Ce travail de thèse poursuit une perspective double dans l'usage conjoint des méthodes de Monte Carlo séquentielles (MMS) et de l'algorithme Espérance-Maximisation (EM) dans le cadre des modèles de Markov cachés présentant une structure de dépendance markovienne d'ordre supérieur à 1 au niveau de la composante inobservée. Tout d'abord, nous commençons par un exposé succinct de l'assise théorique des deux concepts statistiques à travers les chapitres 1 et 2 qui leurs sont consacrés. Dans un second temps, nous nous intéressons à la mise en pratique simultanée des deux concepts au chapitre 3 et ce dans le cadre usuel où la structure de dépendance est d'ordre 1. L'apport des méthodes MMS dans ce travail réside dans leur capacité à approximer efficacement des fonctionnelles conditionnelles bornées, notamment des quantités de filtrage et de lissage dans un cadre non linéaire et non gaussien. Quant à l'algorithme EM, il est motivé par la présence à la fois de variables observables et inobservables (ou partiellement observées) dans les modèles de Markov Cachés et singulièrement les mdèles de volatilité stochastique étudié. Après avoir présenté aussi bien l'algorithme EM que les méthodes MCs ainsi que quelques unes de leurs propriétés dans les chapitres 1 et 2 respectivement, nous illustrons ces deux outils statistiques au travers de la calibration d'un modèle de volatilité stochastique. Cette application est effectuée pour des taux change ainsi que pour quelques indices boursiers au chapitre 3. Nous concluons ce chapitre sur un léger écart du modèle de volatilité stochastique canonique utilisé ainsi que des simulations de Monte Carlo portant sur le modèle résultant. Enfin, nous nous efforçons dans les chapitres 4 et 5 à fournir les assises théoriques et pratiques de l'extension des méthodes Monte Carlo séquentielles notamment le filtrage et le lissage particulaire lorsque la structure markovienne est plus prononcée. En guise d'illustration, nous donnons l'exemple d'un modèle de volatilité stochastique dégénéré dont une approximation présente une telle propriété de dépendance.

GARCH models have been widely used in finance to model volatility ever since the introduction of the ARCH model and its extension to the generalized ARCH (GARCH) model. Lately, there has been growing interest in modelling seasonal volatility, most recently with the introduction of the multiplicative seasonal GARCH models. As an application of the multiplicative seasonal GARCH model with real data, call prices from the major stock market index of India are calculated using estimated parameter values. It is shown that a multiplicative seasonal GARCH option pricing model outperforms the Black-Scholes formula and a GARCH(1,1) option pricing formula. A parametric bootstrap procedure is also employed to obtain an interval approximation of the call price. Narrower confidence intervals are obtained using the multiplicative seasonal GARCH model than the intervals provided by the GARCH(1,1) model for data that exhibits multiplicative seasonal GARCH volatility.

My dissertation consists of three essays in which I document trends in earnings and income volatility, estimate potential causal mechanisms for changing volatility, and examine the long-term consequences of parental income volatility for children. In essay 2 I document trends in earnings and income volatility of individuals and families using matched data in the March Current Population survey from 1973 to 2009. Essay 3 advances the literature on volatility, using matched data from the CPS to identify demographic and labor market correlates of earnings volatility within education-birth year cohorts. This study collapses the cross-sectional CPS into a pseudo-panel and then estimates the association between earnings volatility and race, local economic activity, and industry, accounting for endogeneity and sample selection bias. In essay 4 I use data linked across generations in the Panel Study of Income Dynamics to estimate the relationship between exposure to volatile income during childhood and a set of socioeconomic outcomes in adulthood. The empirical framework is an augmented intergenerational income mobility model that includes controls for income volatility. I find that family income volatility rose by 38 percent over the past four decades, likely driven both by rising volatility of earnings and non means-tested non-labor income. Rising family income volatility occurs across race, education, and family structure. From essay 3, I find that individuals with lower mean earnings have higher earnings volatility. Earnings volatility is also weakly related to race, decreases when young and then rises while workers are still within prime working years. Industry and local economic conditions are significantly related to the occurrence of earnings volatility after accounting for education, though these links differ between men and women. Finally, when examining the intergenerational consequences of volatility, a weak negative association occurs between family income instability during childhood and adult educational outcomes in essay 4.

Modelling the tails of distributions is important in many fields, such as environmental science, hydrology, insurance, engineering and finance, where the risk of unusually large or small events are of interest. This thesis applies extreme value models in neonatal and finance studies and develops novel extreme value modelling for financial applications, to overcome issues associated with the dependence induced by volatility clustering and threshold choice. The instability of preterm infants stimulates the interests in estimating the underlying variability of the physiology measurements typically taken on neonatal intensive care patients. The stochastic volatility model (SVM), fitted using Bayesian inference and a particle filter to capture the on-line latent volatility of oxygen concentration, is used in estimating the variability of medical measurements of preterm infants to highlight instabilities resulting from their under-developed biological systems. Alternative volatility estimators are considered to evaluate the performance of the SVM estimates, the results of which suggest that the stochastic volatility model provides a good estimator of the variability of the oxygen concentration data and therefore may be used to estimate the instantaneous latent volatility for the physiological measurements of preterm infants. The classical extreme value distribution, generalized pareto distribution (GPD), with the peaks-over-threshold (POT) method to ameliorate the impact of dependence in the extremes to infer the extreme quantile of the SVM based variability estimates. Financial returns typically show clusters of observations in the tails, often termed “volatility clustering” which creates challenges when applying extreme value models, since classical extreme value theory assume independence of underlying process. Explicit modelling on GARCH-type dependence behaviour of extremes is developed by implementing GARCH conditional variance structure via the extreme value model parameters. With the combination of GEV and GARCH models, both simulation and empirical results show that the combined model is better suited to explain the extreme quantiles. Another important benefit of the proposed model is that, as a one stage model, it is advantageous in making inferences and accounting for all uncertainties much easier than the traditional two stage approach for capturing this dependence. To tackle the challenge threshold choice in extreme value modelling and the generally asymmetric distribution of financial data, a two tail GPD mixture model is proposed with Bayesian inference to capture both upper and lower tail behaviours simultaneously. The proposed two tail GPD mixture modelling approach can estimate both thresholds, along with other model parameters, and can therefore account for the uncertainty associated with the threshold choice in latter inferences. The two tail GPD mixture model provides a very flexible model for capturing all forms of tail behaviour, potentially allowing for asymmetry in the distribution of two tails, and is demonstrated to be more applicable in financial applications than the one tail GPD mixture models previously proposed in the literature. A new Value-at-Risk (VaR) estimation method is then constructed by adopting the proposed mixture model and two-stage method: where volatility estimation using a latent volatility model (or realized volatility) followed by the two tail GPD mixture model applied to independent innovations to overcome the key issues of dependence, and to account for the uncertainty associated with threshold choice. The proposed method is applied in forecasting VaR for empirical return data during the current financial crisis period.

In this master thesis we have examined the possibility of pricing multiple American options, on an underlying asset with discrete dividends, with a finite difference method. We have found a good and stable way to price one American option by solving the BSM PDE backwards, while also calculating the Greeks of the option with automatic differentiation. The list of Greeks for an option is quite extensive since we have been using a local volatility surface. We have also tried to find a way to price several American options simultaneously by solving a forward PDE. Unfortunately, we haven't found any previous work that we could use with our local volatility surface, while still keeping down the computational time. The closest we got was to calculate the value of a compound option in a forward mode, but in order to use this to value an American option, we needed to go through an iterative process which calculated a forward or backward European PDE in every step.

Im frühen Sonnensystem fanden während der Kondensation der chemischen Elemente volatilitätskontrollierte Fraktionierungsprozesse statt. Gegenstand dieser Doktorarbeit sind Fraktionierungen refraktär-lithophiler Elemente in einzelnen Chondritkomponenten sowie zwischen Bulk-Chondriten, Achondriten und Planeten. Mittels laser ablation inductively coupled plasma mass spectrometry wurden die Gehalte der Seltenen Erden (REE) sowie von Nb, Ta, Zr und Hf analysiert. Einzelne Chondritkomponenten wurden in-situ an dem CV-Chondrit Leoville untersucht. Von den Bulk-Chondriten, Achondriten und terrestrischen Proben wurden Gesamtgesteinsproben durch Laserschmelzen unter aerodynamischer Levitation angefertigt. Die Untersuchung der verschiedenen Bestandteile des Leoville-Chondrits ergab, dass die refraktären Einschlüsse volatilitätskontrollierte fraktionierte REE group-II-Muster und subchondritische Nb/Ta-Verhältnisse aufweisen. Sie sind demzufolge aus einem residualen Gas entstanden, von dem zuvor eine ultrarefraktäre Komponente isoliert worden war. Chondren haben zumeist relativ unfraktionierte REE-Muster sowie unfraktionierte Zr/Hf- und Nb/Ta-Verhältnisse. Einige Typ-1-Chondren, die Al-reichen Chondren und die Chondritmatrix weisen jedoch fraktionierte REE-Muster auf. Dies ist ein Hinweis auf Beimengungen refraktären Materials mit REE group-II-Muster. Die Analysen an Bulk-Chondriten zeigen, dass kohlige Chondrite im Vergleich zu dem CI-Chondrit Orgueil charakteristische volatilitätskontrollierte REE-Muster (ultrarefraktär oder group-II) besitzen, was auf den Einbau refraktärer Komponenten mit fraktionierten Seltenen Erden zurückgeführt wurde. Die Mehrheit der gewöhnlichen, Rumuruti- und Enstatit-Chondrite hat dagegen relativ unfraktionierte REE-Muster. Es konnte gezeigt werden, dass sowohl gewöhnliche, Enstatit- und Rumuruti-Chondrite als auch Proben von Achondriten, Mars, Mond und Erde geringe negative Tm-Anomalien gegenüber dem CI-Chondrit Orgueil aufweisen. Die Objekte des inneren Sonnensystems wurden daher anhand ihrer relativen Gehalte an schweren Seltenen Erden (HREE) in zwei Gruppen eingeteilt: Ein kohliges und ein nichtkohliges Chondrit-Reservoir, dem auch die Achondrite, Mars, Erde und Mond angehören. Es wurde angenommen, dass die Objekte des nichtkohligen Chondrit-Reservoirs die HREE-Verhältnisse des Sonnensystems widerspiegeln; kohlige Chondrite haben dagegen variable Tm-Anomalien, welche durch den Eintrag fraktionierter refraktärer Komponenten in ihre Entstehungsregion zu erklären sind. CI-Chondrite, welche allgemein als die chemisch primitivste Chondritgruppe angesehen werden, hätten in diesem Fall eine positive Tm-Anomalie von 4,8 ± 0,9 % und stimmten somit chemisch nicht mit dem Sonnensystem überein. Durch eine Beimengung von nur 0,2 Gewichtsprozent einer refraktären Komponente mit REE group-II-Muster zu den CI-Chondriten konnte diese Tm-Anomalie erklärt werden.

No description available.

In this thesis we develop and test a new method for interpolating and extrapolating prices of European options. The theoretical base originates from the local variance gamma model developed by Carr (2008), in which the local volatility model by Dupire (1994) is combined with the variance gamma model by Madan and Seneta (1990). By solving a simplied version of the Dupire equation under the assumption of a continuous ve parameter di usion term, we derive a parameterization dened for strikes in an interval of arbitrary size. The parameterization produces positive option prices which satisfy both conditions for absence of arbitrage in a one maturity setting, i.e. all adjacent vertical spreads and buttery spreads are priced non-negatively. The method is implemented and tested in the FX-option market. We suggest two sub-models, one with three and one with ve degrees of freedom. By using a least-square approach, we calibrate the two sub-models against 416 Reuters quoted volatility smiles. Both sub-models succeeds in generating prices within the bid-ask spread for all options in the sample. Compared to the three parameter model, the model with ve parameters calibrates more exactly to market quoted mids but has a longer calibration time. The three parameter model calibrates remarkably quickly; in a MATLAB implementation using a Levenberg-Marquardt algorithm the average calibration time is approximately 1 ms. Both sub-models produce volatility smiles which are C2 and well-behaving. Further, we suggest a technique allowing for arbitrage-free interpolation of calibrated option price functions in the maturity dimension. The interpolation is performed in parameter space, where every set of parameters uniquely determines an option price function. Furthermore, we produce sucient conditions to ensure absence of calendar spread arbitrage when calibrating the proposed model to several maturities. We use this technique to produce implied volatility surfaces which are suciently smooth, satisfy all conditions for absence of arbitrage and fit market quoted volatility surfaces within the bid-ask spread. In the final chapter we use the results for producing Dupire local volatility surfaces and for pricing variance swaps.

In classic GARCH models for financial returns the innovations are usually assumed to be normally distributed. However, it is generally accepted that a non-normal innovation distribution is needed in order to account for the heavier tails often encountered in financial returns. Since the structure of the normal inverse Gaussian (NIG) distribution makes it an attractive alternative innovation distribution for this purpose, we extend the normal GARCH model by assuming that the innovations are NIG-distributed. We use the normal variance mixture interpretation of the NIG distribution to show that a NIG innovation may be interpreted as a normal innovation coupled with a multiplicative random impact factor adjustment of the ordinary GARCH volatility. We relate this new volatility estimate to realised volatility and suggest that the random impact factors are due to a news noise process influencing the underlying returns process. This GARCH model with NIG-distributed innovations leads to more accurate parameter estimates than the normal GARCH model. In order to obtain even more accurate parameter estimates, and since we expect an information gain if we use more data, we further extend the model to cater for high, low and close data, as well as full intraday data, instead of only daily returns. This is achieved by introducing the Brownian inverse Gaussian (BIG) process, which follows naturally from the unit inverse Gaussian distribution and standard Brownian motion. Fitting these models to empirical data, we find that the accuracy of the model fit increases as we move from the models assuming normally distributed innovations and allowing for only daily data to those assuming underlying BIG processes and allowing for full intraday data. However, we do encounter one problematic result, namely that there is empirical evidence of time dependence in the random impact factors. This means that the news noise processes, which we assumed to be independent over time, are indeed time dependent, as can actually be expected. In order to cater for this time dependence, we extend the model still further by allowing for autocorrelation in the random impact factors. The increased complexity that this extension introduces means that we can no longer rely on standard Maximum Likelihood methods, but have to turn to Simulated Maximum Likelihood methods, in conjunction with Efficient Importance Sampling and the Control Variate variance reduction technique, in order to obtain an approximation to the likelihood function and the parameter estimates. We find that this time dependent model assuming an underlying BIG process and catering for full intraday data fits generated data and empirical data very well, as long as enough intraday data is available. / Thesis (Ph.D. (Risk Analysis))--North-West University, Potchefstroom Campus, 2006.

The day-of-the-week effect is a market anomaly that manifests as the cyclical behaviour of traders in the market. This market anomaly was first observed by M.F.M. Osborne (1959). The literature distinguishes between two types of cyclical effects in the market: the cyclical pattern of mean returns and the cyclical pattern of volatility in returns. This dissertation studies and reports on cyclical patterns in the South African market, seeking evidence of the existence of the day-of-the-week effect. In addition, the dissertation aims to investigate the implications of such an effect on hedge fund managers in South Africa. The phenomenon of cyclical volatility and mean returns patterns (day-of-the-week effect) in the South African All-share index returns are investigated by making use of four generalised heteroskedastic conditional autoregressive (GARCH) models. These were based on Nelson's (1991) Exponential GARCH (EGARCH) models. In order to account for the risk taken by investors in the market Engle et al's, (1987) 'in-Mean' (risk factor) effects were also incorporated into the model. To avoid the dummy variable trap, two different approaches were tested for viability in testing for the day-of- the-week effect. In the first approach, one day is omitted from the equation so as to avoid multi-colinearity in the model. The second approach allows for the restriction of the daily dummy variables where all the parameters of the daily dummy variables adds up to zero. This dissertation found evidence of a mean returns effect and a volatility effect (day-of-the- week effect) in South Africa's All-share index returns data (where Wednesdays have been omitted from the GARCH equations). This holds significant implications for hedge fund managers. as hedge funds are very sensitive to volatility patterns in the market, because of their leveraged trading activities. As a result of adverse price movements, hedge fund managers employ strict risk management processes and constantly rebalance their portfolios according to a mandate, to avoid incurring losses. This rebalancing typically involves the simultaneous opening of new positions and closing out of existing positions. Hedge fund managers run the risk of incurring losses should they rebalance their portfolios on days on which the volatility in market returns is high. This study proves the existence of the day-of-the-week effect in the South African market. These results are further confirmed by the evidence of the trading volumes of the JSE's All-share index data for the period of the study. The mean returns effect (high mean returns) and low volatility found on Thursdays, coincide with the evidence that trading volumes on the JSE on Thursdays are the highest of all the days of the week. The volatility effect on Fridays, (high volatility in returns) is similarly correlated with the evidence of the trading volumes found in the JSE's All-share index data for the period of the study. Accordingly. hedge fund managers would be advised to avoid rebalancing their portfolios on Fridays, which show evidence of high volatility patterns. Hedge fund managers are advised to rather rebalance their portfolios on Thursdays, which show evidence of high mean returns patterns, low volatility patterns and high liquidity. / Thesis (M.Com. (Risk Management))--North-West University, Potchefstroom Campus, 2006.