On THe Theory of Pricing of Discrete Time Options of European Type

Abstract

The contemporary nancial market appears to be a part of human activity where ideas of Stochastic Analysis, in particular Martingale Theory and Stochastic Ito's integral, have been implemented in a most complete matter. Creating a link between the theory and practice of nance generates new problems in the Theory of Probability and the Financial Mathematics investigating them. Financial markets consists of two main (primary) assets (securities), bonds (rick- less) and shares (risky). Bonds are debentures issued by a state government or a bank with a goal to accumulate capital. As an example we can consider a bank account or government bonds. Bond owners accept a strictly de ned pro t, conditional on the current interest rate. Shares are equity securities which are issued by companies in order to accumulate capital for further activities. Its price is de ned by the situation at the stock market and by the production activity of the company. Shareholders obtain a pro t according the price of the share. Options belong to derivative (secondary) securities. An option is a security that gives to its owner a right to sell (or to buy) some worth (shares, currency, etc.) by conditions speci ed in advance. In this thesis, we investigate the random behaviour of a share price, creating an \optimal" portfolio of securities and related various problems in Financial Mathe- matics. In particular, we discuss the problem of option pricing. The literature on Financial Mathematics is too large to mention in my theses, consult, for example, the lengthy list of references in the textbook [2]. In our theses, we cite only the articles and books which are important and are directly used in our research. The real breakthrough in the methods of nancial calculations connected with options has been done by Black, Scholes and Merton in 1973 [1], [?]. The theory developed in these manuscripts allows for nding a \fair" price of an option and also provides a guidance to optimal stock transactions that allow for the option writer to guarantee the possible pay o s, which depend on a random behaviour of prices in a nancial market. In this thesis we discuss the theory of pricing options of European type in discrete time setting. Everything is explained from scratch, the reader needs to have only a basic knowledge of Probability Theory at an elementary level. For example, we introduce an advanced and comprehensive notion of a martingale, but we consider measurability by a nite partition, not by a general -algebra. This simpli es the understanding of the theory signi cantly. ii