Option Pricing in Fractional Brownian Markets (eBook)

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