Full Description
This monograph is based on my Ph.D. thesis, which was accepted in Jan- uary 2004 by the faculty of economics at the University of Augsburg. It is a great pleasure to thank my supervisor, Prof. Dr. Manfred Steiner, for his scientific guidance and support throughout my Ph.D. studies. I would also like to express my thanks to Prof. Dr. Gunter Bamberg for his comments and suggestions. To my colleagues at the department of Finance and Banking at the U ni- versity of Augsburg, I express my thanks for their kind support and their helpful comments over the past years. In particular, I would like to thank Dr. Bernhard Brunner for many interesting discussions and also for the careful revision of this manuscript. At risklab germany GmbH, Munich, I would first of alllike to thank Dr. Gerhard Scheuenstuhl and Prof. Dr. Rudi Zagst for creating an ideal environ- ment for research. I would also like to express my thanks to my coIleagues. It has been most enjoyable to work with them. In particular, I would like to thank Dr. Bernd Schmid. Our joint projects on stochastic implied volatil- ity models greatly influenced this work.
I am also indebted to Anja Fischer for valuable contributions during her internship and Didier Vermeiren (from Octanti Associates) for carefuIly reading the manuscript.
Contents
1 Introduction.- 1.1 Motivation and Objectives.- 1.2 Structure of the Work.- 2 Continuous-time Financial Markets.- 2.1 The Financial Market.- 2.1.1 Assets and Trading Strategies.- 2.1.2 Absence of Arbitrage and Martingale Measures.- 2.2 Risk-Neutral Pricing of Contingent Claims.- 2.2.1 Contingent Claims.- 2.2.2 Risk-Neutral Valuation Formula.- 2.2.3 Attainability and Market Completeness.- 3 Implied Volatility.- 3.1 The Black-Scholes Model.- 3.1.1 The Financial Market Model.- 3.1.2 Pricing and Hedging of Contingent Claims.- 3.1.3 The Black-Scholes Option Pricing Formula.- 3.1.4 The Greeks.- 3.2 The Concept ofImplied Volatility.- 3.2.1 Definition.- 3.2.2 Calculation.- 3.2.3 Interpretation.- 3.3 Features of Implied Volatility.- 3.3.1 Volatility Smiles.- 3.3.2 Volatility Term Structures.- 3.3.3 Volatility Surfaces.- 3.4 Modelling Implied Volatility.- 3.4.1 Overview.- 3.4.2 Implied Volatility as an Endogenous Variable.- 3.4.3 Implied Volatility as an Exogenou s Var iabl e.- 3.4.4 Comparison of Approaches.- 4 The General Stochastic Implied Volatility Model.- 4.1 The Financial Market Model.- 4.1.1 Model Specification.- 4.1.2 Movements of the Volatility Surface.- 4.2 Risk-Neutral Implied Volatility Dynamics.- 4.2.1 Change of Measure and Drift Restriction.- 4.2.2 Interpretation of Terms in the Risk-Neutral Drift.- 4.2.3 Existence and Uniqueness of the Risk-Neutral Measure.- 4.1 4.3 Pricing and Hedging of Cont ingent Claims.- 5 Properties of DAX Implied Volatilities.- 5.1 The DAX Option.- 5.1.1 Contract Specifications.- 5.1.2 Previous Studies.- 5.2 Data.- 5.2.1 Raw Data and Data Preparation.- 5.2.2 Correcting for Taxes and Dividends.- 5.2.3 Liquidity Aspects.- 5.3 Structure of DAX Implied Volatilities.- 5.3.1 Estimation of the DAX Volatility Surface.- 5.3.2 Empirical Results.- 5.3.3 Identification and Selection of Volatility Risk Factors.- 5.4 Dynamics of DAX Implied Volatilities.- 5.4.1 Time-Series Properties of DAX Volatility Risk Factors.- 5.4.2 Relating Volatility Risk Factors to Index Returns and other Market Variables.- 5.5 Summary of Empirical Observations.- 6 A Four-Factor Model for DAX Implied Volatilities.- 6.1 The Model under the Objective Measure.- 6.1.1 Model Specification.- 6.1.2 Model Estimation.- 6.1.3 Model Testing.- 6.2 The Model under the Risk-Neutral Measure.- 6.2.1 Risk-Neutral Stock Price and Volatility Dynamics.- 6.2.2 The Market Price of Risk Process.- 6.2.3 Pricing and Hedging of Contingent Claims.- 6.2.4 Model Calibration.- 6.3 Model Review and Conclusion.- 7 Model Applications.- 7.1 Pricing and Hedging of Exotic Derivatives.- 7.1.1 Product Overview.- 7.1.2 Exotic Equity Index Derivatives.- 7.1.3 Volatility Derivatives.- 7.2 Value at Risk for Option Portfolios.- 7.2.1 The Value at Risk Concept.- 7.2.2 Computing VaR for Option Portfolios.- 7.2.3 A Case Study.- 7.2.4 Beyond VaR: Expected Shortfall.- 7.3 Volatility Trading.- 7.3.1 Definition and Motivation.- 7.3.2 Volatility Trade Design.- 7.3.3 Profitability of DAX Volatility Trading Strategies.- 8 Summary and Conclusion.- A Some Mathematical Preliminaries;A.l Probability Theory;A.2 Continuous-time Stochastic Processes.- B Pricing of a Variance Swap via Static Replication.- List of Abbreviations.- List of Symbols.- References.
