Generalised Lévy Processes and their Applications in Insurance and Finance

For insurance risks, jump processes such as homogeneous/non-homogeneous Poisson process and Cox process have been used . In financial modelling, it has been observed that diffusion models are not robust enough to capture the appearance of jumps in underlying asset prices and interest rates. As a result, generalised Lévy processes, which are simply speaking, the combinations of Poisson process and Brownian motion have gained their popularity for modelling in insurance and finance. In this paper, considering a generalised Lévy process, we obtain the explicit expression of the Laplace transform of the distribution of a generalised Lévy process assuming that jump size follows the mixture of two exponential distribution, which is a special case of phase-type distributions. We derive the mean and variance of the aggregate accumulated claim amounts of insurance risk providing their numerical examples. European call option pricing and non-defaultable zero-coupon bond pricing are examined for its application in financial risks. The calculation of call option prices is illustrated using Transform Analysis technique from the financial option pricing literature.