Option Pricing in Fractional Brownian Markets
Option Pricing in Fractional Brownian Markets

Author: Stefan Rostek

Publisher: Springer Science & Business Media

Published: 2009-04-28

Total Pages: 146

ISBN-13: 3642003311

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Mandelbrot and van Ness (1968) suggested fractional Brownian motion as a parsimonious model for the dynamics of ?nancial price data, which allows for dependence between returns over time. Starting with Rogers(1997) there is an ongoing dispute on the proper usage of fractional Brownian motion in option pricing theory. Problems arise because fractional Brownian motion is not a semimartingale and therefore “no arbitrage pricing” cannot be applied. While this is consensus, the consequences are not as clear. The orthodox interpretation is simply that fractional Brownian motion is an inadequate candidate for a price process. However, as shown by Cheridito (2003) any theoretical arbitrage opportunities disappear by assuming that market p- ticipants cannot react instantaneously. This is the point of departure of Rostek’s dissertation. He contributes to this research in several respects: (i) He delivers a thorough introduction to fr- tional integration calculus and uses the binomial approximation of fractional Brownianmotion to give the reader a ?rst idea of this special market setting.

Option Pricing in a Fractional Brownian Motion Environment
Option Pricing in a Fractional Brownian Motion Environment

Author: Ciprian Necula

Publisher:

Published: 2008

Total Pages: 19

ISBN-13:

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In this paper it is developed a framework for evaluating derivatives if the underlying of the derivative contract is supposed to be driven by a fractional Brownian motion with Hurst parameter greater than 0.5. For this purpose we first prove some results regarding the quasi-conditional expectation, especially the behavior to a Girsanov transform. We obtain the risk-neutral valuation formula, the fundamental evaluation equation of a contingent claim, and the formula for the price of a European call option in the case of the fractional Black-Scholes market.

Implied Hurst Exponent and Fractional Implied Volatility
Implied Hurst Exponent and Fractional Implied Volatility

Author: Kinrey Li

Publisher:

Published: 2014

Total Pages: 17

ISBN-13:

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Two methods to derive Hurst exponent from option prices are proposed in this paper. They are based on fractional Brownian market setting. The first method is to use fractional Black-Scholes model inversely to derive implied Hurst exponent. The second one depends on no specific option pricing model. It is a model-free approach which is applicable as long as asset price evolves with on jumps. The difficulty in deriving implied information from fractional Brownian market is due to the fact that both Hurst exponent and volatility are unobservable. So they can be derived as a whole from single-period option prices, but can hardly be separated from each other. In this paper, a method that integrates option prices of different maturities is suggested to solve this problem. We also make a comparison between volatility in classical Brownian market and that in fractional Brownian market, which reveals that variance term structures are fitted differently in two settings. Based on this result, we suggest two potential applications of implied Hurst exponent in this paper.

Pricing European and Barrier Options in the Fractional Black-Scholes Market
Pricing European and Barrier Options in the Fractional Black-Scholes Market

Author: Ciprian Necula

Publisher:

Published: 2008

Total Pages: 0

ISBN-13:

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The aim of this paper is to obtain the valuation formulas for European and barrier options if the underlying of the option contract is supposed to be driven by a fractional Brownian motion with Hurst parameter greater than 0.5. The paper is build upon the framework developed in Necula (2007) for the valuation of derivative products in the fractional Black-Scholes market. We also obtain a reflection principle for the fractional Brownian motion.

Stochastic Calculus for Fractional Brownian Motion and Applications
Stochastic Calculus for Fractional Brownian Motion and Applications

Author: Francesca Biagini

Publisher: Springer Science & Business Media

Published: 2008-02-17

Total Pages: 331

ISBN-13: 1846287979

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The purpose of this book is to present a comprehensive account of the different definitions of stochastic integration for fBm, and to give applications of the resulting theory. Particular emphasis is placed on studying the relations between the different approaches. Readers are assumed to be familiar with probability theory and stochastic analysis, although the mathematical techniques used in the book are thoroughly exposed and some of the necessary prerequisites, such as classical white noise theory and fractional calculus, are recalled in the appendices. This book will be a valuable reference for graduate students and researchers in mathematics, biology, meteorology, physics, engineering and finance.