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

TL;DR

This paper develops coefficient Taylor series approximations for portfolio optimization under local-stochastic volatility, providing explicit formulas for the value function, optimal strategy, and implied Sharpe ratio, with rigorous accuracy bounds and numerical validation.

Contribution

It introduces a systematic approximation method for the value function and strategies in local-stochastic volatility models, extending Merton's classical results.

Findings

First-order correction improves approximation accuracy.

Explicit formulas for implied Sharpe ratio approximations.

Numerical examples confirm the effectiveness of the method.

Abstract

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 the 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 (1969) 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. 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.