Multi-objective Portfolio Optimization with Expected Shortfall under Fractal Brownian Motions

Traditionally, it is assumed that asset returns are typically i.i.d., and the heavy tails and volatility clustering phenomena observed in financial markets are usually ignored. We introduce the fractal Brownian motion (fBm) to describe the dynamic behavior of asset prices with long-range dependence and self-similarity of market prices more reasonably. We also employ the non-dominated sorting genetic algorithm II to solve multi-objective optimization problems by simultaneously optimizing portfolio returns, risks, and transaction costs. The empirical results indicate that by solving the multi-objective portfolio optimization problem by maximizing the mean return, maximizing the expected shortfall (expressed as under zero), and minimizing the transaction cost, we can show the advantage of fBm against multivariate Brownian motion. JEL Codes: G11