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Almost Perfect Shadow Prices

Eberhard Mayerhofer

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.

Almost Perfect Shadow Prices

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.
Paper Structure (10 sections, 13 theorems, 82 equations)

This paper contains 10 sections, 13 theorems, 82 equations.

Table of Contents

  1. Introduction
  2. Materials and Methods
  3. Admissible strategies and their long-run performance
  4. Results
  5. Asymptotically optimal shadow policies
  6. Outperforming the shadow market
  7. The limits of shadow prices
  8. Discussion
  9. The free boundary problem for the shadow price candidate
  10. Asymptotics of the free boundaries

Key Result

Theorem 2.1

Let $\frac{\mu}{\gamma\sigma^2}\neq 1$.

Theorems & Definitions (33)

  • Theorem 2.1
  • Example 2.2
  • Definition 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Definition 2.6
  • Lemma 2.7
  • proof
  • ...and 23 more