Derivatives: Models on Models

Author’s “Disclaimer” ix

Introduction x

Derivatives Models on Models xv

Nassim Taleb on Black Swans 1

Chapter 1 The Discovery of Fat-Tails in Price Data 17

Edward Thorp on Gambling and Trading 27

Chapter 2 Option Pricing and Hedging from Theory to Practice: Know Your Weapon III 33

1 The Partly Ignored and Forgotten History 34

2 Discrete Dynamic Delta Hedging under Geometric Brownian Motion 44

3 Dynamic Delta Hedging Under Jump-Diffusion 50

4 Equilibrium Models 54

5 Portfolio Construction and Options Against Options 55

6 Conclusions 63

Alan Lewis on Stochastic Volatility and Jumps 71

Chapter 3 Back to Basics: A New Approach to the Discrete Dividend Problem 79
Together with Jørgen Haug and Alan Lewis

1 Introduction 79

2 General Solution 82

3 Dividend Models 87

4 Applications 89

Emanuel Derman the Wall Street Quant 101

Chapter 4 Closed Form Valuation of American Barrier Options 115

1 Analytical Valuation of American Barrier Options 115

2 Numerical Comparison 116

3 Conclusion 118

Peter Carr, The Wall Street Wizard of Option Symmetry and Volatility 121

Chapter 5 Valuation of Complex Barrier Options Using Barrier Symmetry 129

1 Plain Vanilla Put–Call Symmetry 129

2 Barrier Put–Call Symmetry 130

3 Simple, Intuitive and Accurate Valuation of Double Barrier Options 132

4 Static Hedging in the Real World 137

5 Conclusion 138

Granger on Cointegration 141

Chapter 6 Knock-in/out Margrabe 145
with Jørgen Haug

1 Margrabe Options 145

2 Knock-in/out Margrabe Options 146

3 Applications 147

Stephen Ross on APT 153

Chapter 7 Resetting Strikes, Barriers and Time 157
with Jørgen Haug

1 Introduction 157

2 Reset Strike Barrier Options 160

3 Reset Barrier Options 161

4 Resetting Time 162

5 Conclusion 163

Bruno Dupire the Stochastic Wall Street Quant 167

Chapter 8 Asian Pyramid Power 177
with Jørgen Haug and William Margrabe

1 Celia in Derivativesland 177

2 Calibrating to the Term Structure of Volatility 180

3 From Geometric to Arithmetic 184

4 The Dollars 185

Eduardo Schwartz: the Yoga Master of Mathematical Finance 191

Chapter 9 Practical Valuation of Power Derivatives 197

1 Introduction 197

2 Energy Swaps/Forwards 199

3 Power Options 202

4 Still, What About Fat-Tails? 209

Aaron Brown on Gambling, Poker and Trading 211

Chapter 10 A Look in the Antimatter Mirror 223

1 Garbage in, Garbage Out? 223

2 Conclusion 227

Knut Aase on Catastrophes and Financial Economics 231

Chapter 11 Negative Volatility and the Survival of the Western Financial Markets 239
Knut K. Aase

1 Introduction 239

2 Negative Volatility – A Direct Approach 240

3 The Value of a European Call Option for any Value – Positive or Negative – of the Volatility 240

4 Negative Volatility – The Haug interpretation 242

5 Chaotic Behavior from Deterministic Dynamics 242

6 Conclusions 243

Elie Ayache on Option Trading and Modeling 247

Chapter 12 Frozen Time Arbitrage 267

1 Time Measure Arbitrage 268

2 Time Travel Arbitrage 269

3 Conclusion 273

Haug on Wilmott and Wilmott on Wilmott 277

Chapter 13 Space-time Finance The Relativity Theory’s Implications for Mathematical Finance 287

1 Introduction 287

2 Time dilation 290

3 Advanced stage of Space-time Finance 292

4 Space-time Uncertainty 293

5 Is High Speed Velocity Possible? 295

6 Black-Scholes in Special Relativity 299

7 Relativity and Fat-Tailed Distributions 301

8 General Relativity and Space-time Finance 302

9 Was Einstein Right? 305

10 Traveling Back in Time Using Wormholes 307

11 Conclusion 308

Andrei Khrennikov on Negative Probabilities 317

Chapter 14 Why so Negative about Negative Probabilities? 323

1 The History of Negative Probability 323

2 Negative Probabilities in Quantitative Finance 324

3 Getting the Negative Probabilities to Really Work in Your Favor 327

4 Hidden Variables in Finance 328

5 The Future of Negative Probabilities in Quantitative Finance 329

6 Appendix: Negative Probabilities in CRR Equivalent Trinomial Tree 330

David Bates on Crash and Jumps 335

Chapter 15 Hidden Conditions and Coin Flip Blow Up’s 343

1 Blowing Up 343

2 Coin Flip Blow Up’s 344

Peter Jáckel on Monte Carlo Simulation 349

Index 359