# Riemann-Stieltjes integrals with respect to fractional Brownian motion and applications

 dc.contributor Aalto-yliopisto fi dc.contributor Aalto University en dc.contributor.author Azmoodeh, Ehsan dc.date.accessioned 2012-08-28T11:35:57Z dc.date.available 2012-08-28T11:35:57Z dc.date.issued 2010 dc.identifier.isbn 978-952-60-3338-9 (electronic) dc.identifier.isbn 978-952-60-3337-2 (printed) #8195; dc.identifier.issn 1797-5867 dc.identifier.uri https://aaltodoc.aalto.fi/handle/123456789/4837 dc.description.abstract In this dissertation we study Riemann-Stieltjes integrals with respect to (geometric) fractional Brownian motion, its financial counterpart and its application in estimation of quadratic variation process. From the point of view of financial mathematics, we study the fractional Black-Scholes model in continuous time. We show that the classical change of variable formula with convex functions holds for the trajectories of fractional Brownian motion. Putting it simply, all European options with convex payoff can be hedged perfectly in such pricing model. This allows us to give new arbitrage examples in the geometric fractional Brownian motion case. Adding proportional transaction costs to the discretized version of the hedging strategy, we study an approximate hedging problem analogous to the corresponding discrete hedging problem in the classical Black-Scholes model. Using the change of variables formula result, one can see that fractional Brownian motion model shares some common properties with continuous functions of bounded variation. We also show a representation for running maximum of continuous functions of bounded variations such that fractional Brownian motion does not enjoy this property. en dc.format.extent Verkkokirja (304 KB, 35 s.) dc.format.mimetype application/pdf dc.language.iso en en dc.publisher Aalto-yliopiston teknillinen korkeakoulu en dc.relation.ispartofseries Research reports. Helsinki University of Technology, Institute of Mathematics, A, 590 en dc.relation.haspart [Publication 1]: Ehsan Azmoodeh, Yuliya Mishura, and Esko Valkeila. 2009. On hedging European options in geometric fractional Brownian motion market model. Statistics & Decisions, volume 27, number 2, pages 129-143. en dc.relation.haspart [Publication 2]: Ehsan Azmoodeh. 2010. On the fractional Black-Scholes market with transaction costs. arXiv:1005.0211v1 [q-fin.PR]. 13 pages. en dc.relation.haspart [Publication 3]: Ehsan Azmoodeh, Heikki Tikanmäki, and Esko Valkeila. 2010. When does fractional Brownian motion not behave as a continuous function with bounded variation? Statistics and Probability Letters, volume 80, numbers 19-20, pages 1543-1550. en dc.relation.haspart [Publication 4]: Ehsan Azmoodeh and Esko Valkeila. 2010. Spectral characterization of the quadratic variation of mixed Brownian fractional Brownian motion. arXiv:1005.4349v1 [math.PR]. 14 pages. en dc.subject.other Mathematics dc.title Riemann-Stieltjes integrals with respect to fractional Brownian motion and applications en dc.type G5 Artikkeliväitöskirja fi dc.contributor.school Aalto-yliopiston teknillinen korkeakoulu fi dc.contributor.department Matematiikan ja systeemianalyysin laitos fi dc.contributor.department Department of Mathematics and Systems Analysis en dc.subject.keyword fractional Brownian motion en dc.subject.keyword pathwise stochastic integral en dc.subject.keyword quadratic variation en dc.subject.keyword functions of bounded variation en dc.subject.keyword arbitrage en dc.subject.keyword pricing by hedging en dc.subject.keyword approximative hedging en dc.subject.keyword proportional transaction costs en dc.identifier.urn URN:ISBN:978-952-60-3338-9 dc.type.dcmitype text en dc.type.ontasot Väitöskirja (artikkeli) fi dc.type.ontasot Doctoral dissertation (article-based) en dc.contributor.supervisor Valkeila, Esko, Prof.
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