Randomized Signature Methods in Optimal Portfolio Selection (2312.16448v1)

Abstract: We present convincing empirical results on the application of Randomized Signature Methods for non-linear, non-parametric drift estimation for a multi-variate financial market. Even though drift estimation is notoriously ill defined due to small signal to noise ratio, one can still try to learn optimal non-linear maps from data to future returns for the purposes of portfolio optimization. Randomized Signatures, in contrast to classical signatures, allow for high dimensional market dimension and provide features on the same scale. We do not contribute to the theory of Randomized Signatures here, but rather present our empirical findings on portfolio selection in real world settings including real market data and transaction costs.

• The paper demonstrates a novel framework leveraging Randomized Signature Methods for non-linear drift estimation in portfolio optimization.
• The methodology outperforms traditional linear models and momentum strategies, achieving higher annualized returns and Sharpe ratios.
• Empirical tests validate the practical utility of the approach under varying market conditions and transaction costs, advancing robust portfolio management.

Empirical Evaluation of Randomized Signature Methods in Portfolio Optimization

Introduction

The cornerstone of quantitative finance has been optimal portfolio construction, a task that involves balancing the expected return against risk. Traditionally, this has been approached through methodologies like Modern Portfolio Theory (MPT) and the Capital Asset Pricing Model (CAPM), which rely on the quantification of expected returns and risk, typically through linear models. However, the assumption of linear relationships and normal distributions of returns often does not hold in practice, particularly given the complexities and non-linear characteristics of financial markets. This paper explores the application of Randomized Signature Methods as a non-linear, non-parametric approach to drift estimation in multi-variate financial markets for portfolio optimization, moving beyond traditional linear models.

Theoretical Background

Randomized Signatures stem from rough path theory and provide a way to handle high-dimensional data by transforming paths into feature sets that capture the non-linear dynamics within the data. Unlike classical signatures, Randomized Signatures can deal with the curse of dimensionality and are robust across different scales, making them particularly suitable for financial markets where asset prices exhibit complex, path-dependent behaviors. This approach does not significantly contribute to new theoretical insights into Randomized Signatures themselves but rather showcases their practical application in portfolio optimization, evidenced through real market data and considering transaction costs.

Methodology and Empirical Findings

The paper introduces a refined framework for optimal portfolio selection employing Randomized Signature Methods for non-linear drift estimation. This involves generating features from historical price data, which are then used to forecast future returns. The novelty lies in the method's ability to exploit the full trajectory of asset prices, capturing critical non-linear dynamics like volatility and autocorrelation, often overlooked by linear models.

The empirical evaluation covers simulated and real-world data, comparing the performance of portfolios optimized using Randomized Signatures against those constructed via traditional linear regression, momentum strategies, and the naive 1/n portfolio. The simulation results indicate a distinctive outperformance of the Randomized Signature approach across a range of metrics, including annualized returns and Sharpe ratios. Furthermore, the method's robustness is tested under varying market conditions and transaction costs, illustrating its practicality for real-world application.

Impacts and Future Directions

The utilization of Randomized Signature Methods in portfolio optimization presents a compelling alternative to traditional linear models, especially in handling non-linear, path-dependent features of asset prices. This research opens avenues for integrating advanced mathematical models into financial decision-making, potentially leading to more robust portfolio construction methodologies that better capture the complexities of the market.

Future work could explore theoretical aspects of how Randomized Signatures capture market dynamics or extend the methodology to incorporate additional market factors or asset classes. Moreover, the adaptability of the approach under different regulatory environments or for varied investment objectives poses an interesting area for further investigation.

Conclusion

This paper successfully demonstrates the efficacy of Randomized Signature Methods in enhancing portfolio optimization through superior non-linear drift estimation. The empirical results underscore the potential of such advanced mathematical approaches in elevating the standards of portfolio management strategies, marking a significant step forward in the intersection of machine learning and finance.