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The CGMYmodel

The CGMYmodel. Finance seminar by Mari Hodnekvam supervised by Prof.Korn. Today . Introduction of the process; parameters, characteristic function, moments, distribution etc. Simulation of the process Application in finance Extended version. Today .

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The CGMYmodel

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

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

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

  4. VG CGMY Relationship to VG

  5. The parameter Y

  6. Interpretation of parameters

  7. Density of the CGMY-model

  8. The characteristic function

  9. Moments • variance = • skewness = • kurtosis =

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

  11. 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

  12. 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

  13. 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

  14. 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

  15. 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

  16. 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

  17. Simulation of the CGMY-process

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

  19. The CGMY stock price process • stock price process: • extended stock price process: • extended CGMY model:

  20. Diffusion term

  21. Density fit

  22. Density fit

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

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

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