  We consider an investor who seeks to maximize her expected utility derived
from her terminal wealth relative to the maximum performance achieved over a
fixed time horizon, and under a portfolio drawdown constraint, in a market with
local stochastic volatility (LSV). In the absence of closed-form formulas for
the value function and optimal portfolio strategy, we obtain approximations for
these quantities through the use of a coefficient expansion technique and
nonlinear transformations. We utilize regularity properties of the risk
tolerance function to numerically compute the estimates for our approximations.
In order to achieve similar value functions, we illustrate that, compared to a
constant volatility model, the investor must deploy a quite different portfolio
strategy which depends on the current level of volatility in the stochastic
volatility model.
