Matematičko modeliranje diferencijalnim jednadžbama u ekonomiji

master's thesis

• Title (english): Mathematical modeling with differential equations in economics
• Author: Nina Ritz
• Mentor: Marko Radulović (mentor)
• Committee member: Marko Radulović (predsjednik povjerenstva)
• Committee member: Igor Pažanin (član povjerenstva)
• Committee member: Boris Muha (član povjerenstva)
• Committee member: Damir Bakić (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2023-09-27, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

Glavna ideja ovog rada bila je modeliranje cijene opcije, financijskog instrumenta koji investitorima na tržištu često služi kao metoda zaštite od rizika. Cilj je bio pomoću diferencijalnih jednadžbi postaviti model te, ako je to moguće, doći do zatvorene formule za cijenu opcije u trenutku t i uz cijenu temeljne imovine S . U prvom smo se poglavlju upoznali s ključnim pojmovima, i financijskim i matematičkim, koji su nam bili potrebni kroz cijeli rad. Tako smo saznali nešto više o opcijama,... More uveli pojmove poput arbitraže i Brownovog gibanja, te na kraju napravili uvod u parcijalne diferencijalne jednadžbe. Drugo je poglavlje bilo posvećeno konkretno europskim opcijama. Nakon uvodenja nekoliko osnovnih pretpostavki, izveli smo jednu od najpoznatijih jednadžbi u svijetu financija, Black-Scholesovu jednadžbu, čijim smo rješavanjem došli do formule za cijenu europske opcije: Black-Scholesova formula. Nakon toga smo malo generalizirali početne pretpostavke kako bismo dobili nešto općenitiji model, primjenjiv u realnom svijetu, te smo u konačnici postavili modele (i izveli formule) za nekoliko vrsta egzotičnih opcija. Rad smo završili s trećim poglavljem gdje smo se bavili isključivo američkim opcijama. Kako je kod američke opcije koncept isplate nešto kompliciraniji, generalno ne postoji zatvorena formula za cijenu. Ipak smo kod perpetualne američke opcije, koja ja nešto jednostavnija, uspjeli izvesti konkretnu formulu, dok smo za opći slučaj razvili model te dokazali formulu za dekompoziciju cijene američke opcije, koja se sastoji od cijene europske opcije te dodatne premije. Less

Abstract (english)

The main idea behind this work was to model the price of an option, a financial instrument that often serves investors as a method of risk protection in the market. The goal was to establish a model using partial differential equations and then, if possible, derive a closed form solution for the option price at time t, with underlying asset price being S. In the first chapter, we defined some of the key concepts, both financial and mathematical, that were necessary throughout the paper. We... More learned more about options, introduced concepts such as arbitrage and Brownian motion, and, in the end, finalized the chapter with an introduction to partial differential equations. The second chapter was dedicated specifically to European options. After introducing few basic assumptions, we derived one of the most famous equations in the world of finance, the Black-Scholes equation, which we then solved to obtain the formula for the price of an European option: the Black-Scholes formula. Following that, we made a few changes to the basic assumptions to obtain a more general model, applicable in the real world, and concluded with deriving models for several types of exotic options. The work was finalized with the third chapter, where we exclusively dealt with American options. As the concept of payout for American options is slightly more complicated, a closed-form solution for the option price generally does not exist. We still managed to derive a specific formula for perpetual American options, which are somewhat simpler, while for the general case, we developed a model and proved a formula for the decomposition of the American option price that consists of the European option price and an additional premium. Less

• Keywords: zaštita od rizika; cijena opcije; Brownovo gibanje; parcijalne diferencijalne jednadžbe
• Keywords (english): risk protection; option price; partial differential equations; Brownian motion
• Language: croatian
• URN:NBN: urn:nbn:hr:217:327282
• 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: 2024-02-05 13:32:28