We investigate global dynamics of the following second order rational difference equation x n + 1 = x n x n − 1 + α x n + β x n − 1 a x n x n − 1 + b x n − 1 , where the parameters α , β , a , b are positive real numbers and initial conditions x − 1 and x 0 are arbitrary positive real numbers. The map associated to the right-hand side of this equation is always decreasing in the second variable and can be either increasing or decreasing in the first variable depending on the corresponding parametric space. In most cases, we prove that local asymptotic stability of the unique equilibrium point implies global asymptotic stability.

In this dissertation convergence of binomial trees for option pricing is investigated. The focus is on American and European put and call options. For that purpose variations of the binomial tree model are reviewed. In the first part of the thesis we investigated the convergence behavior of the already known trees from the literature (CRR, RB, Tian and CP) for the European options. The CRR and the RB tree suffer from irregular convergence, so our first aim is to find a way to get the smooth convergence. We first show what causes these oscillations. That will also help us to improve the rate of convergence. As a result we introduce the Tian and the CP tree and we proved that the order of convergence for these trees is \(O \left(\frac{1}{n} \right)\). Afterwards we introduce the Split tree and explain its properties. We prove the convergence of it and we found an explicit first order error formula. In our setting, the splitting time \(t_{k} = k\Delta t\) is not fixed, i.e. it can be any time between 0 and the maturity time \(T\). This is the main difference compared to the model from the literature. Namely, we show that the good properties of the CRR tree when \(S_{0} = K\) can be preserved even without this condition (which is mainly the case). We achieved the convergence of \(O \left(n^{-\frac{3}{2}} \right)\) and we typically get better results if we split our tree later.

By using the Kolmogorov-Arnold-Moser (KAM) theory, we investigate the stability of the positive elliptic equilibrium point of the difference equation xn+1 = Ax3n +B axn−1 , n = 0, 1, 2, . . . where the parameters A,B, a and the initial conditions x −1, x0 are positive numbers. The specific feature of this difference equation is the fact that we were not able to use the invariant to prove stability or to find feasible periods of the solutions.

C Dedicated to Professor Mustafa Kulenovic on the occasion of his 60th birthday Abstract. We investigate the global asymptotic behavior of solutions of the following anti-competitive system of difference equations xn+1 = 1yn A1 + xn ; yn +1 = 2xn + 2yn yn ; n = 0; 1;:::; where the parameters 1; 2; 2;A1 are positive numbers and the initial conditions x0 0;y0 > 0. We find the basins of attraction of all at- tractors of the system, which are the equilibrium point and period-two solutions.