Multiscale Markowitz

Paper: arXiv 2411.13792

Abstract

Traditional Markowitz portfolio optimization constrains daily portfolio variance to a target value, optimising returns, Sharpe or variance within this constraint. However, this approach overlooks the relationship between variance at different time scales, typically described by $σ(Δt) \propto (Δt)^{“H”}$ where $H$ is the Hurst exponent, most of the time assumed to be (\frac{“1”}{“2”}). This paper introduces a multifrequency optimization framework that allows investors to specify target portfolio variance across a range of frequencies, characterized by a target Hurst exponent $H_{“target”}$, or optimize the portfolio at multiple time scales. By incorporating this scaling behavior, we enable a more nuanced and comprehensive risk management strategy that aligns with investor preferences at various time scales. This approach effectively manages portfolio risk across multiple frequencies and adapts to different market conditions, providing a robust tool for dynamic asset allocation. This overcomes some of the traditional limitations of Markowitz, when it comes to dealing with crashes, regime changes, volatility clustering or multifractality in markets. We illustrate this concept with a toy example and discuss the practical implementation for assets with varying scaling behaviors.

Complexity vs Empirical Score

• Math Complexity: 7.5/10
• Empirical Rigor: 3.0/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper introduces advanced mathematical concepts such as fractional derivatives and scaling laws (Hurst exponent) but lacks concrete backtesting procedures, code examples, or robust empirical validation of its proposed framework.

Research Flowchart

  flowchart TD
  A["Research Goal: Enhance Markowitz<br>by incorporating time-scale<br>variance scaling behavior"] --> B["Methodology: Multifrequency<br>Portfolio Optimization Framework"]
  B --> C["Inputs: Asset Returns &<br>Target Hurst Exponent H_target"]
  C --> D{"Key Process:<br>Optimize across multiple<br>time scales simultaneously"}
  D --> E["Outcome: Robust Dynamic<br>Asset Allocation Strategy"]
  E --> F["Key Findings:<br>1. Better crash/regime risk<br>management via scaling<br>2. Adaptive to market<br>multifractality"]

Cited References (15)

1. Fractional Brownian motions, fractional noises and applications (1968)
2. Volatility is rough (2018)
3. Multifractality in asset returns: Theory and evidence (2002)
4. A closed-form solution for options with stochastic volatility with applications to bond and currency options (1993)
5. Multiscale modeling of financial time series and portfolio optimization (2008)
6. Modelling fluctuations of financial time series: from cascade process to stochastic volatility model (2000)
7. Regime Changes and Financial Markets (2012)
8. Fractal Market Analysis: Applying Chaos Theory to Investment and Economics (1994)

0 of 15 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.