Portfolio benchmarking under drawdown constraint and stochastic sharpe ratio

We consider an investor who seeks to maximize her expected utility of wealth relative to a benchmark, or target over a finite time horizon, and under a portfolio drawdown constraint, in a market with local stochastic volatility. We propose a new investor objective paradigm which allows the investor to target the portfolio benchmark while obeying the constraint, both of which can be characterized in terms of the running maximum wealth process. In the absence of closed-form formulas for the value function and optimal portfolio strategy in the incomplete market models we consider, 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 utility, compared to a constant volatility model, we illustrate that the investor must deploy a quite different portfolio strategy which depends on the current level of volatility.