Option Pricing in Fractional Brownian Markets (eBook)

Stefan Rostek (Autor)

eBook Download: PDF

2009 | 2009
XIV, 137 Seiten
Springer Berlin (Verlag)
978-3-642-00331-8 (ISBN)

Infos

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.

Inhalt

Foreword 6
Acknowledgements 7
Contents 8
Acronyms 10
Chapter 1 Introduction 12
Chapter 2 Fractional Integration Calculus 16
2.1 The Stochastic Process of Fractional Brownian Motion 18
2.2 Serial Correlation: The Role of the Hurst Parameter 24
2.3 The Wick-Based Approach to Fractional Integration 28
2.4 Pathwise and Stratonovich Integrals 33
2.5 Some Important Results of the Wick Type Fractional Integration Calculus 36
2.6 The S-Transform Approach 39
Chapter 3 Fractional Binomial Trees 43
3.1 Binomial Approximation of an Arithmetic Fractional Brownian Motion Process 44
3.2 Binomial Approximation of the Conditional Moments of Fractional Brownian Motion 50
3.3 Binomial Approximation of a Geometric Fractional Price Process 54
3.4 Arbitrage in the Fractional Binomial Market Setting and Its Exclusion 59
Chapter 4 Characteristics of the Fractional Brownian Market: Arbitrage and Its Exclusion 66
4.1 Arbitrage in the Unrestricted Continuous Time Setting 67
4.2 Diverse Approaches to Exclude Arbitrage 71
4.3 On the Non-compatibility of Fractional Brownian Motion and Continuous Tradability 78
4.4 Renouncement of Continuous Tradability, Exclusion of Arbitrage and Transition to Preference-Based Pricing 86
Chapter 5 Risk Preference-Based Option Pricing in a Continuous Time Fractional Brownian Market 88
5.1 Motivation and Setup of the Model 88
5.2 The Conditional Distribution of Fractional Brownian Motion 5.2.1 Prediction Based on an Infinite Knowledge About the Past 90
5.3 A Conditional Fractional Itˆo Theorem 98
5.4 Fractional European Option Prices 101
5.5 The Influence of the Hurst Parameter 108
5.6 The Influence of Maturity and the Term Structure of Volatility 115
Chapter 6 Risk Preference-Based Option Pricing in the Fractional Binomial Setting 120
6.1 The Two-Time Total Equilibrium Approach 122
6.2 The Two-Time Relative Equilibrium Approach 125
6.3 Multi-Time Equilibrium Approaches 130
6.4 Deeper Insights Provided by Discretization: The Continuous Time Case Reconsidered 137
Chapter 7 Conclusion 140
References 144

Produkt-Details

• Erscheint lt. Verlag: 28.4.2009
• Reihe/Serie: Lecture Notes in Economics and Mathematical Systems
• Reihe/Serie: Lecture Notes in Economics and Mathematical Systems
• Zusatzinfo: XIV, 137 p. 36 illus.
• Verlagsort: Berlin
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik
• Themenwelt: Technik
• Themenwelt: Wirtschaft ► Betriebswirtschaft / Management ► Finanzierung
• Themenwelt: Wirtschaft ► Volkswirtschaftslehre
• Schlagworte: Arbitrage • Equilibrium Pricing • Fractional Binomial Trees • Fractional Brownian motion • Hurst Parameter • Modeling • Quantitative Finance • Risk Preference Based Option Pricing
• ISBN-10: 3-642-00331-1 / 3642003311
• ISBN-13: 978-3-642-00331-8 / 9783642003318