Title Statističko modeliranje kopulama Title (english) Statistical modeling of copulas Author Darko Tomljanović Mentor Snježana Lubura Strunjak (mentor)Committee member Snježana Lubura Strunjak (predsjednik povjerenstva)Committee member Nenad Antonić (član povjerenstva)Committee member Eduard Marušić-Paloka (član povjerenstva)Committee member Damir Bakić (član povjerenstva)Granter University of Zagreb Defense date and country 2023-07-19, Croatia Scientific / art field, NATURAL SCIENCES Abstract U ovom radu proučavali smo kopule. Na početku smo definirali kopulu kao funkciju distribucije s uniformnim marginalnim funkcijama distribucije na intervalu \( [0, 1] \). Naveli smo i njezinu karakterizaciju te smo upoznali najjednostavniju kopulu, nezavisnu kopulu. Pokazali smo da gornja i donja Frechet-Hoeffdingova granica omeđuju svaku kopulu te da su, u određenim dimenzijama, i same kopule. Zatim smo se usredotočili na Sklarov teorem, središnji teorem ovog diplomskog rada. Osim što smo ga iskazali i dokazali, dali smo i njegovu interpretaciju. Uveli smo pojmove savršene zavisnosti i nezavisnosti kao i repne zavisnosti. U ovom poglavlju analizirali smo i koeficijente korelacije te smo primijetili da nam linearni koeficijenti korelacije nisu optimalni procjenitelji te smo zbog toga uveli dva koeficijenta korelacije ranga, Spearmanov \(\rho\) i Kendallov \(\tau\). Klase i familije kopula proučavali smo u trećem poglavlju. Naveli smo tri klase kopula: eliptične kopule, Arhimedove kopule i kopule ekstremnih vrijednosti te smo im, izučavajući njihova svojstva, pridružili familije kopula. U sklopu naše analize najveća je pažnja bila usmjerena na normalnu i \(t\)-kopulu kao familijama eliptične klase kopula. U nastavku smo modelirali tri primjera primjene kopula. Prvi primjer govori o modeliranju agregacije rizika pomoću kopula, drugim primjerom pokazali smo da je logaritamske povrate dionica bolje modelirati \(t\)-kopulom nego normalnom kopulom, dok zadnji primjer raspravlja o stajalištu Felixa Salmona da je kopula, zbog pretjerane uporabe, prouzročila financijsku krizu. Na kraju rada se nalazi implementacija primjera. Primjere smo implementirali u programskom jeziku R.
Abstract (english) In this work, we examine copulas. We first define a copula as a distribution function with a uniform marginal distribution function on the interval \( [0, 1] \). We also provide their characteristics and demonstrate the simplest, independent copula. We have shown that each copula is bounded by upper and lower Fréchet-Hoeffding boundaries and that these boundaries are copulas in some dimensions. Next, we focus on Sklar's theorem, which is the central theorem of this work. We not only propose and prove it, but we also provide its interpretation. At the end of the chapter, we introduce the concept of a surviving copula and discuss the properties of radial symmetry and commutativity. The subsequent chapter delved into the concept of dependence. Our analysis comprised of the introduction of perfect dependence, independence, and tail dependence. Additionally, we scrutinized correlation coefficients and arrived at the conclusion that linear correlation coefficients are not the most effective estimators. Therefore, we introduced two alternative rank correlation coefficients: Spearman's \(\rho\) and Kendall's \(\tau\). In the third chapter, our focus was on the study of copulas and their classifications. We identified three main classes of copulas: elliptical copulas, Archimedean copulas, and extreme value copulas. Through the examination of their properties, we were able to associate different copula families with each of these classes. The second instance showcased the superiority of the \(t\)-copula over the normal copula in modeling the logarithmic returns of stocks. Lastly, we explored Felix Salmon's perspective on the financial crisis, which attributes it to the overreliance on copulas. Concluding the paper are the implementation examples, which were executed using the R programming language.
Keywords
nezavisna kopula
Frechet-Hoeffdingova granica
Sklarov teorem
programski jezik R
Keywords (english)
independent copula
Fréchet-Hoeffding boundaries
Sklar's theorem
R programming language.
Language croatian URN:NBN urn:nbn:hr:217:397195 Study programme Title: Financial and Business Mathematics Type of resource Text File origin Born digital Access conditions Open access Terms of use Created on 2024-02-09 14:41:14
