### Refine

#### Document Type

- Report (2)
- Doctoral Thesis (1)

#### Language

- English (3) (remove)

#### Keywords

- Lineare Algebra (2)
- Mathematikunterricht (2)
- Modellierung (2)
- linear algebra (2)
- mathematical education (2)
- modelling (2)
- praxisorientiert (2)
- Approximationsalgorithmus (1)
- Bewertung (1)
- Binomialbaum (1)
- Derivat <Wertpapier> (1)
- Extrapolation (1)
- Finanzmathematik (1)
- Finanznumerik (1)
- Konvergenzrate (1)
- Konvergenzverhalten (1)
- Lineare Optimierung (1)
- Multi-Asset Option (1)
- Option (1)
- Simplex (1)
- Stücklisten (1)
- Sägezahneffekt (1)
- bills of materials (1)
- binomial tree (1)
- computational finance (1)
- convergence behaviour (1)
- linear optimization (1)
- monotone Konvergenz (1)
- multi-asset option (1)
- option valuation (1)
- praxis orientated (1)
- rate of convergence (1)
- sawtooth effect (1)
- simplex (1)

This thesis deals with the application of binomial option pricing in a single-asset Black-Scholes market and its extension to multi-dimensional situations. Although the binomial approach is, in principle, an efficient method for lower dimensional valuation problems, there are at least two main problems regarding its application: Firstly, traded options often exhibit discontinuities, so that the Berry- Esséen inequality is in general tight; i.e. conventional tree methods converge no faster than with order 1/sqrt(N). Furthermore, they suffer from an irregular convergence behaviour that impedes the possibility to achieve a higher order of convergence via extrapolation methods. Secondly, in multi-asset markets conventional tree construction methods cannot ensure well-defined transition probabilities for arbitrary correlation structures between the assets. As a major aim of this thesis, we present two approaches to get binomial trees into shape in order to overcome the main problems in applications; the optimal drift model for the valuation of single-asset options and the decoupling approach to multi-dimensional option pricing. The new valuation methods are embedded into a self-contained survey of binomial option pricing, which focuses on the convergence behaviour of binomial trees. The optimal drift model is a new one-dimensional binomial scheme that can lead to convergence of order o(1/N) by exploiting the specific structure of the valuation problem under consideration. As a consequence, it has the potential to outperform benchmark algorithms. The decoupling approach is presented as a universal construction method for multi-dimensional trees. The corresponding trees are well-defined for an arbitrary correlation structure of the underlying assets. In addition, they yield a more regular convergence behaviour. In fact, the sawtooth effect can even vanish completely, so that extrapolation can be applied.

Linear Optimization is an important area from applied mathematics. A lot of practical problems can be modelled and solved with this technique. This publication shall help to introduce this topic to pupils. The process of modelling, the reduction of problems to their significant attributes shall be described. The linear programms will be solved by using the simplex method. Many examples illustrate the topic.

This publication tries to develop mathematical subjects for school from realistic problems. The center of this report are business planning and decision problems which occur in almost all companies. The main topics are: Calculation of raw material demand for given orders, consumption of existing stock and the lot sizing.