The Normal-Laplace Distribution and Its Relatives

Abstract

The normal-Laplace (NL) distribution results from convolving independent normally distributed and Laplace distributed components. It is the distribution of the stopped state of a Brownian motion with a normally distributed starting value if the stopping hazard rate is constant. Properties of the NL distribution discussed in the article include its shape and tail behaviour (fatter than the normal), its moments, and its infinite divisibility. The double Pareto-lognormal distribution is that of an exponentiated normal-Laplace random variable and provides a useful parametric form for modelling size distributions. The generalized normal-Laplace (GNL) distribution is both infinitely divisible and closed under summation. It is possible to construct a Lévy process whose increments follow the GNL distribution. Such a Lévy motion can be used to model the movement of the logarithmic price of a financial asset. An option pricing formula is derived for such an asset.

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