Optimal portfolios when stock prices follow an exponential Lévy process
The authors investigate portfolio optimization problems when the price processes are governed by general exponential Lévy processes, comparing the optimal solutions with variance and value-at-risk as risk measures, respectively. They introduce a multivariate Lévy Black-Scholes model and calculate the terminal wealth of a portfolio and its moments provided they exist. Then these results are used for a portfolio optimization that consists of maximizing the expected terminal wealth of a portfolio within a well-defined set of strategies under some constraint on the variance. The notion of capital-at-risk (CaR) is introduced, which is defined via a low quantile (value-at-risk) of the wealth process, and methods for its calculation and approximations are discussed. The portfolios, where the variance is replaced by CaR, are optimized. Real life examples as the normal inverse Gaussian and variance gamma model are worked out. Closed form analytic solutions are not obtained, but the optimization problem is solved by an approximation and numerical algorithm.
scientific article; zbMATH DE number 2118687