In this thesis I study alternatives to the geometric Brownian motion where the log-return follow Lévy processes with jumps, in particular the normal invers Gaussian and the CGMY Lévy processes. With focus on Merton's portfolio management problem of deriving the maximum expected utility of consumption over an in finite time horizon I compare the derivation of the optimal portfolio, as well as its behavior, with the GBM and exponential Lévy processes as alternative stock price models. My focus is on model risk; the difference in risk/return due to the different models. As measures of riskI use Value-at-Risk (VaR) and conditional Value-at-Risk (cVaR). I find that the Lévy processes with jumps quickly converge to the normal distribution thus making the estimates of VaR and cVaR similar to the ones with the geometric Brownian motion.