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In the present study I deal with a pricing of derivatives especially with the European option. In the first chapter there are described basic principles of pricing financial derivatives. I focus on the options strategies from the simplest to the more difficult one. The second chapter is dedicated to the Binomial pricing model. It is introduced its derivation, application, its pro and con. Next chapter contains a description of Black-Scholes model. Again it is explained derivation of this model and its properties. At the end of this chapter it is described relationship between Binomial and Black-Scholes models. The forth chapter is consisted of an analysis of real data of stocks company Philip Morris International, Lehman brothers Holding and American Insurance Group. I focus on the relationship between shares and options in time of the financial crisis. Last chapter is dedicated to the description of software concerning options which was created in Microsoft Excel and which is part of this study.

A convertible bond (CB) is a financial derivative, a so called hybrid security. It is an issued contract from a company or a government, which is paid for up-front. The contract yields a known amount at the specified maturity date, unless the holder chooses to convert it into an amount of the underlying asset. This kind of financial products can have complex features affecting the contract price and the optimal exercising situation. The partial differential equation (PDE) approach used for pricing financial derivatives makes it possible to describe convertible bonds with a physical model, a reversed diffusion described by a parabolic PDE. One can sometimes find both analytical and numerical solutions for this type of PDEs and interpret the solutions from a financial point of view, as they suggest predictable behaviour of the contract price.

This thesis is on an advanced method for pricing financial derivatives in a market model,which comprises two stochastic volatilities. Financial derivatives are instruments whosethat is related to any financial asset. Underlying assets in derivatives are mostly financialinstruments; such as security, currency or a commodity. Stochastic volatilities are used infinancial mathematics to assess financial derivative securities; such as contingent claims andoptions for valuation of the derivatives, at the expiration of the contract. This study examinedtheoretical frameworks that evolve around the pricing of financial deriv- atives in a marketmodel and it mainly examines two stochastic volatilities: cubature formula and splittingmethod by analysing how these volatilities affect the pricing of financial derivatives. The studydeveloped an approximation approach with a double stochastic volatilities model in termsof Stratonovich integrals to evaluate the contingent claim, examined the similarities betweenNinomiya–Ninomiya scheme and Ninomiya–Victoir scheme, and rewrite the system of doublestochastic volatility model in terms of the standard Brownian motion.

Thesis is devoted to basic aspects of the reporting of financial derivatives. Mentioned is brief history and determination of term "derivative" from different points of view. Follows diversification of financial derivatives into groups. The main part is focused on reporting of fixed term contracts and options. Finally is described approach of International Financial Reporting Standards.

A design of anautomatic network capacity markets, oftenreferred to as a bandwidth market, is presented. Three topicsare investigated. First, a network model is proposed. Theproposed model is based upon a trisection of the participantroles into network users, network owners, and market middlemen.The network capacity is defined in a way that allows it to betraded, and to have a well defined price. The network devicesare modeled as core nodes, access nodes, and border nodes.Requirements on these are given. It is shown how theirfunctionalities can be implemented in a network. Second, asimulated capacity market is presented, and a statisticalmethod for estimating the price dynamics in the market isproposed. A method for pricing network services based on sharedcapacity is proposed, in which the price of a service isequivalent to that of a financial derivative contract on anumber of simple capacity shares.Third, protocols for theinteraction between the participants are proposed. The marketparticipants need to commit to contracts with an auditableprotocol with a small overhead. The proposed protocol is basedon a public key infrastructure and on known protocols for multiparty contract signing. The proposed model allows networkcapacity to be traded in a manner that utilizes the networkeciently. A new feature of this market model, compared to othernetwork capacity markets, is that the prices are not controlledby the network owners. It is the end-users who, by middlemen,trade capacity among each-other. Therefore, financial, ratherthan control theoretic, methods are used for the pricing ofcapacity. <b>Keywords:</b>Computer network architecture, bandwidthtrading, inter-domain Quality-of-Service, pricing,combinatorial allocation, financial derivative pricing,stochastic modeling

<p>A design of anautomatic network capacity markets, oftenreferred to as a bandwidth market, is presented. Three topicsare investigated. First, a network model is proposed. Theproposed model is based upon a trisection of the participantroles into network users, network owners, and market middlemen.The network capacity is defined in a way that allows it to betraded, and to have a well defined price. The network devicesare modeled as core nodes, access nodes, and border nodes.Requirements on these are given. It is shown how theirfunctionalities can be implemented in a network. Second, asimulated capacity market is presented, and a statisticalmethod for estimating the price dynamics in the market isproposed. A method for pricing network services based on sharedcapacity is proposed, in which the price of a service isequivalent to that of a financial derivative contract on anumber of simple capacity shares.Third, protocols for theinteraction between the participants are proposed. The marketparticipants need to commit to contracts with an auditableprotocol with a small overhead. The proposed protocol is basedon a public key infrastructure and on known protocols for multiparty contract signing. The proposed model allows networkcapacity to be traded in a manner that utilizes the networkeciently. A new feature of this market model, compared to othernetwork capacity markets, is that the prices are not controlledby the network owners. It is the end-users who, by middlemen,trade capacity among each-other. Therefore, financial, ratherthan control theoretic, methods are used for the pricing ofcapacity.</p><p><b>Keywords:</b>Computer network architecture, bandwidthtrading, inter-domain Quality-of-Service, pricing,combinatorial allocation, financial derivative pricing,stochastic modeling</p>

We develop expansion methods as a new computational approach towards high-dimensional partial differential equations (PDEs), particularly of such type as arising in the valuation of financial derivatives. The proposed methods are extended from [41] and use principal component analysis (PCA) of the underlying process in combination with a Taylor expansion of the value function into solutions to low-dimensional PDEs. They enable calculation of highly accurate approximate solutions with computational complexity polynomial in the number of dimensions for PDEs with a low number of dominant principal components. For the case of PDEs with constant coefficients, we show existence of expansion solutions and prove theoretical error bounds. We give a precise characterisation of when our methods can be applied and construct specific examples of a first and second order version. We provide numerical results showing that the empirically observed convergence speeds are in agreement with the theoretical predictions. For the case of PDEs with varying coefficients, we give a heuristic motivation using the Parametrix approach and empirically test the methods' accuracy for a range of variable parameter stock models. We demonstrate the applicability of our expansion methods to real-world securities pricing problems by considering path-dependent and early-exercise options in the LIBOR market model. Using the example of Bermudan swaptions and Ratchet floors, which are considered difficult benchmark problems, we give a careful analysis of the numerical accuracy and computational complexity. We are able to demonstrate that for problems with medium to high dimensionality, around 60-100, and moderate time horizons, the presented PDE methods deliver results comparable in accuracy to benchmark state-of-the-art Monte Carlo methods in similar or (significantly) faster run time.

This dissertation evaluates development of exchange rates and its specific effects on STAP company a.s. The aim of this paper is to present the events that have had influence on the development of the exchange rate between Euro and Czech Crown and precautionary measures taken by STAP a.s. to prevent related risks. The first part comprises a summary of events that had a significant impact on the exchange rate development; the risks created by these events and methods devised to prevent these risks. The second part evaluates the specific financial derivatives used by STAP a.s. and their impact. Finally the recommendation is made for the future more effective usage of the financial instruments.

The goal of thesis is to clarify the nature of the exchange rate and interest rate risk and the possibility to describe the management of these risks. It represents the individual tools used to ensure the exchange rate and interest rate risk and the specific examples explaining the principle of their functioning. The thesis is divided into three parts - the exchange rate hedging, interest rate hedging and risk management, or a summary of each procedure, a brief guide for managers of companies involved in the risk of fluctuations in exchange rates or interest rates touching. Case studies of specific examples shows the possibilities of treatment of exchange rate risk - the exporter trading currency pair EUR / CZK.