In this project, we study an asymptotic expansion method for solving stochastic volatility European option pricing problems. We explain the backgrounds and details associated with the method. Specifically, we present in full detail the arguments behind the derivation of the pricing PDEs and detailed calculation in deriving asymptotic option pricing formulas using our own model specifications. Finally, we discuss potential difficulties and problems in the implementation of the methods.

A thesis submitted for the degree of Doctor of Philosophy 2013 / This thesis presents a range of related pricing kernel models that are driven by incomplete information about a series of future unknowns. These unknowns may, for instance, represent fundamental macroeconomic, political or social random variables that are revealed at future times. They may also represent latent or hidden factors that are revealed asymptotically. We adopt the information-based approach of Brody, Hughston and Macrina (BHM) to model the information processes associated with the random variables. The market filtration is generated collectively by these information processes. By directly modelling the pricing kernel, we generate information-sensitive arbitrage-free models for the term structure of interest rates, the excess rate of return required by investors, and security prices. The pricing kernel is modelled by a supermartingale to ensure that nominal interest rates remain non-negative. To begin with, we primarily investigate finite-time pricing kernel models that are sensitive to Brownian bridge information. The BHM framework for the pricing of credit-risky instruments is extended to a stochastic interest rate setting. In addition, we construct recovery models, which take into consideration information about, for example, the state of the economy at the time of default. We examine various explicit examples of analytically tractable information-driven pricing kernel models. We develop a model that shares many of the features of the rational lognormal model, and investigate examples of heat kernel models. It is shown that these models may result in discount bonds and interest rates being bounded by deterministic functions. In certain situations, incoming information about random variables may exhibit jumps. To this end, we construct a more general class of nite-time pricing kernel models that are driven by Levy random bridges. Finally, we model the aggregate impact of uncertainties on a nancial market by randomised mixtures of Levy and Markov processes respectively. It is assumed that market participants have incomplete information about the underlying random mixture. We apply results from non-linear ltering theory and construct Flesaker-Hughston models and in nite-time heat kernel models based on these randomised mixtures.

Thesis advisor: Nancy Rallis / The theory of Brownian motion is an integral part of statistics and probability, and it also has some of the most diverse applications found in any topic in mathematics. With extensions into fields as vast and different as economics, physics, and management science, Brownian motion has become one of the most studied mathematical phenomena of the late twentieth and early twenty-first centuries. Today, Brownian motion is mostly understood as a type of mathematical process called a stochastic process. The word "stochastic" actually stems from the Greek word for "I guess," implying that stochastic processes tend to produce uncertain results, and Brownian motion is no exception to this, though with the right models, probabilities can be assigned to certain outcomes and we can begin to understand these complicated processes. This work reaches to attain this goal with regard to Brownian motion, and in addition it explores several applications found in the aforementioned fields and beyond. / Thesis (BA) — Boston College, 2007. / Submitted to: Boston College. College of Arts and Sciences. / Discipline: Mathematics. / Discipline: College Honors Program.

Este trabalho aplica metodologia de Wang (2003a, 2003b) para definir uma nova medida do produto de bancos brasileiros. Acredita-se que tal medida seja superior às comumente utilizadas na literatura por se tratar de uma variável de fluxo, incorporar os depósitos bancários de maneira teoricamente embasada e levar em consideração a exposição ao risco. Esta nova variável de produção é utilizada na estimação de fronteiras estocásticas de produção e custo para o setor bancário brasileiro, visando a mensurar sua eficiência técnica e econômica, assim como a magnitude de suas economias de escala. As fronteiras estimadas apresentam especificação dada pela forma funcional flexível de Fourier e incorporam variáveis determinantes da eficiência. Os resultados apontam para uma acentuada ineficiência do setor bancário, assim como para a ocorrência de retornos crescentes de escala. / This work applies the Wang (2003a, 2003b) methodology to define Brazilian banks\' output. It is believed that this new output measure is superior to the ones commonly used by the literature, since it treats output as a flow variable, provides a theoretical basis for defining the role of deposits and takes account of risk exposure in defining output. This new measure is applied to the estimation of stochastic production and cost frontiers for the Brazilian banking sector, aiming at measuring its technical and economic efficiency, and also the size of its scale economies. The frontiers follow a Fourier flexible functional form especification and incorporate efficiency determinants. The results point to the existence of high inneficiency in the banking industry, and also to the ocurrence of increasing returns to scale.

This thesis is composed of three main parts. Each chapter is concerned with characterizing some properties of a random ensemble or stochastic process. The properties of interest and the methods for investigating them di er between chapters. We begin by establishing some asymptotic results regarding zeros of random harmonic mappings, a topic of much interest to mathematicians and astrophysicists alike. We introduce a new model of harmonic polynomials based on the so-called "Weyl ensemble" of random analytic polynomials. Building on the work of Li and Wei [28] we obtain precise asymptotics for the average number of zeros of this model. The primary tools used in this section are the famous Kac-Rice formula as well as classical methods in the asymptotic analysis of integrals such as the Laplace method. Continuing, we characterize several topological properties of this model of harmonic polynomials. In chapter 3 we obtain experimental results concerning the number of connected components of the orientation-reversing region as well as the geometry of the distribution of zeros. The tools used in this section are primarily Monte Carlo estimation and topological data analysis (persistent homology). Simulations in this section are performed within MATLAB with the help of a computational homology software known as Perseus. While the results in this chapter are empirical rather than formal proofs, they lead to several enticing conjectures and open problems. Finally, in chapter 4 we address an industry problem in applied mathematics and machine learning. The analysis in this chapter implements similar techniques to those used in chapter 3. We analyze data obtained by observing CAN tra c. CAN (or Control Area Network) is a network for allowing micro-controllers inside of vehicles to communicate with each other. We propose and demonstrate the e ectiveness of an algorithm for detecting malicious tra c using an approach that discovers and exploits the natural geometry of the CAN surface and its relationship to random walk Markov chains. / Includes bibliography. / Dissertation (Ph.D.)--Florida Atlantic University, 2018. / FAU Electronic Theses and Dissertations Collection

by Shing-chiang Wong. / Bibliography: leaf 73 / Thesis (M.Ph.)--Chinese University of Hong Kong, 1986

Tang Fung Chu. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1999. / Includes bibliographical references (leaves 79-81). / Abstracts in English and Chinese. / Chapter Chapter 1. --- Introduction --- p.1 / Chapter Chapter 2. --- The Multivariate Model and Bayesian Analysis of Stochastic Prior Information --- p.4 / Chapter 2.1 --- The Model --- p.4 / Chapter 2.2 --- Identification of the Model --- p.5 / Chapter 2.3 --- Bayesian Analysis of Stochastic Prior Information --- p.8 / Chapter 2.4 --- Computational Procedure --- p.10 / Chapter 2.4.1 --- Optimization Procedures --- p.11 / Chapter 2.4.2 --- Analytical Expressions --- p.12 / Chapter Chapter 3. --- Example and Simulation Study --- p.18 / Chapter 3.1 --- Example --- p.18 / Chapter 3.2 --- Simulation Study --- p.19 / Chapter 3.2.1 --- Designs --- p.20 / Chapter 3.2.2 --- Results --- p.23 / Chapter Chapter 4. --- Conclusion --- p.26 / Tables --- p.29 / References --- p.79

The theory of large deviations can help to shed light on systems in non-equilibrium statistical mechanics and, more generically, on non-reversible stochastic processes. For this purpose, we target trajectories in space time rather than static configurations and study time-extensive observables. This suggests that the details of the evolution law such as the presence of time correlations take on a major role. In this thesis, we investigate selected models with stochastic dynamics that incorporate memory by means of different mechanisms, devise a numerical approach for such models, and quantify to what extent the memory affects the large deviation functionals. The results are relevant for real-world situations, where simplified memoryless (Markovian) models may not always be appropriate. After an original introduction to the mathematics of stochastic processes, we explore, analytically and numerically, an open-boundary zero-range process which incorporates memory by means of hidden variables that affect particle congestion. We derive the exact solution for the steady state of the one-site system, as well as a mean-field approximation for larger one-dimensional lattices. Then, we focus on the large deviation properties of the particle current in such a system. This reveals that the time correlations can be apparently absorbed in a memoryless description for the steady state and the small fluctuation regime. However, they can dramatically alter the probability of rare currents. Different regimes are separated by dynamical phase transitions. Subsequently, we address systems in which the memory cannot be encoded in hidden variables or the waiting-time distributions depend on the whole trajectory. Here, the difficulty in obtaining exact analytical results is exacerbated. To tackle these systems, we have proposed a version of the so-called 'cloning' algorithm for the evaluation of large deviations that can be applied consistently for both Markovian and non-Markovian dynamics. The efficacy of this approach is confirmed by numerical results for some of the rare non-Markovian models whose large deviation functions can be obtained exactly. We finally adapt this machinery to a technological problem, specifically the performance evaluation of communication systems, where temporal correlations and large deviations are important.

Yang, Nian. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2013. / Includes bibliographical references (leaves 95-100). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstract also in Chinese.

We will begin with a review of key financial topics and outline many of the crucial ideas utilized in the latter half of the paper. Formal notation for important variables will also be established. Then, a derivation of the Black-Scholes equation will lead to a discussion of its shortcomings, and the introduction of stochastic volatility models. Chapter 2 will focus on a variation of the CIR Model using stock price in the volatility driving process, and its behavior to a greater degree. The key area of discussion will be to approximate a hedging function for European call option prices by Taylor Expansion. We will apply this estimation to real data, and analyze the behavior of the price correction. Then make conclusions about whether stock price has any positive effects on the model.