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

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Highly relevant to quantitative trading. The paper provides closed-form optimal trading strategies for mean-reverting assets (common in pairs trading, statistical arbitrage, and fixed-income) under realistic linear price impact. The feedback form of the optimal strategy (Eq. 1.4) is directly implementable. The analysis of how market depth δ affects trading speed and certainty equivalent growth rate (Eq. 1.6) provides actionable insights for execution algorithms. The connection to the frictionless limit and the asymptotic behavior for large T are practically important. However, the one-dimensional setting and lack of numerical validation limit direct applicability.

The optimal strategy formula (1.4) involves nested hyperbolic functions (sinh, cosh, tanh) with parameters depending on market depth δ, time-to-maturity T-t, and the mean-reversion gap (μ - S_t). The value function V(t) (Theorem 1.1) is similarly complex. While all expressions are closed-form and computable, the algebraic complexity is high. The proof methodology (duality, change of measure, Itô representations, Euler-Lagrange equations) is mathematically sophisticated. Implementation requires careful numerical handling of hyperbolic functions for extreme parameter values.

The paper is fully self-contained with complete proofs, explicit formulas for the optimal strategy (Theorem 1.1, Eq. 1.4), the value function (Eq. 1.5), and all auxiliary lemmas (Lemmas 3.1, 3.2). All computations are analytical and can be verified independently. However, no code or numerical examples are provided. The methodology is purely probabilistic and relies on established results from references [5] and [11].