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## The Split tree for option pricing

• 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.

Author: Merima Nurkanović urn:nbn:de:hbz:386-kluedo-46771 Ralf Korn Doctoral Thesis English 2017/06/30 2017 Technische Universität Kaiserslautern Technische Universität Kaiserslautern 2017/05/23 2017/06/30 VI, 92 Fachbereich Mathematik 5 Naturwissenschaften und Mathematik / 510 Mathematik Creative Commons 4.0 - Namensnennung, nicht kommerziell, keine Bearbeitung (CC BY-NC-ND 4.0)