Chapter 1 Mathematical Preliminaries.
Complex Numbers.

Finding Roots of Functions.

OLS and WLS.

Nelder-Mead Algorithm.

Maximum Likelihood Estimation.

Cubic Spline Interpolation.

Exercises and Solutions.

Chapter 2 Numerical Integration.

Newton-Coates Formulas.

Implementing Newton-Coates Formulas in VBA.

Gaussian Quadratures.

Exercises and Solutions.

Chapter 3 Tree-Based Methods.

CRR Binomial Tree.

Leisen-Reimer Binomial Tree.

Edgeworth Binomial Tree.

Flexible Binomial Tree.

Trinomial Tree.

Adaptive Mesh Method.

Comparing Trees.

Implied Volatility Trees.

Allowing for Dividends and the Cost of Carry.

Exercises and Solutions.

Chapter 4 The Black-Scholes, Practitioner Black-Scholes, and Gram-Charlier Models.

Black-Scholes Model.

Implied Volatility and the DVF.

Practitioner Black-Scholes Model.

Gram-Charlier Model.

Exercises and Solutions.

Chapter 5 The Heston Stochastic Volatility Model.

Heston (1993) Model.

Increasing Integration Accuracy.

The Fundamental Tranform.

Sensitivity Analysis.

Exercises and Solutions.

Chapter 6 The Heston and Nandi GARCH Model.

Persistent Volatility in Asset Returns.

GARCH Variance Modeling.

Heston and Nandi (2000) Model.

Exercises and Solutions.

Chapter 7 The Greeks.

Black-Scholes Greeks.

Greeks From the Trees.

Greeks From the Gram-Charlier Model.

Greeks From the Heston (1993) Model.

Greeks From the Heston and Nandi (2000) Model.

Greeks by Finite Differences.

Exercises and Solutions.

Chapter 8 Exotic Options.

Single-Barrier Options.

Digital Options.

Asian Options.

Floating-Strike Lookback Options.

Exercises and Solutions.

Chapter 9 Parameter Estimation.

Unconditional Moments.

Maximum Likelihood for GARCH Models.

Estimation by Loss Functions.

Exercises and Solutions.

Chapter 10 Implied Volatility.

Obtaining Implied Volatility.

Explaining Smiles and Smirks

Fitting the Smile with the Heston (1993) Model.

Exercises and Solutions.

Chapter 11 Model-Free Implied Volatility.

Theoretical Foundation.

Implementation.

Interpolation-Extrapolation Method.

Model-Free Implied Forward Volatility.

The VIX Index.

Exercises and Solutions.

Chapter 12 Model-Free Higher Moments.

Theoretical Foundation.

Implementation.

Verifying Implied Moments.

Gram-Charlier Implied Moments.

Exercises and Solutions.

Chapter 13 Volatility Returns.

Straddle Returns.

Delta-Hedged Gains.

Volatility Exposure.

Variance Swaps.

Exercises and Solutions.

Appendix VBA Primer.