The authors discuss extensions of the concept of normal inverse Gaussian processes [see, for example, O. E. Barndorff-Nielsen, Scand. J. Stat. 24, No. 1, 1-13 (1997; Zbl 0934.62109)] to the concept of normal modified stable processes. The family of normal inverse Gaussian distributions and the normal inverse Gaussian processes have been found to provide accurate modeling of a great variety of empirical data in the physical sciences and in financial econometrics. The wider class of normal generalized inverse Gaussian or generalized hyperbolic processes provides additional possibilities for modeling of data.
The generalization discussed is based on an extension of the family of generalized inverse Gaussian distributions to a class of distributions on \(\mathbb R^+=(0,\infty)\), the modified stable laws. These laws come about in the same way that the generalized inverse Gaussian laws are derived from the inverse Gaussian, namely by exponential and power tempering from one of the positive \(\kappa\)-stable laws. Using the modified stable distributions as mixing distributions for normal variance-mean mixtures yields the class of normal modified stable laws. In the special case of the tempered stable Ornstein-Uhlenbeck process an exact option pricing formula is found extending previous results based on the inverse Gaussian and gamma distributions.