Darbe taikyta Norvaišos ir Salopk (2002) metodologija funkcijos šiurkštumui nagrinėti remiantis modifikuotu funkcijos grafiko dėžučių skaičiaus indeksu. Funkcijos šiurkštumas nusakomas p-variacijos indeksu, kuris prie tam tikrų sąlygų lygus fraktalo dimensijai. Darbe ištirtos tiesinės regresijos, kuri vertina p-variacijos indeksą, liekanos ir pasiūlytas būdas kaip išpildyti balto triukšmo prielaidas. Rezultatai apibendrinti Monte Carlo procedūra. Sukonstruoti p-variacijos indekso pasikliautinieji intevalai -stabiliam ir trupmeniniam Brauno judesio procesams. Ištirtas p-variacijos indekso kintamumas laike „Vallourec” akcijų kainos procesui. / To estimate the roughness of the sample function the methodology introdused in Norvaiša and Salopek (2002) was applied. The roughness is defined as p-variation index of the sample function graph. Methodology is based on linear regression of the oscilation index. This master thesis tests the assumptions of linear regression residuals and constructs estimator which fulfill these assumptions. The model was used for the generated α-stable process and fractional Brownian motion. Conclusions are generalized using Monte-Carlo procedure. The confidence intervals for the p-variation index was constructed making assumption that the process is the realisation of -stable or fractional Brownian motion. The p-variation index was estimated for the „Vallourec” stock price data, sampled at irregular time. In addidion the variability in time of p-variation index was studied for different segments of intervals.

This master thesis should provide reader with an overview of the European spread options evaluation using the fast Fourier transform numerical method. The first and second part of the thesis deal with the theoretical foundations of Fourier analysis and existing approaches of spread option valuation under two and three-factors frameworks (namely GBM - geometric Brown motion and SV - stochastic volatility). The third part describes extention of Hurd-Zhou (2010) valuation method by tool for call and put spread options pricing in case of negative or zero strikes. Extension will be compared with Monte Carlo simulation results from a variety of perspectives, including computing complexity and implementation requirements. Dempster-Hong model, Hurd-Zhou model and Monte Carlo simulation are implemented and tested in R (programming language).