Humboldt-Universität zu Berlin - High Dimensional Nonstationary Time Series

IRTG1792DP2019 004

Constrained Kelly portfolios under alpha-stable laws

Niels Wesselhöfft
Wolfgang K. Härdle

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

growth-optimal, Kelly criterion, protective put, portfolio optimization, stable
distribution, Value at Risk

JEL Classification:
C13, C46, C61, C73, G11