  We consider a portfolio optimisation problem for a utility-maximising
investor who faces convex constraints on his portfolio allocation in Heston's
stochastic volatility model. We apply the duality methods developed in previous
work to obtain a closed-form expression for the optimal portfolio allocation.
In doing so, we observe that allocation constraints impact the optimal
constrained portfolio allocation in a fundamentally different way in Heston's
stochastic volatility model than in the Black Scholes model. In particular, the
optimal constrained portfolio may be different from the naive capped portfolio,
which caps off the optimal unconstrained portfolio at the boundaries of the
constraints. Despite this difference, we illustrate by way of a numerical
analysis that in most realistic scenarios the capped portfolio leads to slim
annual wealth equivalent losses compared to the optimal constrained portfolio.
During a financial crisis, however, a capped solution might lead to compelling
annual wealth equivalent losses.
