Working Papers

The WNE Working Papers series has been published by the Faculty of Economic Sciences at the University of Warsaw since 2008.

The WNE Working Papers series provides a fast, open channel for disseminating research conducted at the Faculty of Economic Sciences, University of Warsaw. The papers hosted here are working versions (pre-prints) and may evolve as authors refine their analyses, incorporate feedback, or progress through formal peer-review. What you read is the current version released by the authors, timestamped and assigned DOI/ISSN identifiers to ensure precise citation and version tracking. Copyright remains with the authors, who may, at any time, upload a revised file or add a note directing readers to a later, peer-reviewed publication.

The Working Papers series accepts articles by research employees of the Faculty and publications from conferences organised at the Faculty of Economic Sciences at the University of Warsaw. Articles should be original research papers which have not been previously published, on the subject of economics.

Please send your paper by e-mail: _6a~d+FV]#zTp0-'2`8lO?Cjhu%]#[W$TjNu*G?v`GdK!qri.dX{~`qfn 

Please send 2 files:

(1) the main text without the title of the article and the authors (DOC/DOCX file) and

(2) the title page including: the title of the paper, the authors and their affiliation (DOC/DOCX file).

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Number of Results 10

WP(25/2025)488. Application of Deep Reinforcement Learning to At-the-Money S&P 500 Options Hedging

Authors: Bracha Zofia, Michańków Jakub, Sakowski Paweł
This paper explores the application of deep Q-learning to hedging at-the-money options on the S&P 500 index. We develop an agent based on the Twin Delayed Deep Deterministic Policy Gradient (TD3) algorithm, trained to simulate hedging decisions w…

WP488. Application of Deep Reinforcement Learning to At-the-Money S&P 500 Options Hedging

This paper explores the application of deep Q-learning to hedging at-the-money options on the S&P 500 index. We develop an agent based on the Twin Delayed Deep Deterministic Policy Gradient (TD3) algorithm, trained to simulate hedging decisions without making explicit model assumptions on price dynamics. The agent was trained on historical intraday prices of S&P 500 call options across years 2004 to 2024, using a single time series of six predictor variables: option price, underlying asset price, moneyness, time to maturity, realized volatility, and current hedge position. A walk-forward procedure was applied for training, which lead to nearly 17 years of out-of-sample evaluation. The performance of the deep reinforcement learning (DRL) agent is benchmarked against the Black–Scholes delta hedging strategy over the same time period. We assess both approaches using metrics such as annualized return, volatility, information ratio, and Sharpe ratio. To test models’ adaptability, we performed simulations across varying market conditions and added constraints such as transaction costs and risk-awareness penalties. Our results show that the DRL agent can outperform traditional hedging methods, particularly in volatile or high-cost environments, highlighting its robustness and flexibility in practical trading contexts. While the agent consistently outperforms delta hedging, its performance deteriorates when the risk-awareness parameter is higher. We also observed that the longer the time interval used for volatility estimation, the more stable the results.


DOI: https://doi.org/10.33138/2957-0506.2025.25.488
Bracha Zofia, Michańków Jakub, Sakowski Paweł
Deep learning Reinforcement learning Double Deep Q-netwoorks options market options hedging deep hedging

WP(22/2025)485. A survey of statistical arbitrage pair trading with machine learning, deep learning, and reinforcement learning methods

Authors: Sun Yufei
Pair trading remains a cornerstone strategy in quantitative finance, having consistently attracted scholarly attention from both economists and computer scientists. Over recent decades, research has expanded beyond traditional linear frameworks&mdash…

WP485. A survey of statistical arbitrage pair trading with machine learning, deep learning, and reinforcement learning methods

Pair trading remains a cornerstone strategy in quantitative finance, having consistently attracted scholarly attention from both economists and computer scientists. Over recent decades, research has expanded beyond traditional linear frameworks—such as regression- and cointegration-based models—to embrace advanced methodologies, including machine learning (ML), deep learning (DL), reinforcement learning (RL), and deep reinforcement learning (DRL). These techniques have demonstrated superior capacity to capture nonlinear dependencies and complex dynamics in financial data, thereby enhancing predictive performance and strategy design.
Building on these academic developments, practitioners are increasingly deploying DL models to forecast asset price movements and volatility in equity and foreign exchange markets, leveraging the advantages of artificial intelligence (AI) for trading. In parallel, DRL has gained prominence in algorithmic trading, where agents can autonomously learn optimal trading policies by interacting with market environments, enabling systems that move beyond price prediction to dynamic signal generation and portfolio allocation.
This paper provides a comprehensive survey of ML-, DL-, RL-, and DRL-based approaches to pair trading within quantitative finance. By systematically reviewing existing studies and highlighting their methodological contributions, it offers researchers a structured foundation for replication and further development. In addition, the paper outlines promising avenues for future research that extend the application of AI-driven methods in statistical arbitrage and market microstructure analysis.


DOI: https://doi.org/10.33138/2957-0506.2025.22.485
Sun Yufei
Pair Trading Machine Learning Deep Learning Reinforcement Learning Deep Reinforcement Learning Artificial Intelligence Quantitative Trading

WP(14/2025)477. Can Artificial Intelligence Trade the Stock Market?

Authors: Jędrzej Maskiewicz, Sakowski Paweł
The paper explores the use of Deep Reinforcement Learning (DRL) in stock market trading, focusing on two algorithms: Double Deep Q-Network (DDQN) and Proximal Policy Optimization (PPO) and compares them with Buy and Hold benchmark. It evaluates these…

WP477. Can Artificial Intelligence Trade the Stock Market?

The paper explores the use of Deep Reinforcement Learning (DRL) in stock market trading, focusing on two algorithms: Double Deep Q-Network (DDQN) and Proximal Policy Optimization (PPO) and compares them with Buy and Hold benchmark. It evaluates these algorithms across three currency pairs, the S&P 500 index and Bitcoin, on the daily data in the period of 2019-2023. The results demonstrate DRL's effectiveness in trading and its ability to manage risk by strategically avoiding trades in unfavorable conditions, providing a substantial edge over classical approaches, based on supervised learning in terms of risk-adjusted returns.


DOI: https://doi.org/10.33138/2957-0506.2025.14.477
Jędrzej Maskiewicz, Sakowski Paweł
Reinforcement Learning Deep Learning stock market algorithmic trading Double Deep Q-Network Proximal Policy Optimization

WP(13/2024)449. The Hybrid Forecast of S&P 500 Volatility ensembled from VIX, GARCH and LSTM models

Authors: Natalia Roszyk, Ślepaczuk Robert
Predicting the S&P 500 index's volatility is crucial for investors and financial analysts as it helps in assessing market risk and making informed investment decisions. Volatility represents the level of uncertainty or risk related to the siz…

WP449. The Hybrid Forecast of S&P 500 Volatility ensembled from VIX, GARCH and LSTM models

Predicting the S&P 500 index's volatility is crucial for investors and financial analysts as it helps in assessing market risk and making informed investment decisions. Volatility represents the level of uncertainty or risk related to the size of changes in a security's value, making it an essential indicator for financial planning. This study explores four methods to improve the accuracy of volatility forecasts for the S&P 500: the established GARCH model, known for capturing historical volatility patterns; an LSTM network that utilizes past volatility and log returns; a hybrid LSTM-GARCH model that combines the strengths of both approaches; and an advanced version of the hybrid model that also factors in the VIX index to gauge market sentiment. This analysis is based on a daily dataset that includes data for S&P 500 and VIX index, covering the period from January 3, 2000, to December 21, 2023. Through rigorous testing and comparison, we found that machine learning approaches, particularly the hybrid LSTM models, significantly outperform the traditional GARCH model. The inclusion of the VIX index in the hybrid model further enhances its forecasting ability by incorporating real-time market sentiment. The results of this study offer valuable insights for achieving more accurate volatility predictions, enabling better risk management and strategic investment decisions in the volatile environment of the S&P 500.


DOI: https://doi.org/10.33138/2957-0506.2024.13.449
Natalia Roszyk, Ślepaczuk Robert
volatility forecasting LSTM-GARCH S&P 500 index hybrid forecasting models VIX index machine learning financial time series analysis walk-forward process hyperparameters tuning deep learning recurrent neural networks

WP(7/2024)443. LSTM-ARIMA as a Hybrid Approach in Algorithmic Investment Strategies

Authors: Kashif Kamil, Ślepaczuk Robert
This study focuses on building an algorithmic investment strategy employing a hybrid approach that combines LSTM and ARIMA models referred to as LSTM-ARIMA. This unique algorithm uses LSTM to produce final predictions but boost results of this RNN by…

WP443. LSTM-ARIMA as a Hybrid Approach in Algorithmic Investment Strategies

This study focuses on building an algorithmic investment strategy employing a hybrid approach that combines LSTM and ARIMA models referred to as LSTM-ARIMA. This unique algorithm uses LSTM to produce final predictions but boost results of this RNN by adding the residuals obtained from ARIMA predictions among other inputs. The algorithm is tested across three equity indices (S&P 500, FTSE 100, and CAC 40) using daily frequency data spanning from January, 2000 to August, 2023. The architecture of testing is based on the walk-forward procedure which is applied for hyperparameter tunning phase that uses using Random Search and backtesting the algorithms. The selection of the optimal model is determined based on adequately selected performance metrics combining focused on risk-adjusted return measures. We considered two strategies for each algorithm: Long-Only and Long-Short in order to present situation of two various groups of investors with different investment policy restrictions. For each strategy and equity index, we compute the performance metrics and visualize the equity curve to identify the best strategy with the highest modified information ratio. The findings conclude that the LSTM-ARIMA algorithm outperforms all the other algorithms across all the equity indices what confirms strong potential behind hybrid ML-TS (machine learning - time series) models in searching for the optimal algorithmic investment strategies.


DOI: https://doi.org/10.33138/2957-0506.2024.7.443
Kashif Kamil, Ślepaczuk Robert
Deep Learning Recurrent Neural Networks Algorithmic Investment Strategy LSTM ARIMA Hybrid/Ensemble Models Walk-Forward Process

WP(5/2023)412. The performance of time series forecasting based on classical and machine learning methods for S&P 500 index

Authors: Uzzal Maudud Hassan, Ślepaczuk Robert
Based on one step ahead forecasts, this study compares the forecasting abilities of the traditional technique (ARIMA) with recurrent neural network (LSTM). In order to check the possible use of these forecasts in different asset management methods, t…

WP412. The performance of time series forecasting based on classical and machine learning methods for S&P 500 index

Based on one step ahead forecasts, this study compares the forecasting abilities of the traditional technique (ARIMA) with recurrent neural network (LSTM). In order to check the possible use of these forecasts in different asset management methods, these forecasts are afterwards included into trading signals of investment strategies. As a benchmark method, the Random Walk model producing naive forecasts has been utilized. This research examines daily data from the S&P 500 index for 20 years, from 2000 to 2020, and it includes information on some significant market turbulence. The methods were tested in terms of robustness to changes in parameters and hyperparameters and evaluated based on various error metrics (MAE, MAPE, RMSE MSE). The results show that ARIMA outperforms LSTM in terms of one step ahead forecasts. Finally, LSTM model with a variety of hyperparameters - including a number of epochs, a loss function, an optimizer, activation functions, a number of units, a batch size, and a learning rate - was tested in order to check its robustness.


Uzzal Maudud Hassan, Ślepaczuk Robert
deep learning recurrent neural networks ARIMA algorithmic investment strategies trading systems LSTM walk-forward process optimization

WP(29/2022)405. The efficiency of various types of input layers of LSTM model in investment strategies on S&P500 index

Authors: Nguyen Thi Thu Giang, Ślepaczuk Robert
The study compares the use of various Long Short-Term Memory (LSTM) variants to conventional technical indicators for trading the S&P 500 index between 2011 and 2022. Two methods were used to test each strategy: a fixed training data set from 200…

WP405. The efficiency of various types of input layers of LSTM model in investment strategies on S&P500 index

The study compares the use of various Long Short-Term Memory (LSTM) variants to conventional technical indicators for trading the S&P 500 index between 2011 and 2022. Two methods were used to test each strategy: a fixed training data set from 2001–2010 and a rolling train–test window. Due to the input sensitivity of LSTM models, we concentrated on data processing and hyperparameter tuning to find the best model. Instead of using the traditional MSE function, we used the Mean Absolute Directional Loss (MADL) function based on recent research to enhance model performance. The models were assessed using the Information Ratio and the Modified Information Ratio, which considers the maximum drawdown and the sign of the annualized return compounded (ARC). LSTM models' performance was compared to benchmark strategies using the SMA, MACD, RSI, and Buy&Hold strategies. We rejected the hypothesis that algorithmic investment strategy using signals from LSTM model consisting only from daily returns in its input layer is more efficient. However, we could not reject the hypothesis that signals generated by LSTM model combining daily returns and technical indicators in its input layer are more efficient. The LSTM Extended model that combined daily returns with MACD and RSI in the input layer generated a better result than Buy&Hold and other strategies using a single technical indicator. The results of the sensitivity analysis show how sensitive this model is to inputs like sequence length, batch size, technical indicators, and the length of the rolling train - test window.


Nguyen Thi Thu Giang, Ślepaczuk Robert
algorithmic investment strategies machine learning testing architecture deep learning recurrent neural networks LSTM technical indicators forecasting financial-time series technical indicators hyperparameter tuning S&P 500 Index

WP(25/2022)401. Daily and intraday application of various architectures of the LSTM model in algorithmic investment strategies on Bitcoin and the S&P 500 Index

Authors: Kryńska Katarzyna, Ślepaczuk Robert
This thesis investigates the use of various architectures of the LSTM model in algorithmic investment strategies. LSTM models are used to generate buy/sell signals, with previous levels of Bitcoin price and the S&P 500 Index value as inputs. Four…

WP401. Daily and intraday application of various architectures of the LSTM model in algorithmic investment strategies on Bitcoin and the S&P 500 Index

This thesis investigates the use of various architectures of the LSTM model in algorithmic investment strategies. LSTM models are used to generate buy/sell signals, with previous levels of Bitcoin price and the S&P 500 Index value as inputs. Four approaches are tested: two are regression problems (price level prediction) and the other two are classification problems (prediction of price direction). All approaches are applied to daily, hourly, and 15-minute data and are using a walk-forward optimization procedure. The out-of-sample period for the S&P 500 Index is from February 6, 2014 to November 27, 2020, and for Bitcoin it is from January 15, 2014 to December 1, 2020. We discover that classification techniques beat regression methods on average, but we cannot determine if intra-day models outperform inter-day models. We come to the conclusion that the ensembling of models does not always have a positive impact on performance. Finally, a sensitivity analysis is performed to determine how changes in the main hyperparameters of the LSTM model affect strategy performance. 


Kryńska Katarzyna, Ślepaczuk Robert
machine learning deep learning recurrent neural networks LSTM algorithmic trading ensemble investment strategy intra-day trading S&P 500 Index Bitcoin

WP(21/2022)397. A comparison of LSTM and GRU architectures with novel walk-forward approach to algorithmic investment strategy

Authors: Baranochnikov Illia, Ślepaczuk Robert
The aim of this work is to build a profitable algorithmic investment strategy on various types of assets. The algorithm is built using recurrent neural networks (LSTM and GRU) as the primary source of  signals to buy/sell financial instruments. …

WP397. A comparison of LSTM and GRU architectures with novel walk-forward approach to algorithmic investment strategy

The aim of this work is to build a profitable algorithmic investment strategy on various types of assets. The algorithm is built using recurrent neural networks (LSTM and GRU) as the primary source of  signals to buy/sell financial instruments. LSTM and GRU architectures are compared in terms of obtaining the best results and beating the market. The algorithm is tested for four financial instruments (Bitcoin, Tesla, Brent Oil and Gold) on daily and hourly data frequencies. The out-of-sample period is from 1 January 2021 to 1 April 2022. A walk-forward process is responsible for training models and selecting the best model to forecast asset prices in the future. Ten model architectures with various hyperparameters are trained during each step of the walk-forward process. The model architecture with the highest Information Ratio (IR*) in the validation period is used for forecasting in the out-of-sample period. For each strategy, the performance metrics are calculated based on which the profitability of the algorithm is evaluated. At the end, a detailed sensitivity analysis with regards to the main hyperparameters is conducted. The results reveal that LSTM outperforms GRU in most of the cases and that investment strategy built based on LSTM/GRU architecture is able to beat the market only on 50% of tested cases. 


Baranochnikov Illia, Ślepaczuk Robert
deep learning recurrent neural networks algorithm trading strategy LSTM GRU walk-forward process

WP(12/2022)388. Investment Portfolio Optimization Based on Modern Portfolio Theory and Deep Learning Models

Authors: Wysocki Maciej, Sakowski Paweł
This paper investigates an important problem of an appropriate variance-covariance matrix estimation in the Modern Portfolio Theory. In this study we propose a novel framework for variance-covariance matrix estimation for purposes of the portfolio op…

WP388. Investment Portfolio Optimization Based on Modern Portfolio Theory and Deep Learning Models

This paper investigates an important problem of an appropriate variance-covariance matrix estimation in the Modern Portfolio Theory. In this study we propose a novel framework for variance-covariance matrix estimation for purposes of the portfolio optimization, which is based on deep learning models. We employ the long short-term memory (LSTM) recurrent neural networks (RNN) along with two probabilistic deep learning models: DeepVAR and GPVAR to the task of one-day ahead multivariate forecasting. We then use these forecasts to optimize portfolios that consisted of stocks and cryptocurrencies. Our analysis presents results across different combinations of observation windows and rebalancing periods to compare performances of classical and deep learning variance-covariance estimation methods. The conclusions of the study are that although the strategies (portfolios) performance differed significantly between different combinations of parameters, generally the best results in terms of the information ratio and annualized returns are obtained using the LSTM-RNN models. Moreover, longer observation windows translate into better performance of the deep learning models indicating that these methods require longer windows to be able to efficiently capture the long-term dependencies of the variance-covariance matrix structure. Strategies with less frequent rebalancing typically perform better than these with the shortest rebalancing windows across all considered methods.


Wysocki Maciej, Sakowski Paweł
Portfolio Optimization Deep Learning Variance-Covariance Matrix Forecasting Investment Strategies Recurrent Neural Networks Long Short-Term Memory Neural Networks
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