The CGMYmodel

Finance seminar by Mari Hodnekvamsupervised 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 < 0 Finite activity

0 ≤ Y ≤ 1 Infinte activity, finite variation

1 ≤ Y < 2 Infinte activity, infintite variation

Interpretation of parameters

C Measure of averall level of activity

G Measure of skewness

M Measure of skewness

Y Measure 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

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