| |

| |

Preface | |

| |

| |

| |

An Introduction to Probability Theory | |

| |

| |

| |

The Notion of a Set and a Sample Space | |

| |

| |

| |

Sigma Algebras or Field | |

| |

| |

| |

Probability Measure and Probability Space | |

| |

| |

| |

Measurable Mapping | |

| |

| |

| |

Cumulative Distribution Functions | |

| |

| |

| |

Convergence in Distribution | |

| |

| |

| |

Random Variables | |

| |

| |

| |

Discrete Random Variables | |

| |

| |

| |

Example of Discrete Random Variables: The Binomial Distribution | |

| |

| |

| |

Hypergeometric Distribution | |

| |

| |

| |

Poisson Distribution | |

| |

| |

| |

Continuous Random Variables | |

| |

| |

| |

Uniform Distribution | |

| |

| |

| |

The Normal Distribution | |

| |

| |

| |

Change of Variable | |

| |

| |

| |

Exponential Distribution | |

| |

| |

| |

Gamma Distribution | |

| |

| |

| |

Measurable Function | |

| |

| |

| |

Cumulative Distribution Function and Probability Density Function | |

| |

| |

| |

Joint, Conditional and Marginal Distributions | |

| |

| |

| |

Expected Values of Random Variables and Moments of a Distribution | |

| |

| |

| |

The Bernoulli Law of Large Numbers | |

| |

| |

| |

Conditional Expectations | |

| |

| |

| |

Stochastic Processes | |

| |

| |

| |

Stochastic Processes | |

| |

| |

| |

Martingales Processes | |

| |

| |

| |

Brownian Motions | |

| |

| |

| |

Brownian Motion and the Reflection Principle | |

| |

| |

| |

Geometric Brownian Motions | |

| |

| |

| |

An Application of Brownian Motions | |

| |

| |

| |

Ito Calculus and Ito Integral | |

| |

| |

| |

Total Variation and Quadratic Variation of Differentiable Functions | |

| |

| |

| |

Quadratic Variation of Brownian Motions | |

| |

| |

| |

The Construction of the Ito Integral | |

| |

| |

| |

Properties of the Ito Integral | |

| |

| |

| |

The General Ito Stochastic Integral | |

| |

| |

| |

Properties of the General Ito Integral | |

| |

| |

| |

Construction of the Ito Integral with Respect to Semi-Martingale Integrators | |

| |

| |

| |

Quadratic Variation of a General Bounded Martingale | |

| |

| |

| |

Quadratic Variation of a General Bounded Martingale | |

| |

| |

| |

Ito Lemma and Ito Formula | |

| |

| |

| |

The Riemann-Stieljes Integral | |

| |

| |

| |

The Black and Scholes Economy | |

| |

| |

| |

Introduction | |

| |

| |

| |

Trading Strategies and Martingale Processes | |

| |

| |

| |

The Fundamental Theorem of Asset Pricing | |

| |

| |

| |

Martingale Measures | |

| |

| |

| |

Girsanov Theorem | |

| |

| |

| |

Risk-Neutral Measures | |

| |

| |

| |

The Randon-Nikodym Condition | |

| |

| |

| |

Geometric Brownian Motion | |

| |

| |

| |

The Black and Scholes Model | |

| |

| |

| |

Introduction | |

| |

| |

| |

The Black and Scholes Model | |

| |

| |

| |

The Black and Scholes Formula | |

| |

| |

| |

Black and Scholes in Practice | |

| |

| |

| |

The Feynman-Kac Formula | |

| |

| |

| |

The Kolmogorov Backward Equation | |

| |

| |

| |

Change of Numeraire | |

| |

| |

| |

Black and Scholes and the Greeks | |

| |

| |

| |

Monte Carlo Methods | |

| |

| |

| |

Introduction | |

| |

| |

| |

The Data Generating Process (DGP) and the Model | |

| |

| |

| |

Pricing European Options | |

| |

| |

| |

Variance Reduction Techniques | |

| |

| |

| |

Antithetic Variate Methods | |

| |

| |

| |

Control Variate Methods | |

| |

| |

| |

Common Random Numbers | |

| |

| |

| |

Importance Sampling | |

| |

| |

| |

Monte Carlo European Options | |

| |

| |

| |

Variance Reduction Techniques - First Part | |

| |

| |

| |

Monte Carlo 1 | |

| |

| |

| |

Monte Carlo 2 | |

| |

| |

| |

Monte Carlo Methods and American Options | |

| |

| |

| |

Introduction | |

| |

| |

| |

Pricing American Options | |

| |

| |

| |

Dynamic Programming Approach and American Option Pricing | |

| |

| |

| |

The Longstaff and Schwartz Least Squares Method | |

| |

| |

| |

The Glasserman and Yu Regression Later Method | |

| |

| |

| |

Upper and Lower Bounds and American Options | |

| |

| |

| |

Multiassets Simulation | |

| |

| |

| |

Pricing a Basket Option Using the Regression Methods | |

| |

| |

| |

American Option Pricing: The Dual Approach | |

| |

| |

| |

Introduction | |

| |

| |

| |

A General Framework for American Option Pricing | |

| |

| |

| |

A Simple Approach to Designing Optimal Martingales | |

| |

| |

| |

Optimal Martingales and American Option Pricing | |

| |

| |

| |

A Simple Algorithm for American Option Pricing | |

| |

| |

| |

Empirical Results | |

| |

| |

| |

Computing Upper Bounds | |

| |

| |

| |

Empirical Results | |

| |

| |

| |

| |

| |

| |

Estimation of Greeks using Monte Carlo Methods | |

| |

| |

| |

Finite Difference Approximations | |

| |

| |

| |

Pathwise Derivatives Estimation | |

| |

| |

| |

Likelihood Ratio Method | |

| |

| |

| |

Discussion | |

| |

| |

| |

Pathwise Greeks using Monte Carlo | |

| |

| |

| |

Exotic Options | |

| |

| |

| |

Introduction | |

| |

| |

| |

Digital Options | |

| |

| |

| |

Asian Options | |

| |

| |

| |

Forward Start Options | |

| |

| |

| |

Barrier Options | |

| |

| |

| |

Hedging Barrier Options | |

| |

| |

| |

Digital Options | |

| |

| |

| |

Pricing and Hedging Exotic Options | |

| |

| |

| |

Introduction | |

| |

| |

| |

Monte Carlo Simulations and Asian Options | |

| |

| |

| |

Simulation of Greeks for Exotic Options | |

| |

| |

| |

Monte Carlo Simulations and Forward Start Options | |

| |

| |

| |

Simulation of the Greeks for Exotic Options | |

| |

| |

| |

Monte Carlo Simulations and Barrier Options | |

| |

| |

| |

The Price and the Delta of Forward Start Options | |

| |

| |

| |

The Price of Barrier Options Using Importance Sampling | |

| |

| |

| |

Stochastic Volatility Models | |

| |

| |

| |

Introduction | |

| |

| |

| |

The Model | |

| |

| |

| |

Square Root Diffusion Process | |

| |

| |

| |

The Heston Stochastic Volatility Model (HSVM) | |

| |

| |

| |

Processes with Jumps | |

| |

| |

| |

Application of the Euler Method to Solve SDEs | |

| |

| |

| |

Exact Simulation Under SV | |

| |

| |

| |

Exact Simulation of Greeks Under SV | |

| |

| |

| |

Stochastic Volatility Using the Heston Model | |

| |

| |

| |

Implied Volatility Models | |

| |

| |

| |

Introduction | |

| |

| |

| |

Modelling Implied Volatility | |

| |

| |

| |

Examples | |

| |

| |

| |

Local Volatility Models | |

| |

| |

| |

An Overview | |

| |

| |

| |

The Model | |

| |

| |

| |

Numerical Methods | |

| |

| |

| |

| |

| |

| |

An Introduction to Interest Rate Modelling | |

| |

| |

| |

A General Framework | |

| |

| |

| |

Affine Models (AMs) | |

| |

| |

| |

The Vasicek Model | |

| |

| |

| |

The Cox, Ingersoll and Ross (CIR) Model | |

| |

| |

| |

The Hull and White (HW) Model | |

| |

| |

| |

The Black Formula and Bond Options | |

| |

| |

| |

Interest Rate Modelling | |

| |

| |

| |

Some Preliminary Definitions | |

| |

| |

| |

Interest Rate Caplets and Floorlets | |

| |

| |

| |

Forward Rates and Numeraire | |

| |

| |

| |

Libor Futures Contracts | |

| |

| |

| |

Martingale Measure | |

| |

| |

| |

Binomial and Finite Difference Methods | |

| |

| |

| |

The Binomial Model | |

| |

| |

| |

Expected Value and Variance in the Black and Scholes and Binomial Models | |

| |

| |

| |

The Cox-Ross-Rubinstein Model | |

| |

| |

| |

Finite Difference Methods | |

| |

| |

| |

The Binomial Method | |

| |

| |

| |

An Introduction to MATLAB | |

| |

| |

| |

What is MATLAB? | |

| |

| |

| |

Starting MATLAB | |

| |

| |

| |

Main Operations in MATLAB | |

| |

| |

| |

Vectors and Matrices | |

| |

| |

| |

Basic Matrix Operations | |

| |

| |

| |

Linear Algebra | |

| |

| |

| |

Basics of Polynomial Evaluations | |

| |

| |

| |

Graphing in MATLAB | |

| |

| |

| |

Several Graphs on One Plot | |

| |

| |

| |

Programming in MATLAB: Basic Loops | |

| |

| |

| |

M-File Functions | |

| |

| |

| |

MATLAB Applications in Risk Management | |

| |

| |

| |

MATLAB Programming: Application in Financial Economics | |

| |

| |

| |

Mortgage Backed Securities | |

| |

| |

| |

Introduction | |

| |

| |

| |

The Mortgage Industry | |

| |

| |

| |

The Mortgage Backed Security (MBS) Model | |

| |

| |

| |

The Term Structure Model | |

| |

| |

| |

Preliminary Numerical Example | |

| |

| |

| |

Dynamic Option Adjusted Spread | |

| |

| |

| |

Numerical Example | |

| |

| |

| |

Practical Numerical Examples | |

| |

| |

| |

Empirical Results | |

| |

| |

| |

The Pre-Payment Model | |

| |

| |

| |

Value at Risk | |

| |

| |

| |

Introduction | |

| |

| |

| |

Value at Risk (VaR) | |

| |

| |

| |

The Main Parameters of a VaR | |

| |

| |

| |

VaR Methodology | |

| |

| |

| |

Historical Simulations | |

| |

| |

| |

Variance-Covariance Method | |

| |

| |

| |

Monte Carlo Method | |

| |

| |

| |

Empirical Applications | |

| |

| |

| |

Historical Simulations | |

| |

| |

| |

Variance-Covariance Method | |

| |

| |

| |

Fat Tails and VaR | |

| |

| |

| |

Generalized Extreme Value and the Pareto Distribution | |

| |

| |

Bibliography | |

| |

| |

References | |

| |

| |

Index | |