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

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

Problem Formulation
This work addresses a continuous-time exponential utility maximization problem within a financial market subject to linear temporary price impact. The market consists of a risk-free asset with zero interest rate and a risky asset $S_t$ following an Ornstein-Uhlenbeck (OU) diffusion process:
$$dS_t = (\mu - S_t) dt + dW_t$$
where $\mu$ is the long-term mean and $W_t$ is a standard Brownian motion.

The model incorporates linear price impact, following the framework of [1]. When an investor trades at a rate $\phi_t = \dot{\Phi}_t$ (where $\Phi_t$ is the number of shares held), the execution price is adjusted to $S_t + \frac{\phi_t}{2\delta}$, where $\delta > 0$ represents the market depth. The resulting terminal wealth $V_T^{\Phi_0, \phi}$, assuming the investor marks to market at time $T$, is given by:
$$V_T^{\Phi_0, \phi} = \Phi_0(S_T - S_0) + \int_0^T \phi_t(S_T - S_t)dt - \frac{1}{2\delta}\int_0^T \phi_t^2 dt$$
The objective is to maximize the expected exponential utility $E_P[-\exp(-\alpha V_T^{\Phi_0, \phi})]$ over admissible strategies $\phi$. Without loss of generality, the risk aversion parameter is normalized to $\alpha=1$, and the initial holding is set to $\Phi_0 = 0$.

Methodology
The solution relies on a purely probabilistic approach utilizing duality theory. The author employs the duality result for exponential hedging with quadratic transaction costs established in Proposition A.2 of [5].

The proof proceeds through the following steps:

1. Change of Measure: The process $B_t = S_t - S_0$ is transformed. Under a new probability measure $\hat{P}$, $B_t$ behaves as a standard Brownian motion. The Radon-Nikodym derivative relating the physical measure $P$ to $\hat{P}$ is explicitly derived.
2. Dual Problem Formulation: The primal maximization problem is converted into a dual minimization problem over the set of equivalent probability measures $\mathcal{Q}$ with finite relative entropy. The dual objective functional involves the relative entropy and a term representing the quadratic cost of trading.
3. Reduction to Deterministic Variational Problems: By analyzing the structure of the dual objective, the author demonstrates that the infimum can be realized by measures whose densities are driven by deterministic functions. This reduces the stochastic control problem to a deterministic variational problem involving ordinary differential equations (ODEs).
4. Explicit Solution: The deterministic problem is solved using calculus of variations (specifically, the Euler-Lagrange equations), yielding explicit formulas for the optimal control and the value function.

Key Results
The paper provides a closed-form solution for the optimal hedging strategy and the corresponding value function.

• Optimal Strategy: The unique optimal trading rate $\hat{\phi}_t$ is given in a feedback form dependent on the current stock price $S_t$, the time to maturity $T-t$, and the market depth $\delta$. The strategy is a mean-reverting process towards a target process $X_t$, which is linear in the deviation $(\mu - S_t)$. The feedback coefficients involve hyperbolic functions (sinh, cosh, tanh) of $\sqrt{1+\delta}(T-t)$.
• Value Function: The maximal expected utility is expressed as:
  $$E_P[-\exp(-V_T^{0, \hat{\phi}})] = -\exp\left( -\frac{1}{2}V(T)(\mu - S_0)^2 - \frac{1}{2}\int_0^T V(t)dt \right)$$
  where $V(t)$ is a deterministic function defined explicitly in terms of $\delta$ and hyperbolic functions.
• Asymptotic Behavior:
  • As market depth $\delta \to 0$ (infinite impact), the value converges to $-1$, implying the zero strategy is optimal.
  • As market depth $\delta \to \infty$ (frictionless limit), the value converges to the known solution for the frictionless OU utility maximization problem.
  • For large time horizons $T$, the certainty equivalent grows linearly with $T$ (specifically, $\lim_{T\to\infty} c(T)/T = (\sqrt{1+\delta}-1)/2$), contrasting with the quadratic growth observed in the frictionless case.

Significance and Claims
The paper claims to provide the first explicit solution for exponential utility maximization with linear temporary price impact for a mean-reverting (Ornstein-Uhlenbeck) asset.

• Methodological Contribution: The work demonstrates that duality methods, previously applied to frictionless or simpler impact models, can be successfully extended to mean-reverting assets with linear price impact, yielding tractable closed-form solutions.
• Economic Insights: The results quantify how market depth $\delta$ affects optimal trading speeds and utility. The analysis highlights that optimal trading slows down significantly when the time to maturity is either very large or very small.
• Limitations and Scope: The author explicitly notes that the solution assumes no liquidation constraint at maturity ($\Phi_T$ is not forced to be zero). The paper acknowledges that imposing a liquidation constraint would introduce significant additional complexity in the dual representation, potentially preventing a closed-form solution for the value function, as seen in related literature (e.g., [7]). Consequently, the assumption $\Phi_0 = 0$ is made to simplify the computations for the value function, though the strategy form is derived generally.

The paper does not propose new experimental applications or future market models but strictly focuses on solving the defined mathematical optimization problem within the specified constraints.