|
| |
|
|
Solving mixed-integer linear programming problems in GAMS
Škoda, Štěpán ; Branda, Martin (advisor) ; Kopa, Miloš (referee)
In the present work we study the problems of integer linear optimi- zation, at first from the theoretical point of view (Part I) and subsequently on the basis of empirical data (Part II). First section explains with what this field deals with and where is applied. Other sections contain annotated mathematical formulation of problems, definitions and theorems needed to understand the general methods for solving integer linear programs. In the last section of Part I, we introduce the two best known groups of algorithms that are used by commercial software. The second part provides more details on an Internet library that contains some practical problems, that has been needed to be solved in the past. Furthermore, there are sections dealing with solvers and the advanced options of GAMS. The last section presents data obtained in the course of solving problems using several codes (solvers) of software. 1
|
|
|
Convexity in chance constraints programming
Olos, Marek ; Kopa, Miloš (advisor) ; Adam, Lukáš (referee)
1 Abstract: This thesis deals with chance constrained stochastic programming problems. We consider several chance constrained models and we focus on their convexity property. The thesis presents the theory of α-concave functions and measures as a basic tool for proving the convexity of the problems. We use the results of the theory to prove the convexity of the models first for the continu- ous distributions, then for the discrete distributions of the random vectors. We characterize a large class of the continuous distributions, that satisfy the suffi- cient conditions for the convexity of the given models and we present solving algorithms for these models. We present sufficient conditions for the convexity of the problems with dicrete distributions, too. We also deal with the algorithms for solving non-convex problems and briefly discuss the difficulties that can occur when using these methods.
|
|
| |
|
|
Stochastic dominance in portfolio optimization
Paulik, Marek ; Kopa, Miloš (advisor) ; Branda, Martin (referee)
The main topic of this thesis is the application of stochastic dominance constrains to portfolio optimization problems. First, we recall Markowitz model. Then we present portfolio selection problems with stochastic dominance constraints. Finally, we compare performance of these two approaches in an empirical study presented in the last chapter.
|
|
|
Combinatorial portfolio optimization
Zákutná, Tatiana ; Kopa, Miloš (advisor) ; Petrásek, Jakub (referee)
In this thesis, a portfolio optimization with integer variables which influence optimal assets allocation, is studied. At the beginning basic terms, measures of risk - variance, Value at Risk (VaR), Conditional Value at Risk (CVaR) are defined and the mean-risk models are derived for a practical application. Heuristics and standard algorithms of software GAMS are used for solving problems of the combinatorial portfolio optimization. Two types of the he- uristics are described: the Threshold Acceptance and the Genetic Algorithm. The heuristics are implemented in the MATLAB, applied on financial data and compared with an output of the software GAMS. 1
|
|
|
Semi - infinite programming: theory and portfolio efficiency application
Klouda, Lukáš ; Kopa, Miloš (advisor) ; Lachout, Petr (referee)
Title: Semi-infinite programming: theory and portfolio efficiency application Author: Bc. Lukáš Klouda Department: Department of Probability and Mathematical Statistics Supervisor: RNDr. Ing. Miloš Kopa, PhD. Supervisor's e-mail address: kopa@karlin.mff.cuni.cz Abstract: The thesis deals with application of semi-infinite programming to a portfolio efficiency testing. The summary of semi-infinite programming, first and second order optimality conditions and duality in linear semi-infinite programming is presented. The optimization problem for a portfolio efficiency testing with respect to the second order stochastic dominance under assumption of discrete, normal, Students and general elliptical distribution is formulated. Conditional value at risk(CVaR) is used as the risk measure, because of its consistency with the second order stochastic dominance relation. Efficiency of index PX with respect to the second order stochastic dominance is tested. The tests are performed using the program GAMS.
|
|
|
Applications of evolution and differential games
Chvoj, Martin ; Lachout, Petr (advisor) ; Kopa, Miloš (referee)
This work deals with advance parts of game theory and their applications in financial area and follows a bachelor work Applications of game theory in economics written by Martin Chvoj at 2008. At the beginning there is a brief introduction into classical game theory, which uses a concept of Nash equilibrium. The work requires knowledge of this theory. In following chapters, modern ways of game theory are described in detail. These chapters deal with basic concepts of evolutionary and differential games which are supported by a number of examples and application possibilities. At the end of the work, there is an original example which models an income tax system as a differential game. This model uses a Czech tax legal framework for year 2009 and statistical data, which can be found at Czech bureau of statistics.
|
|
|
Log-optimal investment
Flimmel, Samuel ; Lachout, Petr (advisor) ; Kopa, Miloš (referee)
The creation of portfolio is an important and frequent task to solve in financial sector. This paper indroduces one of mathematical models used for this problem. For studied market we assume it`s logaritmic utility function and ergodic stationarity only. The low number of assumptions makes this model quite simple and clear. In this paper we describe the model and prove some of it`s features applicable for our model. First, we analyze the case of an known market distribution and suggest an algorithm for obtaining a portfolio. Later, we analyze the case of an unknown market distribution and introduce one of the suitable methods as well. Empirical distribution helps us to gain requiered results asymptotically. Finally, we study behavior of an empirical distribution method for shorter time periods on real data simulation.
|
|
|
Suitable utility function identification
Majerová, Michaela ; Kopa, Miloš (advisor) ; Lachout, Petr (referee)
At the beginning of this work we study basic properties of utility functions and connection between their shape and investor's relation to risk. Then we define risk premium and we recall measure of risk aversion. In the second chapter we study classification of utility functions according to the absolute risk aversion measure and we list some basic types of utility functions. In the third chapter we construct investor's utility function. We use values of insurance premium which we get from questionnaire filled by MFF UK students. We use these utility functions in the last chapter. First we define portfolio selection problem and then we find optimal portfolio for different investors.
|
guest ::
