|
|
Selected applications of mathematics
KRATOCHVÍL, Daniel
The thesis is a contribution to the topic of game theory which belongs to the field of applied mathematics. The work is divided into three parts. In the first part, the basic terminology of game theory is introduced and subsequently applied in conflict mathematical model situations. The second part deals with the history of game theory and presents some historical problems as well as milestones in this area of mathematics. The last part is focused on different variants of NIM games, their analysis and the search for an optimal strategy.
|
|
|
Shadow prices and portfolio management with proportional transaction costs
Klůjová, Jana ; Dostál, Petr (advisor) ; Beneš, Viktor (referee)
The diploma thesis describes portfolio management with proportional transaction costs. The main aim is to describe using of shadow prices to find the optimal Markov policies keeping the proportion of the investor's wealth invested in the risky asset within the corresponding interval in order to maximize the long run geometric growth rate. On the real market, the investor must pay transaction costs when he buys/sells shares. In the diploma thesis we transform these prices into the shadow price; when trading in the shadow price there are no transaction costs. The solution itself is based on Itô formula and the martingal theory. The prices of shares are modeled as geometric Brownian motion. Powered by TCPDF (www.tcpdf.org)
|
|
| |
|
|
The Stigler-Luckock model for a limit order book
Fornůsková, Monika ; Swart, Jan (advisor) ; Večeř, Jan (referee)
THE STIGLER-LUCKOCK MODEL FOR A LIMIT ORDER BOOK Abstract One of the types of modern-day markets are so-called order-driven markets whose core component is a database of all incoming buy and sell orders (order book). The main goal of this thesis is to extend the Stigler-Luckock model for order books to give a better insight into the price forming process and behaviour of the market participants themselves. The model introduced in this thesis focuses on a comparison of behaviour and various strategies of market makers who are sophisticated market participants profiting from extensive trading. The market is described using Markov chains, and the strategies are compared using Monte Carlo simulations and game theory. The results showed that market makers' orders should have small spread and large volumes. The final model compares two strategies in which market makers monitor their portfolio. In case of having more cash than asset (or vice versa), they shift prices of their orders to equalise the portfolio. The model recommends checking the market quite often, but acting conservatively, which means not changing prices that frequently and not jumping to conclusions just from a small imbalance in the portfolio.
|
|
| |
|
| |
|
|
Shadow prices and portfolio management with proportional transaction costs
Klůjová, Jana ; Dostál, Petr (advisor) ; Beneš, Viktor (referee)
The diploma thesis describes portfolio management with proportional transaction costs. The main aim is to describe using of shadow prices to find the optimal Markov policies keeping the proportion of the investor's wealth invested in the risky asset within the corresponding interval in order to maximize the long run geometric growth rate. On the real market, the investor must pay transaction costs when he buys/sells shares. In the diploma thesis we transform these prices into the shadow price; when trading in the shadow price there are no transaction costs. The solution itself is based on Itô formula and the martingal theory. The prices of shares are modeled as geometric Brownian motion. Powered by TCPDF (www.tcpdf.org)
|
|
|
Almost optimal trading strategies for small transaction costs
Jusko, Martin ; Dostál, Petr (advisor) ; Štěpán, Josef (referee)
We consider agent trading futures on a market with small transaction costs. Her capital is deposited on a money market account, where compounding is possible. Arithmetic Brownian motion with random coefficients is considered as a model for futures strike price. The coefficients are assumed to be bounded Itô processes with bounded coefficients. Under these assumptions, an almost optimal interval strategy is derived, which almost maximizes expected utility in certain stopping times under hyperbolic absolute risk aversion utility function. Furthermore, under logarithmic utility function the derived strategy almost maximizes expected utility in wide class of (integrable) stopping times.
|
guest ::
