the model to evaluate the Value at Risk for a portfolio of options.

This paper aims to find numerical solutions of the non-linear Black-Scholes partial differential equation (PDE), which often appears in financial markets, for European option pricing in the appearance of the transaction costs. Here we exploit the transformations for the computational purpose of a non-linear Black-Scholes PDE to modify as a non-linear parabolic type PDE with reliable initial and boundary conditions for call and put options. Several schemes are derived rigorously using the finite volume method (FVM) and finite difference method (FDM), which is the novelty of this paper. Stability and consistency analysis assure the convergence of these schemes. We apply these schemes to various volatility models, such as the Leland, Boyle and Vorst, Barles and Soner, and Risk-adjusted pricing methodology (RAPM). All the schemes are tested numerically. The convergence of the obtained results is observed, and we find that they are also reliable. Finally, we display all the approximate results together with the exact values through graphical and tabular representations.

Cette thèse porte principalement sur deux types de systèmes désordonnés, à savoir les verres de spins et les polymères dirigés en environnement aléatoire. Ces deux thèmes de recherche peuvent s'aborder à l'aide de certains outils communs, même s'ils se distinguent fortement par la nature des interactions envisagées, et des structures géométriques qu'ils engendrent.Voici un résumé succint des résultats obtenus : Pour le modèle de Sherrington-Kirkpatrick de verres de spins, une étude asymptotique des recouvrements multiples, qui généralisent de manière naturelle les recouvrements de deux configurations, couramment étudiés dans ce contexte. Un théorème central de la limite pour la fonction de partition d'un modèle de Sherrington-Kirkpatrick localisé en espace. Une étude fine de la fonction de partition ainsi qu'un résultat de surdiffusivité pour un modèle de polymère dirigé brownien en environnement aléatoire gaussien.This thesis basically study two kinds of dis...

For positive real numbers p and q satisfying 1/p + 1/q > 1, it is known that if f (u) and g(u) have finite mean variations of orders p and q respectively, then an integral t s f (u)dg(u) exists in the Riemann sense. The present paper extends this Stieltjes integration theory, discussed by L. C. Young, to the case where f (u) and g(u) are measurable processes. Moreover, path-by-path piecewise-linear functions are constructed via Riemann-Stieltjes sums for measurable processes, and convergence theorems on such functions are derived.

Financial decisions are made under the state of indeterminacy. Randomness and fuzziness are two basic forms of indeterminacy. Probability theory (Kolmogorov, 1933) models randomness and fuzzy set theory (Zadeh, 1965) deals with fuzziness. However, in some cases, randomness and fuzziness appear simultaneously in a mathematical system. In order to deal with mathematical systems that contain both fuzziness and randomness, chance theory has been developed. Portfolio insurance refers to investment strategies which guarantee that the portfolio value at maturity or at any time before maturity will not go below a stated lower bound (also known as the floor), usually fixed as a fraction of the initial principal investment (Cont and Tankov, 2007). Constant proportion portfolio insurance (CPPI) is a popular example of portfolio insurance techniques. Various research papers have examined CPPI strategies based on probability theory (for example, Neftci, 2008 and Cont and Tankov, 2007) and credib...

Background: A study on the dengue daily counting in São Paulo city in a fixed period of time is assumed considering a new regression model approch. Methods: Under a Bayesian approach, it is introduced a polynomial linear regression model in presence of some covariates which could affect the counts of dengue in São Paulo city considered in the logarithm scale, combined with existing stochastic volatility models usually assumed in financial data analysis. Markov Chain Monte Carlo methods are used to get the posterior summaries of interest. Results: The new model approach showed some advantages when compared to other existing times series models usually used to model epidemics data. Conclusion: The use of the polynomial regression model combined with existing volatility models under a Bayesian approach showed that it is possible to get very accurate fit for the counting dengue data in São Paulo city where it is possible to jointly model the means and volatilities (variances) of the epi...

We propose two new positive weak second-order approximations for the CIR equation dX t = (a − b X t) dt + σ √ X t dB t based on splitting, at each step, the equation into the deterministic part dX t = (a − b X t) dt, which is solved exactly, and the stochastic part dX t = σ √ X t dB t , which is approximated in distribution. The schemes are illustrated by encouraging simulation results.

We introduce a distributionally robust minimium mean square error estimation model with a Wasserstein ambiguity set to recover an unknown signal from a noisy observation. The proposed model can be viewed as a zero-sum game between a statistician choosing an estimator—that is, a measurable function of the observation—and a fictitious adversary choosing a prior—that is, a pair of signal and noise distributions ranging over independent Wasserstein balls—with the goal to minimize and maximize the expected squared estimation error, respectively. We show that, if the Wasserstein balls are centered at normal distributions, then the zero-sum game admits a Nash equilibrium, by which the players’ optimal strategies are given by an affine estimator and a normal prior, respectively. We further prove that this Nash equilibrium can be computed by solving a tractable convex program. Finally, we develop a Frank–Wolfe algorithm that can solve this convex program orders of magnitude faster than state...

We show the advantage of using Google search engine trends to forecast the volatility of the shortterm (weekly) exchange rate between the Mexican peso and United States dollar. We perform a comparison of models in the literature that have used Google Trends to examine explanatory variables. Some ofthe models are based on time series, whereas others are based on the similarity function, which captures the cognitive form of human reasoning. For example, an investor who needs to know the value that a variable will take in the future will take into account relevant, known, and available information, and weigh it to calculate the forecast. We conclude that taking into account the Google Trends variable helps explains partially the behaviour of volatility; and it is necessary to incorporate more aggregation levels. Moreover, to the best of our knowledge, literature on the subject of using Google Trends to explain relevant economic variables is relatively scarce.

The copyright of this thesis rests with the author and is made available under a Creative Commons Attribution Non-Commercial No Derivatives licence. Researchers are free to copy, distribute or transmit the thesis on the condition that they attribute it, that they do not use it for commercial purposes and that they do not alter, transform or build upon it.

We consider option pricing when dynamic portfolios are discretely rebalanced. The portfolio adjustments only occur after fixed relative changes in the stock price. The stock price follows a marked point process (MPP) and the market is incomplete. We first characterise the equivalent martingale measures. An explicit pricing formula based on the minimal martingale measure (MMM) is then provided together with the hedging strategy underlying portfolio adjustments. Two examples illustrate our pricing framework: a jump process driven by a latent geometric Brownian motion and a marked Poisson process. We establish the convergence to the Black-Scholes model when the triggering price increment shrinks to zero. For the empirical application, we use IBM, France Telecom (FT) and CAC 40 intraday transaction data, and compare option prices given by the marked Poisson model, the Black-Scholes model and observed option prices.

We present a new method of pricing plain vanilla call and put options when the underlying asset returns follow a jump-diffusion process. The method is based on stochastic dominance insofar as it does not need any assumption on the utility function of a representative investor apart from risk aversion. It develops discrete time multiperiod reservation write and reservation purchase bounds on option prices. The bounds are valid for any asset dynamics and are such that any risk averse investor improves her expected utility by introducing a short (long) option in her portfolio if the upper (lower) bound is violated by the observed market price. The bounds are evaluated recursively and the limiting forms of the bounds are found as time becomes continuous. It is found that the two bounds tend to the common limit equal to the Black-Scholes-Merton (BSM) price when there is no jump component, but to two different limits when the jump component is present.

In this paper, we prove sufficient conditions for controllability of second order semi-linear neutral impulsive differential inclusions on unbounded domain with infinite delay in Banach spaces using the theory of strongly continuous Cosine families. We shall rely on a fixed point theorem due to Ma for multi-valued maps. The controllability results in infinite dimensional space has been proved without compactness on the family of Cosine operators.

In this paper we consider the mean-variance hedging problem of a continuous state space financial model with the rebalancing strategies for the hedging portfolio taken at discrete times. An expression is derived for the optimal self-financing mean-variance hedging strategy problem, considering any given payoff in an incomplete market environment. To some extent, the paper extends the work of Cerný [1] to the case in which prices may assume any value within a continuous state space, a situation that more closely reflects real market conditions. An expression for the "fair hedging price" for a derivative with any given payoff is derived. Closed-form solutions for both the "fair hedging price" and the optimal control for the case of a European call option are obtained. Numerical results indicate that the proposed method is consistently better than the Black and Scholes approach, often adopted by practitioners.

Most existing hedging approaches are based on neutralizing risk exposures defined under a pre-specified model. This paper proposes a new, simple, and robust hedging approach based on the affinity of the derivative contracts. As a result, the strategy does not depend on assumptions on the underlying risk dynamics. Simulation analysis under commonly proposed security price dynamics shows that the hedging performance of our methodology based on a static position of three options compares favorably against the dynamic delta hedging strategy with daily rebalancing. A historical hedging exercise on S&P 500 index option further highlights the superior performance of our strategy.

We investigate existence and uniqueness of strong solutions of mean-field stochastic differential equations with irregular drift coefficients. Our direct construction of strong solutions is mainly based on a compactness criterion employing Malliavin Calculus together with some local time calculus. Furthermore, we establish regularity properties of the solutions such as Malliavin differentiablility as well as Sobolev differentiability and Hölder continuity in the initial condition. Using this properties we formulate an extension of the Bismut-Elworthy-Li formula to mean-field stochastic differential equations to get a probabilistic representation of the first order derivative of an expectation functional with respect to the initial condition.

En premier lieu, je tiensà remercier de tout coeur ma directrice de thèse, Nicole El Karoui, pour la qualité exceptionnelle de son encadrement. Son savoir, son dynamisme et sa rigueur ont fortement contribuéà la rédaction de cette thèse. C'était un grand honneur d'avoir découvert le monde de la rechercheà ses côtés : elle m'a donné le goût de la recherche en me transmettant sa passion. J'admire son savoir immense mais je pense que j'admire encore plus ses qualités humaines. Nicole a toujoursété disponible pour moi et a réponduà toutes mes questions avec gentillesse et enthousiasme. Je lui suis vraiment reconnaissant pour le soutien et la confiance qu'elle m'a accordé. Mes remerciements respectueuxà Jean Pierre Fouque et Christian Gourieroux de m'avoir fait l'honneur de rapporter ma thèse. Ils m'ont envoyé de nombreuses corrections et un ensemble de conseils précieux qui m'ont permis d'améliorer la rédaction. Je présente mes sincères remerciementsà Sylvie Méléard, Christophe Michel, Nizar Touzi et Alain Trognon de m'avoir fait l'honneur d'accepter de participerà mon jury de thèse. Je tiensà remercier particulièrement Christophe Michel qui m'a accueilli pendant trois années dans sonéquipe de recherche du Crédit Agricole et qui m'a suivi, aidé et conseillé tout au long de cette thèse. Je remercie aussi Anas, membre de l'équipe, d'avoir partagé de nombreuses connaissances et d'avoir accepté de rédiger des articles ensemble. Je remercie Florence pour l'aideà la relecture de la thèse et je remercie vivement Vincent pour ses nombreux conseils avisés et son aide précieuse. Enfin, je remercie toute l'équipe de recherche de CA-CIB pour son accueil chaleureux, et plus particulièrement Vincent, Sophie, Christophe M. et Christophe Z., Kenza, Florence, Thomas, Eric avec qui j'ai pu tisser des liens d'amitié. Mes remerciements d'adressentégalement aux membres du groupe de longévité et plus particulièrement Caroline Hillairet, Stéphane Loisel, Yahia Salhi, Pauline Barrieu, Isabelle Camillier et Trung Lap NGuyen. Ces réunions ontété très enrichissantes et nous avons pu travailler ensemble et partager des discussions sur de nombreux sujets passionnants. Je remercie vivement Chi Tran Viet et Sylvie Méléard pour m'avoir aidéà me plonger dans le domaine de la dynamique de population. Chi m'a aidéà de nombreuses reprises en répondantà mes questions et en me fournissant des pistes de réflexion très intéressantes. Je remercie aussi les membres du groupe de travail de biologie de l'école polytechnique. Un grand mercià Martino Grasseli et José Da Fonseca pour les nombreux conseils qu'ils m'ont donné sur les modèles de Wishart et pour l'intérêt qu'ils ont montré sur mes travaux de recherche. i Je pense aussi aux doctorants du CMAP, et en particulierà Gilles-Edouard, Emilie, Laurent, Isabelle, Mohammed et Trunglap ainsi qu'à l'ensemble des chercheurs du CMAP avec lesquels j'ai pu partager des moments privilégiés. Je pense en particulierà Peter, Emmanuel et Nizar qui ontété disponible et m'ont donné des conseils précieux. Merci au personnel administratif du CMAP, en particulierà Nassera, Anna, Sandra, Nathalie et Alexandra, pour leur gentillesse et leur aide dans leur bonne humeur. Je remercie aussi l'administrateur informatique Sylvain de son assistance ainsi qu'à Aldjia pour m'avoir aidé dans de nombreuses démarches. Enfin, je penseà l'ensemble des membres de l'école doctorale et en particulier Audrey et Fabrice qui m'ont aidé dans toutes les démarches. Je voudrais remercier tous mes amis proches qui ont contribuéà monépanouissement. Mercì a Dany, mon "colloc" adoré avec qui j'ai passé une année géniale, mes "petites soeurs" Cynthia et Karen et l'ensemble de mes amis proches

In this paper, a study of a stochastic volatility model for asset pricing is described. Originally presented by J. Da Fonseca, M. Grasselli and C. Tebaldi, the Wishart volatility model identifies the volatility of the asset as the trace of a Wishart process. Contrary to a classic multifactor Heston model, this model allows to add degrees of freedom with regard to the stochastic correlation. Thanks to its flexibility, this model enables a better fit of market data than the Heston model. Besides, the Wishart volatility model keeps a clear interpretation of its parameters and conserves an efficient tractability. Firstly, we recall the Wishart volatility model and we present a Monte Carlo simulation method in sight of the evaluation of complex options. Regarding stochastic volatility models, implied volatility surfaces of vanilla options have to be obtained for a short time. The aim of this article is to provide an accurate approximation method to deal with asymptotic smiles and to appl...

En premier lieu, je tiensà remercier de tout coeur ma directrice de thèse, Nicole El Karoui, pour la qualité exceptionnelle de son encadrement. Son savoir, son dynamisme et sa rigueur ont fortement contribuéà la rédaction de cette thèse. C'était un grand honneur d'avoir découvert le monde de la rechercheà ses côtés : elle m'a donné le goût de la recherche en me transmettant sa passion. J'admire son savoir immense mais je pense que j'admire encore plus ses qualités humaines. Nicole a toujoursété disponible pour moi et a réponduà toutes mes questions avec gentillesse et enthousiasme. Je lui suis vraiment reconnaissant pour le soutien et la confiance qu'elle m'a accordé. Mes remerciements respectueuxà Jean Pierre Fouque et Christian Gourieroux de m'avoir fait l'honneur de rapporter ma thèse. Ils m'ont envoyé de nombreuses corrections et un ensemble de conseils précieux qui m'ont permis d'améliorer la rédaction. Je présente mes sincères remerciementsà Sylvie Méléard, Christophe Michel, Nizar Touzi et Alain Trognon de m'avoir fait l'honneur d'accepter de participerà mon jury de thèse. Je tiensà remercier particulièrement Christophe Michel qui m'a accueilli pendant trois années dans sonéquipe de recherche du Crédit Agricole et qui m'a suivi, aidé et conseillé tout au long de cette thèse. Je remercie aussi Anas, membre de l'équipe, d'avoir partagé de nombreuses connaissances et d'avoir accepté de rédiger des articles ensemble. Je remercie Florence pour l'aideà la relecture de la thèse et je remercie vivement Vincent pour ses nombreux conseils avisés et son aide précieuse. Enfin, je remercie toute l'équipe de recherche de CA-CIB pour son accueil chaleureux, et plus particulièrement Vincent, Sophie, Christophe M. et Christophe Z., Kenza, Florence, Thomas, Eric avec qui j'ai pu tisser des liens d'amitié. Mes remerciements d'adressentégalement aux membres du groupe de longévité et plus particulièrement Caroline Hillairet, Stéphane Loisel, Yahia Salhi, Pauline Barrieu, Isabelle Camillier et Trung Lap NGuyen. Ces réunions ontété très enrichissantes et nous avons pu travailler ensemble et partager des discussions sur de nombreux sujets passionnants. Je remercie vivement Chi Tran Viet et Sylvie Méléard pour m'avoir aidéà me plonger dans le domaine de la dynamique de population. Chi m'a aidéà de nombreuses reprises en répondantà mes questions et en me fournissant des pistes de réflexion très intéressantes. Je remercie aussi les membres du groupe de travail de biologie de l'école polytechnique. Un grand mercià Martino Grasseli et José Da Fonseca pour les nombreux conseils qu'ils m'ont donné sur les modèles de Wishart et pour l'intérêt qu'ils ont montré sur mes travaux de recherche. i Je pense aussi aux doctorants du CMAP, et en particulierà Gilles-Edouard, Emilie, Laurent, Isabelle, Mohammed et Trunglap ainsi qu'à l'ensemble des chercheurs du CMAP avec lesquels j'ai pu partager des moments privilégiés. Je pense en particulierà Peter, Emmanuel et Nizar qui ontété disponible et m'ont donné des conseils précieux. Merci au personnel administratif du CMAP, en particulierà Nassera, Anna, Sandra, Nathalie et Alexandra, pour leur gentillesse et leur aide dans leur bonne humeur. Je remercie aussi l'administrateur informatique Sylvain de son assistance ainsi qu'à Aldjia pour m'avoir aidé dans de nombreuses démarches. Enfin, je penseà l'ensemble des membres de l'école doctorale et en particulier Audrey et Fabrice qui m'ont aidé dans toutes les démarches. Je voudrais remercier tous mes amis proches qui ont contribuéà monépanouissement. Mercì a Dany, mon "colloc" adoré avec qui j'ai passé une année géniale, mes "petites soeurs" Cynthia et Karen et l'ensemble de mes amis proches

Business activities require to obtain, organize and manage information from large amounts of data. In hedge funds, short selling trade and derivatives valuation, agents change their strategies to improve profits, and therefore to increase their possibilities to remain in the market, as a result of finding more accurate methods to process ever larger volume of information, considering that the information is not evenly distributed among markets participants. In this paper, a hybrid three stage model is formulated consisting of: a high frequency market model through a non-stationary Compound Poisson Process, a multilayer perceptron trained by backpropagation and, finally, estimators based on alpha-stable distributions, as an initial overview to develop a high frequency trading market operating system.

We study the rate of weak convergence of Markov chains to diffusion processes under suitable but quite general assumptions. We give an example in the financial framework, applying the convergence analysis to a multiple jumps tree approximation of the CIR process. Then, we combine the Markov chain approach with other numerical techniques in order to handle the different components in jump-diffusion coupled models. We study the speed of convergence of this hybrid approach and we provide an example in finance, applying our results to a tree-finite difference approximation in the Heston or Bates model.

Focus, in the past four decades, has been obtaining closed-form expressions for the no-arbitrage prices and hedges of modified versions of the Europeanoptions, allowing the dynamic of the underlying assets to have non-constant pa-rameters.In this paper, we obtain a closed-form expression for the price and hedge of an up-and-out European barrier option, assuming that the volatility in the dynamicof the risky asset is an arbitrary deterministic function of time. Setting a con-stant volatility, the formulas recover the Black and Scholes results, which suggestsminimum computational effort.We introduce a novel concept of relative standard deviation for measuring the ex-posure of the practitioner to risk (enforced by a strategy). The notion that is found in the literature is different and looses the correct physical interpreta-tion. The measure serves aiding the practitioner to adjust the number of rebalancesduring the option’s lifetime.

This paper analyzes the factors that contribute to the government obligations yield to maturity on the EU and US markets. Both, the bond characteristics and macroeconomic factors are taken into account, and the magnitude of each of the factor is provided which compose the average yield to maturity. Due to a severe financial crisis in the past years, economies are still recovering from the effects what makes investors to look for the stable investment instruments. Results of this study are a good fundamental for private investors that desire a stable return with a low-risk exposure. The factors and obtained coefficients included in the paper, can be used by investors, both private and institutional, to understand the magnitudes of the premiums that they can take on by varying the bond characteristics which account to the bond yield to maturity. Investors can decide at which premiums they are willing to focus, to obtain the desired expected returns.

The Bates model provides a parsimonious fit to implied volatility surfaces, and its usefulness in developed markets is well documented. However, there is a lack of research assessing its applicability to developing markets. Additionally, research surrounding its usefulness for hedging long term liabilities is limited, despite its frequent use for this purpose. This dissertation dissects the dynamics of the Bates model into the Heston and Merton models in order to separately examine the effects of stochastic volatility and jumps. Challenges surrounding application of this model are investigated through an evaluation of risk-neutral calibration and simulation methods. The model's ability to fit the implied volatility surfaces from the JSE Top 40 equity index is analysed. Lastly, an evaluation of the model's delta and vega hedging performance is presented by comparing it to the hedge performance of other commonly used models. Firstly, I would like to thank my internal supervisors, Obeid Mahomed and Professor David Taylor for their guidance over the course of this dissertation. Thank you to Neil Kennedy for the initial contributions to this report, as well as for the dissertation topic. I thank my family, especially my parents, Jill and Steve Gorven for their endless love and support. I would not have survived my university career without it. More thanks goes to the 2017 M.Phil class for making this tough degree an enjoyable and fulfilling experience. Lastly, thank you to BANKSETA for the funding they provided for this year of study.

Focus, in the past four decades, has been obtaining closed-form expressions for the no-arbitrage prices and hedges of modified versions of the Europeanoptions, allowing the dynamic of the underlying assets to have non-constant pa-rameters.In this paper, we obtain a closed-form expression for the price and hedge of an up-and-out European barrier option, assuming that the volatility in the dynamicof the risky asset is an arbitrary deterministic function of time. Setting a con-stant volatility, the formulas recover the Black and Scholes results, which suggestsminimum computational effort.We introduce a novel concept of relative standard deviation for measuring the ex-posure of the practitioner to risk (enforced by a strategy). The notion that is found in the literature is different and looses the correct physical interpreta-tion. The measure serves aiding the practitioner to adjust the number of rebalancesduring the option’s lifetime.

This research aims to analyze whether the 42 category-specific Equity Market Volatility (EMV) trackers explain the US industrial production index (IPI) including the impact of the Covid-19 crisis. IPI values are forecasted, considering three scenarios of economic recuperation after the end of the Covid-19 pandemic. To achieve this purpose, first an Artificial Neural Network is employed to determine if the EMV categorical tracker elements explain Industrial Production. Once, the incidence of the EMV trackers on IPI is evidenced, an ARIMA model is used to forecast the Industrial Production Index from June 2021 to December 2022. Motivation for this research and its originality is the strong economic and financial links of the Mexican economy with the U.S. Economy. Current economic trends in Mexico, particularly its economic recovery partly linked to the U.S economic recovery.

Business activities require to obtain, organize and manage information from large amounts of data. In hedge funds, short selling trade and derivatives valuation, agents change their strategies to improve profits, and therefore to increase their possibilities to remain in the market, as a result of finding more accurate methods to process ever larger volume of information, considering that the information is not evenly distributed among markets participants. In this paper, a hybrid three stage model is formulated consisting of: a high frequency market model through a non-stationary Compound Poisson Process, a multilayer perceptron trained by backpropagation and, finally, estimators based on alpha-stable distributions, as an initial overview to develop a high frequency trading market operating system.