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- Product code: 12554
- ISBN: 0967637201,
ISBN13: 9780967637204,
350 pages, paperback
Published by Finance Press on 2000
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Description of Option Valuation under Stochastic Volatility |
This book provides an advanced treatment of option pricing for traders, money managers, and researchers. Providing largely original research not available elsewhere, it covers the new generation of option models where both the stock price and its volatility follow diffusion processes.
These new models help explain important features of real-world option pricing that are not captured by the Black-Scholes model. These features include the 'smile' pattern and the term structure of implied volatility. The book includes Mathematica code for the most important formulas and many illustrations.
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Reviews"This exciting book is the first one to focus on the pervasive role of stochastic volatility in option pricing. Since options exist primarily as the fundamental mechanism for trading volatility, students of the fine art of option pricing are advised to pounce."
- Peter Carr, Ph.D., Principal, Banc of America Securities
"I found this book extremely interesting, and valuable for both academics and practitioners. It treats many important aspects of the stochastic volatility problem with novel methods. I especially liked the treatment of the term structure of implied volatility in Chapter 6. This book is a very nice contribution to the literature."
- Prof. Nizar Touzi, Department of Mathematics, University of Paris I, Leading expert on stochastic volatility models
"This book is an impressive collection of methods and results. I found Chapter 7 on equilibrium models particularly helpful, as very often people 'fudge' the discussion of the volatility risk premium by making simple assumptions."
- Prof. Stephen Taylor, Accounting and Finance, Lancaster University
| Contents of Option Valuation under Stochastic Volatility |
1. Introduction and Summary of Results
Summary of Results
The Hedging Argument of Black and Scholes
The Drift Cancellation and Option Sensitivities
The Hedging Argument under Stochastic Volatility
The Martingale Approach
App. 1.1 Parameter Estimators for the GARCH Diffusion Model
App. 1.2 Solutions to PDEs
2. The Fundamental Transform
Assumptions
The Transform-based Solution
Some Models with Closed-form Solutions
Analytic Characteristic Functions
A Bond Price Analogy and Option Price Bound
App. 2.1 Recovery of the Black and Scholes Solution
App. 2.2 Mathematica Code for Chapter 2
App. 2.3 General Properties of Option Prices
3. The Volatility of Volatility Series Expansion
Assumptions
General Steps in the expansion
The Two Series for a Parameterized Model
App. 3.1 Details of the Volatility of Volatility Expansion
4. Mixing Solutions and Applications
The Basic Mixing Solution
Connection between Mixing Densities and the Fundamental Transform 101
A Monte Carlo Application
Arbitrary Payoff Functions
A More General Model without Correlation
5. The Smile
Introduction and Summary of Results
The Symmetric Case
The Correlated Case
Deducing the Risk-adjusted Volatility Process from Option Prices
App. 5.1 Calculating Volatility Moments
App. 5.2 Working with Differential Operators in Mathematica
App. 5.3 Additional Mathematica Code for Chapter 5
App. 5.4 Calculating with the Mixing Theorem
6. The Term Structure of Implied Volatility
Deterministic Volatility
Deterministic Volatility II: a Transform Perspective
Stochastic VolatilityÑThe Eigenvalue Connection
Example I: The Square Root Model
Example II: The 3/2 Model
Example III: The GARCH Diffusion Model
A Variational Principle Method
A Differential Equation (Dsolve) Method
App. 6.1 Mathematica Code for Chapter 6
7. Utility-based Equilibrium Models
A Representative Agent Economy
Examples
The Pure Investment Problem with a Distant Planning Horizon
Preference Adjustments to the Volatility of Volatility Series Expansion 240
The Effect of Risk Attitudes on Option Prices
8. Duality and Changes of Numeraire
Put-Call Duality
Introduction to the Change of Numeraire
Mathematics of the Change of Numeraire
Implications for the Term Structure
9. Volatility Explosions and the
Failure of the Martingale Pricing FormulaIntroduction
The Feller Boundary Classifications
Volatility Explosions I
Volatility Explosions II. Failure of the Martingale Pricing Formula
When Martingale Pricing Fails: Generalized Pricing Formulas
Generalized Pricing Formulas and the Transform-based Solutions
Generalized Pricing Formulas. Example I: the 3/2 Model
Generalized Pricing Formulas. Example II: the CEV Model
10. Option Prices at Large Volatility
Introduction
Asymptotica for the Fundamental Transform
11. Solutions to Models
The Square Root Model
The 3/2 Model
Geometric Brownian Motion
References
Index
Frequent Notations and Abbreviations
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