Egzotične opcije

master's thesis

• Author: Anamarija Užbinec
• Mentor: Zoran Vondraček (mentor)
• Committee member: Bojan Basrak (predsjednik povjerenstva)
• Committee member: Miljenko Huzak (član povjerenstva)
• Committee member: Juraj Šiftar (član povjerenstva)
• Committee member: Pavle Goldstein (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2014-09-25, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

Egzotične opcije jedna su od podvrsta financijskih izvedenica koje su nastale modificiranjem standardnih obilježja europskih i američkih opcija. Postoji velik broj različitih vrsta egzotičnih opcija, a u ovom radu bavimo se onima koje se najčešće koriste. Uvođenje stohastičkog računa u financije i razvoj Black-Scholes-Mertonove formule uvelike su pridonijeli evoluciji tržišta kako financijskih izvedenica tako i samih egzotičnih opcija. U radu objašnjavamo osnove stohastičkog računa i izvodimo Black-Scholes-Mertonovu formulu za cijenu opcije. Također potvrđujemo da se matematička teorija savršeno uklapa u financije. Prilikom izdavanja opcije suočavamo se s problemom određivanja cijene ovoga instrumenta koja ovisi o kretanju cijene imovine na koju je opcija napisana, a istodobno i s problemom onemogućavanja arbitraže na tržištu. Cilj ovog diplomskog rada bio je detaljno prikazati jedan od načina vrednovanja lookback opcija i opcija s barijerom koje se odnose na dionice čije je kretanje modelirano geometrijskim Brownovim gibanjem. U oba slučaja diskontirana cijena opcije je martingal uz vjerojatnost neutralnu na rizik. Izračunali smo formule za cijenu up-and-out opcije s barijerom te lookback opcije s fluktuirajućom cijenom izvršenja preko očekivane vrijednosti isplate opcija.

Abstract (english)

Exotic options are one of the subtypes of derivatives that were created by modifying the standard features of European and American options. There are numerous different types of exotic options. In this paper we deal with those that are most commonly used. The application of stochastic calculus in quantitative finance, and the development of the Black-Scholes-Merton formula have greatly contributed to the evolution of the markets for derivatives and for exotic options. In this paper, we explain the basics of stochastic calculus and derive the Black-Scholes-Merton formula. We also confirm that the mathematical theory fits in perfectly within financial modeling. When issuing options, we are faced with the problem of instrument pricing. Such pricing depends upon the price of the underlying asset on which the instrument is written. At the same time, we also discuss the problem of no arbitrage in financial markets. The aim of this diploma thesis is to describe in detail a method for the pricing of lookback and barrier options. Stock prices are modeled using the Geometric Brownian Motion model for stock prices. In both cases, the discounted prices of options are martingales with a risk-neutral probability. We calculate the formulas for the pricing of up-and-out barrier options, and for the floating strike lookback options using the expected value of option payoff.

• Keywords: egzotične opcije; Black-Scholes-Mertonova formula; geometrijsko Brownovo gibanje; vrednovanje lookback opcija; vrednovanje opcija s barijerom
• Keywords (english): exotic options; Black-Scholes-Merton formula; Geometric Brownian Motion; pricing of lookback options; pricing of barrier options
• Language: croatian
• URN:NBN: urn:nbn:hr:217:435305
• Study programme: Title: Finance and Business Mathematics; Study programme type: university; Study level: graduate; Academic / professional title: magistar/magistra matematike (magistar/magistra matematike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Access restricted to students and staff of home institution
• Created on: 2019-02-18 12:24:52