## 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.

Author: | Stefanie Müller |
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URN (permanent link): | urn:nbn:de:hbz:386-kluedo-24627 |

Advisor: | Ralf Korn |

Document Type: | Doctoral Thesis |

Language of publication: | English |

Year of Completion: | 2009 |

Year of Publication: | 2009 |

Publishing Institute: | Technische Universität Kaiserslautern |

Granting Institute: | Technische Universität Kaiserslautern |

Acceptance Date of the Thesis: | 2009/12/16 |

Date of the Publication (Server): | 2010/01/07 |

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-Keyword: | Approximationsalgorithmus; Bewertung ; Binomialbaum ; Derivat <Wertpapier> ; Finanzmathematik ; Option |

Faculties / Organisational entities: | Fachbereich Mathematik |

DDC-Cassification: | 5 Naturwissenschaften und Mathematik / 51 Mathematik / 510 Mathematik |

MSC-Classification (mathematics): | 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) | |

Licence (German): | Standard gemäß KLUEDO-Leitlinien vor dem 27.05.2011 |