  We introduce a robust variant of the Kelly portfolio optimization model,
called the Wasserstein-Kelly portfolio optimization. Our model, taking a
Wasserstein distributionally robust optimization (DRO) formulation, addresses
the fundamental issue of estimation error in Kelly portfolio optimization by
defining a ``ball" of distributions close to the empirical return distribution
using the Wasserstein metric and seeking a robust log-optimal portfolio against
the worst-case distribution from the Wasserstein ball. Enhancing the Kelly
portfolio using Wasserstein DRO is a natural step to take, given many
successful applications of the latter in areas such as machine learning for
generating robust data-driven solutions. However, naive application of
Wasserstein DRO to the growth-optimal portfolio problem can lead to several
issues, which we resolve through careful modelling. Our proposed model is both
practically motivated and efficiently solvable as a convex program. Using
empirical financial data, our numerical study demonstrates that the
Wasserstein-Kelly portfolio can outperform the Kelly portfolio in out-of-sample
testing across multiple performance metrics and exhibits greater stability.
