On the concept of arbitrage and some applications of arbitrage pricing

Abstract

This thesis will present the concept of arbitrage and some applications of arbitrage pricing. Arbitrage opportunity means that there is a possibility to make money without any initial investment and without a risk of losing money. To start, some definitions are introduced in the fields of measure theory, probability theory and mathematical finance. Then the guidelines of market models considered throughout the thesis will be defined. The mathematical definition of arbitrage and arbitrage pricing are introduced first in simple setting of one period market model and then in multi-period market model.

As a main result of this thesis are introduced and proven The fundamental theorems of arbitrage pricing. The first fundamental theorem of arbitrage pricing shows that a market is arbitrage free if and only if there exists at least one risk neutral probability measure equivalent to original probability measure such that the discounted prices are martingales with respect to this risk neutral measure. This will be proven for multi-period market model. The second fundamental theorem of arbitrage pricing shows that the completeness of a market model is equivalent to existence of unique risk-neutral probability measure. This will be proven for one period market model.

Finally, I look into some investing and hedging strategies replicating payoffs and portfolio insurance. Some examples of commonly used options strategies will be introduced such as butterfly spread and iron condor.