Mathematical Properties of Option Pricing Under Gamma Distribution Process

A problem of risk neutral probability density function estimation for prices of risky assets is discussed when the asset pricing model uses exponential random process with independent increments. The structure of increments consists of two components: systematic drift and a random gamma distributed component. To determine the risk neutral probability density Escher transform is used. Obtained results help to define a formula for European call option pricing. Also we compare the found risk neutral density with the density, when the random component is defined by a Wiener process, driven by gamma process