Optimal Linear Signal - An Unsupervised Machine Learning Framework to Optimize PnL with Linear Signals

Aug 5, 2026

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An unsupervised machine learning framework optimizes Profit and Loss (PnL) in quantitative finance by learning a linear trading signal from exogenous variables. The approach uses a linear relationship between variables and the signal, and another linear link between PnL and the signal, enabling a parameterization of PnL. Parameters are optimized to maximize the empirical Sharpe Ratio, with regularization to reduce overfitting. An application to an ETF tracking U.S. Treasury bonds demonstrates effectiveness, and the study outlines further extensions such as generalized time steps and improved corrective terms.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages10

Reading Time25 min

AccessFree to download

Introduction

Methodology: linear hypotheses and PnL parametrization

Optimization objective: maximizing the Sharpe ratio

Regularization, correction, and strategy construction

Empirical application on an ETF

Conclusion and future work

Optimal Linear Signal: An Unsupervised Machine Learning Framework to Optimize PnL with Linear Signals. Pierre RENUCCI Oct.-Nov. 2023 arXiv :2401 .05337v1 [ q-fin . ST] 22 Nov 2023 Abstract This study presents an unsupervised machine learning approach for optimizing Profit and Loss (PnL) in quantitative finance. Our algorithm, akin to an unsupervised variant of linear regression, maximizes the Sharpe Ratio of PnL generated from signals constructed linearly from exogenous variables. The methodology employs a linear relationship between exogenous variables and the trading signal, with the objective of maximizing the Sharpe Ratio through parameter optimization. Empirical application on an ETF representing U.S. Treasury bonds demonstrates the model’s effectiveness, supported by regularization techniques to mitigate overfitting. The study concludes with potential avenues for further development, including generalized time steps and enhanced corrective terms. The code of the model and the empiric strategy are available on my GitHub: Cnernc/OptimalLinearSignal Introduction In the field of quantitative finance, the creation of signals to generate Profit and Loss (PnL) is a key component. Our initial objective was to find a simple unsupervised machine learning (ML) algorithm capable of exploiting a set of exogenous variables to produce a PnL-effective signal. We observed that the literature in quantitative finance largely favors supervised ML approaches, aiming to predict specific financial magnitudes, such as asset prices, as highlighted in the works of Johnson (2023) [1], Rouf et al. [2], Soni et al. (2022) [3], and Kumar et al. (2022) [4] . There are also unsupervised techniques, as presented by Kelly and Xiu (2023) [5], and Hoang and Wiegratz (2023) [6] . However, these methods are very specific and lack the versatility to be used as general tools, specifically for small dataset. An exception in this category is the generation of signals via Principal Component Analysis (PCA), as explained by Ghorbani and Chong (2020) [7] . Nevertheless, this technique offers little flexibility and does not allow for the explicit optimization of specific criteria within the PnL. We therefore turned our attention to the literature on optimization in finance, with studies such as those by Jurczenko et al. (2019) [8], Huang et al. (2019) [9], and Cornu´ejols et al. (2017, 2018) [10] [11], Reppen et al. (2022) [12], primarily focusing on portfolio optimization. However, these works are more concerned with optimizing portfolios than signal creation. As a result, we decided to construct our own algorithm, developing an unsupervised counterpart to linear regression aimed at optimizing the Sharpe Ratio of a PnL. The model is based on two hypotheses of linearity: the linearity of the relationship between the exogenous variables and the signal; and the linearity of the relationship between the PnL and the signal. Using these, we can deduce a parametric representation of the PnL. We sought to optimize these parameters to maximize the Sharpe Ratio of the obtained PnL. The optimal parameters are calculated over a certain training period and are then used to generate the signal subsequently. Other techniques, mainly regularization and correction of this signal, are then addressed and ultimately enable the creation of a very effective strategy on a backtest of about twenty years. | Executive Summary
• We consider an asset with a specific price series pricet and a set of stationary and homoscedastic variables X1,t, X2,t , ..., Xn,t.
• The objective is to derive the ’optimal’ linear signal extracted from this variable. We work under two linear assumptions:
– We will consider the signal as a linear combination of the exogenous variables:
signalt = α0 + α1 X1,t + α2 X2,t + ... + αnXn,t
– At each time step t, the positions taken is proportional to both the signal and the price post = pricet × signalt (a negative value would

FAQs

What is the main goal of the Optimal Linear Signal framework?

It aims to optimize Profit and Loss (PnL) by learning an unsupervised linear trading signal and maximizing the PnL Sharpe Ratio.

How does the model connect exogenous variables to the trading signal and PnL?

It assumes a linear signal as a combination of exogenous variables and uses a proportional position rule to derive a linear expression of PnL in terms of transformed variables.

How is the model validated in the empirical study?

It is backtested on an ETF representing U.S. Treasury bonds (1-3 year maturities), where the strategy achieves an effective Sharpe ratio of about 1.2 over 2000-2023, supported by regularization to limit overfitting.

Aug 5, 2026

Upload by Hazel