Riemann-Stieltjes integrals with respect to fractional Brownian motion and applications
Loading...
URL
Journal Title
Journal ISSN
Volume Title
Aalto-yliopiston teknillinen korkeakoulu |
Doctoral thesis (article-based)
Checking the digitized thesis and permission for publishing
Instructions for the author
Instructions for the author
Unless otherwise stated, all rights belong to the author. You may download, display and print this publication for Your own personal use. Commercial use is prohibited.
Authors
Date
2010
Major/Subject
Mcode
Degree programme
Language
en
Pages
Verkkokirja (304 KB, 35 s.)
Series
Research reports.
Helsinki University of Technology, Institute of Mathematics,
A,
590
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.Description
Supervising professor
Valkeila, Esko, Prof.Keywords
fractional Brownian motion, pathwise stochastic integral, quadratic variation, functions of bounded variation, arbitrage, pricing by hedging, approximative hedging, proportional transaction costs
Other note
Parts
- [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.
- [Publication 2]: Ehsan Azmoodeh. 2010. On the fractional Black-Scholes market with transaction costs. arXiv:1005.0211v1 [q-fin.PR]. 13 pages.
- [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.
- [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.