Simulating CIR and integrated CIR time change process. The stochastic volatility model simulation. 

The mean-correcting martingale measure - correcting the drift with m_new - for pricing options in exponential Lévy models. 

Pricing options in Lévy models by the Monte-Carlo method. 

Pricing exotic options by the Monte-Carlo method. Example with the up-ad-in Barrier option. 

_General approximation by using Poisson processes 

_Replacing very small jumps by a small Brownian motion

Methods for simulating:

Gamma Process, Variance-Gamma Process, Inverse Gaussian Process, Normal Inverse Gaussian Process, the CIR Process. 

Introduction to the simulation of Lévy processes. 

Simulation of the Brownian motion and of the Poisson Process. 

Numerical approximation of a general Lévy process by a sum of a deterministic drift function, a Brownian motion and a sum of Poisson processes. Replacement of the very small jumps by a Brownian motion component in the approximation. 

Lévy processes and martingales.

Cádlàg functions and processes. 

Definition of the jump process associated to a Lévy process.

The Poisson random measure. 

The Laplace exponent. Examples. 

Time change of Lévy processes using subordinators. Examples.

The Variance-Gamma process and the CGMY process.