Credit risk of derivative securities includes the risk of underlying company and the risk of seller's nonfulfilment of contracts. Take bonds for example, we regard Treasury bills as default-free bonds, and corporate bonds as risky bonds. When the liability of property of derivative securities underlying company is less than 1, we regard the company is of bankruptcy. And then the seller of derivative securities will break the contract. The essay extends two period risky bonds pricing valuation of Jarrow and Turnbull(1995) to multiperiod situation, and derive arbitrage-free condition. Furthermore, we derive formulae of risky bonds prices by assuming the logarithm of the odds ratio of an underlying company's bankruptcy probability satisfies an AR(1) or MA(1) processes. Empirical data of Rebar, Chinarebar, Ceon are studied, time series models are established for logarithm of odds ratios. In most cases, we find that the log odds ratios can be well fitted by AR(1) models.

This thesis focuses on stochastic processes and some of their properties are investigated which are necessary to determine the tools, the extremal index and the extremogram. Both mathematical tools measure extremal dependency within random time series. Two different models are introduced and related properties are discussed. The probability function of the Agent based model is surveyed explicitly and strong stationarity is proven. Data sets for both processes are simulated and clustering of the data is investigated with two different methods. Finally an estimation of the extremogram is used to interpret dependency of extremes within the data.

This article examines the hedging positions derived from the Black-Scholes(B-S) model and the GARCH(1,1) models, respectively, when the log returns of underlying asset exhibits GARCH(1,1) process. The result shows that Black-Scholes and GARCH options deltas, one of the hedging parameters, are similar for near-the-money options, and Black-Scholes options delta is higher then GARCH delta in absolute terms when the options are deep out-of-money, and Black-Scholes options delta is lower then GARCH delta in absolute terms when the options are deep in-the-money. Simulation study of hedging procedure of GARCH(1,1) and B-S models are performed, which also support the above findings.

Social unrest, terrorism and other forms of political violence events are highly unpredictable. These events are driven by human intent and intelligence, both of which are extremely difficult to model accurately. This has resulted in a scarcity of insurance products that cover these types of perils. Links have been found between the incidence of political violence and various economic and socioeconomic variables, but to date no relationships have been identified in South Africa. The aim of this study was to address this. Firstly, by identifying relationships between the incidence of social unrest events and economic and socio-economic variables in South Africa and secondly by using these interactions to model social unrest. Spearman’s rank correlation and trendline analysis were used to compare the direction and strength of the relationships that exist between protests and the economic and socio-economic variables. To gain additional insight with regards to South African protests, daily, monthly, quarterly and annual protest models were created. This was done using four different modelling techniques, namely univariate time series, linear regression, lagged regression and the VAR (1) model. The forecasting abilities of the models were analysed using both a one-step and n-step forecasting procedure. Variations in relationships for different types of protests were also considered for five different subcategories. Spearman’s rank correlation and trendline analysis showed that the relationships between protests and economic and socio-economic variables were sensitive to changes in data frequency and the use of either national or provincial data. The daily, monthly, quarterly and annual models all had power in explaining the variation that was observed in the protest data. The annual univariate model had the highest explanatory power (R2 = 0.8721) this was followed by the quarterly VAR (1) model (R2 = 0.8659), while the monthly lagged regression model had a R2 of 0.8138. The one-step forecasting procedure found that the monthly lagged regression model outperformed the monthly VAR (1) model in the short term. The converse was seen for the short-term performance of the quarterly models. In the long term, the VAR (1) model outperformed the other models. Limitations were identified within the lagged regression model’s forecasting abilities. As a model’s long-term forecasting ability is important in the insurance world, the VAR (1) model was deemed as the best modelling technique for South African social unrest. Further model limitations were identified when the subcategories of protests were considered. This study demonstrates that with the use of the applicable economic and socio-economic variables, social unrest events in South Africa can be modelled. / Dissertation (MSc)--University of Pretoria, 2019. / Absa Chair in Actuarial Science (UP) / South African Department of Science and Technology (DST) Risk Research Platform, under coordination of the North-West University (NWU) / Insurance and Actuarial Science / MSc Actuarial Mathematics / Unrestricted

Count data are often encountered in manufacturing and service industries due to ease of data collection. These counts can be useful in process monitoring to detect shifts of a process from an in-control state to various out-of-control states. It is usually assumed that the observations are independent and identically distributed. However, in practice, observations may be autocorrelated and this may adversely affect the performance of the control charts developed under the assumption of independence. In this thesis, the cumulative sum (CUSUM) control chart for monitoring autocorrelated processes of counts is investigated. To describe the autocorrelation structure of counts, a Poisson integer-valued autoregressive moving average model of order 1, Poisson INAR(1), is employed. Changes in the process mean in both positive and negative directions are taken into account while designing the CUSUM chart. A trivariate Markov Chain approach is utilized for evaluating the performance of the chart.

In the doctoral dissertation, we consider problems of testing and estimating changed segment with unknown starting position and duration of epidemic state in the autoregressive first-order model. The proposed tests are based on partial sums of model residuals and model-parameter partial-estimator polygonal line processes. We derive asymptotic results for these processes in Holder spaces. The behavior of test statistics under the null hypothesis of no change and alternative is provided. Empirical power analysis has shown that tests are more powerful when absolute values of model parameter are quite large or autoregressive process changes from a stationary state to a nonstationary one. We prove the consistency of the least square changed-segment estimators and provide their convergence rates. / Disertacijoje nagrinėjamas pirmos eilės autoregresinio modelio pasikeitusio segmento testavimo ir vertinimo uždavinys. Aprašomo modelio epideminio pasikeitimo pradžia ir ilgis nėra žinomi. Pasiūlyti kriterijai pasikeitusio segmento testavimui, kurie pagrįsti modelio paklaidų įvertinių dalinių sumų ir modelio parametro dalinių įvertinių laužčių procesais. Šiems procesams gautos ribinės teoremos Hiolderio erdvėse. Nurodomas testų statistikų ribinis elgesys esant teisingai nulinei ir alternatyviajai hipotezėms. Iš empirinio kriterijų galios tyrimo rezultatų matyti, kad pasiūlytų testų galia didžiausia aptinkant pasikeitimus iš stacionarios būklės į nestacionarią arba esant artimoms vienetui modelio parametro reikšmėms. Taip pat įrodoma, kad mažiausių kvadratų metodu gauti pasikeitusio segmento pradžios ir ilgio įverčiai bei autoregresinio modelio su pasikeitusiu segmentu parametrų įverčiai yra suderintieji bei pateikiamas jų konvergavimo greitis.

Disertacijoje nagrinėjamas pirmos eilės autoregresinio modelio pasikeitusio segmento testavimo ir vertinimo uždavinys. Aprašomo modelio epideminio pasikeitimo pradžia ir ilgis nėra žinomi. Pasiūlyti kriterijai pasikeitusio segmento testavimui, kurie pagrįsti modelio paklaidų įvertinių dalinių sumų ir modelio parametro dalinių įvertinių laužčių procesais. Šiems procesams gautos ribinės teoremos Hiolderio erdvėse. Nurodomas testų statistikų ribinis elgesys esant teisingai nulinei ir alternatyviajai hipotezėms. Iš empirinio kriterijų galios tyrimo rezultatų matyti, kad pasiūlytų testų galia didžiausia aptinkant pasikeitimus iš stacionarios būklės į nestacionarią arba esant artimoms vienetui modelio parametro reikšmėms. Taip pat įrodoma, kad mažiausių kvadratų metodu gauti pasikeitusio segmento pradžios ir ilgio įverčiai bei autoregresinio modelio su pasikeitusiu segmentu parametrų įverčiai yra suderintieji bei pateikiamas jų konvergavimo greitis. / In the doctoral dissertation, we consider problems of testing and estimating changed segment with unknown starting position and duration of epidemic state in the autoregressive first-order model. The proposed tests are based on partial sums of model residuals and model-parameter partial-estimator polygonal line processes. We derive asymptotic results for these processes in Holder spaces. The behavior of test statistics under the null hypothesis of no change and alternative is provided. Empirical power analysis has shown that tests are more powerful when absolute values of model parameter are quite large or autoregressive process changes from a stationary state to a nonstationary one. We prove the consistency of the least square changed-segment estimators and provide their convergence rates.

The hidden state Markov (HSM) model is introduced as a new conceptual framework for modelling long-term persistence in hydrological time series. Unlike the stochastic models currently used, the conceptual basis of the HSM model can be related to the physical processes that influence long-term hydrological time series in the Australian climatic regime. A Bayesian approach was used for model calibration. This enabled rigourous evaluation of parameter uncertainty, which proved crucial for the interpretation of the results. Applying the single site HSM model to rainfall data from selected Australian capital cities provided some revealing insights. In eastern Australia, where there is a significant influence from the tropical Pacific weather systems, the results showed a weak wet and medium dry state persistence was likely to exist. In southern Australia the results were inconclusive. However, they suggested a weak wet and strong dry persistence structure may exist, possibly due to the infrequent incursion of tropical weather systems in southern Australia. This led to the postulate that the tropical weather systems are the primary cause of two-state long-term persistence. The single and multi-site HSM model results for the Warragamba catchment rainfall data supported this hypothesis. A strong two-state persistence structure was likely to exist in the rainfall regime of this important water supply catchment. In contrast, the single and multi-site results for the Williams River catchment rainfall data were inconsistent. This illustrates further work is required to understand the application of the HSM model. Comparisons with the lag-one autoregressive [AR(1)] model showed that it was not able to reproduce the same long-term persistence as the HSM model. However, with record lengths typical of real data the difference between the two approaches was not statistically significant. Nevertheless, it was concluded that the HSM model provides a conceptually richer framework than the AR(1) model. / PhD Doctorate

Made available in DSpace on 2015-05-14T12:14:07Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 9808595 bytes, checksum: 235eec9737496afa4dfbdf1feafbc21f (MD5) Previous issue date: 2013-02-04 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / The Dark matter (DM) problem constitutes a key question at the interface among Particle Physics, Astrophysics and Cosmology. The observational data which have been accumulated in the last years point to an existence of non baryonic amount of DM. Since the Standard Model (SM) does not provide any candidate for such non-baryonic DM, the evidence of DM is a major indication for new physics beyond the SM. We will study in this work one of the most popular DM candidates, the so called WIMPs (Weakly Interacting Massive Particles) from a direct and indirect detection perspective. In order to approach the direct and indirect dection of DM in the context of Particle Physics in a more pedagogic way, we will begin our discussion talking about a minimal extension of the SM. Later we will work on the subject in a 3-3-1 model. Next, we will study the role of WIMPs in the Big Bang Nucleosynthesis. Lastly, we will look for indirect signals of DM, by looking for gamma-ray excess observed by the NASA Satelite, called Fermi-LAT, in the Galactic Center. Through an analyses of the data events observed by Fermi-LAT and some background models, we will constrain the annihilation cross section-mass relation. As a result of this PhD 9 articles have been done. Two of them are still under the publication process. / O problema da matéria escura (ME) constitui uma questão chave na interface entre física de partículas, astrofísica e cosmologia. O acúmulo de dados observacionais nos últimos anos apontam para uma enorme quantidade de ME não bariônica. Uma vez que o Modelo Padrão (MP), não fornece um candidato para este tipo de matéria, a evidência de ME é uma forte indicação de física nova, além do MP. Estudaremos neste trabalho um dos candidatos à ME mais populares, os chamados WIMPS (partículas massivas que interagem fracamente) sob o ponto de vista de detecção direta e indireta de ME. Para que possamos abordar os meios de detecção direta e indireta de ME no contexto de Física de Partículas de forma didática, iniciaremos nossa discussão apresentando uma extensão mínima do MP. Posteriormente trataremos do assunto no contexto de um modelo 3-3-1. Adiante verificaremos qual o papel da ME no cenário da Nucleossíntese Primordial. Por ´ultimo procuraremos por sinais indiretos de ME, na busca por excessos em raios gama observados pelo satélite da NASA, chamado Fermi-LAT, no centro da nossa galáxia. Através de uma análise dos eventos observados pelo Fermi-LAT e de alguns modelos de background astrofísico iremos impor vínculos com relação à massa e seção de choque de aniquilação. Ao longo desse doutorado foram publicados 9 artigos. Um destes ainda sob o processo de publicação.

This master's thesis deals with queueing theory and its application in designing node models in packet-switched network. There are described general principles of designing queueing theory models and its mathematical background. Further simulator of packet delay in network was created. This application implements two described models - M/M/1 and M/G/1. Application can be used for simulating network nodes and obtaining basic network characteristics like packet delay or packet loss. Next, lab exercise was created, in that exercise students familiarize themselves with basic concepts of queueing theory and examine both analytical and simulation approach to solving queueing systems.