Doç. Dr. Wolfgang Hörmann danışmanlığındaki tezler
11 tez · Boğaziçi University
Variance reduction in Markov chain Monte Carlo algorithms
In a simulation study, the output of the simulation is used to estimate target parameters. The estimated parameters typically has a variance, which impacts the precision of the estimation. In Monte Carlo simulations, the standard approach to increase the precision of the estimation is to increase the sample size. However, we can increase simulation size until a certain point due to resource restrictions. This is where Variance Reduction (VR) techniques help. These techniques use simulations as an input and return another simulation with almost the same expected value and less variance value. VR techniques are widely studied in the context of Monte Carlo simulations. However, the research on the application of these algorithms in Markov Chain Monte Carlo (MCMC) simulations is limited. In this thesis, we study the effectiveness of VR techniques for MCMC algorithms. We created an experiment environment that allows us to control the difficulty of the problems that are solved by MCMC algorithms. In our study, we use the Random-Walk Metropolis-Hastings algorithm with different types of proposal distributions. We first evaluate the convergence of the MCMC algorithms, select chains with good convergence properties, and then apply the VR techniques. The VR techniques of interest are {\it{general}} VR techniques, which are Antithetic Variates, Control Variates, and Latin Hypercube Sampling. We study the effectiveness of these algorithms using problems with varying difficulties. We address the problem of calculating the standard error of the MCMC simulation outputs, evaluate the accuracy of the estimated values, and we evaluate the correctness of the standard error calculations by calculating the coverage probabilities of the confidence intervals and RMSE/MAE values of the estimations. We show that VR techniques are successful at decreasing the variance of the MCMC simulations without introducing any bias.
Ship hull resistance estimation using multivariate statistics
The biggest cost in shipbuilding belongs to the main engine and propeller systems to be used. The selection of these equipment to be used is made according to the resistance values of the ship. Since these are very expensive systems, it is critical to determine the resistance values of the ships precisely. For this reason, many studies have been done on estimating the resistance values of ships. The performances of these studies are measured by comparing the estimations with the towing tank test results. Within the scope of this thesis study, the towing tank test reports of a total of 58 different cargo ships are examined in detail and the needed data are extracted from these reports. Then, the hull resistance values of these ships are estimated with different statistical approaches. First, the problem is handled using a generalized linear model, and then predictions are made with artificial neural networks, which have been used many times in the literature. After all, because of the longitudinal nature of the available data, the problem is addressed using mixed effect linear models. Finally, hull resistance values are estimated by using generalized linear mixed models, which are the combination of generalized linear and mixed effect linear models. Comparisons are made among the completed statistical models using the leave one ship out cross validation technique. In addition, the results of these models are compared with the Holtrop & Mennen method, which is widely used in this field.
Numeric methods for stochastic disease spread models
Disease spread models are important in controlling the new infectious diseases that suddenly threaten the public health. Mathematical tools are especially important to understand the spread of the disease since experiments are not possible in the area. This study focusses on a stochastic SIR (susceptible-infected-recovered) model for a finite population. We first assume a homogeneous population and study Markov modelling of disease spread for an exponential infectious period. We present the algorithms that compute the expected duration of an epidemic, the final outbreak size distribution and the maximum number of simultaneously infected individuals distribution. After stating the problems with exponential infectious period, we assume an Erlang distributed infectious period allowing us to use Markov chains. The Markov disease spread model proposed for it uses the remaining stages as state variables and treats the Erlang distributed infectious period as simply exponential. This enables us to compute the exact final outbreak size distribution for large populations efficiently. Moreover, we propose an approximation for the distribution of the maximum epidemic size using the exact distribution of the remaining stages. We also consider a mixture of Erlangs so that by using the first two moments of an infectious period one can fit a corresponding mixture. Furthermore, by considering a mixture of Erlangs distribution for the infectious period and assuming two types of infected individuals as symptomatic and asymptomatic, our proposed models are implemented with the parameters similar to those reported for COVID-19 spread. Finally, a stochastic SIR model for a non homogeneous population is considered. The notion of R0 for heterogeneous populations is discussed and individual R0 as the expected number of secondary cases produced by a unique given initially infected individual. We propose a general formula for individual R0 and use it for the assessment and development of the intervention methods.
Risk quantification for foreign currency investments
As the economies are globalized, more firms, investors and workers find their fortunes linked to the exchange rates. Therefore, management of exchange rate risk becomes more significant. As the measured risk can be managed better, the aim of this study is to quantify the foreign exchange risk. In this study, six different risk modeling methods - i.i.d Normal, i.i.d Student's t, Normal-GARCH, t-GARCH, unconditional EVT and GARCH filtered EVT - are used to quantify the daily risk of currency pairs. After the daily VaR of the currency pairs has been calculated, an extensive back testing study is performed using 4 different confidence levels to assess the performance of each risk model. The results of the back testing experiment show that the risk modeling methods that rely on the normality assumption, such as the i.i.d Normal and the Normal-GARCH model, result in significant underestimation of daily risk. The sophisticated methods, GARCH filtered EVT and t-GARCH models showed the best overall performance. Besides, it is observed that the performance of i.i.d Student's t is good especially for currency pairs that belong to the countries with strong economical base. The quality of the t-GARCH and GARCH filtered EVT models were observed in our study and it is recommended to use these models to quantify the FX risk. Also i.i.d Student's t model can be the method of choice if the simple implementation of the methods of choice is important.
Input modeling for discrete distributions
In this thesis we developed a new input modeling tool called FitAllDiscrete(x) for the statistical software R. We then tested the success of our modeling tool by comparing it with two popular commercial softwares: Arena input analyzer and EasyFit. Considering limitations of the software packages we mention, we developed a more powerful tool which is based on well-accepted and efficient statistical methods. Input modeling tools model the data to find the most proper distribution. Our input modeling function considers input data from discrete distributions. It tries all distributions automatically and recommends the most proper one for the data. Input modeling is executed by the three essential steps including distribution selection, parameter estimation and goodness of fit testing. In this study, for distribution selection we used Akaike's information criterion (AIC) which is a popular selection criterion. For parameter estimation we used maximum likelihood estimation (MLE). Akaike's information criterion does not guarantee that the selected distribution has a good fit for the data. Therefore, we applied the chi-square test to assess whether the fit is statistically meaningful. According to it, if the fit is good the distribution is accepted and recommended to the user. Otherwise, the empirical distribution is recommended. To decide how reliable our tool is, we first performed an automatic test. Then, we performed a manual test by comparing our input modeling tool with Arena input analyzer and EasyFit. During the tests, we used random discrete data sets generated from the distributions of our pool. The results of the automatic test showed that our input modeling tool is successful in finding the correct distributions for simulated data sets. The comparison test showed that the recommendations by FitAllDiscrete(x) are more successful than those of the commercial products. Hence we succeeded in developing a new high-quality input analyzer.
Bono ve hisse senedi içeren portföylerin risk ölçümü
In this study, we design a method to quantify the expected return and risk of mixed portfolios containing bonds and stocks using the historical data and expert opinion about the future changes in the market. Vasicek model was the model we used as the short-rate model. Short-rate models are single factor models that give the possibility to price bonds with di erent time-to-maturities with a single random variable, the short-rate. We implemented the expert's expectation of future short-rate into Vasicek model to simulate future short-rates and developed a parameter calibration to obtain a yield curve closer to the current market yield curve. We used the lognormal multinormal model to simulate the future log-returns of the stocks. The design of a model for bonds and stock in a single portfolio requires information on the dependence between short-rate changes and bond returns. We calculated the empirical correlation between short-rate and stock returns for a large number of American stocks. The results showed that the empirical correlations were all close to zero. So we assumed that the changes of the short-rate and of the stock prices are independent. It is therefore not di cult to simulate many possible future portfolio values and to represent them in a histogram. It is also easy to use those simulated values to calculate the mean return and the risk measures VaR and CVaR of the portfolio.
Generalized linear modelling for assessing promotions
Price promotions are very important and widely used in marketing and pricing strategy. Therefore, understanding and quantifying the influences of price promotions on demand become interesting. In this study we modelled the demand as a Poisson distributed random variate and built a model using exponential demand function in microeconomics, which considers the variables that have an influence on demand, such as price, yearly seasonality, week days, competition etc. We also generated the necessary data and used our different data sets in the study. Since our main aim is to measure the impacts of price promotions on product demand and forecast the demand, we used generalized linear models, more specifically Poisson regression. GLMs are used for relating random responses to the linear combination of predictor variables. Poisson regression is a GLM for which the response variable is modelled by the Poisson distribution. We found that GLM is very helpful to quantify the effects of price promotions on product demand. If the mean of the daily demand is small, Poisson regression is needed to be used, otherwise ordinary log-level linear regression can be applied instead. We investigated that the results depend very much on the data and number of observations. In addition, we observed that when the price decreases by 20 \%, demand of a product increases about 18 \%. We checked our ``no interaction between covariates'' assumption with the help of AIC-Criterion and found out that the simplest demand model is the best model with the smallest AIC\_Value. We realized that once having reliable and usable real data, the models we suggested in this thesis can be used to quantify the effects of price promotions on demand. These models can help us to plan promotions, forecast the effects of promotions and make revenue maximization. Throughout this thesis R-Software environment has been used for all kind of computations.
Girdi modelleme
In this study input modeling is executed by using the most efficient statistical principles and a new input modeling tool, the Fit All function, is developed in the statistical computing environment R. Moreover a practical comparison between this function and commercial input modeling softwares such as the Arena Input Analyzer and EasyFit is made. The Fit All function automatically tries all distributions for the input data and recommends the most proper model to the user. Twenty-five continuous distributions are included to this function and the maximum likelihood method (MLE) is utilized to estimate the parameters for all these distributions. For model selection, Akaike's Information Criterion (AIC) is used which is a theoretically sound and popular criterion for model selection. To assess the goodness of fit for the best ftting distribution, the Chi-Square test is executed. If a proper fit can be found, this model is recommended to the user. Otherwise the empirical model is recommended. To compare the models recommended by the Fit All function, the Arena Input Analyzer and EasyFit; random samples from common distributions are generated. The results of the simulation study show that the Fit All function identifies the correct distribution clearly more successfully than the other softwares. This means that AIC performs much better than the other model selection procedures. Moreover the parameter estimates that are presented by the Fit All function are closer to the original parameters. This also shows that the MLE obtains high quality parameter estimates.
Efficient stratified sampling for financial risk simulation
Monte Carlo simulation is frequently the only method available for computing financial risk, particularly under the realistic and complex portfolio models. The naive simulation generally leads to large confidence intervals on typical risk measures. Thus, to enhance the efficiency of the estimates, the necessity of variance reduction techniques becomes apparent. In this thesis, we discuss the efficient implementation of stratified sampling technique for Monte Carlo simulation problems. As the application field, we consider the risk evaluation of a linear asset portfolio. For given portfolio and the loss threshold, tail loss probability and conditional excess values are essential. To understand the general risk situation, one needs to efficiently estimate these values for multiple threshold levels in a single simulation. Stratified sampling is especially useful for such a task as the allocation fractions can be used as decision variables to minimize the overall error of all estimates. We develop an efficient simulation algorithm that combines optimal stratification and importance sampling to estimate multiple tail loss probabilities and conditional excess values for linear asset portfolios under the t-copula model. Two different classes of objective functions are proposed to represent the overall error. The first, including the mean squared and the mean squared relative error, allows for a simple closed-form solution. For the second class of error functions, including the maximal absolute and the maximal absolute relative error, a simple and fast heuristic is proposed. The application of the new method, called ``OASIS: optimal allocation stratification and importance sampling", is explained for linear asset portfolios under the t-copula model and its performance is demonstrated with numerical examples.
Hisse senedi ve emtia içeren portföylerin risk ölçümü
In this study, we used copula method in order to model the multivariate return distributions of stock portfolios and, in this manner, to implement this model for risk measure evaluations in practice. Copulas are used to describe the dependence between random variables thus, are enjoyed to model the marginals separately and to represent the dependence structure between them. We also modeled the multivariate return distributions of stock portfolios diversified with commodities, precious metals, crude oil etc. and tted a set of copulasto the joint return data. With this aim, we selected 20 stocks from New York Stock Exchange, gold and crude oil and constructed stock portfolios, stock portfolios with gold, stock portfolios with crude oil and stock portfolios with gold and crude oil inorder to analyze whether the copula method fits the multivariate return distributions of selected portfolios. In order to check the validity of the models, we implemented daily and weeklyback testing using 20 different alpha values. We found that t distribution and generalized hyperbolic distributions are very nice models for modeling individual financial instruments returns and the t copula is the best copula to represent the dependence structure between financial instruments returns. We used this model to calculate the risks of portfolios and observed that adding gold decreases the risk of portfolios wherecrude oil behaves like an ordinary stock.
Simülasyonla opsiyon fiyatlama
The valuation of path dependent and multivariate options require efficient numerical methods, as their prices are not available in closed form. Monte Carlo simulation is one of the widely used techniques. Although simulation is a highly flexible and general method, its efficiency for specific problems depends on exploiting the special features of that problem via variance reduction techniques. The aim in variance reduction is to reduce the variance of the estimator in order to increase the efficiency. This study proposes new variance reduction methods for path dependent and multivariate options under the assumption of geometric Brownian motion. These methods are based on new control variates. Furthermore, a general control variate framework is suggested for Lévy process models. Its application is presented for pricing path dependent options.The options considered in this thesis are European basket, Asian, lookback and barrier options. The method suggested for basket and Asian options combines the use of control variates and conditional Monte Carlo. The new control variate algorithms for lookback and barrier options use the continuously monitored options as external control variates for their discrete counterparts. The general control variate framework for Lévy process models contains special and general control variates, which are path functionals of the original Lévy process and a coupled Brownian motion. The method is based on fast numerical inversion of the cumulative distribution functions.