Monte Carlo and Finite Differences Methods with Applications to Finance

ECTS : 6

Volume horaire : 30

Description du contenu de l'enseignement :

 Chapter 1. Foundations of Monte-Carlo

• Principle of Monte Carlo Methods 
• Random Number Generation
• Inverse Transform Method
• Acceptance-Rejection Method
• Gaussian Distribution

Chapter 2. Variance Reduction Techniques

• Antithetic variable
• Control Variates
• Importance Sampling

Chapter 3. Simulation of Diffusion Processes

• Exact Simulation( Brownian Motion and Black–Scholes Model)
• Euler Scheme (Construction, Strong and week error)  

Chapter 4.  Brownian Bridge Approach

• Brownian Bridge
• Exit Times and Barrier Options (Naive approach and Brownian Bridge Approach)

Chapter 5.  Computation of Sensitivities (Greeks in finance)

• Finite Differences
• Black–Scholes Model
• Pathwise Differentiation 
• Malliavin Differentiation

Chapter 6.  American Options

• Discretization
• Naive Approach 
• Regression Methods

Chapter 7. Finite Difference Method for Linear PDE

• Construction ( Space Discretization, Time Discretization)
• Convergence ( Consistency, Stability, Convergence )

Chapter 8. Finite Difference Method for Non-Linear PDE 

• Non–Linear PDE
• The Linear Case Revisited 
• Variational Inequality 
• Hamilton–Jacobi–Bellman Equation 

Compétence à acquérir :

This course provides an in-depth presentation of the main techniques for the evaluating of options using Monte Carlo techniques.