[Paper Review] Signature Methods in Stochastic Portfolio Theory

Christa Cuchiero, Janka Möller|arXiv (Cornell University)|Oct 3, 2023

Risk and Portfolio Optimization4 citations

TL;DR

This paper introduces linear path-functional portfolios based on the signature of ranked market weights, enabling universal approximation of continuous, path-dependent portfolio functions in stochastic portfolio theory. The key contribution is that signature portfolios can arbitrarily closely approximate the growth-optimal portfolio in non-Markovian models and reduce complex optimization tasks to convex quadratic programs, enabling efficient computation and strong out-of-sample performance, even with transaction costs.

ABSTRACT

In the context of stochastic portfolio theory we introduce a novel class of portfolios which we call linear path-functional portfolios. These are portfolios which are determined by certain transformations of linear functions of a collections of feature maps that are non-anticipative path functionals of an underlying semimartingale. As main example for such feature maps we consider the signature of the (ranked) market weights. We prove that these portfolios are universal in the sense that every continuous, possibly path-dependent, portfolio function of the market weights can be uniformly approximated by signature portfolios. We also show that signature portfolios can approximate the growth-optimal portfolio in several classes of non-Markovian market models arbitrarily well and illustrate numerically that the trained signature portfolios are remarkably close to the theoretical growth-optimal portfolios. Besides these universality features, the main numerical advantage lies in the fact that several optimization tasks like maximizing (expected) logarithmic wealth or mean-variance optimization within the class of linear path-functional portfolios reduce to a convex quadratic optimization problem, thus making it computationally highly tractable. We apply our method also to real market data based on several indices. Our results point towards out-performance on the considered out-of-sample data, also in the presence of transaction costs.

Motivation & Objective

• To develop a novel class of portfolios—linear path-functional portfolios—that generalize functionally generated portfolios by incorporating non-anticipative path functionals of market weights.
• To establish theoretical universality: that any continuous, path-dependent portfolio function of market weights can be uniformly approximated by signature portfolios.
• To demonstrate that signature portfolios can approximate the growth-optimal portfolio arbitrarily well in non-Markovian market models.
• To show that optimization tasks like log-wealth maximization and mean-variance optimization reduce to convex quadratic programs, ensuring computational tractability.
• To validate the approach empirically on simulated and real market data, including S&P500, SMI, and NASDAQ, under transaction costs.

Proposed method

• The authors define linear path-functional portfolios as portfolios determined by linear transformations of the signature of the ranked market weights, a feature map derived from continuous semimartingale processes.
• They employ the universal approximation theorem for linear functions on the signature to prove that such portfolios can uniformly approximate any continuous, path-dependent function of market weights.
• The method leverages the Johnson-Lindenstrauss lemma to reduce dimensionality of the signature while preserving approximation quality.
• Optimization tasks such as maximizing expected log-wealth or performing mean-variance optimization are reformulated as convex quadratic programs over the signature coefficients.
• A regularization scheme is introduced to account for proportional transaction costs, improving performance in out-of-sample tests.
• The approach is applied to real market indices (SMI, S&P500, NASDAQ), with portfolio weights trained via convex optimization and evaluated on out-of-sample wealth processes.

(a) Black-Scholes market: growth-optimal weights

Experimental results

Research questions

• RQ1 Can signature-based portfolios universally approximate any continuous, path-dependent function of market weights in stochastic portfolio theory?
• RQ2 In what classes of non-Markovian market models can the growth-optimal portfolio be represented as a signature portfolio?
• RQ3 Can optimization tasks such as log-wealth maximization and mean-variance optimization be reduced to convex quadratic programs within the class of linear path-functional portfolios?
• RQ4 How do signature portfolios perform in out-of-sample backtests, especially under transaction costs?
• RQ5 Does regularization for transaction costs improve the robustness and performance of signature portfolios in real market data?

Key findings

• Signature portfolios achieve universal approximation of continuous, path-dependent portfolio functions of market weights, as proven by the global universal approximation theorem for linear functions on the signature.
• In non-Markovian market models, the growth-optimal portfolio can be approximated arbitrarily well by signature portfolios, with numerical results showing close proximity to the theoretical optimal.
• For the S&P500 universe, the log-relative wealth of the signature portfolio with regularization at 1% transaction cost achieved a log-wealth of 0.2411, outperforming the equally-weighted portfolio and the universe portfolio under the same conditions.
• On the SMI index, the signature portfolio with 1% transaction cost regularization achieved a log-wealth of 0.0402, significantly outperforming the universe portfolio (log-wealth: -2.5700) under the same cost level.
• The optimal regularization parameter β for 1% transaction costs on the S&P500 was found to be 847.6125, indicating strong sensitivity and effectiveness of the regularization in improving performance.
• In all tested scenarios, including 5% transaction costs, the use of regularization consistently improved portfolio performance, with the signature portfolio outperforming the universe portfolio in every case.

(b) Black-Scholes market: signature weights