Thesis (Ph. D.)--Johns Hopkins University, 2005. / Vita. Includes bibliographical references (leaves 198-210).

Neste trabalho, apresentamos um método de simulação Monte Carlo para o cálculo do hedging dinâmico de opções do tipo europeia em mercados multidimensionais do tipo Browniano e livres de arbitragem. Baseado em aproximações martingales de variação limitada para as decomposições de Galtchouk-Kunita-Watanabe, propomos uma metodologia factível e construtiva que nos permite calcular estratégias de hedging puras com respeito a qualquer opção quadrado integrável em mercados completos e incompletos. Uma vantagem da abordagem apresentada aqui é a flexibilidade de aplicação do método para os critérios quadráticos de minimização do risco local e de variância média de forma geral, sem a necessidade de se considerar hipóteses de suavidade para a função payoff. Em particular, a metodologia pode ser aplicada para calcular estratégias de hedging quadráticas multidimensionais para opções que dependem de toda a trajetória dos ativos subjacentes em modelos de volatilidade estocástica e com funções payoff descontínuas. Ilustramos nossa metodologia, fornecendo exemplos numéricos dos cálculos das estratégias de hedging para opções vanilla e opções exóticas que dependem de toda a trajetória dos ativos subjacentes escritas sobre modelos de volatilidade local e modelos de volatilidade estocástica. Ressaltamos que as simulações são baseadas em aproximações para os processos de preços descontados e, para estas aproximações, utilizamos o método numérico de Euler-Maruyama aplicado em uma discretização aleatória simples. Além disso, fornecemos alguns resultados teóricos acerca da convergência desta aproximação para modelos simples em que podemos considerar a condição de Lipschitz e para o modelo de volatilidade estocástica de Heston. / In this work, we present a Monte Carlo simulation method to compute de dynamic hedging of european-type contingent claims in a multidimensional Brownian-type and arbitrage-free market. Based on bounded variation martingale approximations for the Galtchouk-Kunita- Watanabe decomposition, we propose a feasible and constructive methodology which allows us to compute pure hedging strategies with respect to any square-integrable contingent claim in complete and incomplete markets. An advantage of our approach is the exibility of quadratic hedging in full generality without a priori smoothness assumptions on the payoff function. In particular, the methodology can be applied to compute multidimensional quadratic hedgingtype strategies for fully path-dependent options with stochastic volatility and discontinuous payoffs. We illustrate our methodology, providing some numerical examples of the hedging strategies to vanilla and exotic contingent claims written on local volatility and stochastic volatility models. The simulations are based in approximations to the discounted price processes and, for these approximations, we use an Euler-Maruyama-type method applied to a simple random discretization. We also provide some theoretical results about the convergence of this approximation in simple models where the Lipschitz condition is satisfied and the Heston\'s stochastic volatility model.

Neste trabalho, apresentamos um método de simulação Monte Carlo para o cálculo do hedging dinâmico de opções do tipo europeia em mercados multidimensionais do tipo Browniano e livres de arbitragem. Baseado em aproximações martingales de variação limitada para as decomposições de Galtchouk-Kunita-Watanabe, propomos uma metodologia factível e construtiva que nos permite calcular estratégias de hedging puras com respeito a qualquer opção quadrado integrável em mercados completos e incompletos. Uma vantagem da abordagem apresentada aqui é a flexibilidade de aplicação do método para os critérios quadráticos de minimização do risco local e de variância média de forma geral, sem a necessidade de se considerar hipóteses de suavidade para a função payoff. Em particular, a metodologia pode ser aplicada para calcular estratégias de hedging quadráticas multidimensionais para opções que dependem de toda a trajetória dos ativos subjacentes em modelos de volatilidade estocástica e com funções payoff descontínuas. Ilustramos nossa metodologia, fornecendo exemplos numéricos dos cálculos das estratégias de hedging para opções vanilla e opções exóticas que dependem de toda a trajetória dos ativos subjacentes escritas sobre modelos de volatilidade local e modelos de volatilidade estocástica. Ressaltamos que as simulações são baseadas em aproximações para os processos de preços descontados e, para estas aproximações, utilizamos o método numérico de Euler-Maruyama aplicado em uma discretização aleatória simples. Além disso, fornecemos alguns resultados teóricos acerca da convergência desta aproximação para modelos simples em que podemos considerar a condição de Lipschitz e para o modelo de volatilidade estocástica de Heston. / In this work, we present a Monte Carlo simulation method to compute de dynamic hedging of european-type contingent claims in a multidimensional Brownian-type and arbitrage-free market. Based on bounded variation martingale approximations for the Galtchouk-Kunita- Watanabe decomposition, we propose a feasible and constructive methodology which allows us to compute pure hedging strategies with respect to any square-integrable contingent claim in complete and incomplete markets. An advantage of our approach is the exibility of quadratic hedging in full generality without a priori smoothness assumptions on the payoff function. In particular, the methodology can be applied to compute multidimensional quadratic hedgingtype strategies for fully path-dependent options with stochastic volatility and discontinuous payoffs. We illustrate our methodology, providing some numerical examples of the hedging strategies to vanilla and exotic contingent claims written on local volatility and stochastic volatility models. The simulations are based in approximations to the discounted price processes and, for these approximations, we use an Euler-Maruyama-type method applied to a simple random discretization. We also provide some theoretical results about the convergence of this approximation in simple models where the Lipschitz condition is satisfied and the Heston\'s stochastic volatility model.

This thesis centers on groups of landscape paintings on sliding doors and wall panels for temples in and around Kyoto by the painter Maruyama Ōkyo (1733-1795), dating to the latter half of the eighteenth century. I discuss Snow Landscape of the Chiba City Museum of Art, presumed to have been painted for the temple Enman’in in Ōtsu (Shiga Prefecture), and the former sliding door and wall paintings of Kiun’in, a subtemple of Nanzenji in Kyoto, now in the collection of the Tokyo National Museum. The analysis is embedded into considerations of underlying genre principles in Ōkyo’s art, the reflection on the relevance of “truthfulness to nature” (shasei) and considerations of how his works relate to established painting conventions in early modern Japan. I attempt to frame Ōkyo’s landscapes as an expression of the painter’s Chinese-inspired outlook on painting. Chapter One centers on Snow Landscape. I use stylistic comparison to argue that the paintings do not match other ink landscapes by Ōkyo of the so-called Enman’in period, but resemble closely another set of sliding doors paintings of similar subject matter at Shōkokuji, dated 1790. Snow Landscape can be understood as part of a small group of Ōkyo works that are thematically and formally related, and that all share obscure provenance and previously unaddressed questions of authorship. This includes sliding door paintings of the temple Daijōji (Hyōgo Prefecture), of Nishi Honganji and of the former Hara collection of Toyooka, all of them with their current whereabouts unknown. In Chapter Two, I provide a detailed reconstruction of the original temple spaces based on the features of the extant paintings, then proceed to disentangle the modalities of Ōkyo’s workshop production as the context from which the Kiun’in paintings likely originated. Comparison of large-scale landscape paintings reveals that much of Ōkyo’s approach relied on reuse of complete compositions, or at least, individual motifs. I argue that the Kiun’in paintings were possibly painted by disciples. Chapter Three provides glimpses on primary source material written during Ōkyo’s lifetime by his most important patrons: Banshi (1761-1773) by Prince Abbot Yūjō, the diary Onjiki nikki (1787) by Imperial Prince Shinnin and the records of the temple Myōhōin, Myōhōin hinamiki. I argue that little in Banshi allows to conceive of Ōkyo’s art as “modern;” rather, the documents character is shaped by Yūjō’s interest in technical matters of studio painting. Yūjō and Shinnin are connected through familial ties at the court; in addition, attendance data from the Myōhōin hinamiki foreshadows the later rift into a Maruyama school and a Shijō school after Ōkyo’s death. Chapter Four provides a concluding discussion of the significance and context of Ōkyo’s landscape paintings in Buddhist temples. I argue that Ōkyo’s multi-room ensembles for temple interiors are based on artistic convention and spatial hierarchies that are comparable to approaches of the Kano school, and suggest that response to nature, such as allusion to topographical surroundings of a building, usually played a subordinate role. Ōkyo’s art depended on the appreciation of ancient Chinese culture, and closely related to the intellectual outlook of the court of Emperor Kōkaku.

A standard problem in the field of computational finance is that of pricing derivative securities. This is often accomplished by estimating an expected value of a functional of a stochastic process, defined by a stochastic differential equation (SDE). In such a setting the random sampling algorithm Monte Carlo (MC) is useful, where paths of the process are sampled. However, MC in its standard form (SMC) is inherently slow. Additionally, if the analytical solution to the underlying SDE is not available, a numerical approximation of the process is necessary, adding another layer of computational complexity to the SMC algorithm. Thus, the computational cost of achieving a certain level of accuracy of the estimation using SMC may be relatively high. In this thesis we introduce and review the theory of the SMC method, with and without the need of numerical approximation for path simulation. Two numerical methods for path approximation are introduced: the Euler–Maruyama method and Milstein's method. Moreover, we also introduce and review the theory of a relatively new (2008) MC method – the multilevel Monte Carlo (MLMC) method – which is only applicable when paths are approximated. This method boldly claims that it can – under certain conditions – eradicate the additional complexity stemming from the approximation of paths. With this in mind, we wish to see whether this claim holds when pricing a European call option, where the underlying stock process is modelled by geometric Brownian motion. We also want to compare the performance of MLMC in this scenario to that of SMC, with and without path approximation. Two numerical experiments are performed. The first to determine the optimal implementation of MLMC, a static or adaptive approach. The second to illustrate the difference in performance of adaptive MLMC and SMC – depending on the used numerical method and whether the analytical solution is available. The results show that SMC is inferior to adaptive MLMC if numerical approximation of paths is needed, and that adaptive MLMC seems to meet the complexity of SMC with an analytical solution. However, while the complexity of adaptive MLMC is impressive, it cannot quite compensate for the additional cost of approximating paths, ending up roughly ten times slower than SMC with an analytical solution.

Stochastic delay differential equations (SDDEs) are systems of differential equations with a time lag in a noisy or random environment. There are many nonlinear SDDEs where a linear growth condition is not satisfied, for example, the stochastic delay Lotka-Volterra model of food chain discussed by Xuerong Mao and Martina John Rassias in 2005. Much research has been done using discrete delay where the dynamics of a process at time t depend on the state of the process in the past after a single fixed time lag \tau. We are researching processes with continuously distributed delay which depend on weighted averages of past states over the entire time lag interval [t-\tau,t].By using martingale concepts, we prove sufficient conditions for the existence of a unique solution, ultimate boundedness, and non-extinction of one-dimensional nonlinear SDDE with continuously distributed delay. We give generalized Khasminskii-type conditions which along with local Lipschitz conditions are sufficient to guarantee the existence of a unique global solution of certain n-dimensional nonlinear SDDEs with continuously distributed delay. Further, we give conditions under which Euler-Maruyama numerical approximations of such nonlinear SDDEs converge in probability to their exact solutions.We give some examples of one-dimensional and 2-dimensional stochastic differential equations with continuously distributed delay which satisfy the sufficient conditions of our theorems. Moreover, we simulate their solutions and analyze the error of approximation using MATLAB to implement the Euler-Maruyama algorithm.

Made available in DSpace on 2016-06-02T20:06:10Z (GMT). No. of bitstreams: 1 6520.pdf: 1838901 bytes, checksum: 35b2a71ea573764ae46492a67c0ef3d6 (MD5) Previous issue date: 2015-02-26 / Universidade Federal de Sao Carlos / In this master's thesis we work with Cox-Ingersoll-Ross (CIR) process. This process was originally proposed by John C. Cox, Jonathan E. Ingersoll Jr. and Stephen A. Ross in 1985. Nowadays, this process is widely used in financial modeling, e.g. as a model for short-time interest rates or as volatility process in the Heston model. The stochastic diferential equation (SDE) which defines this model does not have closed form solution, so we need to approximate the process by some numerical method. In the literature, several numerical approximations has been proposed based in interval discretization. We approximate the CIR process by Euler-Maruyama-type method based in random discretization proposed by Leão e Ohashi (2013) under Feller condition. In this context, we obtain an exponential convergence order for this approximation and we use Monte Carlo techniques to compare the numerical results with theoretical values. / Nesta dissertação de mestrado nós trabalhamos com o processo de Cox-Ingersoll- Ross, que foi originalmente proposto por John C. Cox, Jonathan E. Ingersoll Jr. e Stephen A. Ross em 1985. Este processo é amplamente utilizado em modelagem financeira, por exemplo, para descrever a evolução de taxas de juros ou como o processo de volatilidade no modelo de Heston. A equação diferencial estocástica que define este processo não possui solução fechada, logo faz-se necessária a aproximação do processo via algum método numérico. Na literatura diversos trabalhos propõem aproximações baseadas em esquemas de discretização intervalar. Nós aproximamos o processo de Cox-Ingersoll-Ross através de um método numérico do tipo Euler- Maruyama baseado na discretização aleatória proposta por Leão e Ohashi (2013) sob a condição de Feller. Neste contexto, mostramos que esta aproximação possui uma ordem de convergência exponencial e utilizamos técnicas de simulação Monte Carlo para comparar resultados numéricos com valores teóricos.

[pt] Resfriamento é o passo inicial necessário a qualquer experimento optomecânico que tenha como objetivo desbloquear o potencial das pinças ópticas, tanto para a melhoria da sensiblidade a forças em aplicações de sensoreamento quanto para estudos de física quântica fundamental na microescala. O propósito do trabalho descrito nesta dissertação foi o de melhorar a montagem de uma pinça óptica para resfriamento por retroalimentação do movimento translacional de nanopartículas levitadas. Nós implementamos a coleta de luz retroespalhada pela partícula para melhorar a eficiência de detecção do movimento ao longo do eixo óptico. Usando um ambiente de simulação numérica em Python, nós também exploramos o potencial de sistemas optomecânicos como sensores para estados de luz com intensidades muito baixas. / [en] Cooling is the necessary first step for any optomechanical experiment aiming to unleash the full potential of optical tweezers, both in the context of improving force sensitivity in sensor applications and of studying fundamental quantum physics at the microscale. The purpose of the work described in this dissertation was to improve an optical tweezer setup for electrical feedback cooling of the translational motion of levitated nanoparticles. We implement collection of backscattered light from the particle for improved detection efficiency of motion along the optical axis. Using a numerical simulation environment in Python, we also explore the potential of optomechanical systems as sensors for light states with very low intensities.

We suppose that the price of a firm follows a nonlinear stochastic delay differential equation. We also assume that any claim whose value depends on firm value and time follows a nonlinear stochastic delay differential equation. Using self-financed strategy and replication we are able to derive a random partial differential equation (RPDE) satisfied by any corporate claim whose value is a function of firm value and time. Under specific final and boundary conditions, we solve the RPDE for the debt value and loan guarantees within a single period and homogeneous class of debt. We then analyze the risk structure of a levered firm. We also evaluate loan guarantees in the presence of more than one debt. Furthermore, we perform numerical simulations for specific companies and compare our results with existing models.

Nesta dissertação de mestrado nós trabalhamos com o processo de Cox-Ingersoll- Ross, que foi originalmente proposto por John C. Cox, Jonathan E. Ingersoll Jr. e Stephen A. Ross em 1985. Este processo é amplamente utilizado em modelagem financeira, por exemplo, para descrever a evolução de taxas de juros ou como o processo de volatilidade no modelo de Heston. A equação diferencial estocástica que define este processo não possui solução fechada, logo faz-se necessária a aproximação do processo via algum método numérico. Na literatura diversos trabalhos propõem aproximações baseadas em esquemas de discretização intervalar. Nós aproximamos o processo de Cox-Ingersoll-Ross através de um método numérico do tipo Euler- Maruyama baseado na discretização aleatória proposta por Leão e Ohashi (2013) sob a condição de Feller. Neste contexto, mostramos que esta aproximação possui uma ordem de convergência exponencial e utilizamos técnicas de simulação Monte Carlo para comparar resultados numéricos com valores teóricos. / In this master\'s thesis we work with Cox-Ingersoll-Ross (CIR) process. This process was originally proposed by John C. Cox, Jonathan E. Ingersoll Jr. and Stephen A. Ross in 1985. Nowadays, this process is widely used in financial modeling, e.g. as a model for short-time interest rates or as volatility process in the Heston model. The stochastic differential equation (SDE) which defines this model does not have closed form solution, so we need to approximate the process by some numerical method. In the literature, several numerical approximations has been proposed based in interval discretization. We approximate the CIR process by Euler-Maruyama-type method based in random discretization proposed by Leão e Ohashi (2013) under Feller condition. In this context, we obtain an exponential convergence order for this approximation and we use Monte Carlo techniques to compare the numerical results with theoretical values.