An

-dimensional Lévy process

is a stochastic process with time-homogeneous independent increments, such that the sample paths of

are left-continuous with right limits (almost surely), and such that

. By the Lévy-Khinchin theorem, Lévy processes are in a one-to-one correspondence with Lévy triples

, where

is a vector,

is a positive definite matrix, and

is a positive measure on

. The matrix

is the volatility matrix of the Brownian motion component of the Lévy process, the Lévy measure

determines the jump structure of the process, and the vector

, sometimes called the "drift", depends on the choice of truncation function and is not an intrinsic parameter. In this Demonstration we concentrate on the Lévy measure of several well-known one-dimensional Lévy processes. The "jump diffusion" processes have nonzero Brownian component (

), while the pure jump processes have no Brownian component (

).

For a fixed

, you can consider jumps of absolute size less than

as "small jumps" and those of size larger than

as "large jumps". In this Demonstration, as in [1] and [6], we take

. The "total mass" of jumps of a Lévy process is given by

(where

is the density of

) and may be finite or infinite (it is finite for the jump diffusion processes presented here and infinite for the ones that are pure jump) but the quantities

and

are always finite. Thus to compare large jumps of two processes we can always use their mass, that is, the value of

, which we display under the graphs in this Demonstration. However, to compare small jumps, we use the mass

when it is finite (jump diffusions in this Demonstration) and

when the mass is infinite (the pure jump cases in this Demonstration). Note also that the Lévy measures of the Variance Gamma and the NIG process have a singularity at 0. The Lévy measure is not a probability measure. In fact, when the measure is finite (e.g., for jump diffusions), the total mass (i.e., the sum of the masses of small and large jumps) is equal to the jump density.

[1] R. Cont and P. Tankov,

*Financial Modelling with Jump Processes*, Boca Raton: CRC Press, 2004.

[2] O. E. Barndorff–Nielsen, "Normal Inverse Gaussian Distributions and Stochastic

Volatility Modelling,"

*Scandinavian Journal of Statistics ***24**(1), 1997 pp. 1–13.

[3] S. Kou, "A Jump-Diffusion Model for Option Pricing,"

*Management Science* **48**(8), 2002 pp. 1086–1101.

[4] D. B. Madan and E. Seneta, "The Variance Gamma Process (V.G.) Model for Share Market Returns,"

*Journal of Business* **63**(4), pp. 511–524.

[5] R. Merton, "Option Pricing When Underlying Stock Returns Are Discontinuous,"

*J. Financial Economics* **3**, 1976 pp. 125–144.

[6]. J. Bertoin,

*Lévy** Processes*, Cambridge, UK: Cambridge University Press: 1996.