This paper describes methods for pricing European and American options. Monte Carlo simulation and control variates methods are employed to price call options. The binomial model is employed to price American put options. Using daily stock data I am able to compare the model price and market price and speculate as to the cause of difference. Lastly, I build a portfolio in an Interactive Brokers paper trading [1] account using the prices I calculate. This project was done a part of the masters capstone course Math 573: Computational Methods of Financial Mathematics.

A new automatic algorithm for sampling from monotone, unbounded densities is presented. The user has to provide a program to evaluate the density and its derivative and the location of the pole. Then the setup of the new algorithm constructs different hat functions for the pole region and for the tail region, respectively. For the pole region a new method is developed that uses a transformed density rejection hat function of the inverse density. As the order of the pole is calculated in the setup, conditions that guarantee the correctness of the constructed hat functions are provided. Numerical experiments indicate that the new algorithm works correctly and moderately fast for many different unbounded densities. The proposed algorithm is the first black-box method that works for unbounded densities suggested in the literature. (author's abstract) / Series: Research Report Series / Department of Statistics and Mathematics

A new algorithm for sampling from largely abitrary monotone, unbounded densities is presented. The user has to provide a program to evaluate the density and its derivative and the location of the pole. Then the setup of the new algorithm constructs different hat functions for the pole region and for the tail region, respectively. For the pole region a new method is developed that uses a transformed density rejection hat function of the inverse density. As the order of the pole is calculated in the setup, conditions that guarantee the correctness of the constructed hat functions are provided. Numerical experiments indicate that the new algorithm works correctly and moderately fast for many different unbounded densities. (c) ACM, (2007). This is the author's version of the work. It is posted here by permission of ACM for your personal use. Not for redistribution. (author's abstract) / Series: Research Report Series / Department of Statistics and Mathematics

It is well known that the generation of random vectors with non-independent components is difficult. Nevertheless, we propose a new and very simple generation algorithm for multivariate linear densities over point-symmetric domains. Among other applications it can be used to design a simple decomposition-rejection algorithm for multivariate concave distributions. / Series: Research Report Series / Department of Statistics and Mathematics

Snake vertebrae are common in the fossil record, whereas cranial remains are generally fragile and rare. Consequently, vertebrae are the most commonly studied fossil element of snakes. However, identification of snake vertebrae can be problematic due to extensive variation. This study utilizes 2-D geometric morphometrics and canonical variates analysis to 1) reveal variation between genera and species and 2) classify vertebrae of modern and fossil eastern North American Agkistrodon and Crotalus. The results show that vertebrae of Agkistrodon and Crotalus can reliably be classified to genus and species using these methods. Based on the statistical analyses, four of the fossil viperid vertebrae from Hickory Tree Cave were assigned to Crotalus horridus, one to C. adamanteus, and another to Agkistrodon piscivorus. The potential presence of the latter two species could indicate that the deposit is from a warm period during the Quaternary such as a Pleistocene interglacial or Holocene warm interval.

To analyze a communication system without the aid of simulation, the channel noise for the simulation must be assumed to be normal. The assumption is often valid, but the normal distribution may not be able to model the channel noise adequately in some environments. This thesis will explore the generalized exponential distribution for better noise modeling and robustness testing in communication system. When using the generalized exponential distribution for the channel noise, the analysis will become analytically intractable, and simulation becomes mandatory. To generate the noise with the distribution, the rejection method can be used. However, since the distribution can take on different shapes, finding the appropriate Upper Bounding Function (UBF) for the method is very difficult. Thus, two modified versions of the rejection method will be examined. They are the Transformed Density Rejection (TDR) and Ratio-of-Uniform (RoU) method; their quality, efficient, trade-offs, etc will be discussed. Choosing TDR, a simulation of a BPSK communication system will be performed. With the simulation, it can further ascertain that the random variates generated by TDR can be used to model the channel noise and to test the robustness of a communication system. / Master of Science

We present a numerical inversion method for generating random variates from continuous distributions when only the density function is given. The algorithm is based on polynomial interpolation of the inverse CDF and Gauss-Lobatto integration. The user can select the required precision which may be close to machine precision for smooth, bounded densities; the necessary tables have moderate size. Our computational experiments with the classical standard distributions (normal, beta, gamma, t-distributions) and with the noncentral chi-square, hyperbolic, generalized hyperbolic and stable distributions showed that our algorithm always reaches the required precision. The setup time is moderate and the marginal execution time is very fast and the same for all distributions. Thus for the case that large samples with fixed parameters are required the proposed algorithm is the fastest inversion method known. Speed-up factors up to 1000 are obtained when compared to inversion algorithms developed for the specific distributions. This makes our algorithm especially attractive for the simulation of copulas and for quasi-Monte Carlo applications. (author´s abstract) / Series: Research Report Series / Department of Statistics and Mathematics

In this dissertation we developed theoretical models to analyse Gaussian and multinomial distributions. The analysis is focused on classification in high dimensional feature spaces and provides a basis for dealing with issues such as data sparsity and feature selection (for Gaussian and multinomial distributions, two frequently used models for high dimensional applications). A Naïve Bayesian philosophy is followed to deal with issues associated with the curse of dimensionality. The core treatment on Gaussian and multinomial models consists of finding analytical expressions for classification error performances. Exact analytical expressions were found for calculating error rates of binary class systems with Gaussian features of arbitrary dimensionality and using any type of quadratic decision boundary (except for degenerate paraboloidal boundaries). Similarly, computationally inexpensive (and approximate) analytical error rate expressions were derived for classifiers with multinomial models. Additional issues with regards to the curse of dimensionality that are specific to multinomial models (feature sparsity) were dealt with and tested on a text-based language identification problem for all eleven official languages of South Africa. / Thesis (M.Ing. (Computer Engineering))--North-West University, Potchefstroom Campus, 2009.

In this dissertation we developed theoretical models to analyse Gaussian and multinomial distributions. The analysis is focused on classification in high dimensional feature spaces and provides a basis for dealing with issues such as data sparsity and feature selection (for Gaussian and multinomial distributions, two frequently used models for high dimensional applications). A Naïve Bayesian philosophy is followed to deal with issues associated with the curse of dimensionality. The core treatment on Gaussian and multinomial models consists of finding analytical expressions for classification error performances. Exact analytical expressions were found for calculating error rates of binary class systems with Gaussian features of arbitrary dimensionality and using any type of quadratic decision boundary (except for degenerate paraboloidal boundaries). Similarly, computationally inexpensive (and approximate) analytical error rate expressions were derived for classifiers with multinomial models. Additional issues with regards to the curse of dimensionality that are specific to multinomial models (feature sparsity) were dealt with and tested on a text-based language identification problem for all eleven official languages of South Africa. / Thesis (M.Ing. (Computer Engineering))--North-West University, Potchefstroom Campus, 2009.

We present a numerical inversion method for generating random variates from continuous distributions when only the density function is given. The algorithm is based on polynomial interpolation of the inverse CDF and Gauss-Lobatto integration. The user can select the required precision which may be close to machine precision for smooth, bounded densities; the necessary tables have moderate size. Our computational experiments with the classical standard distributions (normal, beta, gamma, t-distributions) and with the noncentral chi-square, hyperbolic, generalized hyperbolic and stable distributions showed that our algorithm always reaches the required precision. The setup time is moderate and the marginal execution time is very fast and nearly the same for all distributions. Thus for the case that large samples with fixed parameters are required the proposed algorithm is the fastest inversion method known. Speed-up factors up to 1000 are obtained when compared to inversion algorithms developed for the specific distributions. This makes our algorithm especially attractive for the simulation of copulas and for quasi-Monte Carlo applications. <P> This paper is the revised final version of the working paper no. 78 of this research report series. / Series: Research Report Series / Department of Statistics and Mathematics