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Numerical methods form an important part of the pricing of financial derivatives and especially in cases where there is no closed form analytical formula. We begin our work with an introduction of the mathematical tools needed in the pricing of financial derivatives. Then, we discuss the assumption of the log-normal returns on stock prices and the stochastic differential equations. These lay the foundation for the derivation of the Black Scholes differential equation, and various Black Scholes formulas are thus obtained. Then, the model is modified to cater for dividend paying stock and for the pricing of options on futures. Multi-period binomial model is very flexible even for the valuation of options that do not have a closed form analytical formula. We consider the pricing of vanilla options both on non dividend and dividend paying stocks. Then show that the model converges to the Black-Scholes value as we increase the number of steps. We discuss the Finite difference methods quite extensively with a focus on the Implicit and Crank-Nicolson methods, and apply these numerical techniques to the pricing of vanilla options. Finally, we compare the convergence of the multi-period binomial model, the Implicit and Crank Nicolson methods to the analytical Black Scholes price of the option. We conclude with the pricing of exotic options with special emphasis on path dependent options. Monte Carlo simulation technique is applied as this method is very versatile in cases where there is no closed form analytical formula. The method is slow and time consuming but very flexible even for multi dimensional problems.

Dimensional measurement plays a critical role in product development and quality control. With the continuously increasing demand for tighter tolerances and more complex workpiece shapes in the industry, dimensional metrology often becomes the bottleneck of taking the quality and performance of manufacturing to the next level. As one kind of the most useful and powerful measuring instruments, coordinate measuring machines (CMMs) are widely employed in manufacturing industries. Since the accuracy and efficiency of a CMM have a vital impact on the product quality, productivity and manufacturing cost, the evaluation and improvement of CMM performance have always been important research topics since the invention of CMM. A novel Advanced Virtual Coordinate Measuring Machine (AVCMM) is proposed against such a background. The proposed AVCMM is a software package that provides an integrated virtual environment, in which user can plan inspection strategy for a given task, carry out virtual measurement, and evaluate the uncertainty associated with the measurement result, all without the need of using a physical machine. The obtained estimate of uncertainty can serve as a rapid feedback for user to optimize the inspection plan in the AVCMM before actual measurement, or as an evaluation of the result of a performed measurement. Without involving a physical CMM in the inspection planning or evaluation of uncertainty, the AVCMM can greatly reduce the time and cost needed for such processes. Furthermore, as the package offers vivid 3D visual representation of the virtual environment and supports operations similar to a physical CMM, it does not only allow the user to easily plan and optimise the inspection strategy, but also provide a cost-effective, risk-free solution for training CMM operators. A modular, multitier architecture has been adopted to develop the AVCMM system, which incorporates a number of functional components covering CMM and workpiece modelling, error simulation, inspection simulation, feature calculation, uncertainty evaluation and 3D representation. A new engine for detecting collision/contact has been developed and utilized, which is suitable for the virtual environment of simulated CMM inspections. A novel approach has been established to calculate errors required for the error simulation, where the data are obtained from FEA simulations in addition to conventional experimental method. Monte Carlo method has been adopted for uncertainty evaluation and has been implemented with multiple options available to meet different requirements. A prototype of the proposed AVCMM system has been developed in this research. Its validity, usability and performance have been verified and evaluated through a set of experiments. The principles for utilising the AVCMM in practical use have also been established and demonstrated. The results have indicated that the proposed AVCMM system has great potentials to improve the functionalities and overall performance of CMMs.

Hill and Monticino (1998) introduced a constructive method for generating random probability measures with a prescribed mean or distribution on the mean. The method involves sequentially generating an array of barycenters that uniquely defines a probability measure. This work analyzes statistical properties of the measures generated by sequential barycenter array constructions. Specifically, this work addresses how changing the base measures of the construction affects the statististics of measures generated by the SBA construction. A relationship between statistics associated with a finite level version of the SBA construction and the full construction is developed. Monte Carlo statistical experiments are used to simulate the effect changing base measures has on the statistics associated with the finite level construction.

Approved for public release; distribution is unlimited / In this thesis, we compare weapon effectiveness methods to determine if current effectiveness models provide accurate results. The United States Military currently adheres to a compilation of data and methodologies named the Joint Munitions Effectiveness Manuals (JMEM) to determine the effectiveness of air delivered weapons against a variety of ground targets. Since the time these manuals were implemented in the 1960s, progress in technology has allowed the weapon/target interaction to be more accurately modeled. This thesis investigates the differences of these high fidelity models for unguided weapons and the JMEM computations in order to determine whether the older, more simplistic, models need to be upgraded. / Ensign, United States Navy

The Monte Carlo (MC) method is an effective approach to simulate the radiative transfer in an inhomogeneous cloudy atmosphere. It is based on the direct physical simulation of the extinction processes that solar and thermal photons incur when traveling through the atmosphere. A detailed description of the MC method is presented in the second chapter. A new three-dimensional Monte Carlo radiative transfer model, based on a pre-existing model (Trautmann et al. [1999]), has been developed. Some outstanding characteristics of this model are discussed in chapter 3. Several simulations of reflectances, transmittances, absorptances and horizontal flux densities have been performed, the results of which have been compared with worldwide accepted codes (chapter 4). The two cases selected for the radiative transfer computations were taken from the Intercomparison of 3D Radiative Codes (I3RC) project: an ARM (Atmospheric Radiation Measurements) reconstructed cloud and a 3D marine boundary layer cloud / Die Monte Carlo (MC) Methode ist ein effektives Verfahren, um den Strahlungstransport in einer inhomogenen bewölkten Atmosphäre zu simulieren. Es begründet sich auf der direkten Simulation der Extinktionsprozesse eines solaren oder thermischen Photons auf seinem Weg durch die Atmosphäre. Eine detallierte Beschreibung der MC Methode erfolgt in Kapitel 2. In Kapitel 3 wird ein neues dreidimensionales MC-Strahlungstranportmodell vorgestellt, das, aufbauend auf einem schon bestehenden Modell (Trautmann et al. [1999]), entwickelt wurde. Mehrere Simulationen von Reflektanzen, Transmittanzen, Absorptanzen und Strahlungsflussdichten für zwei Fälle des \"Intercomparison of 3D radiative Codes\" projektes, nämlich eine ARM rekonstruierte Wolke und eine 3D marine Grenzschichwolke, wurden durchgeführt, und mit den Ergebnissen anderer weltweit akzeptierten Codes verglichen.

O objetivo desta tese de doutorado foi investigar o efeito da flexibilidade na tomada de decisão em uma usina sucroenergética, sob a abordagem da teoria de Opções Reais com simulação de Monte Carlo. Para isso, partiu-se de uma análise econométrica das séries de tempo dos preços do etanol hidratado, da gasolina veicular e do açúcar VHP (Very High Polarized). Dessa análise foi possível verificar que as séries de preços do etanol e do açúcar são modeladas pelo movimento de reversão à média. Além disso, constatou-se que no curto prazo ocorreu uma transmissão de preços do açúcar e da gasolina para o preço do etanol. Por outro lado, a série de preços do açúcar não foi influenciada pelas demais e a da gasolina não foi influenciada pela do açúcar e muito pouco pela do etanol. Nesse sentido, percebe-se que não houve uma interdependência simultânea entre os preços desses três produtos. Adicionalmente, para o restante da análise, foi considerada uma usina fictícia com a hipótese de contar com a flexibilidade para escolher a proporção a ser produzida entre açúcar e etanol, definida como opção de conversão e também com a opção de parada temporária, com a qual é possível interromper a produção na usina, a fim de reduzir perdas caso os custos de produção ultrapassassem os preços do produto a ser vendido. Sob tais hipóteses, buscou-se quantificar o efeito marginal das contribuições das flexibilidades gerenciais na tomada de decisão em investimentos sucroenergéticos. Pelos resultados encontrados, a tomada de decisão que leva em conta as flexibilidades identificadas consegue aumentar em até 88,14% o valor esperado do VPL de uma usina sucroenergética e reduzir o risco de o VPL ser negativo. Assim, esse trabalho propõe que as flexibilidades gerenciais trazem maiores retornos para usinas sucroenergéticas. Não só, mas também, essas flexibilidades mitigam os efeitos de crises no setor. / The goal of this PhD Thesis was to investigate the effect of flexibility in decision making in a sugar cane plant under the Real Options theory with Monte Carlo simulation approach. For this, we started with an econometric analysis of the time series of the prices of hydrous ethanol, vehicular gasoline and VHP (Very High Polarized) sugar. From this analysis it was possible to verify that the ethanol and sugar price series are modeled by the mean reversion movement. In addition, it was found that in the short-term sugar and gasoline prices were transmitted to the price of ethanol. On the other hand, the series of sugar prices was not influenced by the others and that of gasoline was not influenced by sugar and very little by ethanol. In this sense, it can be seen that there was no simultaneous interdependence between the prices of these three products. In addition, for the rest of the analysis, it was considered a fictitious plant with the hypothesis of switch output option and also with the temporary shutdown option. Under these hypotheses, we sought to quantify the marginal effect of the contributions of managerial flexibilities in decision-making on sugar cane investments. Based on the results obtained, decision-making that considers flexibilities can increase the expected value of NPV of a sugar cane plant by up to 88.14% and reduce the risk of NPV being negative. Thus, this work proposes that managerial flexibilities bring greater returns to sugarcane plants. Not only that, but also, these flexibilities mitigate the effects of crises in the sector.

A dissertation submitted in partial ful lment of the degree of Master of Science in Engineering (Electrical) (50/50) in the Faculty of Engineering and the Built Environment Department of Electrical and Information Engineering May 2017 / Numerous applications in a wide variety of elds has resulted in a rich history of research into optimisation for scheduling. Although it is a fundamental form of the problem, the single machine scheduling problem with two or more objectives is known to be NP-hard. For this reason we consider the single machine problem a good test bed for solution algorithms. While there is a plethora of research into various aspects of scheduling problems, little has been done in evaluating the performance of the Simulated Annealing algorithm for the fundamental problem, or using it in combination with other techniques. Speci cally, this has not been done for minimising total weighted earliness and tardiness, which is the optimisation objective of this work. If we consider a mere ten jobs for scheduling, this results in over 3.6 million possible solution schedules. It is thus of de nite practical necessity to reduce the search space in order to nd an optimal or acceptable suboptimal solution in a shorter time, especially when scaling up the problem size. This is of particular importance in the application area of packet scheduling in wireless communications networks where the tolerance for computational delays is very low. The main contribution of this work is to investigate the hypothesis that inserting a step of pre-sampling by Markov Chain Monte Carlo methods before running the Simulated Annealing algorithm on the pruned search space can result in overall reduced running times. The search space is divided into a number of sections and Metropolis-Hastings Markov Chain Monte Carlo is performed over the sections in order to reduce the search space for Simulated Annealing by a factor of 20 to 100. Trade-o s are found between the run time and number of sections of the pre-sampling algorithm, and the run time of Simulated Annealing for minimising the percentage deviation of the nal result from the optimal solution cost. Algorithm performance is determined both by computational complexity and the quality of the solution (i.e. the percentage deviation from the optimal). We nd that the running time can be reduced by a factor of 4.5 to ensure a 2% deviation from the optimal, as compared to the basic Simulated Annealing algorithm on the full search space. More importantly, we are able to reduce the complexity of nding the optimal from O(n:n!) for a complete search to O(nNS) for Simulated Annealing to O(n(NMr +NS)+m) for the input variables n jobs, NS SA iterations, NM Metropolis- Hastings iterations, r inner samples and m sections. / MT 2017