Financial Portfolio Selection in a Nonstationary Gaussian Framework

We introduce the selection of financial portfolios in a nonstationary Gaussian framework that assumes the price process to be modelled by a multifractional Brownian motion (mBm). This process captures the time-changing regularity of the sample paths as a result of the impact of the new information on markets. The key variable is the pointwise Holder exponent, H(t), which summarizes the level of regularity at a given point along the trajectories of the process. Therefore, the exponent H(t) can be viewed as a local (instantaneous) indicator of risk. By the exponents of the individual assets, we derive in closed form the pointwise Hholder exponent of a portfolio and stress the analogies with the classical Markowitz result. Furthermore, we compare the composition of the efficient frontier defined using the new risk measure with respect to Markowitz's one, obtained in the last quarter of the year 2008, a period characterized by a deep financial crisis and unusual movements for the stock prices.