The CGMYmodel Finance seminar by Mari Hodnekvam supervised by Prof.Korn

Introduction of the process; parameters, characteristic function, moments, distribution etc. Simulation of the process Application in finance Extended version Today

Introduction of the process; parameters, characteristic function, moments, distribution etc. Simulation of the process Application in finance Extended version Today

VG CGMY Relationship to VG

The parameter Y Y < 0Finite activity 0 ≤ Y ≤ 1Infinte activity, finite variation 1 ≤ Y < 2Infinte activity, infintite variation

Interpretation of parameters CMeasure of averall level of activity GMeasure of skewness M YMeasure of fine structure

Density of the CGMY-model

The characteristic function

Moments – variance = – skewness = – kurtosis =

Introduction of the process; parameters, characteristic function, moments, distribution etc. Simulation of the process Application in finance Extended version Today

Simulation of the CGMY-process Idea: treat the jumps as compound Poisson process and sample from its Lévy density Problem: for infintite activity Lévy processes the jump arrival rate is infinite

Simulation of the CGMY-process Divide the simulation into three parts: – Negative large jumps, x < -ε – Positive large jumps, x > ε – Small jumps, -ε < x < ε

Simulation of the CGMY-process The algorithm Simulate the number of positive and negative jumps in the time interval by a Poisson process Simulate the large jumps by using the acceptance-rejection method

Simulation of the CGMY-process Acceptance-rejection method: Find a function f(x) whose value is close to those of the Lévy density function for every x Draw samples from the probability distribution function of f(x); F(x) The samples are then either accepted or rejected, when you test them towards a restriction

Simulation of the CGMY-process The algorithm Simulate the number of positive and negative jumps in the time interval by a Poisson process Simulate the large jumps by using the acceptance- rejection method Simulate the small jumps by

Simulation of the CGMY-process The algorithm Simulate the number of positive and negative jumps in the time interval by a poisson process Simulate the large jumps by using the acceptance-rejection method Simulate the small jumps by Return the simulated jumps

Simulation of the CGMY-process

Introduction of the process; parameters, characteristic function, moments, distribution etc. Simulation of the process Application in finance Extended version Today

The CGMY stock price process – stock price process: – extended stock price process: – extended CGMY model:

Diffusion term

Density fit

Introduction of the process; parameters, characteristic function, moments, distribution etc. Simulation of the process Application in finance Extended version Today

Summary Pure jump process Parameter Y Kurtosis, skewness Time change, volatility clustering