2

Foreword

This book is an outgrowth of a number of activities of the author in the fieldsof Derivatives Analytics and Python Programming at Visixion GmbH on theone hand and of teaching Mathematical Finance at Saarland University onthe other hand.

It is the second preliminary edition of the book, finished in March 2013(the first was finished in January 2012), and it will most probably be cor-rected, revised and amended in a number of minor and even major ways.The author is thankful for any hint with regard to errors, inconsistencies,potential clarifications and improvements in general.

The book is targeted at practitioners, researchers and students interestedin the market-based valuation of options from a practical perspective, i.e. thesingle implementation steps that make up such an effort. It is also for thosewho want to learn how Python can be used for Derivatives Analytics andFinancial Engineering. However, apart from being primarily practical andimplementation-oriented, the book also provides the necessary theoreticalfoundations and numerical tools.

My hope is that the book will contribute to the increasing acceptance ofPython in the financial community. If you are interested in getting the Pythonscripts accompanying the book, send an email to contact@visixion.com.

I thank my familyand in particular my wife Sandrafor their supportand understanding that such a project requires many hours of solitude. Ialso want to thank my colleague Michael Schwed for his continuous help andsupport. Discussions with participants of seminars and of my lectures atSaarland University also helped the project significantly. Parts of this bookhave benefited from talks I have given at Python conferences in Florence,Paris and Leipzig during 2011 and 2012.

3

4I dedicate this book to my lovely son Henry Nikolaus whose direct ap-proach to living and clear view of the world I admire.

Volklingen, Saarland, Germany27. March 2013

Yves Hilpisch

Contents

1 A Quick Tour 171.1 Market-Based Valuation . . . . . . . . . . . . . . . . . . . . . 171.2 Structure of the Book . . . . . . . . . . . . . . . . . . . . . . . 181.3 Why Python? . . . . . . . . . . . . . . . . . . . . . . . . . . . 201.4 Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . 21

I The Market 23

2 What is Market-Based Valuation? 252.1 Options and their Value . . . . . . . . . . . . . . . . . . . . . 252.2 Vanilla vs. Exotic Instruments . . . . . . . . . . . . . . . . . . 292.3 Risks Affecting Equity Derivatives . . . . . . . . . . . . . . . . 31

2.3.1 Market Risks . . . . . . . . . . . . . . . . . . . . . . . 312.3.2 Other Risks . . . . . . . . . . . . . . . . . . . . . . . . 33

2.4 Hedging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332.5 Market-Based Valuation as a Process . . . . . . . . . . . . . . 34

3 Equity and Option Markets 373.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.2 Volatility, Correlation & Co. . . . . . . . . . . . . . . . . . . . 383.3 Normal Returns as the Benchmark Case . . . . . . . . . . . . 403.4 Indices and Stocks . . . . . . . . . . . . . . . . . . . . . . . . 44

3.4.1 Stylized Facts . . . . . . . . . . . . . . . . . . . . . . . 443.4.2 DAX Index Returns . . . . . . . . . . . . . . . . . . . 45

3.5 Option Markets . . . . . . . . . . . . . . . . . . . . . . . . . . 503.5.1 Bid/Ask Spreads . . . . . . . . . . . . . . . . . . . . . 503.5.2 Implied Volatility Surface . . . . . . . . . . . . . . . . 52

3.6 Short Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573.8 Python Scripts . . . . . . . . . . . . . . . . . . . . . . . . . . 58

5

6 CONTENTS

3.8.1 GBM Analysis . . . . . . . . . . . . . . . . . . . . . . 583.8.2 DAX Analysis . . . . . . . . . . . . . . . . . . . . . . 613.8.3 BSM Implied Volatilities . . . . . . . . . . . . . . . . 633.8.4 DAX Implied Volatilities . . . . . . . . . . . . . . . . 643.8.5 EURIBOR Analysis . . . . . . . . . . . . . . . . . . . 65

II Theoretical Valuation 69

4 Risk-Neutral Valuation 714.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 714.2 Discrete-Time Uncertainty . . . . . . . . . . . . . . . . . . . . 724.3 Discrete Market Model . . . . . . . . . . . . . . . . . . . . . . 77

4.3.1 Primitives . . . . . . . . . . . . . . . . . . . . . . . . . 774.3.2 Basic Definitions . . . . . . . . . . . . . . . . . . . . . 78

4.4 Central Results in Discrete Time . . . . . . . . . . . . . . . . 804.5 Continuous-Time Case . . . . . . . . . . . . . . . . . . . . . . 854.6 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

4.6.1 Proof of Lemma 1 . . . . . . . . . . . . . . . . . . . . . 914.6.2 Proof of Proposition 1 . . . . . . . . . . . . . . . . . . 924.6.3 Proof of Theorem 1 . . . . . . . . . . . . . . . . . . . . 93

5 Complete Market Models 955.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 955.2 Black-Scholes-Merton Model . . . . . . . . . . . . . . . . . . . 96

5.2.1 Market Model . . . . . . . . . . . . . . . . . . . . . . . 965.2.2 The Fundamental PDE . . . . . . . . . . . . . . . . . . 975.2.3 European options . . . . . . . . . . . . . . . . . . . . . 98

5.3 Greeks in the BSM Model . . . . . . . . . . . . . . . . . . . . 1025.4 Cox-Ross-Rubinstein Model . . . . . . . . . . . . . . . . . . . 1065.5 Proofs and Python Scripts . . . . . . . . . . . . . . . . . . . . 109

5.5.1 Itos Lemma . . . . . . . . . . . . . . . . . . . . . . . . 1095.5.2 Python Script for BSM Option Valuation . . . . . . . . 1105.5.3 Python Script for BSM Call Greeks . . . . . . . . . . . 1125.5.4 Python Script for CRR Option Valuation . . . . . . . . 113

6 Fourier-based Option Pricing 1156.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1156.2 The Pricing Problem . . . . . . . . . . . . . . . . . . . . . . . 1166.3 Fourier Transforms . . . . . . . . . . . . . . . . . . . . . . . . 1176.4 Fourier-based Option Pricing . . . . . . . . . . . . . . . . . . . 119

CONTENTS 7

6.4.1 Lewis (2001) Approach . . . . . . . . . . . . . . . . . . 1196.4.2 Carr-Madan (1999) Approach . . . . . . . . . . . . . . 122

6.5 Numerical Evaluation . . . . . . . . . . . . . . . . . . . . . . . 1246.5.1 Fourier Series . . . . . . . . . . . . . . . . . . . . . . . 1246.5.2 Fast Fourier Transform . . . . . . . . . . . . . . . . . . 126

6.6 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1276.6.1 Continuous Market Model . . . . . . . . . . . . . . . . 1276.6.2 Discrete Market Model . . . . . . . . . . . . . . . . . . 128

6.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1326.8 Python Scripts . . . . . . . . . . . . . . . . . . . . . . . . . . 133

6.8.1 Fourier Series . . . . . . . . . . . . . . . . . . . . . . . 1336.8.2 Roots of Unity . . . . . . . . . . . . . . . . . . . . . . 1336.8.3 Convolution . . . . . . . . . . . . . . . . . . . . . . . . 1346.8.4 Module with Parameters . . . . . . . . . . . . . . . . . 1356.8.5 Call Value by Convolution . . . . . . . . . . . . . . . . 1356.8.6 Option Pricing by Convolution . . . . . . . . . . . . . 1366.8.7 Option Pricing by DFT . . . . . . . . . . . . . . . . . 1366.8.8 Speed Test of DFT . . . . . . . . . . . . . . . . . . . . 137

7 Valuation of American Options 1397.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1397.2 Financial Model . . . . . . . . . . . . . . . . . . . . . . . . . . 1407.3 American Option Valuation . . . . . . . . . . . . . . . . . . . 141

7.3.1 Problem Formulations . . . . . . . . . . . . . . . . . . 1417.3.2 Valuation Algorithms . . . . . . . . . . . . . . . . . . . 143

7.4 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . 1467.4.1 American Put Option . . . . . . . . . . . . . . . . . . . 1467.4.2 American Short Condor Spread . . . . . . . . . . . . . 147

7.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1497.6 Python Scripts . . . . . . . . . . . . . . . . . . . . . . . . . . 150

7.6.1 Binomial Valuation . . . . . . . . . . . . . . . . . . . . 1507.6.2 Monte Carlo Valuation with LSM . . . . . . . . . . . . 151

III Market-Based Valuation 157

8 A First Example 1598.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1598.2 Market Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 1598.3 Valuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1608.4 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

8 CONTENTS

8.5 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

8.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

8.7 Python Scripts . . . . . . . . . . . . . . . . . . . . . . . . . . 167

8.7.1 Valuation by Numerical Integration . . . . . . . . . . . 167

8.7.2 Valuation by FFT . . . . . . . . . . . . . . . . . . . . . 168

8.7.3 Calibration to Three Maturities . . . . . . . . . . . . . 170

8.7.4 Calibration to Short Maturity . . . . . . . . . . . . . . 171

8.7.5 Valuation by MCS . . . . . . . . . . . . . . . . . . . . 173

9 General Model Framework 175

9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

9.2 The Framework . . . . . . . . . . . . . . . . . . . . . . . . . . 175

9.3 Features of the Framework . . . . . . . . . . . . . . . . . . . . 177

9.4 Zero-Coupon Bond Valuation . . . . . . . . . . . . . . . . . . 180

9.5 European Option Valuation . . . . . . . . . . . . . . . . . . . 180

9.5.1 PDE Approach . . . . . . . . . . . . . . . . . . . . . . 181

9.5.2 Transform Methods . . . . . . . . . . . . . . . . . . . . 183

9.5.3 Monte Carlo Simulation . . . . . . . . . . . . . . . . . 184

9.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

9.7 Proofs and Python Scripts . . . . . . . . . . . . . . . . . . . . 185

9.7.1 Itos Lemma . . . . . . . . . . . . . . . . . . . . . . . . 185

9.7.2 Python Script for Bond Valuation . . . . . . . . . . . . 186

9.7.3 Python Script for European Call Valuation . . . . . . . 187

10 Monte Carlo Simulation 191

10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

10.2 Valuation of Zero-Coupon Bonds . . . . . . . . . . . . . . . . 192

10.3 Valuation of European Options . . . . . . . . . . . . . . . . . 196

10.4 Valuation of American Options . . . . . . . . . . . . . . . . . 201

10.4.1 Numerical Results . . . . . . . . . . . . . . . . . . . . 203

10.4.2 Higher Accuracy vs. Lower Speed . . . . . . . . . . . . 208

10.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210

10.6 Python Scripts . . . . . . . . . . . . . . . . . . . . . . . . . . 210

10.6.1 General Zero-Coupon Bond Valuation . . . . . . . . . . 210

10.6.2 CIR85 Simulation and Valuation . . . . . . . . . . . . 211

10.6.3 PyTables Database for European Options . . . . . . . 214

10.6.4 Automated Valuation of European Options by MCS . . 219

10.6.5 PyTables Database for American Put Options . . . . . 224

10.6.6 Automated Valuation of American Put Options by MCS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229

CONTENTS 9

11 Model Calibration 237

11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237

11.2 General Considerations . . . . . . . . . . . . . . . . . . . . . . 238

11.2.1 Why Calibration At All? . . . . . . . . . . . . . . . . . 238

11.2.2 Which Role Play Different Model Components? . . . . 240

11.2.3 What Objective Function? . . . . . . . . . . . . . . . . 241

11.2.4 What Market Data? . . . . . . . . . . . . . . . . . . . 243

11.2.5 What Optimization Algorithm? . . . . . . . . . . . . . 244

11.3 Calibration of Short Rate Component . . . . . . . . . . . . . . 244

11.3.1 Theoretical Foundations . . . . . . . . . . . . . . . . . 244

11.3.2 Calibration to German Bund Yield Curve . . . . . . . 246

11.4 Calibration of Equity Component . . . . . . . . . . . . . . . . 249

11.4.1 Valuation via Fourier Transform Method . . . . . . . . 249

11.4.2 Calibration to DAX Call Option Quotes . . . . . . . . 253

11.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256

11.6 Python Scripts for Cox-Ingersoll-Ross Model . . . . . . . . . . 258

11.6.1 Calibration of CIR85 . . . . . . . . . . . . . . . . . . . 258

11.6.2 Calibration Results for CIR85 Model . . . . . . . . . . 260

11.6.3 Bond Valuation in CIR85 Model . . . . . . . . . . . . . 261

11.7 Python Scripts for Bakshi-Cao-Chen Model . . . . . . . . . . . 262

11.7.1 Call Option Valuation in BCC97 Model . . . . . . . . . 262

11.7.2 BSM Valuation Algorithms and Implied Volatility . . . 265

11.7.3 Option Data and Calibration Results . . . . . . . . . . 269

11.7.4 Calibration of Jump-Diffusion Part of BCC97 . . . . . 270

11.7.5 Calibration of Complete Model of BCC97 . . . . . . . 272

11.7.6 Short Rate Calculation . . . . . . . . . . . . . . . . . . 274

11.7.7 Comparison of Implied Volatilities . . . . . . . . . . . . 274

12 Simulation and Valuation 277

12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277

12.2 Simulation of BCC97 Model . . . . . . . . . . . . . . . . . . . 277

12.3 Valuation of Equity Options . . . . . . . . . . . . . . . . . . . 281

12.3.1 European Options . . . . . . . . . . . . . . . . . . . . 281

12.3.2 American Options . . . . . . . . . . . . . . . . . . . . . 282

12.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282

12.5 Python Scripts . . . . . . . . . . . . . . . . . . . . . . . . . . 283

12.5.1 Simulating the BCC97 Model . . . . . . . . . . . . . . 283

12.5.2 Valuation of European Call Options by MCS . . . . . . 285

12.5.3 Valuation of American Call Options by MCS . . . . . . 286

10 CONTENTS

13 Dynamic Hedging 28913.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28913.2 Hedging Study for BSM Model . . . . . . . . . . . . . . . . . 29013.3 Hedging Study for BCC97 Model . . . . . . . . . . . . . . . . 29513.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29913.5 Python Scripts . . . . . . . . . . . . . . . . . . . . . . . . . . 300

13.5.1 LSM Delta Hedging in BSM (Single Path) . . . . . . . 30013.5.2 LSM Delta Hedging in BSM (Multiple Paths) . . . . . 30213.5.3 LSM Algorithm for American Put in BCC97 . . . . . . 30413.5.4 LSM Delta Hedging in BCC97 (Single Path) . . . . . . 30513.5.5 LSM Delta Hedging in BCC97 (Multiple Paths) . . . . 308

14 Executive Summary 311

A Python in a Nutshell 313A.1 Python Fundamentals . . . . . . . . . . . . . . . . . . . . . . . 313

A.1.1 Installing Python Packages . . . . . . . . . . . . . . . . 313A.1.2 First Steps with Python . . . . . . . . . . . . . . . . . 314A.1.3 Array Operations . . . . . . . . . . . . . . . . . . . . . 317A.1.4 Random Numbers . . . . . . . . . . . . . . . . . . . . . 320A.1.5 Plotting . . . . . . . . . . . . . . . . . . . . . . . . . . 320

A.2 European Option Pricing . . . . . . . . . . . . . . . . . . . . . 323A.2.1 Black-Scholes-Merton Approach . . . . . . . . . . . . . 323A.2.2 Cox-Ross-Rubinstein Approach . . . . . . . . . . . . . 324A.2.3 Monte Carlo Approach . . . . . . . . . . . . . . . . . . 329

A.3 Selected Financial Topics . . . . . . . . . . . . . . . . . . . . . 331A.3.1 Approximation . . . . . . . . . . . . . . . . . . . . . . 331A.3.2 Optimization . . . . . . . . . . . . . . . . . . . . . . . 334A.3.3 Numerical Integration . . . . . . . . . . . . . . . . . . 335

A.4 Advanced Python Topics . . . . . . . . . . . . . . . . . . . . . 336A.4.1 Classes and Objects . . . . . . . . . . . . . . . . . . . 336A.4.2 Data Import and Export . . . . . . . . . . . . . . . . . 337

A.5 Rapid Financial Engineering . . . . . . . . . . . . . . . . . . . 341

Bibliography 345

Index 354

List of Tables

3.1 Option bid/ask spreads for call options on the DJIA . . . . . . 513.2 Option bid/ask spreads for put options on the DJIA . . . . . . 51

5.1 Valuation results from the CRR binomial algorithm for theEuropean call option . . . . . . . . . . . . . . . . . . . . . . . 108

7.1 Valuation results from the LSM and DUAL Monte Carlo al-gorithms for the American put option . . . . . . . . . . . . . . 149

7.2 Valuation results from the LSM and DUAL Monte Carlo al-gorithms for the Short Condor Spread . . . . . . . . . . . . . 150

10.1 Monte Carlo valuation results for European call and put op-tions in Hestons stochastic volatility model (I) . . . . . . . . . 200

10.2 Monte Carlo valuation results for European call and put op-tions in Hestons stochastic volatility model (II) . . . . . . . . 202

10.3 Monte Carlo valuation results for American put options inHeston-Cox-Ingersoll-Ross stochastic volatility and short ratemodel (I) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206

10.4 Monte Carlo valuation results for American put options inHeston-Cox-Ingersoll-Ross stochastic volatility and short ratemodel (II) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208

10.5 Monte Carlo valuation results for American put options inHeston-Cox-Ingersoll-Ross stochastic volatility and short ratemodel (III) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209

11

12 LIST OF TABLES

List of Figures

2.1 Inner value of a European call option . . . . . . . . . . . . . . 26

2.2 Black-Scholes-Merton value of a European call option . . . . . 28

3.1 A single simulated path for the geometric Brownian motion . . 41

3.2 Histogram of the daily log returns of the geometric Brownianmotion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.3 Realized volatility for the simulated path of the geometricBrownian motion . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.4 Rolling mean log return, volatility and correlation (261 days)for Geometric Brownian motion . . . . . . . . . . . . . . . . . 43

3.5 DAX daily quotes as downloaded from http://finance.yahoo.com into a spreadsheet . . . . . . . . . . . . . . . . . . . . . . 46

3.6 DAX index level quotes and daily log returns over the periodfrom 26. Nov 1990 to 01. Feb 2013 . . . . . . . . . . . . . . . 46

3.7 Histogram of the daily log returns of the DAX over the periodfrom 26. Nov 1990 to 01. Feb 2013 . . . . . . . . . . . . . . . 47

3.8 Left tail of histogram of the daily log returns of the DAX overthe period from 26. Nov 1990 to 01. Feb 2013 . . . . . . . . . 47

3.9 Realized volatility for the DAX over the period from 26. Nov1990 to 01. Feb 2013 . . . . . . . . . . . . . . . . . . . . . . . 49

3.10 Rolling mean log return, volatility and correlation (252 days)for the DAX . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.11 Implied volatilities from European call options on the DAXon 29. Apr 2011 (I) . . . . . . . . . . . . . . . . . . . . . . . . 53

3.12 Implied volatilities from European call options on the DAXon 29. Apr 2011 (II) . . . . . . . . . . . . . . . . . . . . . . . 53

3.13 Daily quotes of 1 week EURIBOR and daily log changes overthe period from 01. Jan 1999 to 01. Feb 2013 . . . . . . . . . 55

3.14 Histogram of daily log changes in 1 week EURIBOR . . . . . . 56

3.15 Right tail of histogram of daily log changes in 1 week EURIBOR 56

13

14 LIST OF FIGURES

3.16 Daily quotes of 1 week, 1 month, 6 months and 1 year EURI-BOR over the period from 01. Jan 1999 to 01. Feb 2013 . . . 57

5.1 Value of the example European call option for varying strikeK, maturity date T , short rate r and volatility (vol) . . . . 100

5.2 Value of the example European put option for varying strikeK, maturity date T , short rate r and volatility (vol) . . . . 101

5.3 The delta of the European call option with respect to matu-rity date T and strike K . . . . . . . . . . . . . . . . . . . . . 103

5.4 The gamma of the European call option with respect to ma-turity date T and strike K . . . . . . . . . . . . . . . . . . . . 104

5.5 The theta of the European call option with respect to matu-rity date T and strike K . . . . . . . . . . . . . . . . . . . . . 104

5.6 The rho of the European call option with respect to maturitydate T and strike K . . . . . . . . . . . . . . . . . . . . . . . 105

5.7 The vega of the European call option with respect to maturitydate T and strike K . . . . . . . . . . . . . . . . . . . . . . . 105

6.1 Fourier series approximation of function f(x) = |x| of order 1and of order 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

6.2 Series with roots of unity for n = 5 and n = 30 plotted in theimaginary plane . . . . . . . . . . . . . . . . . . . . . . . . . . 129

7.1 Valuation results for the American put option from 25 sim-ulation runs with M = 100 time intervals . . . . . . . . . . . . 148

7.2 Valuation results for the American Short Condor Spreadfrom 25 simulation runs with M = 100 time intervals . . . . . 148

8.1 Results of the calibration of Mertons jump-diffusion model tomarket quotesthree maturities . . . . . . . . . . . . . . . . . 163

8.2 Results of the calibration of Mertons jump-diffusion model tomarket quotesshort maturity only . . . . . . . . . . . . . . . 163

8.3 Comparison of European call option values from Lewis for-mula, Carr-Madan formula and Monte Carlo simulation . . . . 165

10.1 20 simulated paths for the Cox-Ingersoll-Ross short rate process194

10.2 Monte Carlo values for a ZCB maturing at T = 2exact scheme195

10.3 Monte Carlo values for a ZCB maturing at T = 2Euler scheme195

11.1 German Bund yields for maturities up to 10 years . . . . . . . 247

11.2 Market implied forward rates from German Bunds comparedto model forward rates . . . . . . . . . . . . . . . . . . . . . . 248

LIST OF FIGURES 15

11.3 Bond values at time t for a unit ZCB maturing at time T = 10 248

11.4 Valuation accuracy of Lewis integral approach in comparisonto Black-Scholes-Merton analytical formula . . . . . . . . . . . 251

11.5 Valuation accuracy of Carr-Madan FFT approach in compar-ison to Black-Scholes-Merton analytical formula . . . . . . . . 251

11.6 Results of jump-diffusion part calibration to five European calloptions on the DAX with 29 days maturity . . . . . . . . . . . 255

11.7 Results of simultaneous jump-diffusion part and stochasticvolatility part calibration . . . . . . . . . . . . . . . . . . . . . 256

11.8 Implied volatilities of European call option market quotes (29days maturity) compared to model implied volatilities aftercalibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257

11.9 Implied volatilities of European call option market quotes (239days maturity) compared to model implied volatilities aftercalibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257

11.10Implied volatilities of European call option market quotes (603days maturity) compared to model implied volatilities aftercalibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258

12.1 10 simulated short rate paths from calibrated Cox-Ingersoll-Ross short rate model for a time horizon of 1 year . . . . . . . 279

12.2 10 simulated volatility paths from calibrated general marketmodel for a time horizon of 1 year . . . . . . . . . . . . . . . . 279

12.3 10 simulated DAX level paths from calibrated general marketmodel for a time horizon of 1 year . . . . . . . . . . . . . . . . 280

12.4 Histogram of simulated DAX levels from calibrated generalmarket model after a time period of 1 year . . . . . . . . . . . 280

13.1 Dynamic replication of American put option in Black-Scholes-Merton model with profit at exercise . . . . . . . . . . . . . . 291

13.2 Dynamic replication of American put option in Black-Scholes-Merton model with loss at exercise . . . . . . . . . . . . . . . 294

13.3 Frequency distribution of (discounted) P&L at exercise date of10,000 dynamic replications of American put option in Black-Scholes-Merton model . . . . . . . . . . . . . . . . . . . . . . 295

13.4 Dynamic replication of American put option in general marketmodel with profit at maturity . . . . . . . . . . . . . . . . . . 297

13.5 Dynamic replication of American put option in general marketmodel with loss at maturity . . . . . . . . . . . . . . . . . . . 297

16 LIST OF FIGURES

13.6 Frequency distribution of (discounted) P&L at exercise date of10,000 dynamic replications of American put option in generalmarket model . . . . . . . . . . . . . . . . . . . . . . . . . . . 298

13.7 Dynamic replication of American put option in general marketmodel with huge loss at exercise due to an index jump . . . . 299

A.1 Example of figure with matplotlibhere: line . . . . . . . . 321A.2 Example of figure with matplotlibhere: dots & bars . . . . 322A.3 Histogram of simulated stock index levels at T . . . . . . . . . 331A.4 Approximation of cosine function by regression . . . . . . . . . 333A.5 Approximation of cosine function by cubic splines . . . . . . . 333A.6 Excel sample sheet with DAX quotes . . . . . . . . . . . . . . 339A.7 Historic DAX index levels from 04. Jan 2010 to 07. Feb 2013 . 341A.8 DAX index quotes from 03. Jan 2005 to 07. Feb 2013 and

daily log returns; source: finance.yahoo.com . . . . . . . . . 344

Chapter 1

A Quick Tour

1.1 Market-Based Valuation

This book is about the market-based valuation of (stock) index options. Inthe domain of Derivatives Analytics this is an important task which everymajor investment bank and buy-side decision maker in this market is con-cerned with on a daily basis. While theoretical valuation approaches developa model, parametrize it and then derive values for options, the market-basedapproach works the other way round. Prices from liquidly traded optionsare taken as given (i.e. they are input instead of output) and one tries toparametrize a market model in a way that replicates the observed optionprices as good as possible. This activity is generally referred to as modelcalibration. Being equipped with a calibrated model, one then proceeds withthe task at hand, be it valuation, trading, investing, hedging or risk manage-ment. A bit more specific, one might be interested in pricing and hedgingan exotic derivative instrument with such a modelhoping that the resultsare in line with the overall market due to the previous calibration to moresimple, vanilla instruments.

To accomplish a market-based valuation, four areas have to be covered:

1. market: knowledge about market realities is a conditio sine qua nonfor any sincere attempt to develop market-consistent models and toaccomplish market-based valuation

2. theory: every valuation must be grounded on a sound market model,ensuring, for example, the absence of arbitrage opportunities and pro-viding means to derive option values from observed quantities

3. numerics: one can not hope to work with analytical results only; nu-merical techniques, like Monte Carlo simulation, are generally required

17

18 CHAPTER 1. A QUICK TOUR

in different steps of a market-based valuation process

4. technology: to implement numerical techniques efficiently one is de-pendent on technology

This book covers all of these areas in an integrated manner. It uses equityindex options as the prime example for derivative instruments throughout.

1.2 Structure of the Book

The book is divided in three parts. The first part is concerned with market-based valuation as a process and empirical findings about market realities.The second part covers a number of topics for the theoretical valuation of op-tions and derivatives. It also develops tools fiercely needed during a market-based valuation. Part three finally covers the major aspects related to amarket-based valuation and also of hedging strategies in such a context.

Part I The Market comprises two chapters:

Chapter 2: this chapter contains a discussion of topics related tomarket-based valuation, like risks affecting the value of equity indexoptions

Chapter 3: this chapter mainly documents empirical and anecdo-tal facts about stocks, stock indices and in particular volatility (e.g.stochasticity, clustering, smiles)

Part II Theoretical Valuation comprises four chapters:

Chapter 4: this chapter covers arbitrage pricing theory and risk-neutral valuation in discrete time (in some detail) and continuous time(on a higher level) according to the Harrison-Kreps-Pliska paradigm(cf. [57] and [58])

Chapter 5: the topic of this chapter are the complete market modelsof Black-Scholes-Merton (BSM, cf. [8], [81]) and Cox-Ross-Rubinstein(CRR, cf. [34]) that are generally considered benchmarks for optionvaluation

Chapter 6: Fourier-based approaches allow to derive semi-analyticalvaluation formulas for European options in market models more com-plex and realistic than the BSM/CRR models; this chapter introducesthe two main approaches of Carr-Madan (cf. [18])and Lewis (cf. [73])

1.2. STRUCTURE OF THE BOOK 19

Chapter 7: the valuation of American options is more involved thanthe one of European options; this chapter analyzes the respective prob-lem and introduces algorithms for their valution via binomial trees andMonte Carlo simulation; at the center stands the Least-Squares MonteCarlo algorithm of Longstaff-Schwartz (cf. [76])

Finally, Part III Market-Based Valuation has seven chapters:

Chapter 8: before going into details, this chapter illustrates the wholeprocess of a market-based valuation effort in the simple, but neverthe-less still useful, setting of Mertons jump diffusion model (cf. [80])

Chapter 9: this chapter introduces the general market model usedhenceforth, which is from Bakshi-Cao-Chen (cf. [4]) and which accountsfor stochastic volatility, jumps and stochastic short rates

Chapter 10: Monte Carlo simulation is generally the method of choicefor the valuation of exotic/complex index options and derivatives; thischapter therefore discusses in some detail the discretization and simu-lation of the stochastic volatility model by Heston (cf. [61]) with con-stant as well as stochastic short rates according to Cox-Ingersoll-Ross(cf. [33])

Chapter 11: model calibration stays at the center of market-basedvaluation; the chapter considers several general aspects associated withthis topic and then proceeds with the numerical calibration of the gen-eral market model to real market data

Chapter 12: this chapter combines the results from the previous twoto value European and American index options via Monte Carlo simu-lation in the calibrated general market model

Chapter 13: this chapter analyzes dynamic delta hedging strategiesfor American options by Monte Carlo simulation in different settings,from a simple one to the calibrated market model

Chapter 14: this brief chapter provides a concise summary of centralaspects related to the market-based valuation of index options

In addition, the book has an Appendix with one chapter:

Appendix A: this appendix introduces into some of the most impor-tant Python concepts and libraries in a nutshell; the selection of topicsis clearly influenced by the requirements of the rest of the book

20 CHAPTER 1. A QUICK TOUR

1.3 Why Python?

Currently, and for the foreseeable future as well, Python does not belong tothe most important programming languages in the financial services industryin general or in Derivatives Analytics in particular. Languages like C++, C,C#, VBA or Java and toolboxes like Matlab or R dominate the scene. How-ever, we see a number of good reasons to choose Python, the most importantones we want to mention in no particular order:

Open Source: Python and the majority of available libraries are com-pletely open source; this allows an entry to this technology at no costs,something particularly important for students, academics or other in-dividuals

Syntax: Python programming is easy to learn, the code is quite com-pact and in general highly readable; at universities it is increasinglyused to introduce to programming in general

Multi-Paradigm: Python is as good at procedural programming (whichsuffices for the purposes of this book) as well as at object-oriented pro-gramming (which is necessary in more complex/professional contexts)

Interpreted: Python is an interpreted language which makes rapidprototyping and development in general a bit more convenient, espe-cially for beginners

Libraries: nowadays, there is a wealth of powerful libraries availableand the supply grows steadily; there is hardly a problem which cannotbe easily attacked with an existing library, be it a numerical problem,a graphical one or a data-related problem

Speed: a common prejudice with regard to interpreted languagescompared to compiled ones like C++ or Cis the slow speed of codeexecution; however, financial applications are more or less all aboutmatrix and array manipulations and operations which can be done atthe speed of C code with the essential Python library NumPy

Market: in the London area (mainly financial services) the number ofPython developer contract offerings was 644 in the three months ending22. April 2011; the comparable figure in the same period 2010 was 245and in 2009 only 142; this implies a more than fourfold demand for the

1.4. FURTHER READING 21

Python skill in 2011 as compared to 2009; over the same period theaverage daily rate increased from 400 GBP to 475 GBP1

All in all, Python seems to be a good choice for our purposes. And that itis possible to develop a commercial, full-fledged Derivatives Analytics suitewith Python is illustrated by the solution DEXISION of Visixion GmbH. Itis the first derivatives analytics suite whose core is completely built in thislanguage.2 The cover story Python Takes a Bite of the March 2010 issueof Wilmott Magazine (cf. [72]) also illustrates that Python is gaining groundin the financial world.

1.4 Further Reading

The book covers a great variety of aspects which comes at the cost of depthof exposition and analysis. Our aim is to emphasize the red line and to guidethe reader easily through the different topics. However, this inevitably leadsto uncovered aspects, omitted proofs and unanswered questions. Fortunately,a number of good sources in book form is available which may be consultedon the different topics.

market: cf. [11] to learn about options fundamentals, the main mi-crostructure elements of their markets and the specific lingo; [50] is aconcise reference about option and volatility modeling in practice; [88]is a book that covers comprehensively option markets, their empiricalspecialities and the models used in theory and practice

theory: [85] is a comprehensive source for discrete market models; [36]covers the general arbitrage theory in continuous time and is quite ad-vanced; less advanced, but still comprehensive, treatments of arbitragepricing are [12] for continuous processes based on Brownian motionand [28] for continuous processes with jumps; [100] offers a detaileddiscussion of the seminal Black-Scholes-Merton model

numerics: [23] is a book-length treatment of the Fourier-based optionpricing approach; [52] is the standard textbook on Monte Carlo simu-lation in financial applications; [14] covers a wide range of numericaltechniques regularly applied in mathematical finance and offers imple-mentation examples in Matlab3

1 Source: www.itjobswatch.co.uk/contracts/london/python.do on 24. April 2011.2 See the Web site www.dexision.com where free trials are available.3 Python in combination with NumPy comes quite close to the syntax of Matlab such that

translations are generally straightforward.

22 CHAPTER 1. A QUICK TOUR

implementation: probably the best introduction into Python for thepurposes of this book is the excellent book [78]; this book covers themain tools and libraries needed for this book, like IPython, NumPy andpandas, in a detailed fashion and with a wealth of practical examples;good general introductions to Python from a scientific perspective are[55] and [71]; [46] introduces into the language with the specific purposeof building financial models; [75] is a huge book that covers a greatvariety of financial models and topics and shows how to implementthem in C++; in addition, there is a wealth of Python documentationavailable for free in the Internet

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Index

algebra, 72algebra generation, 75American options, 139, 201

Bakshi-Cao-Chen (1997), 282Black-Scholes-Merton (1973), 146,

290Cox-Ross-Rubinstein (1979), 142dual problem, 142early exercise premium, 282least-squares algorithm, 203primal problem, 141

American put option, 146American short condor spread, 147approximation, 331

cubic splines, 332first order, 290ordinary least-squares regression

(OLS), 331regression, 331second order, 295

arbitrage opportunitycontinuous time, 88discrete time, 81weak, 81

Arrow-Debreu security, 84ask quote, 50at-the-money, 26

Bachelier model, 87bid quote, 50bid/ask spread, 50bond valuation

Cox-Ingersoll-Ross (1985), 180, 246Monte Carlo simulation, 192

Brownian motion, 87

call option, 25call option transform, 119characteristic function, 118

Bakshi-Cao-Chen (1997), 184, 252Black-Scholes-Merton (1973), 249Heston (1993), 183, 252Merton (1976), 128, 160Merton (1976) jump part, 184, 252

contingent claim, 80attainable, 80

convergencebinomial model, 109

convolution, 130correlation

historical, 39instantaneous, 40

correlation risk, 31

DAX index, 45, 243, 339, 340leverage effect, 48normality test, 45realized volatility, 48volatility clustering, 45

default risk, 31delta, 103, 289

hedging, 289discretization

absorption, 198Bakshi-Cao-Chen (1997), 278Black-Scholes-Merton (1973), 140exact log Euler scheme, 197full truncation, 197

354

INDEX 355

Higham-Mao, 198partial truncation, 197reflection, 197simple Euler scheme, 197simple reflection, 198truncation, 197

dynamic hedging, 289, 312Bakshi-Cao-Chen (1997), 295Black-Scholes-Merton (1973), 290

Euler formulas, 124EURIBOR, 55

stochasticity, 55term structure, 57

EURO STOXX 50, 161, 243European options, 25, 196

Bakshi-Cao-Chen (1997), 253, 281Black-Scholes-Merton (1973), 98,

99, 249, 323, 336Cox-Ross-Rubinstein (1979), 324Monte Carlo simulation, 184, 329PDE approach, 181Python classes, 336transform methods, 183

exotic instruments, 29expectation, 74

conditional, 76

fat tails, 45filtration, 74forward rates, 245

Cox-Ingersoll-Ross (1985), 245Fourier approach, 311

advantages, 115Bakshi-Cao-Chen (1997), 183, 249Black-Scholes-Merton (1973), 249Carr-Madan (1999), 122, 161, 183fast Fourier transform (FFT), 126,

250, 328Lewis (2001), 119, 120, 161, 183,

249

valuation accuracy, 250Fourier inversion, 117Fourier series, 124Fourier transform, 117

discrete, 129fundamental theorem of asset pricing

continuous time, 89discrete time, 82

gains processcontinuous time, 86discrete time, 79

gamma, 103Greeks, 102

delta, 102gamma, 102rho, 102theta, 102vega, 102

hedging, 33, 289

in-the-money, 26inner value, 25interest rate

mean reversion, 54positivity, 54stochasticity, 54term structure, 55

interest rate market, 54interest rate risk, 31interpolation, 331IPython, 22, 313Itos lemma

Bakshi-Cao-Chen (1997), 185Black-Scholes-Merton, 109

jump risk, 31jump-diffusion model

Bates (1996), 176Merton (1976), 127, 178

jumps, 45

356 INDEX

kurtosis, 39

Levy process, 87leverage effect, 45liquidity risk, 31

market completenesscontinuous time, 91discrete time, 84

market incompletenessnarrow sense, 238numerical example, 239wider sense, 238

market microstructure, 50market model

Bakshi-Cao-Chen (1997), 176, 249,252, 277

Bates (1996), 176, 178binomial model, 106, 324Black-Scholes-Merton (1973), 96,

140, 178, 323complete, 95continuous time, 85, 89, 140, 159,

176Cox-Ingersoll-Ross (1985), 176, 192,

201, 244Cox-Ross-Rubinstein (1979), 106,

324discrete time, 77, 79, 324Heston (1993), 178, 196, 201, 252Merton (1976), 159, 178, 252requirements, 57

market realities, 17, 21market risks, 311market-based valuation, 17, 30, 159,

166American options, 282European options, 281process, 34

martingale, 76martingale measure, 76

non-uniqueness, 240uniqueness, 84

matplotlib, 313, 320bars, 322dots, 322lines, 321

model calibration, 161, 237, 311DAX call option quotes, 253degeneracy, 253equity index component, 249German Bund yield curve, 246implied volatilities, 256indeterminacy, 253market data, 243market incompleteness, 238mean squared error (MSE), 242numerical results, 254objective function, 241optimization algorithm, 244role of model components, 240short rate component, 244Tikhonov regularization, 254

model risk, 33Monte Carlo simulation, 277, 311, 341

accuracy vs. speed, 208algorithm features, 205Bakshi-Cao-Chen (1997), 277, 295Black-Scholes-Merton (1973), 290,

329dual algorithm, 145Heston (1993), 197least-squares algorithm, 139, 143,

201, 205, 282, 290, 311Merton (1976), 162recipe, 184square-root diffusion, 192variance reduction, 199, 201, 278

no arbitrage condition (NA), 88no free lunches with vanishing risk

condition (NFLVR), 88

INDEX 357

numerical integration, 335numerical methods, 18, 21NumPy, 22, 313

arange, 319array operations, 318arrays, 317iterations, 327loops, 327polyfit, 332polyval, 332random, 320

objective functiondegeneracy, 253indeterminacy, 253mean squared error (MSE), 242

operational risk, 33optimization

brute force algorithm, 162global, 162, 244, 334local, 162, 244, 334simplex algorithm, 162

option definition, 25option markets, 50option price

moneyness, 101short rate, 101time-to-maturity, 101volatility, 102

option quote spread, 50option value factors, 27out-of-the-money, 27

pandas, 22, 313, 341data storage, 341DataFrame, 340Excel spreadsheets, 340financial data gathering, 341plotting, 340, 343

Parsevals relation, 118partial differential equation

Bakshi-Cao-Chen (1997), 181Black-Scholes-Merton (1973), 98

portfolio, 78value, 78

predictability, 78price risk, 31price system

linear, 81probability measure, 73probability space, 73

filtered, 74PyTables, 198, 313Python, 313

arrays, 317class, 336class attributes, 336class methods, 336data, 337dynamic typing, 315Excel spreadsheets, 338function definition, 316integrated development environ-

ment (IDE), 314iterations, 326loops, 326math, 316module, 317object, 336random numbers, 320range, 319string replacement, 319xlrd, 313xlwt, 313

Python programming language, 20Python(x,y), 314

qualitative model features, 30quantitative model features, 30

Radon-Nikodym derivative, 77random variable, 73

358 INDEX

rapid financial engineering, 341rho, 106risk-neutral valuation, 71

continuous time, 85, 117Cox-Ross-Rubinstein (1979), 107,

325discrete time, 82

root of unity, 128

SciPy, 313brute, 334fmin, 334integrate, 335interpolate, 332optimize, 334quad, 335

semimartingale, 86total, 86

short rate, 54skewness, 39square-root diffusion, 192

Euler discretization, 193Euler schemes, 197exact discretization, 192short rate process, 176, 201, 245variance process, 196, 201

state price, 84stochastic differential equation

Bakshi-Cao-Chen (1997), 176Black-Scholes-Merton (1973), 96Cox-Ingersoll-Ross (1985), 192Merton (1976), 159

stochastic process, 75decomposable, 86

stochastic volatility modelBates (1996), 176, 178Heston (1993), 178, 196, 201

stopping time, 75super-replication, 81

technology, 18, 22

Tikhonov regularization, 254trading strategy, 79

admissible, 79, 87self-financing, 79, 88simple predicable, 85

uncertaintycontinuous time, 85discrete time, 72

valuation speed, 328valuation theory, 17, 21, 71value process, 79vanilla instruments, 29variance reduction, 278

antithetic paths, 199, 278control variates, 201importance sampling, 278moment matching, 199, 278

vega, 106volatility

clustering, 44historical, 38implied, 38, 256instantaneous, 38leverage effect, 45mean reversion, 44stochastic, 44

volatility risk, 31