Stohastički račun u financijskoj matematici

master's thesis

• Title (english): Stochastic calculus in financial mathematics
• Author: Sara Mihaela Prpić
• Mentor: Marko Radulović (mentor)
• Committee member: Marko Radulović (predsjednik povjerenstva)
• Committee member: Hrvoje Šikić (član povjerenstva)
• Committee member: Vanja Wagner (član povjerenstva)
• Committee member: Marko Vrdoljak (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2025-02-26, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

U ovom radu objasnili smo veliku važnost stohastičkog računa u financijskoj matematici, te pokazali neke od njegovih najbitnijih upotreba i utjecaj na razvoj tržišta. U prvom smo poglavlju dali kratak uvod u svijet opcija i portfelja, te se upoznali s ključnim pojmovima koji su nam potrebni kroz cijeli rad, kao što su Brownovo gibanje te Itôv integral i njegova formula. Oni nam daju osnovu za uvođenje jednog od najvažnijih modela za vrednovanje financijske imovine, a to je... More Black-Scholes-Mertonov model kojeg promatramo u drugom poglavlju. Prije samog izvoda modela definiramo pojmove ekvivalentne martingalne mjere i navodimo Girsanovljev teorem koji omogućuje promjenu mjere, kako za jednodimenzionalni tako i za višedimenzionalni slučaj. U trećem poglavlju analiziramo povezanost stohastičkih i parcijalnih diferencijalnih jednadžbi pomoću Feynman-Kac teorema i uvodimo Hull-White model. Na kraju, proučavamo opcije koje se sastoje od više osnovnih imovina te se na početku poglavlja upoznajemo s višedimenzionalnom Black-Scholesovom funkcijom pomoću koje u ostatku poglavlja računamo cijene tih opcija. Opisujemo tri opcije s više osnovnih imovina, a to su rainbow, basket i quanto opcije, a one igraju ključnu ulogu u smislu upravljanja rizicima, te diverzifikaciji i optimizaciji portfelja. Less

Abstract (english)

This thesis explains the great importance of stochastic calculus in financial mathematics, and shows some of its most important applications and its influence on market development. In the first chapter, we gave a brief introduction to the world of options and portfolios, and got acquainted with the key concepts necessary throughout the paper, such as Brownian motion and It\^o integral along with its formula. They give us the foundation for introducing one of the most... More important models for the valuation of financial assets, the Black-Scholes-Merton model, which we analyze in the second chapter. Before deriving the model itself, we define the terms of equivalent martingale measure and state Girsanov's theorem, which allows a measure change, both for the one-dimensional and for the multidimensional case. In the third chapter, we analyze the connection between stochastic and partial differential equations using the Feynman-Kac theorem and introduce the Hull-White model. Finally, we study options that consist of several underlying assets, and at the beginning of the chapter we are introduced to the multidimensional Black-Scholes function, which we use to calculate the prices of these options in the rest of the chapter. We describe three types of multi-asset options: rainbow, basket and quanto options, which play a key role in terms of risk management, porfolio diversification and optimization. Less

• Keywords: stohastički račun; opcije; Brownovo gibanje; Itôv integral; Itôva formula; ekvivalentna martingalna mjera; Girsanovljev teorem; Black-Scholes-Mertonova funkcija; Feynman-Kac; Hull-White; rainbow opcije; basket opcije; quanto opcije
• Keywords (english): stochastic calculus; options; Brownian motion; Itô's integral; Itô's formula; equivalent martingale measure; Girsanov's theorem; Black-Scholes-Merton function; Feynman-Kac; Hull-White; rainbow options; basket options; quanto options
• Language: croatian
• URN:NBN: urn:nbn:hr:217:082034
• Study programme: Title: Financial and Business Mathematics; Study programme type: university; Study level: graduate; Academic / professional title: sveučilišni magistar matematike (sveučilišni magistar matematike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2025-02-10 23:37:09