OPTION PRICING WITH THE BINOMIAL TREE MODEL

Matematična obravnava ekonomskih zakonitosti je v sodobnem času postala nepogrešljiva. Ekonomska stroka se namreč ukvarja z merljivimi količinami, med katerimi so določene medsebojne odvisnosti. Matematični modeli opisujejo in ponazarjajo te odvisnosti ter omogočajo po eni strani analizo teh odnosov, po drugi strani pa nudijo možnost, da predvidimo, kako se bodo, ob izbranih predpostavkah, vrednosti spremenljivk obnašale v prihodnosti. Namen in cilj diplomskega dela je opisati model binomskega drevesa, s katerim lahko vrednotimo (izvedene) finančne instrumente. Glavno pozornost smo posvetili vrednotenju evropskih in ameriških opcij. V uvodnem poglavju smo predstavili definicije, trditve in izreke, ki smo jih potrebovali v nadaljevanju. V drugem poglavju so prikazane osnove vrednotenja netveganih sredstev kot sta obveznica in bančni depozit. Opisane so osnove obrestnega obrestovanja (letna, izpodletna kapitalizacija, zvezno obrestovanje). V tretjem poglavju so zapisana temeljna načela, na katerih slonijo diskretni modeli denarnega trga. Poseben poudarek je dan načelu nearbitražnosti, ki je osnova pri dokazih trditev in izrekov. Prikazan in opisan je model binomskega drevesa in vrednotenje delnic v okvirih tega modela. V drugem delu diplomskega dela so opisane osnove vrednotenja nekaterih izvedenih finančnih instrumentov v okvirih modela binomskega drevesa. Tako je v četrtem poglavju obravnavano vrednotenje terminskih pogodb. Peto poglavje je v celoti namenjeno opcijam. V tem poglavju smo preučili nekatere lastnosti opcij in zapisali razlike med prodajnimi in nakupnimi opcijami ter med evropskimi in ameriškimi opcijami. V zadnjem poglavju je predstavljeno vrednotenje (evropskih in ameriških) opcij v modelu binomskega drevesa z enim, dvema in več koraki.Mathematical discussion of economic mechanisms in modern times has become indispensable. Economics is the discipline concerned with the measurable quantities, in which certain co-dependence exists. Mathematical models are describing and illustrating this co-dependence, what enables on the one hand analysis of these relations and on the other hand offers the possibility (upon given assumptions) to predict the behaviour of the values of variables in the future. The purpose and the goal of the thesis is to describe the binomial tree model, which can be used to evaluate derivative securities. The main attention is given to the pricing of European and American options. First, in the introduction, we present definitions, arguments and theorems which we need in the continuation. In the second chapter we show the basics of valuation of risk-free assets such as bonds and bank deposits. We describe the basics of compounding (annual, periodic compounding, continuous compounding). In the third chapter fundamental principles of discrete money market models are briefly discussed. Special emphasis is laid on the no-arbitrage principle, which is the basic principle for proving propositions and theorems. We also show and describe the binomial tree model and stock pricing with this model. In the second part of the thesis the basics of valuation of certain derivative securities with the binomial tree model are explained. Consequently, in the fourth chapter the valuation of forward contracts is debated. The fifth chapter is entirely dedicated to options. In this chapter we study some properties of options and list differences between put and call options and between European and American options. In the last chapter we present pricing of (European and American) options with the binomial tree model with one, two and multiple steps