Table Of ContentLecture N otes in Economies
and Mathematical Systems 545
Founding Editors:
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Reinhold Hafner
Stochastic
Implied Volatility
A Factor-Based Model
~Springer
Author
Dr. Reinhold Hafner
risklab germany GmbH
Nymphenburger Straße 112-116
80636 München
Germany
Library of Congress Control Number: 2004109369
ISSN 0075-8442
ISBN 978-3-540-22183-8 ISBN 978-3-642-17117-8 (eBook)
DOI 10.1007/978-3-642-17117-8
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Für meine Eltern
Preface
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. Günter 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.
Further, I am extremely grateful to Prof. Dr. Martin WaIlmeier for his
continuous support and advice, his thorough revision of the manuscript, as
weIl as for many fruitful discussions. The results of our joint projects on the
estimation and explanation of implied volatility structures entered this work.
Most of aIl, I want to thank my girlfriend Heike for endless patience,
encouragement, and support, and also my mother Lieselotte and my brot her
Jürgen for being there aIl the times.
Mering, May 2004 Reinhold Hafner
Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Motivation and Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Structure of the Work. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2 Continuous-time Financial Markets . . . . . . . . . . . . . . . . . . . . . . . . 9
2.1 The Financial Market . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 10
2.1.1 Assets and Trading Strategies . . . . . . . . . . . . . . . . . . . . . .. 10
2.1.2 Absence of Arbitrage and Martingale Measures. . . . . . .. 13
2.2 Risk-Neutral Pricing of Contingent Claims. . . . . . . . . . . . . . . . .. 15
2.2.1 Contingent Claims. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 15
2.2.2 Risk-Neutral Valuation Formula. . . . . . . . . . . . . . . . . . . .. 18
2.2.3 Attainability and Market Completeness .............. 20
3 Implied Volatility. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 23
3.1 The Black-Scholes Model ................................. 24
3.1.1 The Financial Market Model ........................ 24
3.1.2 Pricing and Hedging of Contingent Claims. . . . . . . . . . .. 25
3.1.3 The Black-Scholes Option Pricing Formula ........... 27
3.1.4 The Greeks ....................................... 29
3.2 The Concept ofImplied Volatility ......................... 32
3.2.1 Definition........................................ 32
3.2.2 Calculation....................................... 34
3.2.3 Interpretation..................................... 35
3.3 Features of Implied Volatility ............................. 38
3.3.1 Volatility Smiles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 38
3.3.2 Volatility Term Structures . . . . . . . . . . . . . . . . . . . . . . . . .. 39
3.3.3 Volatility Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 41
3.4 Modelling Implied Volatility ............... . . . . . . . . . . . . . .. 43
3.4.1 Overview......................................... 43
3.4.2 Implied Volatility as an Endogenous Variable ......... 45
3.4.3 Implied Volatility as an Exogenous Variable .......... 51
X Contents
3.4.4 Comparison of Approaches ......................... 56
4 The General Stochastic Implied Volatility Model. . . . . . . . . .. 59
4.1 The Financial Market Model .............................. 60
4.1.1 Model Specification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 60
4.1.2 Movements of the Volatility Surface ................. 61
4.2 Risk-Neutral Implied Volatility Dynamics. . . . . . . . . . . . . . . . . .. 63
4.2.1 Change of Measure and Drift Restriction ............. 63
4.2.2 Interpretation of Terms in the Risk-Neutral Drift. . . . .. 68
4.2.3 Existence and Uniqueness of the Risk-Neutral Measure. 68
4.3 Pricing and Hedging of Contingent Claims. . . . . . . . . . . . . . . . .. 70
5 Properties of DAX Implied Volatilities .................... 73
5.1 The DAX Option ........................................ 73
5.1.1 Contract Specifications. . . . . . . . . . . . . . . . . . . . . . . . . . . .. 73
5.1.2 Previous Studies .................................. 75
5.2 Data ................................................... 76
5.2.1 Raw Data and Data Preparation .................... 76
5.2.2 Correcting for Taxes and Dividends. . . . . . . . . . . . . . . . .. 78
5.2.3 Liquidity Aspects ................................. 82
5.3 Structure of DAX Implied Volatilities . . . . . . . . . . . . . . . . . . . . .. 83
5.3.1 Estimation of the DAX Volatility Surface. . . . . . . . . . . .. 83
5.3.2 Empirical Results ................................. 92
5.3.3 Identification and Selection of Volatility Risk Factors .. 99
5.4 Dynamics of DAX Implied Volatilities ...................... 102
5.4.1 Time-Series Properties of DAX Volatility Risk Factors . 102
5.4.2 Relating Volatility Risk Factors to Index Returns and
other Market Variables ............................. 109
5.5 Summary of Empirical Observations ....................... 113
6 A Four-Factor Model for DAX Implied Volatilities ......... 115
6.1 The Model under the Objective Measure ................... 115
6.1.1 Model Specification ................................ 115
6.1.2 Model Estimation ................................. 118
6.1.3 Model Testing .................................... 124
6.2 The Model under the Risk-Neutral Measure ................. 131
6.2.1 Risk-Neutral Stock Price and Volatility Dynamics ..... 131
6.2.2 The Market Price of Risk Process ................... 133
6.2.3 Pricing and Hedging of Contingent Claims ............ 137
6.2.4 Model Calibration ................................. 140
6.3 Model Review and Conclusion ............................ 144
Contents XI
7 Model Applieations ........................................ 145
7.1 Pricing and Hedging of Exotic Derivatives .................. 145
7.1.1 Product Overview ................................. 145
7.1.2 Exotic Equity Index Derivatives ..................... 147
7.1.3 Volatility Derivatives .............................. 153
7.2 Value at Risk for Option Portfolios ........................ 158
7.2.1 The Value at Risk Concept ......................... 158
7.2.2 Computing VaR for Option Portfolios ................ 160
7.2.3 A Case Study ..................................... 162
7.2.4 Beyond VaR: Expected Shortfall .................... 167
7.3 Volatility Trading ....................................... 170
7.3.1 Definition and Motivation .......................... 170
7.3.2 Volatility Trade Design ............................. 171
7.3.3 Profit ability of DAX Volatility Trading Strategies ..... 178
8 Summary and Conclusion .................................. 187
A Some Mathematical Preliminaries ......................... 193
A.l Probability Theory ...................................... 193
A.2 Continuous-time Stochastic Processes ...................... 194
B Pricing of a Varianee Swap via Statie Replieation .......... 201
List of Abbreviations .......................................... 205
List of Symbols ................................................ 207
Referenees ..................................................... 215
Index .......................................................... 225
1
Introduction
1.1 Motivation and Objectives
Financial derivatives or contingent claims are specialized contracts whose in
tention is to transfer risk from those who are exposed to risk to those who are
willing to bear risk for a price. Derivatives are heavily used by different groups
of market participants, including financial institutions, fund managers (most
notably hedge funds), and corporations. While speculators intend to benefit
from the derivative's leverage to make large profits, hedgers want to ins ure
their positions against adverse price movements in the derivative's underlying
asset, and arbitrageurs are willing to exploit price inefficiencies between the
derivative and the underlying asset. During the last two decades the market far
financial derivatives has experienced rapid growth. From 2000 to 2002 alone,
global exchange-traded derivatives volume nearly doubled, to reach almost 6
billion contracts traded in 2002. With a market share of approximately 50%,
equity index derivatives are thereby the most actively traded contracts.1 Huge
volumes of derivatives are also traded over the counter (OTC). In addition
to standard products, the OTC market offers a wide variety of different con
tracts, including so-called exotic derivatives. Exotic derivatives were developed
as advancements to standard derivative products with specific characteristics
tailared to particular investors' needs. The latest development in this area are
volatility derivatives. These contracts, written on realized or implied volatil
ity, provide direct exposure to volatility without inducing additional exposure
to the underlying asset.
The increasing use and complexity of derivatives raises the need for a
framework that enables for the accurate and consistent pricing and hedging,
risk management, and trading of a wide range of derivative products, including
all kinds of exotic derivatives. The first important attempt in this direction was
the Black-Scholes option pricing model, developed by BlackjScholes (1973),
formalized and extended in the same year by Merton (1973). It builds a cor-
lSee FrA (2003).
R. Hafner, Stochastic Implied Volatility
© Springer-Verlag Berlin Heidelberg 2004
2 1 Introduction
nerstone in the theory of modern finance, and has led to many insights into
the valuation of derivative securities. In 1997, the importance of the model
was recognized when Myron Scholes and Robert Merton received the Nobel
Price for Economics. The Black-Scholes model provides a unique "fair price"
for a (European) option that is traded on a frictionless market and whose
underlying asset exhibits lognormally distributed prices. Under the model's
assumptions, an option's return stream can be perfectly replicated by con
tinuously rebalancing a self-financing portfolio involving stocks2 and risk-free
bonds. In the absence of arbitrage, the price of an option equals the initial
value of the portfolio that exactly matches the option's payoff.
The Black-Scholes model is often applied as a starting point for valuing
options. However, the empirical investigation of the Black-Scholes model re
vealed statistically significant and economically relevant deviations between
market prices and model prices. A convenient way of illustrating these dis
crepancies is to express the option price in terms of its implied volatility,
i.e. as a number that, when plugged into the Black-Scholes formula for the
volatility parameter, results in a model price equal to the market price. If the
Black-Scholes model holds exactly, then all options on the same underlying
asset should provide the same implied volatility. Yet, as is well known, on
many markets, Black-Scholes implied volatilities tend to differ across exercise
prices and times to maturity. The relationship between implied volatilities and
exercise prices is commonly referred to as the "volatility smile" and the rela
tionship between implied volatilities and times to maturity as the "volatility
term structure". Volatility surfaces combine the volatility smile with the term
structure of volatility. The existence of volatility surfaces implies that the im
plied volatility of an option is not necessarily equal to the expected volatility
of the underlying asset's rate of return. It rather also reflects determinants of
the option's value that are neglected in the Black-Scholes formula.
The obvious shortcomings of the Black-Scholes model have led to the de
velopment of a considerable literature on alternative option pricing models,
which attempt to identify and model the financial mechanisms that give rise to
volatility surfaces, in particular to smiles. One strand of the literature concen
trates on the nature of the underlying asset price process which was assumed
to be a geometric Brownian motion with constant volatility in the Black
Scholes framework. Here the main focus is on models which assume that the
volatility of the underlying asset varies over time, either deterministically or
stochastically. Derman/Kani (1994b), Derman/Kani (1994a), Dupire (1994),
and Rubinstein (1994) were the first to model volatility as a deterministic func
tion of time and stock price, known as local volatility. The unknown volatility
function can be fitted to observed option prices to obtain an implied price
process for the underlying asset. In an empirical study Dumas et al. (1998)
2We use the term "stock" as a general expression for the underlying asset of a
derivative security, although it could as well be an equity index, an exchange rate,
or the price of a commodity.
