Portfolio Optimization under Local-Stochastic Volatility: Coefficient Taylor Series Approximations and Implied Sharpe Ratio

We study the finite horizon Merton portfolio optimization problem in a general local-stochastic volatility setting. Using model coefficient expansion techniques, we derive approximations for both the value function and the optimal investment strategy. We also analyze the "implied Sharpe ratio" and derive a series approximation for this quantity. The zeroth order approximation of the value function and optimal investment strategy correspond to those obtained by [Merton, Rev. Econ. Statist., 51, pp. 247-257] when the risky asset follows a geometric Brownian motion. The first order correction of the value function can, for general utility functions, be expressed as a differential operator acting on the zeroth order term. For power utility functions, higher order terms can also be computed as a differential operator acting on the zeroth order term. While our approximations are derived formally, we give a rigorous accuracy bound for the higher order approximations in this case in pure stochastic volatility models. A number of examples are provided in order to demonstrate numerically the accuracy of our approximations.