Utility maximization under endogenous pricing

TL;DR

This paper models a large investor’s utility maximization in an incomplete market with endogenous prices, characterized by coupled FBSDEs and BSPDEs, providing existence and smoothness results under various conditions.

Contribution

It introduces a novel framework using coupled FBSDEs and BSPDEs to characterize optimal strategies with endogenous market impacts, extending existing models.

Findings

Existence of solutions under quadratic growth conditions.

Smoothness results for solutions of BSPDEs.

Examples include complete markets and exponential utility.

Abstract

We study the expected utility maximization problem of a large investor who is allowed to make transactions on tradable assets in an incomplete financial market with endogenous permanent market impacts. The asset prices are assumed to follow a nonlinear price curve quoted in the market as the utility indifference curve of a representative liquidity supplier. Using generalized subgradients, we show that optimality can be fully characterized via a system of coupled forward-backward stochastic differential equations (FBSDEs) which corresponds to a non-linear backward stochastic partial differential equation (BSPDE). We show existence of solutions to the optimal investment problem and the FBSDEs in the case where the driver function of the representative market maker grows at least quadratically or the utility function of the large investor falls faster than quadratically or is exponential. Furthermore, we derive smoothness results for the existence of solutions of BSPDEs. Examples are provided when the market is complete, the driver is positively homogeneous or the utility function is exponential.