Introduction

Stefan Rostek ()
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Stefan Rostek: University of Tübingen

Chapter 1 in Option Pricing in Fractional Brownian Markets, 2009, pp 1-4 from Springer

Abstract: The vast majority of approaches towards option pricing deals with Brownian motion as a source of randomness. The seminal articles by Black and Scholes (1973) as well as by Merton (1973) crowned this evolution but did not conclude it by any means. Right up to today, the favorable properties and the well-developed stochastic calculus of classical Brownian motion attract both scientists and practitioners.However, there was early evidence about some incompatibilities with regard to real market data. Concerning the stochastic process of Brownian motion, the main critique drawn from empiricism is at least two-fold: On the one hand, real market distributions were shown to be not Gaussian (see e.g. Fama (1965)). The debate of recent years has put a great deal of effort on correcting this problem. Particularly the theory of Lévy processes allows it to incorporate a wide range of distributions into financial models. However, despite the large set of Lévy type stochastic processes, closed-form solutions are still limited to specific cases of non-Gaussian distributions. For more details about Lévy processes we refer the interested reader to the monograph of Cont and Tankov (2004) who provide a distinguished starting point to the topic. On the other hand, the processes of observable market values seem to exhibit serial correlation (see e.g. Lo and MacKinley (1988)). Much less endeavor has been made to get a grip on this problem by factoring in aspects of persistence. However, at least there is one stochastic process that has often been proposed for mapping this kind of behavior: the very candidate is called fractional Brownian motion.

Keywords: Brownian Motion; Option Price; Fractional Brownian Motion; Implied Volatility; Hurst Parameter (search for similar items in EconPapers)
Date: 2009
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DOI: 10.1007/978-3-642-00331-8_1

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