Dissertations / Theses on the topic 'Risk-neutral probability density function'

The target of the study is to find out if the direct methodology could provide same information about the parameters of the risk neutral probability density function (RND) than the reference RND methodologies. The direct methodology is based on for defining the parameters of the RND from underlying asset by using futures contracts and only few at-the-money (ATM) and/or close at-the-money (ATM) options on asset. Of course for enabling the analysis of the feasibility of the direct methodology the reference RNDs must be estimated from the option data. Finally the results of estimating the parameters by the direct methodology are compared to the results of estimating the parameters by the selected reference methodologies for understanding if the direct methodology can be used for understanding the key parameters of the RND. The study is based on S&P 500 index option data from year 2008 for estimating the reference RNDs and for defining the reference moments from the reference RNDs. The S&P 500 futures contract data is necessary for finding the expectation value estimation for the direct methodology. Only few ATM and/or close ATM options from the S&P 500 index option data are necessary for getting the standard deviation estimation for the direct methodology. Both parametric and non-parametric methods were implemented for defining reference RNDs. The reference RND estimation results are presented so that the reference RND estimation methodologies can be compared to each other. The moments of the reference RNDs were calculated from the RND estimation results so that the moments of the direct methodology can be compared to the moments of the reference methodologies. The futures contracts are used in the direct methodology for getting the expectation value estimation of the RND. Only few ATM and/or close ATM options are used in the direct methodology for getting the standard deviation estimation of the RND. The implied volatility is calculated from option prices using ATM and/or close ATM options only. Based on implied volatility the standard deviation can be calculated directly using time scaling equations. Skewness and kurtosis can be calculated from the estimated expectation value and the estimated standard deviation by using the assumption of the lognormal distribution. Based on the results the direct methodology is acceptable for getting the expectation value estimation using the futures contract value directly instead of the expectation value, which is calculated from the RND of full option data, if and only if the time to maturity is relative short. The standard deviation estimation can be calculated from few ATM and/or at close ATM options instead of calculating the RND from full option data only if the time to maturity is relative short. Skewness and kurtosis were calculated from the expectation value estimation and the standard deviation estimation by using the assumption of the lognormal distribution. Skewness and kurtosis could not be estimated by using the assumption of the lognormal distribution because the lognormal distribution is not correct generic assumption for the RND distributions.

Mestrado em Mathematical Finance
O petróleo simboliza uma das mais importantes commodities trocadas mundialmente, apresentando-se como a fonte primária de produção de energia. Este pode ser trocado pelo seu spot price ou pelo recurso a contratos de derivados financeiros, sendo a troca de opções de petróleo um exemplo do segundo tipo de investimento. O estudo desta variável revela-se pertinente para compreender o seu comportamento e estimar previsões possíveis. Tal estudo pode ser conduzido através de uma função densidade de probabilidade. A análise da função densidade de probabilidade risk-neutral urge, pelas suas falhas em abraçar todas as hipóteses representativas do investidor comum, o estudo de uma função densidade de probabilidade suavizada/subjetiva. Considerando apenas a função densidade de probabilidade risk-neutral para uma análise ao preço das opções de petróleo, assumir-se-ia que todos os investidores são indiferentes ao risco, independentemente da circunstância, ou seja, colocariam as suas decisões de investimento unicamente nos eventuais retornos associados a esse investimento. No entanto, tal não vai de encontro com a realidade, o que apela ao cálculo e análise dos valores de aversão ao risco dos investidores respeitantes à variável em estudo. Através do presente ensaio, consegue-se concluir que a aversão ao risco é bastante elevada em períodos de grande volatilidade e incerteza, e consideravelmente menor em períodos de maior estabilidade económica e financeira.
Oil is considered to be one of the most important commodities traded worldwide and is deemed to be the primary source in energy production. It can be traded for its spot price or by using a financial derivatives contract, being oil option trading an example of the second type of investment mentioned. The study of this variable is relevant in order to comprehend its behaviour and estimate possible forecasts. Said study can be conducted through a probability density function. The analysis of a risk-neutral probability density function requires, because of its flaws in embracing all the hypothesis that portray the common investor, the study of a smoothed/subjective probability density function. Considering only a risk-neutral probability density function for the analysis of oil option prices, it would be assuming all investors are indifferent to risk, whichever the circumstance, which means they would place their investment decisions solely on the possible outcomes associated with that investment. However, that does not meet reality, which calls for the computation and analysis of the investors risk aversion's values with respect to the variable at play. From the present essay, it is possible to conclude that risk aversion is rather high in periods of great volatility and uncertainty, and considerably smaller in periods of greater economic and financial stability.
info:eu-repo/semantics/publishedVersion

Un des dangers chimiques les plus importants relevés par l'OMS concerne la contamination des céréales par les mycotoxines et notamment les aflatoxines et les fumonisines. La réglementation recommande des contaminations maximales d'aflatoxines dans les céréales inférieures à 20 mg/kg ; cependant on relève couramment des taux supérieurs à 200 mg/kg dans le maïs au Mexique. Bien qu'il ait été évalué que le processus de nixtamalisation détruit plus de 90 % des fumonisines et de 80 à 95 % des aflatoxines, le taux résiduel peut encore être élevé : certaines publications rapportent des concentrations jusqu'à 100 mg/kg dans les tortillas, ce qui représente un risque avéré vue la grande consommation de tortillas au Mexique (325g/j). Le JECFA (2001) a établi une dose maximale acceptable de 2μg/kg pc/j pour la fumonisine et recommande de réduire l'exposition aux aflatoxines au plus faible niveau possible en considérant que le seuil de 1 ng/kg pc/j ne devrait pas être atteint. Au cours de cette thèse 3 enquêtes aléatoires et représentatives ont été menées dans 40 tortillerias de la ville de Veracruz. La consommation de maïs de la population a été évaluée à partir d'un questionnaire de consommation. L'analyse des mycotoxines a été réalisée par HPLC-FD par utilisation de colonnes à immunoaffinité selon la réglementation européenne (CIRAD-Montpellier). L'analyse des données obtenues a été effectuée selon une méthode probabiliste permettant de construire une fonction de distribution de probabilités à partir de la méthode de Monte Carlo (UBO). La représentativité de la population a été validée à partir d'évaluation de quotas de population après échantillonnage aléatoire initial. La contamination des tortillas a été mesurée à 0.54-1.96 mg/kg pour les aflatoxines et à 65-136 mg/kg pour les fumonisines. La consommation moyenne de tortillas a été mesurée à 148 g de maïs par jour. L'exposition de la population aux aflatoxines apparaît alors comprise entre 0,94 et 3,14 ng/kg pc/j et celle aux fumonisines entre 146 et 315 ng/kg pc/j. Les échantillons les plus contaminés proviennent des tortillerias réalisant elles-mêmes leur procédure de nixtamalisation. L'analyse des résultats montre que 60 % de la population de Veracruz serait à risque selon les préconisations du JECFA. L'exposition aux fumonisines atteint 5 % de la dose maximale acceptable, du fait d'une relativement faible contamination du maïs à cette mycotoxine. Les résultats montrent donc un risque sanitaire pour la population de la ville de Veracruz. Une extension de ce travail à la totalité de l’Etat de Veracruz, incluant la population rurale, devrait être menée du fait du risque probablement accru de cette dernière catégorie de population en lien avec sa plus forte consommation de maïs
One of the chemical hazards that WHO has reported more frequently is cereals contamination with mycotoxins, mainly aflatoxins and fumonisins. NOM-188-SSA1-2002 establishes that aflatoxin concentration in grain should not exceed 20 mg kg-1 ; however, there are reported concentrations > 200 mg kg-1 in maize. Although it has been documented that nixtamalizacion removes more than 90% of fumonisins and between 80 and 95% of aflatoxins, the residual amount could be important, finding reports concentrations higher than 100 mg kg-1 of aflatoxin in tortilla, representing a risk due to the high consumption of tortillas in Mexico (325 g d-1). The JECFA (2001) establishes a maximum intake of 2 mg kg-1 pc d-1 for fumonisin and aflatoxin recommends reducing “as low as reasonably achievable” levels. 3 random and representative sampling in Veracruz city, each in 40 tortillerias, were made. Corn intake and weight of the population were estimated using a consumption questionnaire. Mycotoxins analysis were performed by HPLC-FD using immunoaffinity columns according to European standard UNE-EN ISO 14123 : 2008 for aflatoxins and UNE-EN 13585 : 2002 for fumonisin in the CIRAD (Montpellier, France). Statistical analysis were performed under a probabilistic approach in collaboration with the University of Bretagne Occidentale (Brest, France), building probability density function (PDF) and using the Monte Carlo method. PDF parameters of the weight of the population was 74.15kg for men (which coincides with reported by CANAIVE) and 65.83kg for women ; the pollution aflatoxin tortilla was 0.54 – 1.96mg kg-1 and fumonisin from 65.46 – 136.00mg kg-1 ; the tortilla consumption was 148.3g of corn per person per day ; the daily intake of aflatoxins was 0.94 – 3.14ng kg-1 bw d-1 and fumonisin of 146.24 – 314.99ng kg-1 bw d-1. Samples with higher aflatoxin contamination came from tortillerias that make the nixtamalization in situ. In assessing exposure it was found that up to 60% of the population could be consuming more than the recommended by JECFA (2001) for aflatoxin dose (1ng kg-1 bw d-1). Exposure to fumonisins intake was < 5% due to low contamination by these mycotoxins. The results suggest that the population of the city of Veracruz could be in food risk by eating contaminated corn tortillas AFT. It is advisable to extend this study in rural communities, where risk factors could increase

Mestrado em Finanças
Este trabalho tem o objectivo de facilitar a previsão para investidores em mercados financeiros. Embora possa ser usado em acções e futuros de petróleo, o principal objectivo é o mercado cambial, mais especificamente, opções de moeda, extraindo com risco neutro a densidade de probabilidade da função através de uma abordagem paramétrica e não paramétrica. Consequentemente, tal foi aplicado a um caso muito recente, em 2019, entre o dólar Norte americano e a libra inglesa, tornando assim mais atractiva a leitura do comportamento da densidade, especialmente com a saída do Reino unido da União Europeia.
This work has the purpose of easing the forecast for financial market investors. Although it can be used on equities and oil futures, the main aim is the Foreign exchange. More so, it is specialized on currency options, extracting then the closer Risk Neutral Probability Density Function through a parametric approach and a nonparametric approach. Subsequently, this was applied to a very recent case, in 2019, between the United States of America dollar and United Kingdom pound, making it more attractive to assess the behaviour of the density, specially linked to the withdrawal of United Kingdom from the European Union.
info:eu-repo/semantics/publishedVersion

This thesis models the Australian-US Dollar (AUD/USD) exchange rate with particular attention being paid to investor risk aversion. Accounting for investor risk aversion in AUD/USD exchange rate modelling is novel, so too is the method used to measure risk aversion in this thesis. Investor risk aversion is measured using a technique developed in Bliss and Panigirtzoglou (2004), which makes use of Probability Density Functions (PDFs) extracted from option markets. More conventional approaches use forward-market pricing or Uncovered Interest Parity. Several methods of estimating PDFs from option and spot markets are examined, with the estimations from currency spot-markets representing an original application of an arbitrage technique developed in Stutzer (1996) to the AUD/USD exchange rate. The option and spot-market PDFs are compared using their first four moments and if estimated judiciously, the spot-market PDFs are found to have similar shapes to the option-market PDFs. So in the absence of an AUD/USD exchange rate options market, spot-market PDFs can act as a reasonable substitute for option-market PDFs for the purpose of examining market sentiment. The Relative Risk Aversion (RRA) attached to the AUD/USD, the US Dollar-Japanese Yen, the US Dollar-Swiss Franc and the US-Canadian Dollar exchange rates is measured using the Bliss and Panigirtzoglou (2004) technique. Amongst these exchange rates, only the AUD/USD exchange rate demonstrates a significant level of investor RRA and only over a weekly forecast horizon. The Bliss and Panigirtzoglou (2004) technique is also used to approximate a time-varying risk premium for the AUD/USD exchange rate. This risk premium is added to the cointegrating vectors of fixed-price and asset monetary models of the AUD/USD exchange rate. An index of Australia???s export commodity prices is also added. The out-of-sample forecasting ability of these cointegrating vectors is tested relative to a random walk using an error-correction framework. While adding the time-varying risk premium improves this forecasting ability, adding export commodity prices does so by more. Further, including both the time-varying risk premium and export commodity prices in the cointegrating vectors reduces their forecasting ability. So the time-varying risk premium is important for AUD/USD exchange rate modelling, but not as important as export commodity prices.

Mestrado em Economia Monetária e Financeira
It's usually accepted that financial asset prices reflect market participants' expectations concerning the evolution of certain important economic and financial variables. In comparison with the majority of the other financial assets, in particular against the futures or forwards, option prices have an additional interest resulting from their ability to provide information, not only in terms of the expected future value of an asset, but also about the higher moments of the probability distribution perceived by economic agents. Regarding the relationship among option prices and their strike prices it's possible to estimate the risk-neutral probability density function (RNPDF), which allow us to characterise almost completely, market expectations regarding the price of the underlying asset at the maturity date of the option. In the Black-Sholes framework, RNPDF estimation would be a trivial issue only consisting in implied volatility estimation. However, the systematic differences observed between theoretical model prices and those observed in the market, raise some suspicions about the reality adjustment of some of its premises, namely the lognormality for the underlying price distribution and the constant volatility assumed across the time and different strike prices. In this work, a set of alternative approaches for the RNPDF estimation is presented, along with its advantages and drawbacks, as a way to characterise the current state of the art. Using EURIBOR 3-month interest rate future option prices, and considering a mixture of lognormal distributions, we estimate RNPDF for some days around ECB Council meetings, in order to analyse market views about possible changes in ECB reference interest rates in these meetings. We also study the impact in market expectations, regarding the evolution of shorter-term interest rates, from the events that occurred in the USA on the 11th of September 2001, and the "time-to-maturity" effect on RNPDF volatility.
É geralmente aceite que os preços dos activos financeiros reflectem as expectativas dos participantes nos mercados. Face à maioria dos activos financeiros, em particular aos futuros ou "forwards", os preços das opções possuem um interesse adicional oriundo da capacidade que têm de fornecer informação relativa, não só, ao valor médio esperado pelo mercado para o preço futuro do activo subjacente, mas também, sobre os momentos de ordem mais elevada da distribuição de probabilidade percepcionada pelos agentes económicos. Com base na relação entre os preços das opções e respectivos preços de exercício é possível estimar a função de densidade de probabilidade neutra ao risco (FDPNR) que permite caracterizar, de uma forma quase completa, o perfil de expectativas dos agentes relativamente ao preço do activo subjacente na maturidade opção. No universo do modelo Black-Sholes, a estimação da FDPNR seria um assunto trivial consistindo apenas na estimação da volatilidade implícita. Porém, as diferenças sistemáticas observadas, entre os preços gerados pelo modelo e os observados no mercado, levantam a suspeita que algumas das suas hipóteses são pouco realísticas, nomeadamente a lognormalidade para a distribuição do preço do activo subjacente e a volatilidade constante assumida para os diferentes preços de exercício e ao longo do tempo. Neste trabalho é apresentado um conjunto de abordagens alternativas para a estimação de FDPNR, suas vantagens e desvantagens relativas, procurando-se caracterizar o actual estado da arte. Recorrendo aos preços de opções sobre futuros da taxa de juro EURIBOR a três meses e com base numa mistura de distribuições lognormais, estimamos as FDPNR para algumas datas em torno das reuniões do Conselho do BCE, utilizando-as na análise das expectativas do mercado relativamente a possíveis alterações das taxas de juro directoras nessas reuniões. Analisamos ainda o impacto nas expectativas dos agentes, relativamente à evolução das taxas de juro, dos acontecimentos ocorridos nos EUA, em 11 de Setembro de 2001 e o efeito "time-to-maturity" na volatilidade das FDPNR.

In this thesis, we study the estimation of the nonlinear conditionalrisk-neutral density function (RND) in discrete time. Specifically, weevaluate the extent to which the estimated nonlinear conditional RNDvaluable insights to answer relevant economic questions regarding to optionpricing, the measurement of invertors' preferences and portfolio choice.We make use of large dataset of options contracts written on the S&P 500index from 1996 to 2011, to estimate the parameters of the conditional RNDfunctions by minimizing the squared option pricing errors delivered by thenonlinear models studied in the thesis.In the first essay, we show that a semi-nonparametric option pricing modelwith GARCH variance outperforms several benchmarks models in-sample andout-of-sample. In the second essay, we show that a simple two-state regimeswitching model in volatility is not able to fully account for the pricingkernel and the risk aversion puzzle; however, it provides a reasonablecharacterisation of the time-series properties of the estimated riskaversion.In the third essay, we evaluate linear stochastic discount factormodels using an out-of-sample financial metric. We find that multifactormodels outperform the CAPM when this metric is used, and that modelsproducing the best fit in-sample are also those exhibiting the bestperformance out-of-sample.

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior
In Brazil over recent years the interest increased on the sunflower cultivation. Sunflower yield can be decreased by water deficit. To solve this problem, is necessary to calculate the probable water deficit in critical sunflower sub-phases and in the whole development cycle at different sowing dates. The objective of this study was to determine the probable duration values of sub-phases and the developmental cycle and get sowing dates with lower risk of water deficit and the occurrence probability in different levels of water deficit during the developmental sub-phases of sunflower crop sown at different sowing dates, considering the water storage capacity in different soils in the Central Region of Rio Grande do Sul. Also determine the probability of occurrence of water stress for different years classified as the El Niño Southern Oscilation (ENSO). Crop development was simulated using the thermal time method, for 14 sowing dates, from August until mid-February, for every year during the period from 1968 to 2011, covered by database of Meteorological Station of Santa Maria, RS. For calculating the water deficit, the 13 soils were grouped into six groups with similar water storage capacity (CAD) and infiltration capacity. The water deficit was calculated from daily water balance. Data analysis consisted of analysis of variance, means comparison tests and analysis of probability distribution for the variables: duration of crop developmental sub-phases and the whole developmental cycle of the sunflower, water deficit in the sub-phases and whole developmental cycle. The length of the sub-phases and the development cycle of the sunflower crop are variable depending on sowing date. The length of the developmental sub-phases that occur from sowing until flower bud visible of sunflower are higher in the earliest sowing date (01/08). After anthesis, the longer length of developmental sub-phases occurs in the latest sowing (16/02). The lognormal, normal and gamma distributions represent better the development of sunflower to estimate the length of the phases and the whole cycle. At sowing date of 16/12, for 90% probability level, sunflower has the shortest length of the developmental cycle ending the cycle in a maximum of 96 days. The longer length of the sunflower cycle occurs at sowing date of 01/08, which reaches 132 days, at 90% level of occurrence probability. The sowing dates from early October until early November are the ones with the highest water deficit, considering the whole development cycle of the sunflower regardless of soil, a different choice on sowing date reduces the risk and the level of water deficit in sunflower cycle. In the soils in which the water storage capacity is lower, water deficit is greater in sub-phases as in the full cycle of the sunflower compared to other soils and is little variable among the sowing dates. Sunflower Sowings in the first half of August and since December are the ones with the lowest risk occurrence of water deficit during the more critical sub-phase of sunflower crop, at least there are favorable conditions for sowing and initial establishment of plants.
No Brasil nos últimos anos elevou-se o interesse pelo cultivo do girassol. Quando submetida à deficiência hídrica a cultura do girassol apresenta redução na produtividade. Para contornar esse problema, é necessário calcular a provável deficiência hídrica nos subperíodos críticos e no ciclo de desenvolvimento do girassol para cada uma das diferentes datas de semeadura. O objetivo desse trabalho foi determinar os valores prováveis de duração dos subperíodos e do ciclo de desenvolvimento e obter as datas de semeadura com menor risco de deficiência hídrica e a probabilidade de ocorrência de diferentes níveis de déficit hídrico durante os subperíodos de desenvolvimento do girassol semeado em datas de semeadura distintas, considerando a capacidade de armazenamento de água nos diferentes solos da região central do RS. Também determinar a probabilidade de ocorrência de deficiência hídrica para os diferentes anos classificados conforme o fenômeno El Niño Oscilação Sul (ENOS). O desenvolvimento da cultura foi simulado por meio do método da soma térmica, para 14 datas de semeadura, do início do mês de agosto até meados de fevereiro, para cada ano do banco de dados da Estação Meteorológica Principal de Santa Maria, RS, utilizando o período de 1968 a 2011. Para calcular a deficiência hídrica, os 13 solos da região foram agrupados em seis grupos que apresentam características semelhantes de capacidade de armazenamento de água disponível (CAD) e capacidade de infiltração. As deficiências hídricas foram determinadas a partir do balanço hídrico diário. A análise dos dados consistiu na análise da variância, teste de comparação de médias e análise de distribuição de probabilidade para as variáveis: duração dos subperíodos e do ciclo de desenvolvimento do girassol, deficiência hídrica nos subperíodos e no ciclo do girassol. A duração dos subperíodos e do ciclo de desenvolvimento do girassol é variável conforme a data de semeadura. A duração dos subperíodos que ocorrem da semeadura até o botão floral visível do girassol são maiores na primeira data de semeadura (01/08). Após a antese a maior duração dos subperíodos ocorre na semeadura mais tardia (16/02). As distribuições lognormal, normal e gama representam melhor o desenvolvimento do girassol para estimar a duração dos subperíodos e do ciclo. Na data de semeadura de 16/12, ao nível de 90% de probabilidade de ocorrência, o girassol tem a menor duração do ciclo, completando o ciclo em no máximo de 96 dias. A maior duração do ciclo do girassol ocorre na data de semeadura de 01/08, na qual alcança 132 dias, em nível de 90% de probabilidade de ocorrência. As datas de semeadura de início de outubro até o início de novembro são as que apresentam a maior deficiência hídrica, considerando todo o ciclo de desenvolvimento do girassol independente do solo; a escolha de outra data de semeadura reduz o risco e o nível de deficiência hídrica durante o ciclo do girassol. Nos solos em que a capacidade de armazenamento de água disponível é menor, a deficiência hídrica é maior tanto nos subperíodos quanto no ciclo do girassol em relação aos demais solos e é pouco variável ao longo das datas de semeadura. Semeaduras de girassol na primeira quinzena de agosto e a partir do mês de dezembro são as que apresentam os menores riscos de ocorrer deficiência hídrica no transcorrer do subperíodo mais crítico do girassol, desde que se tenham condições favoráveis para a semeadura e o estabelecimento inicial das plantas.

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior
The expansion of soybean production area in Planosols is rather limited by the high frequency of occurrence of excess water, leading to reduced availability of oxygen in the root zone, reduced photosynthesis, as well as productivity, depending on its duration and developmental phase of plants it occurs. The aim of this study was to identify sowing dates with smaller risk of excess water to the subperiods and crop cycle, taking into account three relative maturity groups of soybean cultivars and water storage capacity of Planosols in the central region of Rio Grande do Sul State. The simulation of soybean development and the calculation of crop daily sequential water balance were performed at different sowing dates in each year from August 1968 to July 2012. Thus, the change of soil water storage and the water surpluses in the different soybean developmental phases were quantified for each sowing date. Data from days with excess water were submitted to analysis of variance and Scott-Knott test at 5% probability, and the sources of variation were sowing dates, soils and their interaction. These data also were submitted to the probability distribution analysis, using the chi-square and Kolmogorov-Smirnov tests to verify the probability density function that best fit the data distribution. The greatest number of fittings for the development cycle and subperiods were obtained by the Gamma and Weibull functions, respectively. October's sowings have the highest risk of excess water during the crop cycle. Subperiod sowing-emergence shows up as the most limiting to define the sowing date. Due to the lowest risk of excess water in this sub-period, the sowing carried out after November 1st are the most favorable for soybean sowing in Planosols.
A expansão da área de produção de soja em Planossolos é bastante limitada pela elevada frequência de ocorrência de excesso hídrico, ocasionando redução na disponibilidade de oxigênio na zona radicular, redução da fotossíntese, assim como da produtividade, dependendo da duração do excesso e do subperíodo de desenvolvimento das plantas em que ocorre. O objetivo deste trabalho foi identificar datas de semeadura com menor risco de ocorrência de excesso hídrico para os subperíodos e ciclo da cultura, considerando três grupos de maturidade relativa de cultivares de soja e a capacidade de armazenamento de água dos Planossolos da região central do Rio Grande do Sul. A simulação do desenvolvimento da soja e o cálculo do balanço hídrico sequencial diário da cultura foram realizados em diferentes datas de semeadura de cada ano do período de agosto de 1968 a julho de 2012. Assim, a variação do armazenamento hídrico no solo e a ocorrência de excedentes hídricos nos diferentes subperíodos de desenvolvimento da soja foram quantificadas para cada data de semeadura. Os dados de dias de excesso hídrico foram submetidos à análise de variância e teste de Scott-Knott, a 5% de probabilidade de erro, sendo que as fontes de variação constaram das datas de semeadura, os solos e a sua interação. Os dados também foram submetidos à análise de distribuição de probabilidades, utilizando-se os testes qui-quadrado e Kolmogorov-Smirnov para verificar a função densidade probabilidade que melhor se ajustou à distribuição dos dados. O maior número de ajustes para o ciclo de desenvolvimento e para os subperíodos foram obtidos para as funções gama e weibull, respectivamente. As semeaduras realizadas no mês de outubro são as de maior risco de ocorrência de excesso hídrico ao longo do ciclo da cultura. O subperíodo semeadura-emergência mostra-se como o mais limitante para a definição da data de semeadura. Devido ao menor risco de ocorrência de excesso hídrico neste subperíodo as semeaduras realizadas após o dia primeiro de novembro são as mais favoráveis para a semeadura da soja em Planossolos.

Nesta dissertação, o modelo de seleção de carteiras de Markowitz será estendido com uma análise em tempo discreto e hipóteses mais realísticas. Um produto tensorial finito de densidades Erlang será usado para aproximar a densidade de probabilidade multivariada dos retornos discretos uniperiódicos de ativos dependentes. A Erlang é um caso particular da distribuição Gama. Uma mistura finita pode gerar densidades multimodais não-simétricas e o produto tensorial generaliza este conceito para dimensões maiores. Assumindo que a densidade multivariada foi independente e identicamente distribuída (i.i.d.) no passado, a aproximação pode ser calibrada com dados históricos usando o critério da máxima verossimilhança. Este é um problema de otimização em larga escala, mas com uma estrutura especial. Assumindo que esta densidade multivariada será i.i.d. no futuro, então a densidade dos retornos discretos de uma carteira de ativos com pesos não-negativos será uma mistura finita de densidades Erlang. O risco será calculado com a medida Downside Risk, que é convexa para determinados parâmetros, não é baseada em quantis, não causa a subestimação do risco e torna os problemas de otimização uni e multiperiódico convexos. O retorno discreto é uma variável aleatória multiplicativa ao longo do tempo. A distribuição multiperiódica dos retornos discretos de uma seqüência de T carteiras será uma mistura finita de distribuições Meijer G. Após uma mudança na medida de probabilidade para a composta média, é possível calcular o risco e o retorno, que levará à fronteira eficiente multiperiódica, na qual cada ponto representa uma ou mais seqüências ordenadas de T carteiras. As carteiras de cada seqüência devem ser calculadas do futuro para o presente, mantendo o retorno esperado no nível desejado, o qual pode ser função do tempo. Uma estratégia de alocação dinâmica de ativos é refazer os cálculos a cada período, usando as novas informações disponíveis. Se o horizonte de tempo tender a infinito, então a fronteira eficiente, na medida de probabilidade composta média, tenderá a um único ponto, dado pela carteira de Kelly, qualquer que seja a medida de risco. Para selecionar um dentre vários modelos de otimização de carteira, é necessário comparar seus desempenhos relativos. A fronteira eficiente de cada modelo deve ser traçada em seu respectivo gráfico. Como os pesos dos ativos das carteiras sobre estas curvas são conhecidos, é possível traçar todas as curvas em um mesmo gráfico. Para um dado retorno esperado, as carteiras eficientes dos modelos podem ser calculadas, e os retornos realizados e suas diferenças ao longo de um backtest podem ser comparados.
In this thesis, Markowitz’s portfolio selection model will be extended by means of a discrete time analysis and more realistic hypotheses. A finite tensor product of Erlang densities will be used to approximate the multivariate probability density function of the single-period discrete returns of dependent assets. The Erlang is a particular case of the Gamma distribution. A finite mixture can generate multimodal asymmetric densities and the tensor product generalizes this concept to higher dimensions. Assuming that the multivariate density was independent and identically distributed (i.i.d.) in the past, the approximation can be calibrated with historical data using the maximum likelihood criterion. This is a large-scale optimization problem, but with a special structure. Assuming that this multivariate density will be i.i.d. in the future, then the density of the discrete returns of a portfolio of assets with nonnegative weights will be a finite mixture of Erlang densities. The risk will be calculated with the Downside Risk measure, which is convex for certain parameters, is not based on quantiles, does not cause risk underestimation and makes the single and multiperiod optimization problems convex. The discrete return is a multiplicative random variable along the time. The multiperiod distribution of the discrete returns of a sequence of T portfolios will be a finite mixture of Meijer G distributions. After a change of the distribution to the average compound, it is possible to calculate the risk and the return, which will lead to the multiperiod efficient frontier, where each point represents one or more ordered sequences of T portfolios. The portfolios of each sequence must be calculated from the future to the present, keeping the expected return at the desired level, which can be a function of time. A dynamic asset allocation strategy is to redo the calculations at each period, using new available information. If the time horizon tends to infinite, then the efficient frontier, in the average compound probability measure, will tend to only one point, given by the Kelly’s portfolio, whatever the risk measure is. To select one among several portfolio optimization models, it is necessary to compare their relative performances. The efficient frontier of each model must be plotted in its respective graph. As the weights of the assets of the portfolios on these curves are known, it is possible to plot all curves in the same graph. For a given expected return, the efficient portfolios of the models can be calculated, and the realized returns and their differences along a backtest can be compared.

博士
國立中正大學
財務金融研究所
93
The Studies On Taiwan’s OTC Market Using Value-at-Risk of Probability Density Function Abstract This paper provides a different approach to capture the effects of the Probability Density Function with “Value-at-Risk” (VaR). We first apply the Knight, Satchell and Tran’s (1995) Double-gamma distribution to measure VaR and examine models including EWMA, Mixture Normal, Student’s t, Gaussian Kernel, Epanechnikov Kernel, GEV and GPD. The results show that the EWMA is the best performer with one-day holding period and the Student’s t model tends to outperform the others with five-day holding period. In addition, the Double-gamma model also performs well in asset portfolio. Keywords: Value at Risk, Probability Density Function, Kernel Density Function, Extreme Value Theory

This Dissertation builds a routine to extract inflation expectations from options through the risk-neutral distribution methodology, based on the Black-Scholes pricing model. The data includes cap and floor options, equivalents of call and put options with inflation as the underlying, with some technical differences. The methodology includes extracting implied volatilities from observed prices of caps and floors, then use a cubic interpolation on implied volatilities, to be able to fit to the volatility smile. Further, taking advantage of Breeden and Litzenberger (1976), the initial density functions are calculated as the second derivative of the option price in the Black-Scholes model. For a more encapsulating approach, the cap and floor density functions were aggregated, thereby being able to compare net effects with objectives of the monetary policy. A robustness test was also performed to analyze price variations along the two days prior, and after, the announcements of each policy, using floor options and inflation-linked swaps. The results show that, the ABSPP & CBPP3 and the TLTRO I announcements registered decreases, although the variations in prices and densities are small, therefore suggesting the importance of anticipation of measures and of other transmission channels to assess the effects of these policies. This is in line with the September 2015 report by the Vice-President of the ECB, Vítor Constâncio, that acclaims the effectiveness of the non-conventional monetary policies in controlling low inflation expectations, although affirming that the below, but close to, 2% goal of inflation has not been met.
Esta Dissertação elabora uma rotina para extrair expetativas de inflação através de uma metodologia de distribuição neutra de risco, baseada no modelo de preços de Black-Scholes. A data utilizada inclui opções cap e floor, equivalentes às calls e puts mas com inflação subjacente, com certas diferenças tecnicas. A metodologia engloba extrair volatilidades implicitas dos preços das opções observadas, procedendo uma interpolação cúbica nas volatilidades implicitas, para ser possível aplicar um fit ao grafico da volatilidade. Tomando partido do trabalho realisado por Breenden e Litzenberger (1978), as funções de densidade iniciais são calculadas apartir da segunda derivada da função do preço de opções do modelo Black-Scholes. Para uma perspectiva completa, procedeu-se ao agregamento das funções de densidade dos caps e floors numa única garantindo também as comparações com objectivos da politica monetária. Um teste de rebustez foi também conduzido através das variações dos preços de floors e swaps, dois dias antes e depois dos anuncios das politicas. Os resultados indicam que para quantificar com sucesso os efeitos destas politicas, um periodo de tempo maior deve ser considerado, sugerindo ainda a importância da antecipação das medidas e outros canais de transmissão para averiguar os efeitos da mesma. Isto está em linha com o documento apresentado em Setembro de 2015 por Vítor Constâncio, VP do ECB, que enalteceu a efetividade das medidas de politica monetária não convencionais no combate às expectativas de baixa inflação, acrescentado ainda que o objectivo do banco central de inflação inferior, mas perto de, 2% ainda não foi atingido.

碩士
國立中正大學
財務金融所
94
The main point is to discuss the character of the implied risk neutral probability density functions about the Taiwan stock index option market. We adopt the Black option pricing model, LAP option pricing model, Corrado and Su option pricing model, asymmetry ch-square option pricing model, and the mixed normal option pricing model to undertake our empirical test. In the empirical analysis about these models, we undertake the in-sample analysis and the out of sample analysis to compare the pricing performance among these option pricing models. In addition, we undertake the analysis of the hedging performance about these option pricing models. We attempt to provide investors one of the best option pricing models so that investors can use the option pricing model to proceed pricing and hedging in the option market.

Dissertação de Mestrado em Métodos Quantitativos em Finanças apresentada à Faculdade de Ciências e Tecnologia
Com esta dissertação pretende-se estudar e implementar vários métodos de estimação de funções densidade de probabilidade neutras face ao risco, identificando as perspectivas dos investidores quanto à evolução do mercado. Assim, este trabalho incide sobre preços teóricos de opções europeias, gerados pelo modelo de Black-Scholes, e sobre preços de mercado de opções do mesmo estilo sobre o índice S&P 500, transacionadas no Chicago Board Options Exchange (CBOE) e relativas ao dia 4 de janeiro de 2016 para diferentes maturidades.Inicialmente consideraram-se preços teóricos, gerados pelo modelo de Black-Scholes, e compararam-se, através da análise gráfica e do cálculo do root mean square error (RMSE), com os preços estimados por diferentes métodos. Os métodos utilizados neste estudo foram a mistura de duas distribuições lognormais, a expansão de Edgeworth e o método da volatilidade implícita proposto por Shimko. Seguidamente, estimaram-se os parâmetros relativos aos diferentes métodos e através deles produziram-se as funções densidade de probabilidade neutras face ao risco. Após esta estimação, efetuou-se a análise das perspectivas de mercado, utilizando opções sobre o índice S&P 500 para sete maturidades diferentes. Deste modo, compararam-se os preços das opções de mercado com os preços estimados pelo modelo de Black-Scholes, pela mistura de duas distribuições lognormais, pela expansão de Edgeworth e pelo método da volatilidade implícita proposto por Shimko. De seguida, estimaram-se os parâmetros dos diferentes métodos e, através destes, produziram-se as funções densidade de probabilidade neutras face ao risco. Por fim, analisaram-se algumas variáveis estatísticas das funções densidade de probabilidade obtidas, tais como a média, a variância, o coeficiente de assimetria e o excesso de curtose, de modo a retirar conclusões acerca das expectativas dos investidores em relação à evolução do mercado.
This dissertation intends to study and implement several methods of estimating risk-neutral probability density functions, identifying investors' perspectives on market evolution. Thus, this work focuses on the theoretical prices of European options, generated by the Black-Scholes model, and on market prices of similar options on the S& P 500 index, relative to January 4, 2016 for different maturities, which are traded on the Chicago Board Options Exchange (CBOE).Initially, theoretical prices, generated by the Black-Scholes model, were compared, using the graphical analysis and the calculation of the root mean square error (RMSE), with the prices estimated by different methods. The methods used in this study are a mixture of two lognormal distributions, an Edgeworth expansion and the implicit volatility method proposed by Shimko. Then, the parameters related to different methods were estimated and through them, risk-neutral probability density functions are produced. After this estimation, a market outlook analysis was performed using S&P 500 index options for seven different maturities. The market option prices were then compared with the prices estimated by the Black-Scholes model, the combination of two lognormal distributions, the Edgeworth expansion and the implicit volatility method proposed by Shimko. Afterwards, the parameters of the different methods are estimated and, through these parameters, risk-neutral probability density functions were produced. Finally, some statistical variables of the probability density functions obtained were analyzed, such as the mean, the variance, the asymmetry coefficient and the excess kurtosis, in order to draw conclusions about the investors' expectations regarding market development.

This study is designed to understand how investors risk preferences change, when faced by financial crisis. Option prices implied densities provide information about market movements and risk preferences. The data used are European call options prices on the DJIA with a time to maturity of four weeks. This paper obtains the risk aversion estimates by the extraction of options implied risk neutral densities and their translation to real world densities, applied to three crisis : Dotcom bubble in 2001; Subprime mortgage in 2008 and European sovereign debt in 2011. RND is achieved by the use of two parametric methods: mixture of lognormal densities (MLN); and generalised beta distribution of the second kind (GB2). The risk transformation procedure from RND to RWD (real world densities) is achieved by applying the power utility function. The RND empirical results imply that the GB2 method is disregarded due to inferior quality whilst the MLN produces results of higher uncertainty and expected future results of index levels corrected downward. The risk aversion estimates obtained from the RWD generation process do not present any evident pattern of evolution from a stable financial period to one of financial shock. It is also important to mention that for some periods the risk aversion reached negative values. This is a surprising result due to existent theoretical assumption of positive risk aversion. Overall, this study documents inconclusive results. Nevertheless, several important topics are left for future research. Interesting developments would consist: either replicating this study considering more expiration dates, bearing in mind that too many periods would imply an extremely generalised risk aversion estimate that would be counterproductive for achieving the objective of this paper; apply another method from the available literature; or testing the negative estimates with more sophisticated models that are beyond the focus of this paper.

Dans cette thèse, j’exploite le cadre d’analyse de données fonctionnelles et développe l’analyse d’inférence et de prédiction, avec une application à des sujets sur les marchés financiers. Cette thèse est organisée en trois chapitres. Le premier chapitre est un article co-écrit avec Marine Carrasco. Dans ce chapitre, nous considérons un modèle de régression linéaire fonctionnelle avec une variable prédictive fonctionnelle et une réponse scalaire. Nous effectuons une comparaison théorique des techniques d’analyse des composantes principales fonctionnelles (FPCA) et des moindres carrés partiels fonctionnels (FPLS). Nous déterminons la vitesse de convergence de l’erreur quadratique moyen d’estimation (MSE) pour ces méthodes. Aussi, nous montrons cette vitesse est sharp. Nous découvrons également que le biais de régularisation de la méthode FPLS est plus petit que celui de FPCA, tandis que son erreur d’estimation a tendance à être plus grande que celle de FPCA. De plus, nous montrons que le FPLS surpasse le FPCA en termes de prédiction avec moins de composantes. Le deuxième chapitre considère un modèle autorégressif entièrement fonctionnel (FAR) pour prèvoir toute la courbe de rendement du S&P 500 a la prochaine journée. Je mène une analyse comparative de quatre techniques de Big Data, dont la méthode de Tikhonov fonctionnelle (FT), la technique de Landweber-Fridman fonctionnelle (FLF), la coupure spectrale fonctionnelle (FSC) et les moindres carrés partiels fonctionnels (FPLS). La vitesse de convergence, la distribution asymptotique et une stratégie de test statistique pour sélectionner le nombre de retard sont fournis. Les simulations et les données réelles montrent que les méthode FPLS performe mieux les autres en terme d’estimation du paramètre tandis que toutes ces méthodes affichent des performances similaires en termes de prédiction. Le troisième chapitre propose d’estimer la densité de neutralité au risque (RND) dans le contexte de la tarification des options, à l’aide d’un modèle fonctionnel. L’avantage de cette approche est qu’elle exploite la théorie d’absence d’arbitrage et qu’il est possible d’éviter toute sorte de paramétrisation. L’estimation conduit à un problème d’inversibilité et la technique fonctionnelle de Landweber-Fridman (FLF) est utilisée pour le surmonter.
In this thesis, I exploit the functional data analysis framework and develop inference, prediction and forecasting analysis, with an application to topics in the financial market. This thesis is organized in three chapters. The first chapter is a paper co-authored with Marine Carrasco. In this chapter, we consider a functional linear regression model with a functional predictor variable and a scalar response. We develop a theoretical comparison of the Functional Principal Component Analysis (FPCA) and Functional Partial Least Squares (FPLS) techniques. We derive the convergence rate of the Mean Squared Error (MSE) for these methods. We show that this rate of convergence is sharp. We also find that the regularization bias of the FPLS method is smaller than the one of FPCA, while its estimation error tends to be larger than that of FPCA. Additionally, we show that FPLS outperforms FPCA in terms of prediction accuracy with a fewer number of components. The second chapter considers a fully functional autoregressive model (FAR) to forecast the next day’s return curve of the S&P 500. In contrast to the standard AR(1) model where each observation is a scalar, in this research each daily return curve is a collection of 390 points and is considered as one observation. I conduct a comparative analysis of four big data techniques including Functional Tikhonov method (FT), Functional Landweber-Fridman technique (FLF), Functional spectral-cut off (FSC), and Functional Partial Least Squares (FPLS). The convergence rate, asymptotic distribution, and a test-based strategy to select the lag number are provided. Simulations and real data show that FPLS method tends to outperform the other in terms of estimation accuracy while all the considered methods display almost the same predictive performance. The third chapter proposes to estimate the risk neutral density (RND) for options pricing with a functional linear model. The benefit of this approach is that it exploits directly the fundamental arbitrage-free equation and it is possible to avoid any additional density parametrization. The estimation problem leads to an inverse problem and the functional Landweber-Fridman (FLF) technique is used to overcome this issue.