A p�nz�gyi eszk�z�k nagyon fontos csoportj�t alkotj�k a sz�rmaztatott �gyletek, derivat�vok. Ezek olyan �gyletek, amelyek �rt�k�t m�s �rt�kpap�rok �rfolyama hat�rozza meg.
Az egyik legfontosabb ilyen sz�rmaztatott �gylet az opci�k. A dolgozat t�rgya az opci��raz�s, azaz arra keresi a v�laszt, hogy mennyit �r egy opci�, �s melyek azok a t�nyez�k, amelyek a r�szv�nyopci�k �r�ra kihat�ssal vannak.
Indul�sk�ppen kivizsg�l�sra ker�l az �n. egyperi�dusos binomi�lis fa modellje, amely csak a legsz�ks�gesebbeket tartalmazza: egy r�szv�nyt �s egy kincst�rjegyet. Arbitr�zsmentess�gen alapul� �rt�kel�s mellett meghat�roz�sra ker�l az opci� �ra. A munka m�sodik r�sz�ben m�dosulnak a felt�telek. A lej�rati id� t�bb azonos r�szre oszt�dik �s minden id�szak v�g�n a r�szv�ny �rfolyama k�tf�lek�ppen v�ltozhat. Ily m�don j�n l�tre a t�bbperi�dusos binomi�lis fa modellje. A fa szerkezete biztos�tja azt, hogy b�rmely derivat�v term�knek a fa minden egyes pontj�ban egy�rtelm� �rt�ke legyen, hiszen b�rmely m�s �rt�k arbitr�zshelyzetet teremtene. A kifizet�sek a megfelel� visz-szasz�m�tott �rt�keken kereszt�l a fa teljes kit�lt�s�vel elvezetnek a derivat�v jelenbeli �rt�k�hez.
V�g�l levezet�sre ker�l a nevezetes Black-Scholes opci��raz�si formula. A dolgozatot megfelel� p�ld�k teszik sz�nesebb�.

Kulcsszavak: opci�k, binomi�lis fa, arbitr�zs, Black-Scholes formula

OPTION PRICING
Author: Anik� LUKITY, 5th year student
Supervisor: Dr. �rp�d TAK�CSY, university professor
Institution: University of Novi Sad, Faculty of Science, Department of Mathematics and Informatics, Novi Sad
Hungarian College For Higher Education in Vojvodina

One of the often used financial instruments are the contingent claims (derivatives). A derivative is a security whose value depends on the value of some underlying security. Perhaps the most important contingent claims are the options. The topic of this paper is the option pricing, i.e., to find the value of an option and determine the factors that influence its price.
At the beginning, we analyze the model of the one-periodic binomial tree, which contains only the necessary components: a stock and a bond. Beside the arbitrage-free evaluation, one has to determine the price of the option. In the second part of the paper, we impose more involved assumptions. Namely, time interval is divided into several identical parts, and at the end of each time period the value of the stock can change in two ways. Thus the multiperiod binomial tree model is obtained. The construction of the tree insures that any derivative has a unique value at every point of the tree, since any other value would ask for an arbitrage. The outpayments, together with the adequate discount values, lead to the present value of the derivative.
Finally, we apply the well known Black-Scholes option pricing formula for some examples.

Keywords: options, binomial tree, arbitrage, Black-Scholes formula

CENE OPCIJE
Autor: Aniko LUKI�, apsolvent
Mentor: Dr Arpad TAKA�I, redovni profesor
Institucija: Univerzitet u Novom Sadu, Prirodno-matemati�ki fakultet, Odsek za matematiku i informatiku, Novi Sad
Kolegijum za visoko obrazovanje vojvo�anskih Ma�ara

Derivativi �ine najva�niju grupu finansijskih instrumenata. To su takvi finansijski poslovi, �iju cenu odre�uju vrednosti drugih vrednosnih papira.
Jedan od najva�nijih derivativa �ine opcije. Predmet ovog rada su cene opcija, tj. odre�uju se njihova vrednost i glavni faktori koji uti�u na formiranje njihove cene.
Na po�etku, ispitujemo tzv. jednoperiodi�ni binomialni model, koji sadr�i samo najosnovnije elemente: jednu akciju i jedan bankarski ulog. Cena opcije se odre�uje pod uslovom da nema mogu�nosti arbitra�e. U drugom delu rada modifikujemo uslove. Naime, vremenski intevral se deli na vi�e jednakih delova, a na kraju svakog perioda se cena opcije mo�e menjati na dva na�ina. Ovako se dolazi do vi�eperiodi�nog binomialnog modela. Takva konstrukcija omogu�ava da se u bilo kojoj ta�ki dobijenog drveta jednozna�no odredi cena opcije, jer bi se u protivnom otvorila mogu�nost arbitra�e. Diskontiranjem isplata se dobija sada�nja vrednost derivativa.
Na kraju, izvodi se i primenjuje poznata Black-Scholes-ova formula za odre�ivanje cene opcija, �to ilustrujemo na primerima.

Klju�ne re�i: opcije, binomialno drvo, arbitra�a, formula Black-Scholes-a