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Utility maximization under endogenous pricing

Thai Nguyen, Mitja Stadje

TL;DR

This work addresses utility maximization for a large investor in a market with endogenous permanent price impact by formulating the problem in terms of a $g$-expectation evaluated by a Market Maker. The core methodology develops coupled forward-backward stochastic differential equations ($FBSDEs$) and their connection to backward SPDEs ($BSPDEs$), enabling a rigorous characterization of optimal wealth and the value function under broad conditions, including quadratic and exponential utilities. Existence and regularity results are established for the coupled systems, with explicit treatment of complete markets and exponential utilities, and the framework is illustrated through concrete examples. The findings advance the theoretical understanding of liquidity risk and price impact in continuous time and pave the way for numerical schemes via BSDE/FBSDE methods, with potential implications for robust pricing and risk management in illiquid markets.

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.

Utility maximization under endogenous pricing

TL;DR

This work addresses utility maximization for a large investor in a market with endogenous permanent price impact by formulating the problem in terms of a -expectation evaluated by a Market Maker. The core methodology develops coupled forward-backward stochastic differential equations () and their connection to backward SPDEs (), enabling a rigorous characterization of optimal wealth and the value function under broad conditions, including quadratic and exponential utilities. Existence and regularity results are established for the coupled systems, with explicit treatment of complete markets and exponential utilities, and the framework is illustrated through concrete examples. The findings advance the theoretical understanding of liquidity risk and price impact in continuous time and pave the way for numerical schemes via BSDE/FBSDE methods, with potential implications for robust pricing and risk management in illiquid markets.

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.

Paper Structure

This paper contains 19 sections, 30 theorems, 155 equations.

Table of Contents

  1. Introduction
  2. The model setting
  3. The basic setting
  4. The evaluation method of the market maker and examples
  5. The pricing rule and the trading with permanent market impact
  6. Expected utility maximization
  7. An FBSDE approach
  8. Background: Subgradients for convex functions on convex or non-convex domains
  9. Necessary and sufficient conditions
  10. Examples
  11. Existence results for coupled FBSDEs
  12. Connection with BSPDEs
  13. Regularity of the value function
  14. Conclusion
  15. BMO's
  16. ...and 4 more sections

Key Result

Proposition 2.1

Under Assumption (B), there exists a $\mathcal{P} \otimes {\mathcal{B}} (\mathbb{R}^n)$- measurable, mapping $Z:\Omega\times [0,T]\times \mathbb{R}^n \mapsto \mathbb{R}^d$ such that ${\cal Z}^y_t(\omega)=Z(\omega,t,y)$$\mathrm{d} \mathbf{P}\times \mathrm{d} t$ almost surely for each fixed $y\in\math

Theorems & Definitions (43)

  • Example 1: $g$-expectation
  • Proposition 2.1
  • Proposition 2.2
  • Remark 2.3
  • Remark 2.4
  • Proposition 2.5
  • Proposition 3.1
  • Theorem 3.2
  • Remark 3.3
  • Theorem 3.4
  • ...and 33 more