Uncertain Volatility Models - Theory and Application : Theory and Application - Robert Buff

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Uncertain Volatility Models - Theory and Application

Theory and Application

By: Robert Buff, R. Buff

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Paperback | 1 May 2002

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256 Pages

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23.5 x 15.24 x 1.27

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Many introductory books on mathematical finance also outline some com- puter algorithms. My goal is to contribute a closer look at algorithmic issues that arise from complex forms of the underlying pricing models-issues many practitioners need to solve sooner or later in their careers. This book takes such a close look at uncertain volatility models, an exten- sion of Black-Scholes theory.It discusses applications to exotic option portfo- lios with barriers and early exercise features. It describes an object-oriented C++ solution, included in source code on the accompanying CD. Practitioners and students who need to build analytic software libraries may benefit from reading this book and studying the software. The book focuses on a family of mathematical models, while in the field one encounters greater variation in instrument properties. In both cases mathematical and financial knowledge must be complemented by good programming skills to produce the best system. Analytic software needs design-a central message of the later chapters of this book. This book has come out of my Ph.D. thesis. I am very grateful to my academic advisor, Marco Avellaneda of New York University, who taught me mathematical finance and uncertain volatility. Computational finance be- came exciting for me because Marco encouraged an algorithmic approach to uncertain volatility. I thank Afshin Bayrooti, Vladimir Finkelstein, and Antonio Paras for giving valuable feedback. Antonio is the co-inventor of the original uncertain volatility model, A-UVM. Richard Holmes has found a crucial bug in an early implementation of the software.

Industry Reviews

From the reviews:

MATHEMATICAL REVIEWS

"The book bridges theory and real-world problems in a clear and pragmatic fashion. It can be useful both for academics and professionals in the financial community."

"This book, which comes out of the author's Ph.D. thesis, introduces uncertain volatility models. ... The formal results are illustrated by many empirical examples. ... The book bridges theory and real-world problems in a clear and pragmatic fashion. It can be useful both for academics and for professionals in the financial community." (Damir Filipovic, Mathematical Reviews, 2003 i)

"The book is devoted to the study of uncertain volatility models that evaluate option portfolios ... . The author travels in this book the entire road from innovative mathematical finance to a working software system ... . Practitioners and students who need to build analytic software libraries may benefit from reading this book ... . This book is also for graduate students and researchers who wish to study advanced aspects of volatility risk in portfolios of vanilla and exotic options." (Anatoliy Swishchuk, Zentralblatt MATH, Vol. 1004 (4), 2003)

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You Can Find This Book In

Non-Fiction Mathematics Applied Mathematics Economics Accounting & Finance Finance Investment & Securities Computing & I.T.Graphical & Digital Media Applications

Introduction	p. 1
Uncertain Volatility Scenarios and Exotic Options	p. 2
Volatility Shock Scenarios	p. 3
Object-Oriented Implementation	p. 4
User Interfaces: Scripting and Mathematica	p. 5
How to Best Read This Book	p. 6
Computational Finance: Theory
Notation and Basic Definitions	p. 11
Linear Algebra	p. 11
Probability and Stochastic Processes	p. 11
Partial Portfolios and Positions	p. 12
Accents, Superscript, Subscript	p. 13
Continuous Time Finance	p. 15
Deterministic Volatility	p. 15
One-Factor Black-Scholes Analysis	p. 15
Hedging with Black-Scholes	p. 17
Interest Rate Models	p. 19
Stochastic Volatility	p. 24
Tradable and Nontradable Factors	p. 24
Some Concrete One-Dimensional Models	p. 25
Model Calibration	p. 30
Parametric Methods	p. 30
Non-Parametric Methods	p. 31
Scenario-Based Evaluation and Uncertainty	p. 33
Preliminaries	p. 33
The Worst-Case Volatility Scenario	p. 35
Worst-Case Pricing	p. 36
The Optimal Hedge Portfolio	p. 38
Calibration to the Worst Case	p. 39
Minimum-Entropy Calibration	p. 41
Scenarios and Nonlinearity	p. 43
Algorithms for Uncertain Volatility Models
A Lattice Framework	p. 47
Multi-Lattice Dynamic Programming	p. 48
Data Structures	p. 49
Dataflow for Explicit Methods	p. 50
Dataflow for Mixed Explicit/Implicit Methods	p. 51
Numerical Issues	p. 51
Algorithms for Vanilla Options	p. 57
Algorithms for Barrier Options	p. 61
The Hierarchy of PDEÆs	p. 63
Construction	p. 63
Complexity	p. 66
Empirical Results	p. 71
Numerical Convergence	p. 71
Introducing Uncertainty	p. 73
Algorithms for American Options	p. 77
Early Exercise Combinations	p. 78
Long and Short Positions	p. 78
Best Worst-Case Evaluation Formalized	p. 81
Speedup Techniques	p. 90
Maintaining the Corridor of Uncertainty	p. 93
Collapsing the Corridor of Uncertainty	p. 98
Miscellaneous Issues	p. 105
Empirical Results	p. 110
Computational Complexity	p. 110
Stress Tests	p. 115
American Options and Calibration	p. 122
Exotic Volatility Scenarios	p. 123
Volatility Shocks for Portfolios of Vanilla Options	p. 123
Worst-Case Volatility Shocks	p. 125
Empirical Results	p. 133
Volatility Shocks and Exotic Options	p. 138
Object-Oriented Implementation
The Architecture of Mtg	p. 143
The Class Hierarchy of MtgLib-External	p. 145
Instruments	p. 146
Portfolios	p. 151
Models	p. 152
Model Coefficients	p. 158
The Base Class tTermStruct	p. 159
Classes Derived from tTermStruct	p. 162
Classes with tTermStruct Components	p. 162
Scenarios	p. 166
The Base Class tScenario	p. 167
Classes Derived from tScenario	p. 170
Numerical Methods	p. 171
Lattice Templates and Instances	p. 172
Finite Difference Solvers	p. 179
Evaluators	p. 182
The Class Hierarchy of MtgLib-Internal	p. 185
Compute Engines	p. 185
The Abstract Class tEngine	p. 186
The Abstract Class tFDEngine	p. 188
The Abstract Class tOFEngine	p. 190
The Concrete Class tGeoEngine	p. 193
The Concrete Class tShockEngine	p. 193
Other Groups of Classes	p. 194
Extensions for Monte-Carlo Pricing and Calibration	p. 195
The Network Application MtgClt/MtgSvr	p. 197
The Scripting Language MtgScript	p. 203
Factor Objects	p. 204
Claim and Portfolio Objects	p. 205
Model, Drift and Volatility Objects	p. 208
Lattice and Path Space Objects	p. 212
Bootstrapping, Curve and Image Objects	p. 213
Scenario and Optimizer Objects	p. 216
Evaluation Objects and Examples	p. 217
Mathematica Extensions	p. 227
The Syntax of Object Expressions	p. 227
Turning Scripts into Functions	p. 230
Profiling and Diagrams	p. 231
References	p. 233
Index	p. 237
Table of Contents provided by Publisher. All Rights Reserved.

Uncertain Volatility Models - Theory and Application

Theory and Application

At a Glance

Paperback

Industry Reviews

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