Statistical Methods for Financial Engineering

Black-Scholes Model
The Black-Scholes Model
Dynamic Model for an Asset
Estimation of Parameters
Estimation Errors
Black-Scholes Formula
Greeks
Estimation of Greeks using the Broadie-Glasserman Methodologies

Multivariate Black-Scholes Model
Black-Scholes Model for Several Assets
Estimation of Parameters
Estimation Errors
Evaluation of Options on Several Assets
Greeks

Discussion of the Black-Scholes Model
Critiques of the Model
Some Extensions of the Black-Scholes Model
Discrete Time Hedging
Optimal Quadratic Mean Hedging

Measures of Risk and Performance
Measures of Risk
Estimation of Measures of Risk by Monte Carlo Methods
Measures of Risk and the Delta-Gamma Approximation
Performance Measures

Modeling Interest Rates
Introduction
Vasicek Model
Cox-Ingersoll-Ross (CIR) Model
Other Models for the Spot Rates

Lévy Models
Complete Models
Stochastic Processes with Jumps
Lévy Processes
Examples of Lévy Processes
Change of Distribution
Model Implementation and Estimation of Parameters

Stochastic Volatility Models
GARCH Models
Estimation of Parameters
Duan Methodology of Option Pricing
Stochastic Volatility Model of Hull-White
Stochastic Volatility Model of Heston

Copulas and Applications
Weak Replication of Hedge Funds
Default Risk
Modeling Dependence
Bivariate Copulas
Measures of Dependence
Multivariate Copulas
Families of Copulas
Estimation of the Parameters of Copula Models
Tests of Independence
Tests of Goodness-of-Fit
Example of Implementation of a Copula Model

Filtering
Description of the Filtering Problem
Kalman Filter
IMM Filter
General Filtering Problem
Computation of the Conditional Densities
Particle Filters

Applications of Filtering
Estimation of ARMA Models
Regime-Switching Markov Models
Replication of Hedge Funds

Appendix A: Probability Distributions
Appendix B: Estimation of Parameters

Index