Asymptotic analysis of the expected utility maximization problem with respect to perturbations of the num\'eraire

TL;DR

This paper analyzes how small changes in the numéraire affect the expected utility maximization problem in incomplete markets, providing quadratic and first-order approximations of the value function and optimal strategies.

Contribution

It introduces a second-order asymptotic expansion framework for the utility maximization problem under numéraire perturbations, linking it to the return process characteristics.

Findings

Quadratic approximation of the value function.

First-order expansion of terminal wealth.

Construction of nearly optimal strategies matching the primal value up to second order.

Abstract

In an incomplete model, where under an appropriate num\'eraire, the stock price process is driven by a sigma-bounded semimartingale, we investigate the behavior of the expected utility maximization problem under small perturbations of the num\'eraire. We establish a quadratic approximation of the value function and a first-order expansion of the terminal wealth. Relying on a description of the base return process in terms of its semimartingale characteristics, we also construct wealth processes and nearly optimal strategies that allow for matching the primal value function up to the second order. We also link perturbations of the num\'eraire to distortions of the finite-variation part and martingale part of the stock price return and characterize the asymptotic expansions in terms of the risk-tolerance wealth process.