Merton and Kou use compound Poisson jumps — finitely many per unit time. But the empirical return distribution has a kurtosis and tail structure that can be better matched by models with infinitely many small jumps per unit time: the infinite-activity Lévy models. These models have Lévy measures with infinite total mass near the origin ( for all ) but still have finite quadratic variation.
Three models are standard: Variance Gamma (VG) (Madan-Seneta, 1990), Normal Inverse Gaussian (NIG) (Barndorff-Nielsen, 1998), and CGMY (Carr-Geman-Madan-Yor, 2002). Each is characterised by its Lévy triple and calibrates to the volatility surface better than a pure diffusion — at the cost of losing the Merton series formula, but gained by the Fourier pricing approach (covered in the next lesson).
Learning objectives. By the end of this note you should be able to:
- explain what "infinite activity" means and why it is consistent with finite quadratic variation;
- state the VG process as a time-changed Brownian motion and compute its Lévy measure;
- describe the NIG process and the CGMY tail-index parameter ;
- compute the characteristic function of the log-stock under each model.
In this lesson
- 5.1 Infinite activity vs infinite variation
- 5.2 The Variance Gamma (VG) model
- 5.3 The Normal Inverse Gaussian (NIG) model
- 5.4 The CGMY model
Before you start
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