### Abstract

We study the finite horizon Merton portfolio optimization problem in a general local-stochastic volatility setting. Using model coefficient expan sion techniques, we derive approximations for both the value function and the optimal investment s trategy. We also analyze the "implied Sharpe ratio" and derive a series approximation for this quan tity. 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.

Original language | English (US) |
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Pages (from-to) | 418-447 |

Number of pages | 30 |

Journal | SIAM Journal on Financial Mathematics |

Volume | 7 |

Issue number | 1 |

DOIs | |

State | Published - 2016 |

### All Science Journal Classification (ASJC) codes

- Numerical Analysis
- Finance
- Applied Mathematics

### Keywords

- Local volatility
- Merton pr oblem
- Portfolio optimization
- Stochastic volatility