Primjena modela volatilnosti na izvedenice

master's thesis

• Title (english): Application of volatility model to derivatives
• Author: Ema Marija Yeoman
• Mentor: Azra Tafro (mentor)
• Committee member: Azra Tafro (predsjednik povjerenstva)
• Committee member: Hrvoje Šikić (član povjerenstva)
• Committee member: Hrvoje Planinić (član povjerenstva)
• Committee member: Sonja Štimac (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2022-05-05, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

U ovom diplomskom radu predstavljamo jednu od brojnih primjena modela volatilnosti na izvedenice. U samom središtu je GARCH model koji je jedan od najznačajnijih i često korištenih modela volatilnosti za financijske vremenske nizove. Na početku, iznosimo već poznate rezultate i alate za vremenske nizove i njihovu primjenu u financijama. Dajemo uvid u modele volatilnosti kroz razumijevanje osnovnog koncepta volatilnosti, te motivaciju iza proučavanja takvih modela kao i primjenu istih za odredivanje cijena opcija. Takoder, ukratko uvodimo ARCH model koji je preteča GARCH modelu kojemu smo se detaljnije posvetili. Između ostalog, iskazujemo teorem o nužnom i dovoljnom uvjetu pri kojemu je GARCH(p,q) model slabo stacionaran, te u nastavku promatramo neke modifikacije samog modela i njegova proširenja. Slijedi okosnica ovog rada u kojoj promatramo model određivanja cijena opcija gdje uvjetna varijanca log-povrata financijske imovine prati GARCH(p,q) proces. Na kraju rad zaokružujemo praktičnim primjerom ilustriranim u programskom jeziku R gdje teorijske rezultate primjenjujemo na stvarnim podacima s tržišta za određivanja cijene europske call opcije i odgovarajuće delte.

Abstract (english)

In this Master’s thesis, we present one of the many applications of the volatility model on derivatives. The focus is on the GARCH model as one of the most significant and frequently used volatility models for financial time series. In the beginning, we present the already known results and tools for time series and their application in finance. We provide insight into volatility models through understanding the basic concept of volatility, the motivation behind studying such models, as well as the application in option pricing. Also, we briefly introduce the ARCH model which is the forerunner of the GARCH model that we observe in more detail. Moreover, we present the theorem that gives the necessary and sufficient conditions under which the GARCH (p, q) model is weakly stationary. Furthermore, we observe some modifications of the model itself and its extensions. The following chapter is the backbone of this paper in which we present the option pricing model where the conditional variance of the log-return of financial assets follows the GARCH (p, q) process. Finally, we conclude the thesis with a practical example illustrated in the programming language R, where we apply the theoretical results to actual data for the european call option pricing and the corresponding delta.

• Keywords: GARCH model; ARCH model; programski jezik R; europska call opcija
• Keywords (english): GARCH model; ARCH model; programming language R; european call option pricing
• Language: croatian
• URN:NBN: urn:nbn:hr:217:381764
• 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: Open access
• Created on: 2022-06-09 10:13:08