Financial Derivatives Modeling
•
Christian Ekstrand
Financial Derivatives
Modeling
123
Christian Ekstrand
Stockholm
Sweden
christian.ekstrand@seb.se
ISBN 978-3-642-22154-5 e-ISBN 978-3-642-22155-2
DOI 10.1007/978-3-642-22155-2
Springer Heidelberg Dordrecht London New York
Library of Congress Control Number: 2011936378
c
Springer-Verlag Berlin Heidelberg 2011
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Preface
The purpose of this book is to give a comprehensive introduction to the modeling of
financial derivatives, covering the major asset classes and stretching from Black and
Scholes’ lognormal modeling to current-day research on skew and smile models.
The intended reader has a solid mathematical background and works, or plans to
work, at a financial institution such as an investment bank or a hedge fund. The aim
of the book is to equip the reader with modeling tools that can be used in the (future)
work involving derivatives pricing, trading, or risk management.
The field of derivatives modeling is extensive and to keep the book within a
reasonable size, certain sacrifices have been made. For instance, the implementation
of models is not discussed as this can be viewed as an art rather than science and is
therefore an ungrateful subject for a text book. Minor asset classes, such as inflation
products, and asset classes that require specific mathematical tools, e.g., credit and
mortgage products, have been left out. Furthermore, the financial basics are covered
at a faster pace than in other introductory books to the area. For example, the
martingale theory is summarized in a compact appendix, and the introduction to
the Black–Scholes model is done by working directly in continuous space-time, in
contrast to the pedagogical approach of initially reviewing the binomial model. This
enables us to quickly go beyond the Black–Scholes framework and thereby focus
on skew and smile models and on derivatives in specific asset classes.
The book is divided into four parts. The first part consists of Chaps. 1–4 and
contains the general framework of derivatives pricing. This part is essential for the
understanding of the rest of the book. An exception is Chap. 4 which a novice
reader might find too abstract and is advised to skip and come back to later
when the necessary financial maturity has been reached. The rest of the book
consists of chapters that can be read independently. Chapters 5–8 cover skew and
smile modeling. The pricing of exotic derivatives is the subject of the third part,
Chaps. 9–10. The concluding fourth part comprises Chaps. 11–14 and applies the
pricing methods to specific asset classes.
Stockholm Christian Ekstrand
v
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Contents
Part I Derivatives Pricing Basics
1 Pricing by Replication ..................................................... 3
1.1 Underlyings and Derivatives ....................................... 3
1.2 Assumptions ........................................................ 4
1.3 The No-Arbitrage Assumption ..................................... 5
1.4 Replication .......................................................... 6
2 Static Replication ........................................................... 9
2.1 Forward Contracts .................................................. 9
2.2 European Options ................................................... 10
2.3 Non-Linear Payoffs ................................................. 10
2.4 European Option Price Constraints ................................ 13
2.5 American and Bermudan Options.................................. 15
2.6 Barrier Options...................................................... 16
2.7 Model-Dependent Pricing .......................................... 17
3 Dynamic Replication ....................................................... 19
3.1 Naive Replication of European Options ........................... 19
3.2 Dynamic Strategies ................................................. 21
3.3 Replication of Fixed-Time Payoffs................................. 24
3.4 The Black–Scholes Formula ....................................... 24
3.5 Analysis of the Black–Scholes Formula ........................... 28
3.6 Implied Volatility ................................................... 30
3.7 Relations between PDEs and SDEs ................................ 32
3.8 The Fundamental Theorem of Asset Pricing ...................... 34
3.9 Expectation of Non-Linear Payoffs ................................ 36
3.10 Futures Contracts ................................................... 37
3.11 Settlement Lag ...................................................... 40
Bibliography ................................................................. 42
vii
viii Contents
4 Derivatives Modeling in Practice ......................................... 43
4.1 Model Applications ................................................. 43
4.2 Calibration .......................................................... 45
4.3 Risk Management................................................... 53
4.4 Model Limitations .................................................. 69
4.5 Testing ............................................................... 73
Bibliography ................................................................. 78
Part II Skew and Smile Techniques
5 Continuous Stochastic Processes .......................................... 81
5.1 The Linear SDE ..................................................... 82
5.2 The Lognormal SDE ................................................ 82
5.3 The Normal SDE.................................................... 84
5.4 The Shifted Lognormal SDE ....................................... 86
5.5 The Quadratic SDE ................................................. 88
5.6 The Ornstein-Uhlenbeck Process .................................. 90
5.7 The Brownian Bridge ............................................... 91
5.8 The CEV Process ................................................... 93
5.9 The Bessel Process.................................................. 98
5.10 Non-Analytic SDEs................................................. 102
Bibliography ................................................................. 105
6 Local Volatility Models..................................................... 107
6.1 ATM Perturbation ................................................... 108
6.2 Dupire’s Equation ................................................... 113
6.3 Short Maturity Expansion .......................................... 114
6.4 Dynamics............................................................ 117
Bibliography ................................................................. 117
7 Stochastic Volatility Models ............................................... 119
7.1 Skew and Smile ..................................................... 120
7.2 Perturbation for Small Volatility of Volatility ..................... 123
7.3 Conditional Expectation Approach ................................ 129
7.4 Fourier Transform Approach ....................................... 130
7.5 Comparison of Methods ............................................ 134
7.6 Relations to Implied and Local Volatility .......................... 135
7.7 Dynamics............................................................ 136
7.8 Local Stochastic Volatility.......................................... 138
Bibliography ................................................................. 138
8L
´
evy Models ................................................................. 139
8.1 L´evy Processes ...................................................... 140
8.2 L´evy-Ito Decomposition............................................ 141
8.3 Stochastic Calculus ................................................. 144
8.4 Examples of L´evy Processes ....................................... 147
Contents ix
8.5 Pricing ............................................................... 154
8.6 Dynamics............................................................ 154
Bibliography ................................................................. 155
Part III Exotic Derivatives
9 Path-Dependent Derivatives ............................................... 159
9.1 Barrier Options...................................................... 161
9.2 Volatility Products .................................................. 168
9.3 American Options................................................... 170
9.4 Callable Products ................................................... 174
Bibliography ................................................................. 176
10 High-Dimensional Derivatives ............................................ 177
10.1 Copulas .............................................................. 177
10.2 Variable Freezing ................................................... 180
10.3 Moment Matching .................................................. 181
10.4 Quadratic Functional Modeling .................................... 182
10.5 Change of Measure ................................................. 186
10.6 Digital Options ...................................................... 186
10.7 Spread Options ...................................................... 188
10.8 Correlations ......................................................... 189
10.9 Calibration .......................................................... 190
Bibliography ................................................................. 191
Part IV Asset Class Specific Modeling
11 Equities ...................................................................... 195
11.1 Stylized Facts ....................................................... 196
11.2 Dividends............................................................ 196
11.3 More Advanced Models ............................................ 199
11.4 Volatilities and Correlations ........................................ 200
12 Commodities ................................................................ 201
12.1 Commodities Trading and Investment ............................. 202
12.2 Commodity Price Characteristics .................................. 205
12.3 Commodities Derivatives Modeling ............................... 209
12.4 Volatilities and Correlations ........................................ 214
Bibliography ................................................................. 219
13 Interest Rates ............................................................... 221
13.1 Interest Rates and Conventions..................................... 222
13.2 Static Replication ................................................... 224
13.3 Caps, Floors and Swaptions ........................................ 226
13.4 Convexity Adjustment .............................................. 231
13.5 The Yield Curve..................................................... 236
