# The Esscher Transform of the Densities of a Symmetric NIG Lévy Process

The Esscher Transform of the Densities of a Symmetric NIG Lévy Process

This Demonstration illustrates the effect of applying the Esscher transform to one-dimensional densities when the logarithm of the price process is the so-called Normal Inverse Gaussian (NIG) Lévy process. The distribution of this process is determined by four parameters , , , and : steepness, asymmetry, scale, and location. The location parameter is set to 0 (in other words, the process is centered), which makes the implementation of the Esscher transfer particularly simple (while at the same time being quite acceptable, on empirical grounds, for modeling stock prices). We display on the same graph the density of the logarithm of the price process at a given time , and the density of the process obtained by applying to it the Esscher transform. It is the latter density that is used in computing option prices.

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In addition to the three model parameters, you can vary the interest rate, the moment in time for which the density of the log of the price is shown, and the range of values over which the densities are shown. You can identify the graphs by hovering over them.