The doctoral dissertations of the former Helsinki University of Technology (TKK) and Aalto University Schools of Technology (CHEM, ELEC, ENG, SCI) published in electronic format are available in the electronic publications archive of Aalto University - Aaltodoc.
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Riemann-Stieltjes Integrals with Respect to Fractional Brownian Motion and Applications

Ehsan Azmoodeh

Doctoral dissertation for the degree of Doctor of Science in Technology to be presented with due permission of the Faculty of Information and Natural Sciences for public examination and debate in Auditorium G at the Aalto University School of Science and Technology (Espoo, Finland) on the 8th of October 2010 at 12 noon.

Overview in PDF format (ISBN 978-952-60-3338-9)   [304 KB]
Dissertation is also available in print (ISBN 978-952-60-3337-2)

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.

This thesis consists of an overview and of the following 4 publications:

  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.
  2. Ehsan Azmoodeh. 2010. On the fractional Black-Scholes market with transaction costs. arXiv:1005.0211v1 [q-fin.PR]. 13 pages.
  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.
  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.

Keywords: fractional Brownian motion, pathwise stochastic integral, quadratic variation, functions of bounded variation, arbitrage, pricing by hedging, approximative hedging, proportional transaction costs

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Last update 2011-05-26