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Model-free portfolio theory: A rough path approach.

Andrew L Allan1, Christa Cuchiero2, Chong Liu3

  • 1Durham University Durham UK.

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Summary
This summary is machine-generated.

This study introduces a new model-free approach to stochastic portfolio theory using rough path integration. This method offers a more general way to analyze portfolios without assuming underlying probability models.

Keywords:
Cover's universal portfoliolog‐optimal portfoliomodel uncertaintypathwise integrationrough pathstochastic portfolio theory

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Area of Science:

  • Quantitative Finance
  • Stochastic Analysis
  • Mathematical Finance

Background:

  • Stochastic Portfolio Theory (SPT) traditionally relies on specific probabilistic models.
  • Existing model-free approaches, like Föllmer integration, have limitations in portfolio generality.
  • Rough integration offers a novel framework for analyzing stochastic processes.

Purpose of the Study:

  • To develop a model-free approach to Stochastic Portfolio Theory (SPT) grounded in rough path integration.
  • To extend the applicability of model-free methods to more general portfolio types.
  • To establish new theoretical results in rough integration and its financial applications.

Main Methods:

  • Development of a novel model-free framework for SPT based on rough path foundations.
  • Pathwise analysis of the relative wealth process without assuming underlying probabilistic models.
  • Generalization of Cover's universal portfolio using controlled paths.

Main Results:

  • A pathwise formula for the relative wealth process is derived, reducing to a master formula for functionally generated portfolios.
  • The asymptotic growth rate of a generalized universal portfolio matches the best retrospectively chosen portfolio.
  • Log-optimal portfolios in ergodic Itô diffusion settings exhibit the same asymptotic growth rate as universal portfolios.

Conclusions:

  • The rough path approach provides a powerful, model-free methodology for advanced portfolio analysis.
  • This framework accommodates more general portfolios than previous model-free methods.
  • The study demonstrates the equivalence of asymptotic growth rates for different optimal portfolio strategies.