Constrained Kelly portfolios under alpha-stable laws

This paper provides a detailed framework for modeling portfolios, achieving the highest growth rate under subjective risk constraints such as Value at Risk (VaR) in the presence of stable laws. Although the maximization of the expected logarithm of wealth induces outperforming any other significantly different strategy, the Kelly Criterion implies larger bets than a risk-averse investor would accept. Restricting the Kelly optimization by spectral risk measures, the authors provide a generalized mapping for different measures of growth and security. Analyzing over 30 years of S&P 500 returns for different sampling frequencies, the authors find evidence for leptokurtic behavior for all respective sampling frequencies. Given that lower sampling frequencies imply a smaller number of data points, this paper argues in favor of α-stable laws and its scaling behavior to model financial market returns for a given horizon in an i.i.d. world. Instead of simulating from the class of elliptically stable distributions, a nonparametric scaling approximation, based on the data-set itself, is proposed. Our paper also uncovers that including long put options into the portfolio optimization, improves the growth criterion for a given security level, leading to a new Kelly portfolio providing the highest geometric mean.