The Binomial Approach to Option Valuation: Getting Binomial Trees into Shape
- 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.
Verfasser*innenangaben: | Stefanie Müller |
---|---|
URN: | urn:nbn:de:hbz:386-kluedo-24627 |
Betreuer*in: | Ralf Korn |
Dokumentart: | Dissertation |
Sprache der Veröffentlichung: | Englisch |
Jahr der Fertigstellung: | 2009 |
Jahr der Erstveröffentlichung: | 2009 |
Veröffentlichende Institution: | Technische Universität Kaiserslautern |
Titel verleihende Institution: | Technische Universität Kaiserslautern |
Datum der Annahme der Abschlussarbeit: | 16.12.2009 |
Datum der Publikation (Server): | 07.01.2010 |
Freies Schlagwort / Tag: | Extrapolation; Finanznumerik; Konvergenzrate; Konvergenzverhalten; Multi-Asset Option; Sägezahneffekt; monotone Konvergenz binomial tree; computational finance; convergence behaviour; multi-asset option; option valuation; rate of convergence; sawtooth effect |
GND-Schlagwort: | Finanzmathematik; Option; Derivat <Wertpapier>; Bewertung; Binomialbaum; Approximationsalgorithmus |
Fachbereiche / Organisatorische Einheiten: | Kaiserslautern - Fachbereich Mathematik |
DDC-Sachgruppen: | 5 Naturwissenschaften und Mathematik / 510 Mathematik |
MSC-Klassifikation (Mathematik): | 65-XX NUMERICAL ANALYSIS / 65Cxx Probabilistic methods, simulation and stochastic differential equations (For theoretical aspects, see 68U20 and 60H35) / 65C40 Computational Markov chains |
91-XX GAME THEORY, ECONOMICS, SOCIAL AND BEHAVIORAL SCIENCES / 91Gxx Mathematical finance / 91G20 Derivative securities | |
91-XX GAME THEORY, ECONOMICS, SOCIAL AND BEHAVIORAL SCIENCES / 91Gxx Mathematical finance / 91G60 Numerical methods (including Monte Carlo methods) | |
Lizenz (Deutsch): | Standard gemäß KLUEDO-Leitlinien vor dem 27.05.2011 |