Submitted by Cristiane Shirayama (cristiane.shirayama@fgv.br) on 2011-06-03T16:41:12Z No. of bitstreams: 1 66080100251.pdf: 1071566 bytes, checksum: 633248672cb6ac94f704bfeda06b29d3 (MD5) / Approved for entry into archive by Vera Lúcia Mourão(vera.mourao@fgv.br) on 2011-06-03T16:46:36Z (GMT) No. of bitstreams: 1 66080100251.pdf: 1071566 bytes, checksum: 633248672cb6ac94f704bfeda06b29d3 (MD5) / Approved for entry into archive by Vera Lúcia Mourão(vera.mourao@fgv.br) on 2011-06-03T17:00:17Z (GMT) No. of bitstreams: 1 66080100251.pdf: 1071566 bytes, checksum: 633248672cb6ac94f704bfeda06b29d3 (MD5) / Made available in DSpace on 2011-06-03T18:49:55Z (GMT). No. of bitstreams: 1 66080100251.pdf: 1071566 bytes, checksum: 633248672cb6ac94f704bfeda06b29d3 (MD5) Previous issue date: 2011-02-11 / Nos últimos anos, o mercado brasileiro de opções apresentou um forte crescimento, principalmente com o aparecimento da figura dos High Frequency Traders (HFT) em busca de oportunidades de arbitragem, de modo que a escolha adequada do modelo de estimação de volatilidade pode tornar-se um diferencial competitivo entre esses participantes. Este trabalho apresenta as vantagens da adoção do modelo de volatilidade estocástica de Heston (1993) na construção de superfície de volatilidade para o mercado brasileiro de opções de dólar, bem como a facilidade e o ganho computacional da utilização da técnica da Transformada Rápida de Fourier na resolução das equações diferenciais do modelo. Além disso, a partir da calibração dos parâmetros do modelo com os dados de mercado, consegue-se trazer a propriedade de não-arbitragem para a superfície de volatilidade. Os resultados, portanto, são positivos e motivam estudos futuros sobre o tema. / In recent years, the Brazilian option market has grown considerable, especially with the emergence of the High Frequency Traders (HFT) in search of arbitrage opportunities, so that the appropriate choice of a volatility estimation model should become a competitive differentiator among these participants. This paper presents the advantages of adopting the Heston stochastic volatility model on the construction of the volatility surface for the Brazilian US Dollar option market, as well as the easiness and the computational gain by applying the Fast Fourier Transform technique on the models differential equations resolution. Furthermore, from calibration of the model parameters to market data, it is possible to bring the no-arbitrage property to the volatility surface. The results, therefore, are positive and motivate further studies on the subject.

Cette thèse a pour objectif la compréhension de plusieurs aspects du caractère rugueux de la volatilité observé de manière universelle sur les actifs financiers. Ceci est fait en six étapes. Dans une première partie, on explique cette propriété à partir des comportements typiques des agents sur le marché. Plus précisément, on construit un modèle de prix microscopique basé sur les processus de Hawkes reproduisant les faits stylisés importants de la microstructure des marchés. En étudiant le comportement du prix à long terme, on montre l’émergence d’une version rugueuse du modèle de Heston (appelé modèle rough Heston) avec effet de levier. En utilisant ce lien original entre les processus de Hawkes et les modèles de Heston, on calcule dans la deuxième partie de cette thèse la fonction caractéristique du log-prix du modèle rough Heston. Cette fonction caractéristique est donnée en terme d’une solution d’une équation de Riccati dans le cas du modèle de Heston classique. On montre la validité d’une formule similaire dans le cas du modèle rough Heston, où l’équation de Riccati est remplacée par sa version fractionnaire. Cette formule nous permet de surmonter les difficultés techniques dues au caractère non markovien du modèle afin de valoriser des produits dérivés. Dans la troisième partie, on aborde la question de la gestion des risques des produits dérivés dans le modèle rough Heston. On présente des stratégies de couverture utilisant comme instruments l’actif sous-jacent et la courbe variance forward. Ceci est fait en spécifiant la structure markovienne infini-dimensionnelle du modèle. Étant capable de valoriser et couvrir les produits dérivés dans le modèle rough Heston, nous confrontons ce modèle à la réalité des marchés financiers dans la quatrième partie. Plus précisément, on montre qu’il reproduit le comportement de la volatilité implicite et historique. On montre également qu’il génère l’effet Zumbach qui est une asymétrie par inversion du temps observée empiriquement sur les données financières. On étudie dans la cinquième partie le comportement limite de la volatilité implicite à la monnaie à faible maturité dans le cadre d’un modèle à volatilité stochastique général (incluant le modèle rough Bergomi), en appliquant un développement de la densité du prix de l’actif. Alors que l’approximation basée sur les processus de Hawkes a permis de traiter plusieurs questions relatives au modèle rough Heston, nous examinons dans la sixième partie une approximation markovienne s’appliquant sur une classe plus générale de modèles à volatilité rugueuse. En utilisant cette approximation dans le cas particulier du modèle rough Heston, on obtient une méthode numérique pour résoudre les équations de Riccati fractionnaires. Enfin, nous terminons cette thèse en étudiant un problème non lié à la littérature sur la volatilité rugueuse. Nous considérons le cas d’une plateforme cherchant le meilleur système de make-take fees pour attirer de la liquidité. En utilisant le cadre principal-agent, on décrit le meilleur contrat à proposer au market maker ainsi que les cotations optimales affichées par ce dernier. Nous montrons également que cette politique conduit à une meilleure liquidité et à une baisse des coûts de transaction pour les investisseurs. / The aim of this thesis is to study various aspects of the rough behavior of the volatility observed universally on financial assets. This is done in six steps. In the first part, we investigate how rough volatility can naturally emerge from typical behav- iors of market participants. To do so, we build a microscopic price model based on Hawkes processes in which we encode the main features of the market microstructure. By studying the asymptotic behavior of the price on the long run, we obtain a rough version of the Heston model exhibiting rough volatility and leverage effect. Using this original link between Hawkes processes and the Heston framework, we compute in the second part of the thesis the characteristic function of the log-price in the rough Heston model. In the classical Heston model, the characteristic function is expressed in terms of a solution of a Riccati equation. We show that rough Heston models enjoy a similar formula, the Riccati equation being replaced by its fractional version. This formula enables us to overcome the non-Markovian nature of the model in order to deal with derivatives pricing. In the third part, we tackle the issue of managing derivatives risks under the rough Heston model. We establish explicit hedging strategies using as instruments the underlying asset and the forward variance curve. This is done by specifying the infinite-dimensional Markovian structure of the rough Heston model. Being able to price and hedge derivatives in the rough Heston model, we challenge the model to practice in the fourth part. More precisely, we show the excellent fit of the model to historical and implied volatilities. We also show that the model reproduces the Zumbach’s effect, that is a time reversal asymmetry which is observed empirically on financial data. While the Hawkes approximation enabled us to solve the pricing and hedging issues under the rough Heston model, this approach cannot be extended to an arbitrary rough volatility model. We study in the fifth part the behavior of the at-the-money implied volatility for small maturity under general stochastic volatility models. In the same spirit as the Hawkes approximation, we look in the sixth part of this thesis for a tractable Markovian approximation that holds for a general class of rough volatility models. By applying this approximation on the specific case of the rough Heston model, we derive a numerical scheme for solving fractional Riccati equations. Finally, we end this thesis by studying a problem unrelated to rough volatility. We consider an exchange looking for the best make-take fees system to attract liquidity in its platform. Using a principal-agent framework, we describe the best contract that the exchange should propose to the market maker and provide the optimal quotes displayed by the latter. We also argue that this policy leads to higher quality of liquidity and lower trading costs for investors.

Black and Scholes have proposed a model for pricing European options where the underlying asset follows a so-called geometric Brownian motion which assumes constant volatility. The proposed Black-Scholes model has an exact solution. However, it has been shown that such an assumption of constant volatility is not realistic, and numerous extensions have been developed. In addition, models usually do not have a closed-form solution which makes pricing a challenging task. The thesis focuses on pricing Bermudan options under two stochastic volatility Heston-type models using an Almost-Exact scheme for simulation. Namely, we focus on deriving the Almost-Exact scheme for Heston and Double Heston model and numerically study the behaviour of the scheme. We show that the AES works well when the number of simulated steps is equal to the number of exercise dates which makes it efficient.

This investigation undertakes to re-centre the figure of the film star and their film appearances in the field of star study. To this end it uses Charlton Heston as its focus in a re-appraisal of existing methods of accounting for the star phenomenon in cinema. It also concomitantly re-assesses existing accounts of the significance of Charlton Heston as a film star. This thesis posits a robust method for identifying the specificities of the star’s contribution to a film’s meanings and effects across the body of their work by drawing on Andrew Britton’s understanding of the ‘star enactment’. Present approaches through which to engage with the details of a star’s performance are considered in detail and the weaknesses of those that seek to impose external schemas onto such discussions are highlighted. The difficulties with approaches that attempt to account for the star as a signifying phenomenon through the concepts of acting and performance are also considered. Existing methods which may allow for a fruitful investigation into the significance of the star enactment, such as the commutation test, are re- formulated in this study and their benefits are demonstrated through their application to key Heston star enactments. These new understandings are also made possible through the application of an ‘ekphrastic’ method of rendering film moments. Previous readings of Heston’s star figure are also re- appraised, and their conclusions questioned, through closer reference to the evidence of details from films. The fruitfulness of this method for analysing and commenting on film is thus demonstrated and Heston’s relationship to genre and its effect on performance style is also considered in order to be able to confidently assert the specific features of the Heston aesthetic.

MASTER THESIS ABSTRACT TITLE: Trading Strategy in Incomplete Market AUTHOR: Tomáš Bunčák DEPARTMENT: Department of Probability and Mathematical Statistics, Charles University in Prague SUPERVISOR: Andrea Karlová We focus on the problem of finding optimal trading strategies (in a meaning corresponding to hedging of a contingent claim) in the realm of incomplete markets mainly. Although various ways of hedging and pricing of contingent claims are outlined, main subject of our study is the so-called mean-variance hedging (MVH). Sundry techniques used to treat this problem can be categorized into two approaches, namely a projection approach (PA) and a stochastic control approach (SCA). We review the methodologies used within PA in diversely general market models. In our research concerning SCA, we examine the possibility of using the methods of optimal stochastic control in MVH, and we study the problem of our interest in several settings of market models; involving cases of pure diffusion models and a jump- diffusion case. In order to reach an exemplary comparison, we provide solutions of the MVH problem in the setting of the Heston model via techniques of both of the approaches. Some parts of the thesis are accompanied with numerical illustrations.

<p>In the Black-Scholes model, the volatility considered being deterministic and it causes some</p><p>inefficiencies and trends in pricing options. It has been proposed by many authors that the</p><p>volatility should be modelled by a stochastic process. Heston Model is one solution to this</p><p>problem. To simulate the Heston Model we should be able to overcome the correlation</p><p>between asset price and the stochastic volatility. This paper considers a solution to this issue.</p><p>A review of the Heston Model presented in this paper and after modelling some investigations</p><p>are done on the applet.</p><p>Also the application of this model on some type of options has programmed by MATLAB</p><p>Graphical User Interface (GUI).</p>

In the Black-Scholes model, the volatility considered being deterministic and it causes some inefficiencies and trends in pricing options. It has been proposed by many authors that the volatility should be modelled by a stochastic process. Heston Model is one solution to this problem. To simulate the Heston Model we should be able to overcome the correlation between asset price and the stochastic volatility. This paper considers a solution to this issue. A review of the Heston Model presented in this paper and after modelling some investigations are done on the applet. Also the application of this model on some type of options has programmed by MATLAB Graphical User Interface (GUI).

In this paper, we investigate the problem of estimating the autocorrelation of squared returns modeled by diffusion processes with data observed at non-equi-spaced discrete times. Throughout, we will suppose that the stock price processes evolve in continuous time as the Heston-type stochastic volatility processes and the transactions arrive randomly according to a Poisson process. In order to estimate the autocorrelation at a fixed delay, the original non-equispaced data will be synchronized. When imputing missing data, we adopt the previous-tick interpolation scheme. Asymptotic property of the sample autocorrelation of squared returns based on the previous-tick synchronized data will be investigated. Simulation studies are performed and applications to real examples are illustrated.

Submitted by Fernando Teixeira (fernote7@gmail.com) on 2017-12-08T15:48:21Z No. of bitstreams: 1 Download File (1).pdf: 1437428 bytes, checksum: d6dfbfe41919a0cdd657900b6784f310 (MD5) / Approved for entry into archive by Janete de Oliveira Feitosa (janete.feitosa@fgv.br) on 2017-12-08T16:04:57Z (GMT) No. of bitstreams: 1 Download File (1).pdf: 1437428 bytes, checksum: d6dfbfe41919a0cdd657900b6784f310 (MD5) / Made available in DSpace on 2017-12-22T17:16:31Z (GMT). No. of bitstreams: 1 Download File (1).pdf: 1437428 bytes, checksum: d6dfbfe41919a0cdd657900b6784f310 (MD5) Previous issue date: 2017-09-29 / In this thesis we revisit numerical methods for the simulation of the Heston model’sEuropean call. Specifically, we study the Euler, the Kahl-Jackel an two versions of theexact algorithm schemes. To perform this task, firstly we present a literature reviewwhich brings stochastic calculus, the Black-Scholes (BS) model and its limitations,the stochastic volatility methods and why they resolve the issues of the BS model,and the peculiarities of the numerical methods. We provide recommendations whenwe acknowledge that the reader might need more specifics and might need to divedeeper into a given topic. We introduce the methods aforementioned providing all ourimplementations in R language within a package.

The thesis focuses on methods of option prices calculations using two different pricing models which are Heston and Black-Scholes models. The first part describes theory of these two models and conlcudes with a comparison of the risk-neutral measures of these two models. In the second part, the relations between input parameters and the option price generated by these models are clarified. This part ends up with an analysis of the market data and it answers the question which model predicts better.