Almost Perfect Shadow Prices

ArXiv ID: 2401.00970 “View on arXiv”

Authors: Unknown

Abstract

Shadow prices simplify the derivation of optimal trading strategies in markets with transaction costs by transferring optimization into a more tractable, frictionless market. This paper establishes that a naïve shadow price Ansatz for maximizing long term returns given average volatility yields a strategy that is, for small bid-ask-spreads, asymptotically optimal at third order. Considering the second-order impact of transaction costs, such a strategy is essentially optimal. However, for risk aversion different from one, we devise alternative strategies that outperform the shadow market at fourth order. Finally, it is shown that the risk-neutral objective rules out the existence of shadow prices.

Keywords: Shadow prices, Transaction costs, Asymptotic optimization, Risk aversion, Bid-ask spread, General equities/financial assets

Complexity vs Empirical Score

• Math Complexity: 9.0/10
• Empirical Rigor: 1.0/10
• Quadrant: Lab Rats
• Why: The paper is heavily mathematical, featuring advanced stochastic calculus, asymptotic expansions, PDE solutions, and rigorous proofs (Theorem 2.1, free boundary problems). Empirical rigor is low, focusing on theoretical asymptotics and proofs without any backtesting, real data, or implementation details.

  flowchart TD
    A["Research Goal:<br>Shadow Prices in Markets<br>with Transaction Costs"] --> B["Methodology:<br>Asymptotic Expansion<br>(Small Bid-Ask Spreads)"]
    B --> C["Model: Maximize Long-Term<br>Return given Average Volatility"]
    C --> D["Computation:<br>Naïve Shadow Price Ansatz<br>(Frictionless Transformation)"]
    D --> E{"Outcomes & Findings"}
    E --> F["1. Asymptotic Optimality<br>3rd Order for Risk Aversion = 1"]
    E --> G["2. Superior Strategies<br>4th Order for Risk Aversion ≠ 1"]
    E --> H["3. Negative Result:<br>No Shadow Prices for<br>Risk-Neutral Objective"]