Quantitative Finance

5.

Numerical Methods in Finance

5.1.

Monte Carlo Simulation

5.1.1.

Random Path Generation

5.1.1.1.

Pseudorandom Number Generation

5.1.1.2.

Quasi-Monte Carlo Methods

5.1.1.3.

Low-Discrepancy Sequences

5.1.2.

Variance Reduction Techniques

5.1.2.1.

Antithetic Variates

5.1.2.2.

Control Variates

5.1.2.3.

Importance Sampling

5.1.2.4.

Stratified Sampling

5.1.3.

Path-Dependent Option Pricing

5.1.3.1.

Asian Option Simulation

5.1.3.2.

Barrier Option Simulation

5.1.3.3.

Lookback Option Simulation

5.1.4.

5.1.4.1.

Euler-Maruyama Scheme

5.1.4.2.

Discretization Methods

5.1.4.3.

Convergence Analysis

5.1.4.4.

Stability Analysis

5.2.

Finite Difference Methods

5.2.1.

Discretization Schemes

5.2.1.1.

Explicit Schemes

5.2.1.2.

Implicit Schemes

5.2.1.3.

Crank-Nicolson Schemes

5.2.1.4.

Stability Analysis

5.2.1.5.

Accuracy Analysis

5.2.2.

PDE Solving for Option Pricing

5.2.2.1.

Black-Scholes PDE

5.2.2.2.

American Option Pricing

5.2.2.3.

Early Exercise Features

5.2.3.

Boundary Conditions

5.2.3.1.

Dirichlet Conditions

5.2.3.2.

Neumann Conditions

5.2.3.3.

Free Boundary Problems

5.2.3.4.

Optimal Stopping Problems

5.3.

Fourier Transform Methods

5.3.1.

Fourier Transform Basics

5.3.1.1.

Discrete Fourier Transform

5.3.1.2.

Fast Fourier Transform

5.3.1.3.

Computational Efficiency

5.3.2.

Characteristic Functions in Pricing

5.3.2.1.

Inversion Techniques

5.3.2.2.

Lévy Process Applications

5.3.3.

FFT-Based Pricing

5.3.3.1.

Carr-Madan Method

5.3.3.2.

Implementation Techniques

5.3.3.3.

Computational Advantages

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