  We study the problem of active portfolio management where an investor aims to
outperform a benchmark strategy's risk profile while not deviating too far from
it. Specifically, an investor considers alternative strategies whose terminal
wealth lie within a Wasserstein ball surrounding a benchmark's -- being
distributionally close -- and that have a specified dependence/copula -- tying
state-by-state outcomes -- to it. The investor then chooses the alternative
strategy that minimises a distortion risk measure of terminal wealth. In a
general (complete) market model, we prove that an optimal dynamic strategy
exists and provide its characterisation through the notion of isotonic
projections.
  We further propose a simulation approach to calculate the optimal strategy's
terminal wealth, making our approach applicable to a wide range of market
models. Finally, we illustrate how investors with different copula and risk
preferences invest and improve upon the benchmark using the Tail Value-at-Risk,
inverse S-shaped, and lower- and upper-tail distortion risk measures as
examples. We find that investors' optimal terminal wealth distribution has
larger probability masses in regions that reduce their risk measure relative to
the benchmark while preserving the benchmark's structure.
