Paper: arXiv 2506.07993

Authors: David Itkin

Abstract

We develop a framework for stochastic portfolio theory (SPT), which incorporates modern nonlinear price impact and impact decay models. Our main result is the derivation of the celebrated master formula for additive functional generation of trading strategies in a general high-dimensional market model with price impact. We also derive formulas for an investor’s relative wealth with respect to the market portfolio, conditions that guarantee positive observed market prices and a stochastic differential equation governing the dynamics of the observed price, the investor’s holdings and the price impact state processes. As an application of these results, we develop conditions for relative arbitrage in the price impact setting analogous to previously obtained results for the frictionless setting. We then apply our framework to backtest the quadratic and entropy generating functions on historical US equity data, illustrating how price impact can negatively affect portfolio performance.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 6.5/10
  • Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper provides a significant theoretical breakthrough by extending Stochastic Portfolio Theory (SPT) to include nonlinear price impact models with rigorous SDE derivations and master formulas. It complements the deep mathematical formulation with empirical backtesting on historical US equity data using functional portfolio generation.

Research Flowchart

  flowchart TD
    A["Research Goal: Incorporate price impact & decay into Stochastic Portfolio Theory (SPT)"] --> B["Framework Development: Formulate market model with nonlinear price impact state processes"]
    B --> C["Theoretical Derivation: Extend SPT Master Formula & derive Relative Wealth / Price SDEs"]
    C --> D["Arbitrage Analysis: Establish conditions for relative arbitrage under price impact"]
    E["Historical US Equity Data"] --> F["Backtesting: Apply Quadratic & Entropy generating functions"]
    C --> F
    D --> F
    F --> G["Key Outcome: Demonstrated negative impact of price execution costs on portfolio performance"]

Cited References (44)

  1. Fredholm Approach to Nonlinear Propagator Models — → 2503.04323
  2. Statistical Arbitrage in Rank Space — → 2410.06568
  3. Macroscopic properties of equity markets: stylized facts and portfolio performance — → 2409.10859
  4. Signature Methods in Stochastic Portfolio Theory — → 2310.02322
  5. Optimal Liquidation With Signals: The General Propagator Case (2022) — link
  6. Open Markets and Hybrid Jacobi Processes (2021) — link
  7. Functional Portfolio Optimization in Stochastic Portfolio Theory (2021) — link
  8. Càdlàg semimartingale strategies for optimal trade execution in stochastic order book models (2020) — link
and 36 more
  1. Relative arbitrage: Sharp time horizons and motion by curvature (2020) — link
  2. The Impact of Proportional Transaction Costs on Systematically Generated Portfolios (2019) — link
  3. Dynamics of observables in rank-based models and performance of functionally generated portfolios (2018) — link
  4. Polynomial processes in stochastic portfolio theory (2017) — link
  5. Dynamic portfolio choice with frictions (2016) — link
  6. Volatility and Arbitrage (2016) — link
  7. Trading strategies generated by Lyapunov functions (2016) — link
  8. Algorithmic and High Frequency Trading (2016) — link
  9. Hedging, Arbitrage, and Optimality with Superlinear Frictions (2015) — link
  10. Beyond the Square Root: Evidence for Logarithmic Dependence of Market Impact on Size and Participation Rate (2014) — link
  11. Slow Decay of Impact in Equity Markets (2014) — link
  12. The geometry of relative arbitrage (2014) — link
  13. The non-linear market impact of large trades: evidence from buy-side order flow (2013) — link
  14. HYBRID ATLAS MODELS (2009) — link
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  17. Atlas models of equity markets (2005) — link
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  27. Optimal trading strategy and supply/demand dynamics (2013)
  28. Optimal execution for portfolio transactions (2006)
  29. Direct Estimation of Equity Market Impact (2005)
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  31. Matrix analysis (1987) — link
  32. Efficient trading with price impact
  33. Definition 5.6 (Floor constraints)
  34. ii) The market weights are said to satisfy a floor constraint with bound ℓ > 0 on [0 , T ] if min i ∈{ 1 ,…,d } µ i ( t ) ≥ ℓ almost surely for every t ∈ [0 , T ]
  35. Trading with concave price impact – theory and evidence
  36. Q is additively functionally generated by G on [0 , ξ ) and its wealth process V Q satisfies (4.9

4 of 44 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.