Optimal investment and price dependence in a semi-static market

TL;DR

This paper investigates optimal investment strategies in a semi-static market with derivatives and stocks, analyzing how utility maximization depends on derivative prices, including stability and mathematical properties.

Contribution

It provides a comprehensive analysis of the existence, uniqueness, and sensitivity of utility maximization solutions in semi-static markets with derivative dependence.

Findings

Established conditions for existence and uniqueness of solutions.

Analyzed stability, differentiability, and monotonicity of utility with respect to derivative prices.

Explored limiting behavior of the utility maximization problem.

Abstract

This paper studies the problem of maximizing expected utility from terminal wealth in a semi-static market composed of derivative securities, which we assume can be traded only at time zero, and of stocks, which can be traded continuously in time and are modeled as locally-bounded semi-martingales. Using a general utility function defined on the positive real line, we first study existence and uniqueness of the solution, and then we consider the dependence of the outputs of the utility maximization problem on the price of the derivatives, investigating not only stability but also differentiability, monotonicity, convexity and limiting properties.