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Exponential Hedging for the Ornstein-Uhlenbeck Process in the Presence of Linear Price Impact

Yan Dolinsky

TL;DR

This work studies exponential utility maximization with a linear temporary price impact in a market where the risky asset follows an Ornstein–Uhlenbeck process. The authors adopt a purely probabilistic duality approach to derive a unique optimal hedging strategy in feedback form and an explicit value, with the optimal trading rate depending on the distance to the mean reversion level and the remaining time to maturity. Central to the analysis is a dual problem over a family of equivalent measures, whose unique minimizer yields the primal strategy via an Itô-representation, while auxiliary variational lemmas provide closed-form expressions for the necessary optimizations. The results illuminate how liquidity constraints shape optimal trading toward a mean-reverting target and converge to the frictionless benchmark as the market depth grows, with the horizon length enhancing the value of hedging.

Abstract

In this work we study a continuous time exponential utility maximization problem in the presence of a linear temporary price impact. More precisely, for the case where the risky asset is given by the Ornstein-Uhlenbeck diffusion process we compute the optimal portfolio strategy and the corresponding value. Our method of solution relies on duality, and it is purely probabilistic.

Exponential Hedging for the Ornstein-Uhlenbeck Process in the Presence of Linear Price Impact

TL;DR

This work studies exponential utility maximization with a linear temporary price impact in a market where the risky asset follows an Ornstein–Uhlenbeck process. The authors adopt a purely probabilistic duality approach to derive a unique optimal hedging strategy in feedback form and an explicit value, with the optimal trading rate depending on the distance to the mean reversion level and the remaining time to maturity. Central to the analysis is a dual problem over a family of equivalent measures, whose unique minimizer yields the primal strategy via an Itô-representation, while auxiliary variational lemmas provide closed-form expressions for the necessary optimizations. The results illuminate how liquidity constraints shape optimal trading toward a mean-reverting target and converge to the frictionless benchmark as the market depth grows, with the horizon length enhancing the value of hedging.

Abstract

In this work we study a continuous time exponential utility maximization problem in the presence of a linear temporary price impact. More precisely, for the case where the risky asset is given by the Ornstein-Uhlenbeck diffusion process we compute the optimal portfolio strategy and the corresponding value. Our method of solution relies on duality, and it is purely probabilistic.

Paper Structure

This paper contains 3 sections, 4 theorems, 37 equations.

Table of Contents

  1. Introduction and the Main Result
  2. Proof of Theorem \ref{['thm1.1']}
  3. Auxiliary Results

Key Result

Theorem 1.2

Set, The utility maximization problem problem has a unique optimal hedging strategy $\hat{\phi}_t$, $t\in [0,T)$ which is given by the feedback form where $\hat{\Phi}_t:=\int_{0}^t\hat{\phi}_s ds$, $t\in [0,T].$ The corresponding value is equal to

Theorems & Definitions (9)

  • Remark 1.1
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • proof
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof