Zhu Applications of Fourier Transform to Smile Modeling

Theory and Implementation
2. Auflage 2009
ISBN: 978-3-642-01808-4
Verlag: Springer
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 330 Seiten

This book addresses the applications of Fourier transform to smile modeling. Smile effect is used generically by ?nancial engineers and risk managers to refer to the inconsistences of quoted implied volatilities in ?nancial markets, or more mat- matically, to the leptokurtic distributions of ?nancial assets and indices. Therefore, a sound modeling of smile effect is the central challenge in quantitative ?nance. Since more than one decade, Fourier transform has triggered a technical revolution in option pricing theory. Almost all new developed option pricing models, es- cially in connection with stochastic volatility and random jump, have extensively applied Fourier transform and the corresponding inverse transform to express - tion pricing formulas. The large accommodation of the Fourier transform allows for a very convenient modeling with a general class of stochastic processes and d- tributions. This book is then intended to present a comprehensive treatment of the Fourier transform in the option valuation, covering the most stochastic factors such as stochastic volatilities and interest rates, Poisson and Levy ´ jumps, including some asset classes such as equity, FX and interest rates, and providing numerical ex- ples and prototype programming codes. I hope that readers will bene?t from this book not only by gaining an overview of the advanced theory and the vast large l- erature on these topics, but also by gaining a ?rst-hand feedback from the practice on the applications and implementations of the theory.

Autoren/Hrsg.

Zhu, Jianwei

Inhaltsverzeichnis

1;Preface;7
2;Contents;11
3;Option Valuation and the Volatility Smile;16
3.1;Stochastic Processes for Stocks;16
3.1.1;Brownian Motion;16
3.1.2;Stock Price as Geometric Brownian Motion;18
3.1.3;Itô Process and Itô's Lemma;19
3.2;The Black-Scholes Model;20
3.2.1;Options and Dynamic Hedging;20
3.2.2;Risk-Neutral Valuation;22
3.2.3;Self-financing and No Arbitrage;23
3.2.4;Equivalent Martingale Measures;26
3.3;Volatility Quotations in Markets;30
3.3.1;Implied Volatilities;30
3.3.2;Market Quotations;31
3.3.3;Special Case: FX Market;32
4;Characteristic Functions in Option Pricing;35
4.1;Constructing Characteristic Functions (CFs);36
4.1.1;A General Process for Stock Price;36
4.1.2;Valuation of European-style Options via CFs;37
4.1.3;Special Case: FX Options;41
4.2;Understanding Characteristic Functions;43
4.2.1;Properties of Characteristic Functions;43
4.2.2;Economic Interpretation of CFs;46
4.2.3;Examination of Existing Option Models;49
4.2.4;Relationship between CF to PDE;53
4.2.5;Advantages of CF and Modular Pricing;56
5;Stochastic Volatility Models;58
5.1;Introduction;58
5.2;The Heston Model;61
5.2.1;Model Setup and Properties;61
5.2.2;PDE Approach to Pricing Formula;63
5.2.3;Expectation Approach to Pricing Formula;65
5.2.4;Various Representations of CFs;67
5.3;The Schöbel-Zhu Model;68
5.3.1;Model Setup and Properties;68
5.3.2;Derivation of CFs;71
5.3.3;Numerical Examples;73
5.4;Double Square Root Model;76
5.4.1;Model Setup and Properties;76
5.4.2;Numerical Examples;82
5.5;Other Stochastic Volatility Models;83
5.6;Appendices;86
6;Numerical Issues of Stochastic Volatility Models;90
6.1;Alternative Pricing Formulas with CFs;91
6.1.1;The Formula á la Black-Scholes;91
6.1.2;The Carr and Madan Formula;91
6.1.3;The Attari Formula;93
6.2;Risk Sensitivities;93
6.2.1;Delta and Gamma;94
6.2.2;Various Vegas;95
6.2.3;Curvature and Slope;97
6.2.4;Volga and Vanna;97
6.3;Direct Integration (DI);99
6.3.1;The Gaussian Integration;99
6.3.2;Multi-Domain Integration;100
6.3.3;Strike Vector Computation;101
6.4;Fast Fourier Transform (FFT);102
6.4.1;Algorithms of FFT;102
6.4.2;Restrictions;104
6.5;Direct Integration vs. FFT;105
6.5.1;Computation Speed;106
6.5.2;Computation Accuracy;107
6.5.3;Matching Market Data;107
6.5.4;Calculation of Greeks;108
6.5.5;Implementation;108
6.6;Logarithm of Complex Number;112
6.6.1;Definition;112
6.6.2;Three Algorithms Dealing with Branch Cut;114
6.6.3;When Main Argument Is Appropriate;116
6.7;Calibration to Market Data;117
6.7.1;General Procedure;117
6.7.2;Fixing Velocity Parameter;118
6.7.3;Fixing Spot Volatility;119
6.8;Markovian Projection;121
7;Simulating Stochastic Volatility Models;125
7.1;Simulation Scheme;126
7.1.1;Discretization;126
7.1.2;Moment-Matching;127
7.2;Problems in the Heston Model;128
7.2.1;Negative Values in Paths;128
7.2.2;Log-normal Scheme;129
7.2.3;Transformed Volatility Scheme;130
7.2.4;QE Scheme;132
7.2.5;The Broadie-Kaya Scheme;134
7.2.6;Some Other Schemes;136
7.3;Simulation Examples;137
7.4;Maximum and Minimum;138
7.5;Multi-Asset Model;142
8;Stochastic Interest Models;146
8.1;Introduction;146
8.2;The Cox-Ingosoll-Ross Model;149
8.2.1;The Zero-Correlation Case;149
8.2.2;The Correlation Case;151
8.3;The Vasicek Model;153
8.4;The Longstaff Model;155
8.4.1;The Zero-Correlation Case;156
8.4.2;The Correlation Case;157
8.5;Correlations with Stock Returns: SI versus SV;159
9;Poisson Jumps;164
9.1;Introduction;164
9.2;Simple Jumps;169
9.3;Lognormal Jumps;171
9.4;Pareto Jumps;174
9.5;The Kou Model: An Equivalence to Pareto Jumps;176
9.6;Affine Jump-Diffusions;179
10;Lévy Jumps;184
10.1;Introduction;185
10.2;Stochastic Clock Models;188
10.2.1;Variance-Gamma Model;190
10.2.2;Normal Inverse Gaussian Model;192
10.3;Time-Changed Lévy Process;194
10.3.1;Uncorrelated Time-Change;195
10.3.2;Correlated Time-Change;199
10.4;The Barndorff-Nielsen and Shephard Model;204
10.5;Alpha Log-Stable Model;205
10.6;Empirical Performance of Various Lévy Processes;207
10.7;Monte-Carlo Simulation;209
10.7.1;Generating Random Variables;209
10.7.2;Simulation of Lévy Process;212
11;Integrating Various Stochastic Factors;214
11.1;Stochastic Factors as Modules;214
11.2;Integration Approaches;217
11.2.1;Modular Approach;217
11.2.2;Time-Change Approach;223
11.3;Pricing Kernels for Options and Bonds;228
11.4;Criterions for Model Choice;229
12;Exotic Options with Stochastic Volatilities;233
12.1;Forward-Starting Options;234
12.2;Barrier Options;236
12.2.1;Introduction;236
12.2.2;Two Special Cases;238
12.2.3;Numerical Examples;243
12.3;Lookback Options;245
12.3.1;Introduction;245
12.3.2;Pricing Formulas with Stochastic Factors;248
12.4;Asian Options;254
12.4.1;Introduction;254
12.4.2;The Black-Scholes World;255
12.4.3;Asian Options in a Stochastic World;259
12.4.4;Approximations for Arithmetic Average Asian Options;262
12.4.5;A Model for Asian Interest Rate Options;264
12.5;Correlation Options;267
12.5.1;Introduction;267
12.5.2;Exchange Options;270
12.5.3;Quotient Options;273
12.5.4;Product Options;274
12.6;Other Exotic Options;276
12.7;Appendices;277
13;Libor Market Model with Stochastic Volatilities;282
13.1;Introduction;282
13.2;Standard Libor Market Model;284
13.2.1;Model Setup;284
13.2.2;Term Structure and Smile of Volatility;289
13.3;Swap Market Model;291
13.3.1;Model Setup;291
13.3.2;Correlation Structure;295
13.3.3;Convexity Adjustments for CMS ;300
13.4;Incorporating Stochastic Volatility;303
13.4.1;The Andersen and Brotherton-Ratcliffe Model;304
13.4.2;The Piterbarg Model;307
13.4.3;The Wu and Zhang Model;311
13.4.4;The Zhu Model;314
13.4.5;The Belomestny, Matthew and Schoenmakers Model;321
13.5;Conclusive Remarks;325
14;References;327
15;Index;335