The Black-Scholes option pricing model has several well recognized deficiencies, one of which is its assumption of a constant and time-homogeneous stock return volatility term. The implied volatility smile has been studied by subsequent researchers and various models have been developed in an attempt to reproduce this phenomenon from within the models. However, few of these models yield closed-form pricing formulas that are easy to implement in practice. In this thesis, we study a Mixture Lognormal model (MLN) for European option pricing, which assumes that future stock prices are conditionally described by a mixture of lognormal distributions. The ability of mixture models in generating volatility smiles as well as delivering pricing improvement over the traditional Black-Scholes framework have been much researched under multi-component mixtures for many derivatives and high-volatility individual stock options. In this thesis, we investigate the performance of the model under the simplest two-component mixture in a market characterized by relative tranquillity and over a relatively stable period for broad-based index options. A careful interpretation is given to the model and the results obtained in the thesis. This di erentiates our study from many previous studies on this subject. Throughout the thesis, we establish the unique advantage of the MLN model, which is having closed-form option pricing formulas equal to the weighted mixture of Black-Scholes option prices. We also propose a robust calibration methodology to fit the model to market data. Extreme market states, in particular the so-called crash-o-phobia effect, are shown to be well captured by the calibrated model, albeit small pricing improvements are made over a relatively stable period of index option market. As a major contribution of this thesis, we extend the MLN model to price exotic options including binary, Asian, and barrier options. Closed-form formulas are derived for binary and continuously monitored barrier options and simulation-based pricing techniques are proposed for Asian and discretely monitored barrier options. Lastly, comparative results are analysed for various strike-maturity combinations, which provides insights into the formulation of hedging and risk management strategies.

La tesis tiene como objeto el estudio del aseguramiento (reaseguramiento) de un Plan de Pensiones que asume el riesgo derivado de las desviaciones de mortalidad en un colectivo de partícipes (personas físicas en cuyo interés se crea el Plan) y beneficiarios (personas físicas con derecho a la percepción de prestaciones del Plan).Las modalidades de reaseguro que hemos planteado han sido las siguientes:a) Reaseguro del Percentil, contrato de reaseguro por el cual el reasegurador se obliga a cubrir la pérdida que puede tener plan como consecuencia en las desviaciones en la mortalidad de los partícipes del colectivo. En esta modalidad hemos supuesto que el Plan de Pensiones parte con un nivel de solvencia prefijado.b) Reaseguro de diferencia de siniestralidad, contrato por el cual el reasegurador se encarga de cubrir en cada período las provisiones matemáticas y márgenes de solvencia que pudieran establecerse, y el riesgo de fallecimiento.Dentro de esta modalidad hemos planteado dos tipos de reaseguro:- Tipo A: el reasegurador interviene siempre y cuando con las primas recargadas cobradas no tenga suficiente para poder hacer frente a las contingencias indicadas.- Tipo B: se diferencia del anterior en que en aquellos períodos donde el Plan requiera de financiación externa para cubrir la provisión y el margen de solvencia correspondiente, éste cede al reasegurador aquella parte de sus fondos que exceden de la provisión matemática y margen de solvencia correspondiente. De esta forma el plan, consigue tener garantizados exactamente en cada período la provisión matemática y el margen de solvencia a un coste menor al que tendría con el reaseguro Tipo A.Para cada modalidad de reaseguro hemos contemplado la posibilidad de ceder al reaseguro el posible beneficio del plan en el momento de extinción del colectivo (sólo posible en colectivos cerrados). Lo cual ha permitido reducir el coste de la operación sin menoscabo en los objetivos del reaseguro.La consideración del recargo de seguridad del plan como variable independiente que puede manejar el plan, nos ha permitido encontrar combinaciones óptimas recargo-reaseguro, en el sentido que hacen mínimo el coste total de la operación, dadas el resto de variables que interviene en el modelo.El modelo aplicado en la tesis para calcular la prima de reaseguro se caracteriza por ser:a) Actuarial: En la medida que utiliza como instrumento la matemática actuarial.b) Estocástico: ya que trabajamos con toda la aleatoriedad de las variables aleatorias que intervienen en el modelo.c) General: puesto que el mismo nos permite obtener las primas de reaseguro para las dos modalidades anteriormente descritas.d) Discreto: puesto que trabajamos con variables aleatorias discretas.Las hipótesis del modelo son:- Las prestaciones se relacionan únicamente con la vida y fallecimiento del partícipe.- El único riesgo es el derivado de las fluctuaciones aleatorias de la mortalidad respecto de su valor esperado. - El tipo de interés está determinado y es determinista.- El Plan de Pensiones no persigue ánimo de lucro.- El Plan de Pensiones una vez reasegurado no asume ningún riesgo.- El Plan de Pensiones carece de reservas iniciales y sólo se financia de las aportaciones de los partícipes. El modelo ha sido aplicado al caso tanto de colectivos cerrados como abiertos, utilizando como instrumento para poder determinar la evolución del colectivo el método de simulación de Monte Carlo.El cálculo de la prima de reaseguro de cada elemento del colectivo aparece como un problema de reparto del coste del reaseguro entre los partícipes que forma el colectivo. Este reparto, que puede realizarse de distintas formas, ha dado lugar a criterios en la determinación de la prima de reaseguro.La conclusión a la que llegamos es que no podemos establecer una prima de reaseguro general y válida en cada modalidad planteada y para cualquier Plan de Pensiones, ya que ésta depende del tipo de prestaciones y contraprestaciones del colectivo, de las bases técnicas utilizadas y de la composición del colectivo.Por tanto, cada Plan de Pensiones requerirá de un análisis particular para determinar la prima de reaseguro. / The objective of this thesis is the insuring (or reinsuring) of a Pension Plan that assumes the risk derived from the deviations of mortality in a collective of participants.The proposed types of reinsurance are the following:a) Reinsurance of Percentile: in which the reinsurance covers the loss of the plan as a consequence of deviations in the mortality of the participants of the collective. In this type, we have supposed that the Pensions Plan starts from a given level of solvency.b) Reinsurance of the difference in a misfortune: in which the reinsurance assumes responsability for covering, in each period the mathematical provisions and margins of solvency that could be establisited and the risk of death.In particular, we have supposed that:b-l) The margin of solvency in each period is a given as a result of the reserves necessary to the mathematical provisions of each period so that the level of insolvency of the plan is maintained in a predetermined.b-2) The margin of solvency in each period is a given as result of a percentage of the mathematical provisions.The fact that the Plans aren't intended for profit has allowed us to consider a new possibility of a contract of reinsurance based in the payment to the reinsurance of the expected profit that the plan can have at the moment of the extinction of the collective.We will propose the problem and the calculation of the premium of reinsurance for the stated types in closed collectives as well as in open collectives.The conclusion we will reach is that we cannot establish a general and valid premium of reinsurance in each proposed type and for any pension plan in as much as this depends on the type of payments and counter-payments of the collective, the technical bases used and the composition of the collective. Therefore each pension plan will require a special analysis to determine the premium of the reinsurance.

Diss. (sammanfattning) Stockholm : Karolinska institutet, 2002. / Härtill 5 uppsatser.

M.Sc. / Please refer to full text to view abstract

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With the issuing of Pension Funds circular PF117 in 2004 in South Africa, regulation required valuation assumptions for defined benefit pension funds to be on a best-estimate basis. Allowance for prudence was to be made through explicit contingency reserves, in order to increase reporting transparency. These reserves for prudence, however, were not permitted to put the fund into deficit (the no-deficit clause). Analysis is conducted to understand the risk that PF117 poses to pension fund sponsors and members under two key measures: contribution rate risk and solvency risk. A stochastic model of a typical South African defined benefit fund is constructed with simulations run to determine the impact of the PF117 requirements. Findings show that a best-estimate funding basis, coupled with the no-deficit clause, results in significant risk under both contribution rate and solvency risk measures, particularly in the short-term. To mitigate these risks, alternative ways of introducing conservatism into the funding basis are required, with possible options including incorporating margins into investment return assumptions or the removal of the no-deficit clause.

There has been a rising concern surrounding risk within society. This increasing concern has dominated almost all aspects of human life and more specifically the way in which citizens are governed. How risk is addressed in general has shifted significantly; given this, the criminal justice system has also seen an escalation in concerns surrounding risk. Subsequently, there has been a push towards evaluating said risks through the use of actuarial assessment tools. Research has shown that with the rising reliance on actuarial assessment tools came the decrease in practitioner’s ability to rely on their professional judgement when conducting their work. However, there has been a gap identified in the literature. This gap pertains to how practitioners, particularly, probation officers perceive the impact of these actuarial tools on their work. This study aims to analyse how probation officers, within the province of Ontario, view the impact of actuarial assessment tools on their work. This study is guided by the theory of governmentality, as coined by Michel Foucault. In order to explore the impact of actuarial assessment tools on the practice of probation, seven semi-structured interviews were conducted with former probation officers. The perceptions varied and participants did not provide a unique and monolithic response; rather, the voices of all participants were shared to create a larger picture of how actuarial assessment tools impact the work of practitioners in the practice of probation.

Research investigating clinical versus actuarial prediction and judgment has consistently demonstrated the superiority of actuarial (statistical) methods. Little research to date has directly compared clinical and actuarial predictions in the context of patient-focused psychotherapy outcomes. The most relevant study on this issue was completed with an adult population and results indicated that the actuarial method was significantly more accurate at predicting client treatment failure compared to clinician’s predictions. This study examined clinical versus actuarial prediction of client deterioration in a sample of children and adolescents receiving treatment in a managed care and community mental health setting. Predictions of treatment failure made by the actuarial method were found to be significantly more accurate than predictions of treatment failure made by clinicians. More specifically, participating clinicians did not make a single prediction of treatment failure. These findings add further evidence to support the use of actuarial methods in enhancing clinical decision-making in community-based mental health services for children and adolescents.

Thesis (PhD)--Stellenbosch University, 2012. / ENGLISH ABSTRACT: Multi-state models are used in this dissertation to model panel data, also known as longitudinal or cross-sectional time-series data. These are data sets which include units that are observed across two or more points in time. These models have been used extensively in medical studies where the disease states of patients are recorded over time. A theoretical overview of the current multi-state Markov models when applied to panel data is presented and based on this theory, a simulation procedure is developed to generate panel data sets for given Markov models. Through the use of this procedure a simulation study is undertaken to investigate the properties of the standard likelihood approach when fitting Markov models and then to assess its shortcomings. One of the main shortcomings highlighted by the simulation study, is the unstable estimates obtained by the standard likelihood models, especially when fitted to small data sets. A Bayesian approach is introduced to develop multi-state models that can overcome these unstable estimates by incorporating prior knowledge into the modelling process. Two Bayesian techniques are developed and presented, and their properties are assessed through the use of extensive simulation studies. Firstly, Bayesian multi-state models are developed by specifying prior distributions for the transition rates, constructing a likelihood using standard Markov theory and then obtaining the posterior distributions of the transition rates. A selected few priors are used in these models. Secondly, Bayesian multi-state imputation techniques are presented that make use of suitable prior information to impute missing observations in the panel data sets. Once imputed, standard likelihood-based Markov models are fitted to the imputed data sets to estimate the transition rates. Two different Bayesian imputation techniques are presented. The first approach makes use of the Dirichlet distribution and imputes the unknown states at all time points with missing observations. The second approach uses a Dirichlet process to estimate the time at which a transition occurred between two known observations and then a state is imputed at that estimated transition time. The simulation studies show that these Bayesian methods resulted in more stable results, even when small samples are available. / AFRIKAANSE OPSOMMING: Meerstadium-modelle word in hierdie verhandeling gebruik om paneeldata, ook bekend as longitudinale of deursnee tydreeksdata, te modelleer. Hierdie is datastelle wat eenhede insluit wat oor twee of meer punte in tyd waargeneem word. Hierdie tipe modelle word dikwels in mediese studies gebruik indien verskillende stadiums van ’n siekte oor tyd waargeneem word. ’n Teoretiese oorsig van die huidige meerstadium Markov-modelle toegepas op paneeldata word gegee. Gebaseer op hierdie teorie word ’n simulasieprosedure ontwikkel om paneeldatastelle te simuleer vir gegewe Markov-modelle. Hierdie prosedure word dan gebruik in ’n simulasiestudie om die eienskappe van die standaard aanneemlikheidsbenadering tot die pas vanMarkov modelle te ondersoek en dan enige tekortkominge hieruit te beoordeel. Een van die hoof tekortkominge wat uitgewys word deur die simulasiestudie, is die onstabiele beramings wat verkry word indien dit gepas word op veral klein datastelle. ’n Bayes-benadering tot die modellering van meerstadiumpaneeldata word ontwikkel omhierdie onstabiliteit te oorkom deur a priori-inligting in die modelleringsproses te inkorporeer. Twee Bayes-tegnieke word ontwikkel en aangebied, en hulle eienskappe word ondersoek deur ’n omvattende simulasiestudie. Eerstens word Bayes-meerstadium-modelle ontwikkel deur a priori-verdelings vir die oorgangskoerse te spesifiseer en dan die aanneemlikheidsfunksie te konstrueer deur van standaard Markov-teorie gebruik te maak en die a posteriori-verdelings van die oorgangskoerse te bepaal. ’n Gekose aantal a priori-verdelings word gebruik in hierdie modelle. Tweedens word Bayesmeerstadium invul tegnieke voorgestel wat gebruik maak van a priori-inligting om ontbrekende waardes in die paneeldatastelle in te vul of te imputeer. Nadat die waardes ge-imputeer is, word standaard Markov-modelle gepas op die ge-imputeerde datastel om die oorgangskoerse te beraam. Twee verskillende Bayes-meerstadium imputasie tegnieke word bespreek. Die eerste tegniek maak gebruik van ’n Dirichletverdeling om die ontbrekende stadium te imputeer by alle tydspunte met ’n ontbrekende waarneming. Die tweede benadering gebruik ’n Dirichlet-proses om die oorgangstyd tussen twee waarnemings te beraam en dan die ontbrekende stadium te imputeer op daardie beraamde oorgangstyd. Die simulasiestudies toon dat die Bayes-metodes resultate oplewer wat meer stabiel is, selfs wanneer klein datastelle beskikbaar is.

Thesis (MComm)--Stellenbosch University, 2011. / ENGLISH ABSTRACT: The analysis of multidimensional (multivariate) data sets is a very important area of research in applied statistics. Over the decades many techniques have been developed to deal with such datasets. The multivariate techniques that have been developed include inferential analysis, regression analysis, discriminant analysis, cluster analysis and many more exploratory methods. Most of these methods deal with cases where the data contain numerical variables. However, there are powerful methods in the literature that also deal with multidimensional binary and count data. The primary purpose of this thesis is to discuss the exploratory and inferential techniques that can be used for binary and count data. In Chapter 2 of this thesis we give the detail of correspondence analysis and canonical correspondence analysis. These methods are used to analyze the data in contingency tables. Chapter 3 is devoted to cluster analysis. In this chapter we explain four well-known clustering methods and we also discuss the distance (dissimilarity) measures available in the literature for binary and count data. Chapter 4 contains an explanation of metric and non-metric multidimensional scaling. These methods can be used to represent binary or count data in a lower dimensional Euclidean space. In Chapter 5 we give a method for inferential analysis called the analysis of distance. This method use a similar reasoning as the analysis of variance, but the inference is based on a pseudo F-statistic with the p-value obtained using permutations of the data. Chapter 6 contains real-world applications of these above methods on two special data sets called the Biolog data and Barents Fish data. The secondary purpose of the thesis is to demonstrate how the above techniques can be performed in the software package R. Several R packages and functions are discussed throughout this thesis. The usage of these functions is also demonstrated with appropriate examples. Attention is also given to the interpretation of the output and graphics. The thesis ends with some general conclusions and ideas for further research. / AFRIKAANSE OPSOMMING: Die analise van meerdimensionele (meerveranderlike) datastelle is ’n belangrike area van navorsing in toegepaste statistiek. Oor die afgelope dekades is daar verskeie tegnieke ontwikkel om sulke data te ontleed. Die meerveranderlike tegnieke wat ontwikkel is sluit in inferensie analise, regressie analise, diskriminant analise, tros analise en vele meer verkennende data analise tegnieke. Die meerderheid van hierdie metodes hanteer gevalle waar die data numeriese veranderlikes bevat. Daar bestaan ook kragtige metodes in die literatuur vir die analise van meerdimensionele binêre en telling data. Die primêre doel van hierdie tesis is om tegnieke vir verkennende en inferensiële analise van binêre en telling data te bespreek. In Hoofstuk 2 van hierdie tesis bespreek ons ooreenkoms analise en kanoniese ooreenkoms analise. Hierdie metodes word gebruik om data in gebeurlikheidstabelle te analiseer. Hoofstuk 3 bevat tegnieke vir tros analise. In hierdie hoofstuk verduidelik ons vier gewilde tros analise metodes. Ons bespreek ook die afstand maatstawwe wat beskikbaar is in die literatuur vir binêre en telling data. Hoofstuk 4 bevat ’n verduideliking van metriese en nie-metriese meerdimensionele skalering. Hierdie metodes kan gebruik word om binêre of telling data in ‘n lae dimensionele Euclidiese ruimte voor te stel. In Hoofstuk 5 beskryf ons ’n inferensie metode wat bekend staan as die analise van afstande. Hierdie metode gebruik ’n soortgelyke redenasie as die analise van variansie. Die inferensie hier is gebaseer op ’n pseudo F-toetsstatistiek en die p-waardes word verkry deur gebruik te maak van permutasies van die data. Hoofstuk 6 bevat toepassings van bogenoemde tegnieke op werklike datastelle wat bekend staan as die Biolog data en die Barents Fish data. Die sekondêre doel van die tesis is om te demonstreer hoe hierdie tegnieke uitgevoer word in the R sagteware. Verskeie R pakette en funksies word deurgaans bespreek in die tesis. Die gebruik van die funksies word gedemonstreer met toepaslike voorbeelde. Aandag word ook gegee aan die interpretasie van die afvoer en die grafieke. Die tesis sluit af met algemene gevolgtrekkings en voorstelle vir verdere navorsing.