Value-at-Risk (VaR) model fails to predict financial risk accurately especially during financial crises. This is mainly due to the model’s inability to calibrate new market information and the fact that the risk measure is characterised by poor tail risk quantification. An alternative approach which comprises of the Expected Shortfall measure and the Lognormal Jump-Diffusion (LJD) model has been developed to address the aforementioned shortcomings of VaR. This model is called the Simulated-Expected-Shortfall (SES) model. The Maximum Likelihood Estimation (MLE) approach is used in determining the parameters of the LJD model since it’s more reliable and authenticable when compared to other nonconventional parameters estimation approaches mentioned in other literature studies. These parameters are then plugged into the LJD model, which is simulated multiple times in generating the new loss dataset used in the developed model. This SES model is statistically conservative when compared to peers which means it’s more reliable in predicting financial risk especially during a financial crisis. / Statistics / M.Sc. (Statistics)

In the present thesis we deal with several possible approaches to financial de- rivatives pricing. In the first part, we introduce the basic types of derivatives and the methods of trading. Furthermore, we present several models for the valuati- on of specific financial derivative, i.e. options. Firstly we describe Black-Scholes model in detail, which considers that the development of the underlying asset price is governed by Wiener process. Following are the jumps diffusion models that are extension of the Black-Scholes model with jumps. Then we get to jump models, which are based on Lévy processes. Finally, we will deal with the model, which considers that the development of the underlying asset price is governed by fractional Brownian motion with Hurst's coefficient greater than 1/2. All models are suplemented with sample examples. 1

In life insurance contracts, actuaries generally value premiums using deterministic mortality rates and interest rates. They have ignored them stochastically in most of the studies. However it is known that neither interest rates nor mortality rates are constant. It is also known that companies may encounter insolvency problems such as ruin, so the ruin probability need to be added to the valuation of the life insurance contracts process. Insurance companies should model their surplus processes to price some types of life insurance contracts and to see risk position. In this study, mortality rates and surplus processes are modeled and financial strength of companies are utilized when pricing life insurance contracts.

In this thesis, we investigate the potential of improving demand management activities in the global supply chains. In the increasingly global world, commerce is becoming more complex with an incredible amount of internal and external information available for businesses to select, analyze, understand and react. We identify opportunities for companies to convert data and business information into actionable intelligence. We first study the logistics industry with real data. In the Less-than-Truckload (LTL) market, we analyze an extensive historical shipment database to identify important factors to estimate LTL market rates. Quantifying critical expert knowledge, we develop a price estimation model to help shippers reduce their logistics cost and carriers to better manage their demand. In our second study, we analyze a global supply chain in the high tech industry. Using the demand dependency structure of certain products, we identify collaboration opportunities in the ordering practices that results in increased forecast accuracy. In our third study, we focus on using historical product adoption patterns for developing good pre-launch forecasts for new product introductions. Through a normalization approach and algebraic estimation procedures that use intuitive parameters, our models provide opportunities to significantly improve pre-launch forecast accuracy. Finally, in our fourth study, we develop novel approaches for modeling and mitigating the impact of demand seasonality in new product diffusion context. Focusing mainly on practical applications, our research shows that companies can find innovative ways for turning raw data into valuable insights leading to better demand management activities.

Doctor Scientiae - DSc / Robust Spectral Methods for Solving Option Pricing Problems by Edson Pindza PhD thesis, Department of Mathematics and Applied Mathematics, Faculty of Natural Sciences, University of the Western Cape Ever since the invention of the classical Black-Scholes formula to price the financial derivatives, a number of mathematical models have been proposed by numerous researchers in this direction. Many of these models are in general very complex, thus closed form analytical solutions are rarely obtainable. In view of this, we present a class of efficient spectral methods to numerically solve several mathematical models of pricing options. We begin with solving European options. Then we move to solve their American counterparts which involve a free boundary and therefore normally difficult to price by other conventional numerical methods. We obtain very promising results for the above two types of options and therefore we extend this approach to solve some more difficult problems for pricing options, viz., jump-diffusion models and local volatility models. The numerical methods involve solving partial differential equations, partial integro-differential equations and associated complementary problems which are used to model the financial derivatives. In order to retain their exponential accuracy, we discuss the necessary modification of the spectral methods. Finally, we present several comparative numerical results showing the superiority of our spectral methods.

In this masterthesis I have rewied basic approaches to volatility estimating. These approaches are based on classical and Bayesian statistics. I have applied the volatility models for the purpose of volatility forecasting of a different foreign exchange (EURUSD, GBPUSD and CZKEUR) in the different period (from a second period to a day period). I formulate the models EWMA, GARCH, EGARCH, IGARCH, GJRGARCH, jump diffuison with constant volatility and jump diffusion model with stochastic volatility. I also proposed an MCMC algorithm in order to estimate the Bayesian models. All the models we estimated as univariate models. I compared the models according to Mincer Zarnowitz regression. The most successfull model is the jump diffusion model with a stochastic volatility. On the second place they were the GJR- GARCH model and the jump diffusion model with a constant volatility. But the jump diffusion model with a constat volatilit provided much more overvalued results.The rest of the models were even worse. From the rest the IGARCH model is the best but provided undervalued results. All these findings correspond with R squared coefficient.

L’homme est un animal social. La majorité des décisions que nous prenons se font dans un contexte social et dépendent d’autrui, ce qui implique des calculs cérébraux complexes qui incluent tous les facteurs contextuels et environnementaux. La majorité des études ultérieures de la prise en compte d’autrui dans la décision ont utilisé des tâches de partage de récompenses entre soi et autrui. Les choix possibles amènent le décideur à considérer autrui, mais dans le but de gagner soi-même une récompense ; donc dans un contexte où les récompenses liées à soi et les récompenses liées à autrui sont confondues. Le travail présenté dans cette thèse avait pour but une meilleure compréhension des mécanismes cérébraux soutenant l’intégration d’autrui dans la prise de décision, sans que la récompense pour autrui n’interfère directement avec soi. Nous nous sommes appuyés sur le cadre théorique de la décision perceptuelle et des modèles de diffusion pour l'étude i) des modifications du processus décisionnel induites par une récompense monétaire allant à autrui et ii) de l’impact de l’effet d’audience (le fait de se sentir observé) sur la décision. Nos résultats computationnels montrent qu'une récompense pour autrui, par rapport à une récompense pour soi, et une audience, par rapport au secret, modifient le taux de dérive de la variable de décision. En magnétoencéphalographie, nos résultats indiquent que les décisions pour soi et pour autrui diffèrent pendant, mais aussi après, la prise de décision dans des zones cérébrales associées avec la transformation sensori-motrice, l'ajustement du compromis entre rapidité et justesse et avec la cognition sociale. Ainsi, le cortex temporal montre des différences de -1170 millisecondes (ms) à -1023 ms, de -993 ms à -915 ms et de -343 ms à -188 ms en amont de la réponse. Ce qui suppose une influence sur l’intégration des preuves sensorielles. Après la décision, les régions frontales ont également montré des différences entre soi et autrui, de 153 ms à 303 ms post-réponse, suggérant une différence entre soi et autrui dans l’ajustement du compris entre justesse et rapidité. Le bénéficiaire de la récompense associée à la décision modifie les paramètres décisionnels et les corrélats cérébraux de la décision perceptuelle, démontrant l’importance du contexte social dans l’implémentation de la prise de décision chez l’Homme. Ce travail appuie également l’utilité des modèles mathématiques tels que les modèles de diffusion dans la compréhension des processus décisionnels, même de ceux découlant de la cognition sociale / Humans are inherently social: most of human’s decisions are within a social context and depend on others. For more than a century, researchers explore aspects of social cognition. Aiming to understand human behavior in social contexts, neuro-economic researches showed that taking others into account involve complex brain computations that include all environmental and contextual factors. However, most of the work was made using money allocation tasks; mixing self-affecting and other-affecting rewards into the decision making process. The present work intended the understanding of the brain mechanisms underpinning the integration of others into the decision making process for decisions that include others and do not interfere with self-rewards.Taking advantage of mathematical models from the drift diffusion models framework, we conducted experiments investigating how others influence the mechanistic of perceptual decisions and their correlates in the human brain. We showed that taking rewards for others into account and being observed by others influence the drift rate of the decision variable. The drift rate is higher in audience than in secret and higher for self-rewards than for other-rewards. These results indicate that others are integrated into the accumulation process together with the evidence available for making a decision. At the brain level, we found difference between self and other decisions over the anterior temporal and centro-frontal cortices during decision making. This suggests that the beneficiary of a decision modifies sensory-motor transformation processes. In addition, self- and other-affecting difference showed difference over the medial frontal sensors after the decision making process, indicating a variation in the speed-accuracy tradeoff adjustment process

>Magister Scientiae - MSc / Warrant pricing has become very crucial in the present market scenario. See, for example, M. Hanke and K. Potzelberger, Consistent pricing of warrants and traded options, Review Financial Economics 11(1) (2002) 63-77 where the authors indicate that warrants issuance affects the stock price process of the issuing company. This change in the stock price process leads to subsequent changes in the prices of options written on the issuing company's stocks. Another notable work is W.G. Zhang, W.L. Xiao and C.X. He, Equity warrant pricing model under Fractional Brownian motion and an empirical study, Expert System with Applications 36(2) (2009) 3056-3065 where the authors construct equity warrants pricing model under Fractional Brownian motion and deduce the European options pricing formula with a simple method. We study this paper in details in this mini-thesis. We also study some of the mathematical models on warrant pricing using the Black-Scholes framework. The relationship between the price of the warrants and the price of the call accounts for the dilution effect is also studied mathematically. Finally we do some numerical simulations to derive the value of warrants.

Magnetic resonance (MR) images together with computed tomography (CT) images are used in many medical practices, such as radiation therapy. To capture those images, patients have to undergo two separate scans: one for the MR image, which involves using strong magnetic fields, and one for the CT image which involves using radiation (x-rays). Another approach is to generate synthetic CT (sCT) images from MR images, thus the patients only have to take one image (the MR image), making the whole process easier and more effcient. One way of generating sCT images is by using generative diffusion models which are a relatively new class in generative models. To this end, this project aims to enquire whether generative diffusion models are capable of generating viable and realistic sCT images from MR images. Firstly, a denoising diffusion probabilistic model (DDPM) with a U-Net backbone neural network is implemented and tested on the MNIST dataset, then it is implemented on a pelvis dataset consisting of 41600 pairs of images, where each pair is made up of an MR image with its respective CT image. The MR images were added at each sampling step in order to condition the sampled sCT images on the MR images. After successful implementation and training, the developed diffusion model got a Fréchet inception distance (FID) score of 14.45, and performed as good as the current state-of-the-art model without any major optimizations to the hyperparameters or to the model itself. The results are very promising and demonstrate the capabilities of this new generative modelling framework.

Объект исследования: процесс аугментации изображений для решения задачи сегментации. Предмет исследования: методы аугментации и машинного обучения, с помощью которых осуществляется сегментация изображений. Цель работы: исследование эффективности генеративной аугментации изображений, выполненной с помощью диффузионной модели Stable Diffusion на примере задачи семантической сегментации. В процессе исследования проводились: рассмотрение основных подходов сегментации изображений и методов аугментации данных, разработка и реализация экспериментов для оценки эффективности генеративной аугментации изображений. В работе продемонстрирована эффективность подхода аугментации изображений, реализованного за счет расширения части исходного датасета путем генерирования новых данных с помощью диффузионной модели. Область практического применения: предложенный подход может быть использован для улучшения качества работы моделей семантической сегментации изображений в условиях ограниченного количества исходных данных, дефицита размеченных данных или дисбаланса данных. / Object of study: the process of image augmentation to solve the segmentation problem. Subject of research: augmentation and machine learning methods used for image segmentation. Purpose of the work: to study the effectiveness of generative image augmentation performed using the Stable Diffusion model using the example of a semantic segmentation task. During the research process, the following was carried out: consideration of the main approaches to image segmentation and data augmentation methods, development and implementation of experiments to evaluate the effectiveness of generative image augmentation. The work demonstrates the effectiveness of the image augmentation approach, implemented by expanding part of the original dataset by generating new data using a diffusion model. Area of practical application: the proposed approach can be used to improve the quality of work of semantic image segmentation models in conditions of a limited amount of source data, a shortage of labeled data, or data imbalance.