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Stochastic Models in Financial Mathematics
Waczulík, Oliver ; Hurt, Jan (advisor) ; Večeř, Jan (referee)
Title: Stochastic Models in Financial Mathematics Author: Bc. Oliver Waczulík Department: Department of Probability and Mathematical Statistics Supervisor: doc. RNDr. Jan Hurt, CSc., Department of Probability and Mathe- matical Statistics Abstract: This thesis looks into the problems of ordinary stochastic models used in financial mathematics, which are often influenced by unrealistic assumptions of Brownian motion. The thesis deals with and suggests more sophisticated alternatives to Brownian motion models. By applying the fractional Brownian motion we derive a modification of the Black-Scholes pricing formula for a mixed fractional Bro- wnian motion. We use Lévy processes to introduce subordinated stable process of Ornstein-Uhlenbeck type serving for modeling interest rates. We present the calibration procedures for these models along with a simulation study for estima- tion of Hurst parameter. To illustrate the practical use of the models introduced in the paper we have used real financial data and custom procedures program- med in the system Wolfram Mathematica. We have achieved almost 90% decline in the value of Kolmogorov-Smirnov statistics by the application of subordinated stable process of Ornstein-Uhlenbeck type for the historical values of the monthly PRIBOR (Prague Interbank Offered Rate) rates in...
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Stochastic Models in Financial Mathematics
Waczulík, Oliver ; Hurt, Jan (advisor) ; Večeř, Jan (referee)
Title: Stochastic Models in Financial Mathematics Author: Bc. Oliver Waczulík Department: Department of Probability and Mathematical Statistics Supervisor: doc. RNDr. Jan Hurt, CSc., Department of Probability and Mathe- matical Statistics Abstract: This thesis looks into the problems of ordinary stochastic models used in financial mathematics, which are often influenced by unrealistic assumptions of Brownian motion. The thesis deals with and suggests more sophisticated alternatives to Brownian motion models. By applying the fractional Brownian motion we derive a modification of the Black-Scholes pricing formula for a mixed fractional Bro- wnian motion. We use Lévy processes to introduce subordinated stable process of Ornstein-Uhlenbeck type serving for modeling interest rates. We present the calibration procedures for these models along with a simulation study for estima- tion of Hurst parameter. To illustrate the practical use of the models introduced in the paper we have used real financial data and custom procedures program- med in the system Wolfram Mathematica. We have achieved almost 90% decline in the value of Kolmogorov-Smirnov statistics by the application of subordinated stable process of Ornstein-Uhlenbeck type for the historical values of the monthly PRIBOR (Prague Interbank Offered Rate) rates in...
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Multiple Testing Problem
Waczulík, Oliver ; Komárek, Arnošt (advisor) ; Anděl, Jiří (referee)
Title: Multiple Testing Problem Author: Oliver Waczulík Department: Department of Probability and Mathematical Statistics Supervisor: RNDr. Arnošt Komárek, Ph.D., Department of Probability and Mat- hematical Statistics Abstract: This thesis looks into the different approaches to control the type-1 error inflation, more specifically, it looks into different procedures that control familywise error rate (FWER), when testing more than one elementary hypothesis simultaneously. It focuses on two basic types of problems: comparing the mean of more than one independently normally distributed random sample with the control and com- paring the means of more than one independently normally distributed random sample between each other. To further illustrate the mechanics of the multiple comparison procedures stated in this thesis we use a real measurements of the brain volume of four independent groups of males, where we anticipate normal distribution with the same variance. Global null hypothesis is always tested as first, followed by testing of elementary hypothesis in the case of rejection. To tackle the first problem, we use Bonfferoni and Holm multiple comparison pro- cedures along with Dunnett method and closure principle. Unlike Dunnett with Bonfferoni, Holm and Bonfferoni's closed test provide different conclusion...
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