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The Heston Model and its Extensions in Matlab and C#, + Website

Book Description

Tap into the power of the most popular stochastic volatility model for pricing equity derivatives

Since its introduction in 1993, the Heston model has become a popular model for pricing equity derivatives, and the most popular stochastic volatility model in financial engineering. This vital resource provides a thorough derivation of the original model, and includes the most important extensions and refinements that have allowed the model to produce option prices that are more accurate and volatility surfaces that better reflect market conditions. The book's material is drawn from research papers and many of the models covered and the computer codes are unavailable from other sources.

The book is light on theory and instead highlights the implementation of the models. All of the models found here have been coded in Matlab and C#. This reliable resource offers an understanding of how the original model was derived from Ricatti equations, and shows how to implement implied and local volatility, Fourier methods applied to the model, numerical integration schemes, parameter estimation, simulation schemes, American options, the Heston model with time-dependent parameters, finite difference methods for the Heston PDE, the Greeks, and the double Heston model.

  • A groundbreaking book dedicated to the exploration of the Heston model—a popular model for pricing equity derivatives

  • Includes a companion website, which explores the Heston model and its extensions all coded in Matlab and C#

  • Written by Fabrice Douglas Rouah a quantitative analyst who specializes in financial modeling for derivatives for pricing and risk management

Engaging and informative, this is the first book to deal exclusively with the Heston Model and includes code in Matlab and C# for pricing under the model, as well as code for parameter estimation, simulation, finite difference methods, American options, and more.

Note: The ebook version does not provide access to the companion files.

Table of Contents

  1. Coverpage
  2. Half Title Page
  3. Authors Page
  4. Title Page
  5. Copyright
  6. Contents
  7. Foreword
  8. Preface
  9. Acknowledgments
  10. 1 The Heston Model for European Options
    1. Model Dynamics
    2. The European Call Price
    3. The Heston PDE
    4. Obtaining the Heston Characteristic Functions
    5. Solving the Heston Riccati Equation
    6. Dividend Yield and the Put Price
    7. Consolidating the Integrals
    8. Black-Scholes as a Special Case
    9. Summary of the Call Price
    10. Conclusion
  11. 2 Integration Issues, Parameter Effects, and Variance Modeling
    1. Remarks on the Characteristic Functions
    2. Problems With the Integrand
    3. The Little Heston Trap
    4. Effect of the Heston Parameters
    5. Variance Modeling in the Heston Model
    6. Moment Explosions
    7. Bounds on Implied Volatility Slope
    8. Conclusion
  12. 3 Derivations Using the Fourier Transform
    1. The Fourier Transform
    2. Recovery of Probabilities With Gil-Pelaez Fourier Inversion
    3. Derivation of Gatheral (2006)
    4. Attari (2004) Representation
    5. Carr and Madan (1999) Representation
    6. Bounds on the Carr-Madan Damping Factor and Optimal Value
    7. The Carr-Madan Representation for Puts
    8. The Representation for OTM Options
    9. Conclusion
  13. 4 The Fundamental Transform for Pricing Options
    1. The Payoff Transform
    2. The Fundamental Transform and the Option Price
    3. The Fundamental Transform for the Heston Model
    4. Option Prices Using Parseval’s Identity
    5. Volatility of Volatility Series Expansion
    6. Conclusion
  14. 5 Numerical Integration Schemes
    1. The Integrand in Numerical Integration
    2. Newton-Cotes Formulas
    3. Gaussian Quadrature
    4. Integration Limits and Kahl and Jackel Transformation
    5. Illustration of Numerical Integration
    6. Fast Fourier Transform
    7. Fractional Fast Fourier Transform
    8. Conclusion
  15. 6 Parameter Estimation
    1. Estimation Using Loss Functions
    2. Speeding up the Estimation
    3. Differential Evolution
    4. Maximum Likelihood Estimation
    5. Risk-Neutral Density and Arbitrage-Free Volatility Surface
    6. Conclusion
  16. 7 Simulation in the Heston Model
    1. General Setup
    2. Euler Scheme
    3. Milstein Scheme
    4. Milstein Scheme for the Heston Model
    5. Implicit Milstein Scheme
    6. Transformed Volatility Scheme
    7. Balanced, Pathwise, and IJK Schemes
    8. Quadratic-Exponential Scheme
    9. Alfonsi Scheme for the Variance
    10. Moment Matching Scheme
    11. Conclusion
  17. 8 American Options
    1. Least-Squares Monte Carlo
    2. The Explicit Method
    3. Beliaeva-Nawalkha Bivariate Tree
    4. Medvedev-Scaillet Expansion
    5. Chiarella and Ziogas American Call
    6. Conclusion
  18. 9 Time-Dependent Heston Models
    1. Least-Squares Monte Carlo
    2. The Explicit Method
    3. Beliaeva-Nawalkha Bivariate Tree
    4. Medvedev-Scaillet Expansion
    5. Chiarella and Ziogas American Call
    6. Conclusion
  19. 10 Time-Dependent Heston Models
    1. Generalization of the Riccati Equation
    2. Bivariate Characteristic Function
    3. Linking the Bivariate CF and the General Riccati Equation
    4. Mikhailov and Nogel Model
    5. Elices Model
    6. Benhamou-Miri-Gobet Model
    7. Black-Scholes Derivatives
    8. Conclusion
  20. 11 The Heston Greeks
    1. Analytic Expressions for European Greeks
    2. Finite Differences for the Greeks
    3. Numerical Implementation of the Greeks
    4. Greeks Under the Attari and Carr-Madan Formulations
    5. Greeks Under the Lewis Formulations
    6. Greeks Using the FFT and FRFT
    7. American Greeks Using Simulation
    8. American Greeks Using the Explicit Method
    9. American Greeks from Medvedev and Scaillet
    10. Conclusion
  21. 12 The Double Heston Model
    1. Multi-Dimensional Feynman-KAC Theorem
    2. Double Heston Call Price
    3. Double Heston Greeks
    4. Parameter Estimation
    5. Simulation in the Double Heston Model
    6. American Options in the Double Heston Model
  22. Bibliography
  23. About the Website
  24. Index