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Crafting Portfolios Tailored to Investor Preferences with Generative AI

Submitted: 30 April, 2024

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Authors: Jean-Jacques Ohana, Eric Benhamou, Beatrice Guez, Baptiste Lefort, David Saltiel, and Thomas Jacquot

Abstract: Investors often face challenges aligning portfolios with specific preferences or regulatory constraints, such as avoiding commodities. Using Graphical Models, we frame this as an inference problem to replicate a desired fund within set constraints. Our approach achieves a high correlation with the target portfolio, allowing cost-effective replication. An example to decoding global macro hedge funds is provided. It shows very stable correlation emphasizing that the method is able to replicate the strategy even under some strong investment constraints.   Read more

Small Triumphs Over Large: Instances Where BERT-Based Fine-Tuned Models Surpass GPT-4 in Classification Tasks

Submitted: 26 April, 2024

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Authors: Baptiste Lefort, Eric Benhamou, Jean-Jacques Ohana, Beatrice Guez, and David Saltiel

Abstract: This paper demonstrates that in sentiment classification tasks, non-generative BERT models with reduced numbers of parameters exhibit similar performance as generative Large Language Models (LLMs). More specifically, we compared FinBERT (110 million parameters) and FinDROBERTA (82.1 million parameters) with GPT-4 (at least more than 1 trillion parameters) on financial text classification, using our specially developed market-based dataset to fine-tune the models. This approach yielded significant and uniform performance improvements across all models, indicating that correctly fine-tuned small models perform even better than a large generative model like GPT-4. Furthermore, the application of a bagging majority vote classifier did not lead to any significant performance gains, suggesting similarities (or mathematically an absence of independence) of our models. This could indicate that these models process and classify financial sentiment similarly as the Condorcet Jury theorem is not validated. Our fine-tuned models are publicly accessible on Hugging Face for further research.   Read more

Uncertainty in Sentiment Analysis with LLMs using QCM (Quantiles of Correlation Matrices) - Distance

Submitted: 28 April, 2024

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Authors: Baptiste Lefort, Eric Benhamou, Jean-Jacques Ohana, Beatrice Guez, and David Saltiel

Abstract: In this paper, we investigate how to better understand uncertainty in sentiment score when using Natural Language Processing (NLP) and Large Language Models (LLMs) to derive from a sentiment score from a text. After explaining why uncertainty on sentiment score is challenging, especially in the case of news sentiment analysis, we present multiple prompts and an on-going research of a novel way to identify uncertainty in sentiment analysis. Our method computes the correlation between the cumulative sentiment score over different time windows with important influenced factors and validate that a similar proportion of patterns exists, using a measure based on the distance of Quantiles of Correlation Matrices, called QCMDistance. We apply this approach to see how sentiments in financial news are connected with changes in stock markets and to assess that a common pattern exists with the produced sentiment score, establishing a moderate uncertainty of the resulted sentiment score.   Read more

GRIP: Graphical Models Revealing Insights for Portfolio Replication - A Learning Approach

Submitted: 15 April, 2024

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Authors: Eric Benhamou, Jean-Jacques Ohana, and Beatrice Guez

Abstract: This paper presents a new and effective methodology for decoding strategies in the context of investment portfolios. The proposed approach relies on Dynamic Bayesian Graphical Models, which are powerful tools for capturing complex relationships and dependencies in data over time. Using these models, we can accurately decode the hidden strategy within the investment universe. By leveraging Dynamic Bayesian Graphical Models, we calculate dynamic weights that exhibit the most stable allocation rules. Through extensive experimentation on various investment scenarios, we demonstrate that our approach achieves high accuracy in decoding strategies. The method’s reliance on Dynamic Bayesian Graphical Models enables it to effectively uncover hidden patterns and relationships within the investment data, leading to improved portfolio allocation decisions and robust generalization across different market conditions.   Read more

Stress Index Strategy Enhanced With Financial News Sentiment Analysis for the Equity Markets

Submitted: 15 April, 2024

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Authors: Baptiste Lefort, Eric Benhamou, Jean-Jacques Ohana, Beatrice Guez, David Saltiel, and Thomas Jacquot

Abstract: This paper introduces a new risk-on risk-off strategy for the stock market, which combines a financial stress indicator with a sentiment analysis done by ChatGPT reading and interpreting Bloomberg daily market summaries. Forecasts of market stress derived from volatility and credit spreads are enhanced when combined with the financial news sentiment derived from GPT4. As a result, the strategy shows improved performance, evidenced by higher Sharpe ratio and reduced maximum drawdowns. The improved performance is consistent across the NASDAQ, the SP 500 and the six major equity markets, indicating that the method generalises across equities markets.   Read more

Sentiment Analysis of Bloomberg Markets Wrap Using ChatGPT: Application to the NASDAQ

Submitted: 15 April, 2024

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Authors: Baptiste Lefort, Eric Benhamou, Jean-Jacques Ohana, David Saltiel, Beatrice Guez, and Thomas Jacquot

Abstract: In this paper, we use a comprehensive dataset of daily Bloomberg Financial Market Summaries spanning from 2010 to 2023, published by Yahoo Finance, CNews and multiple large financial medias, to determine how global news headlines may affect stock market movements. To make this analysis more effective, we employed ChatGPT. First, from the vast pool of daily financial updates, we identify the top global news headlines that could potentially have a significant influence on stock prices. Second, for each headline, we question ChatGPT to answer whether the news might lead to a rise, fall in stock prices or is indecisive. This two-stage method proves more effective than posing a direct question to the entire text. By gathering ChatGPT’s predictions day by day, we formed an overall market sentiment score. We transform this score into a practical investment strategy in the NASDAQ index, demonstrating the significance of minimizing noise in sentiment scores by initially accumulating and then detrending them. This approach showcases that ChatGPT’s analysis of news headlines can provide valuable insights into future stock market behaviors and be a valuable tool to develop intuitive NLP-driven investment strategies leveraging news predictive power.   Read more

Mixing Financial Stress with GPT-4 News Sentiment Analysis for Optimal Risk-On/Risk-Off Decisions

Submitted: 27 April, 2024

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Authors: Baptiste Lefort, Eric Benhamou, Jean-Jacques Ohana, David Saltiel, Beatrice Guez, and Thomas Jacquot

Abstract: This paper introduces a new risk-on risk-off strategy for the stock market, which combines a financial stress indicator with a sentiment analysis done by ChatGPT reading and interpreting Bloomberg daily market summaries. Forecasts of market stress derived from volatility and credit spreads are enhanced when combined with the financial news sentiment derived from GPT4. As a result, the strategy shows improved performance, evidenced by higher Sharpe ratio and reduced maximum drawdowns. The improved performance is consistent across the NASDAQ, the SP 500 and the six major equity markets, indicating that the method generalises across equities markets.   Read more

Deep Reinforcement Learning: Extending Traditional Financial Portfolio Methods

Submitted: 15 April, 2024

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Authors: Eric Benhamou, Beatrice Guez, and Jean-Jacques Ohana

Abstract: Portfolio allocation, a key part of investment management, aims to balance risk and return. Traditional methodologies, rooted in modern portfolio theory, have been widely used for this purpose. Recently, deep reinforcement learning (DRL) has emerged as a powerful tool to tackle these complex problems, allowing finding new solutions through a trial-and-error process. The central idea of this paper is to demonstrate that traditional portfolio allocation strategies can be reframed in the DRL framework. It shows that a short-sighted agent, driven by immediate rewards and only considering the first two moments, converges to the Markowitz portfolio. By supplying this agent with more information, such as contextual data and additional future rewards, the DRL model can outperform traditional methods, though this comes with added complexity. Experiments confirm the usefulness of contextual data and show that DRL can improve traditional financial methods.   Read more

Generative AI: Crafting Portfolios Tailored to Investor Preferences

Submitted: 15 April, 2024

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Authors: Eric Benhamou, Jean-Jacques Ohana, and Beatrice Guez

Abstract: Investors often face challenges aligning portfolios with specific preferences or regulatory constraints, such as avoiding commodities. Can machine learning offer a solution? Using Graphical Models, we frame this as an inference problem to replicate a desired fund within set constraints. Our approach achieves a high correlation with the target portfolio, allowing cost-effective replication. By harnessing generative AI, we can even enhance the original portfolio’s performance. Applied to hedge funds, our method outperforms benchmarks while maintaining a strong correlation with the original strategy.   Read more

When Small Wins Big: Classification Tasks Where Compact Models Outperform Original GPT-4

Submitted: 15 April, 2024

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Authors: Baptiste Lefort, Eric Benhamou, Jean-Jacques Ohana, David Saltiel, Beatrice Guez, and Damien Challet

Abstract: This paper evaluates Large Language Models (LLMs) on financial text classification, comparing GPT-4 (1.76 trillion parameters) against FinBERT (110 million parameters) and FinDROBERTA (82.1 million parameters). We achieved a classification task on short financial sentences involving multiple divergent insights with both textual and numerical data. We developed a market-based large dataset that enabled us to fine-tune the models on a real-world ground truth. Utilizing a marketbased dataset for fine-tuning on extensive datasets, we achieved significant enhancements with FinBERT and FinDROBERTA over GPT-4. However, the use of a bagging majority classifier did not yield performance improvements, demonstrating that the principles of Condorcet’s jury Theorem do not apply, suggesting a lack of independence among the models and similar behavior patterns across all evaluated models. Our results indicate that for complex sentiment classification, compact models match larger models, even with fine-tuning. The fine-tuned models are made available as opensource for additional research.   Read more

Can ChatGPT Compute Trustworthy Sentiment Scores from Bloomberg Market Wraps?

Submitted: 5 February, 2024

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Authors: Baptiste Lefort, Eric Benhamou, David Saltiel, Beatrice Guez, Jean-Jacques Ohana and Damien Challet

Abstract: We used a dataset of daily Bloomberg Financial Market Summaries from 2010 to 2023, reposted on large financial media, to determine how global news headlines may affect stock market movements using ChatGPT and a two-stage prompt approach. We document a statistically significant positive correlation between the sentiment score and future equity market returns over short to medium term, which reverts to a negative correlation over longer horizons. Validation of this correlation pattern across multiple equity markets indicates its robustness across equity regions and resilience to non-linearity, evidenced by comparison of Pearson and Spearman correlations. Finally, we provide an estimate of the optimal horizon that strikes a balance between reactivity to new information and correlation.   Read more

Using ChatGPT to Predict the Equity Markets - PSL Week Slides

Submitted: 5 January, 2024

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Authors: Eric Benhamou, Beatrice Guez, Jean-Jacques Ohana, David Saltiel, and Baptiste Lefort

Abstract: In the evolving landscape of natural language processing, the capability to accurately gauge sentiment in text data is crucial for numerous applications, ranging from market analysis to customer feedback interpretation. This presentation will introduce a novel approach to sentiment analysis, utilizing OpenAI's ChatGPT, a variant of the powerful GPT model. We will delve into how ChatGPT, when used appropriately, can surpass the performance of specialized models like FinBERT in sentiment detection tasks.   Read more

Can Deep Reinforcement Learning Solve the Portfolio Allocation Problem?

Submitted: 9 November, 2023

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Authors: Eric Benhamou

Abstract: The promise of deep reinforcement learning (DRL) is to make no initial assumptions in terms of decisions or rules and let the machine find the best solution. In this thesis, we show that this type of machine learning method provides a new solution for portfolio allocation. In the first part, we present how to apply DRL to portfolio allocation and take into account the specific nature of financial time series. We introduce the concept of contextual variables allowing better learning. We change the cross-validation to a stepwise approach to ensure that the training data does not contain any future points compared to the validation and test data. We empirically show that DRL makes it possible to go beyond the state of the art of portfolio allocation methods by finding portfolios better adapted to market conditions, thanks to layers of convolutions allowing us to capture the dependence between market data and allocation decisions. We conclude this part with a model-based approach where DRL selects volatility-targeting models. In the second part, we present theoretical results justifying the DRL approach. We show how the DRL approach generalises classical portfolio theories. We study how DRL methods achieve variance reduction. We find similarities between reinforcement learning and supervised learning.   Read more

Comparing Deep RL and Traditional Financial Portfolio Methods

Submitted: 21 October, 2023

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Authors: Eric Benhamou, Beatrice Guez, Jean-Jacques Ohana, David Saltiel, Rida Laraki, and Jamal Atif

Abstract: Traditional portfolio allocation methods are based on the assumption of known risk tolerance and parametric models of market returns. Deep reinforcement learning (DRL) is a non-parametric machine learning technique that can learn optimal policies for complex sequential decision-making problems. This paper conducts a comprehensive comparative analysis of traditional portfolio allocation methods and DRL. The paper finds that DRL outperforms traditional methods in terms of both risk-adjusted return and Sharpe ratio in a simulation study. The authors argue that DRL is a promising new approach to portfolio management with the potential to outperform traditional methods in a variety of market conditions.   Read more

FSDA: Tackling Tail-Event Analysis in Imbalanced Time Series Data with Feature Selection and Data Augmentation

Submitted: 21 October, 2023

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Authors: Raphael Krief, Eric Benhamou, Beatrice Guez, Jean-Jacques Ohana, David Saltiel, Rida Laraki, and Jamal Atif

Abstract: Financial time series data is often imbalanced, with the majority of data points representing normal conditions and a small minority representing tail events, such as market crashes. This imbalance can make it difficult to train machine learning models to accurately predict tail events. FSDA (Feature Selection and Data Augmentation) is a novel approach to tackling tail-event analysis in imbalanced time series data. FSDA first uses a feature selection algorithm to identify the most predictive features for tail data. These features may exhibit different sensitivities compared to the rest of the dataset, so it is important to select them carefully. Once the most predictive features have been identified, FSDA uses data augmentation to generate synthetic tail data. Data augmentation is a conventional technique for handling imbalanced data, and it can be used to enhance the accuracy of machine learning regression methods. FSDA has been evaluated on a financial time series dataset, and it has been shown to consistently outperform feature selection (FS) and data augmentation (DA) methods across all percentiles ranging from 85 to 99. This demonstrates the efficacy of FSDA in managing imbalanced time series data and improving predictive accuracy.   Read more

Deep Decoding of Strategies

Submitted: 6 June 2022

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Authors: Eric Benhamou, David Saltiel, Beatrice Guez and Jean-Jacques Ohana

Abstract: To the best of our knowledge, the application of machine learning and in particular graphical models in the field of quantitative risk management is still a relatively recent and new phenomenon. This paper presents a new and effective methodology for decoding strategies. Given an investment universe, we calculate dynamic weights for a sparse portfolio whose aim is to replicate the strategy with the most stable allocation rules. Naturally, this can be formulated as a reinforcement learning problem whose reward is a weighted sum of tracking error and turnover. We show on stylized examples that we can accurately decode strategies or funds with meaningful factors and allocations.   Read more

Adaptive Supervised Learning for Volatility Targeting Models

Submitted: 15 September, 2021

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Authors: Eric Benhamou, David Saltiel, Serge Tabachnik, Corentin Bourdeix, François Chareyron, and Beatrice Guez

Abstract: In the context of risk-based portfolio construction and pro-active risk management, finding robust predictors of future realised volatility is paramount to achieving optimal performance. Volatility has been documented in economics literature to exhibit pronounced persistence with clusters of high or low volatility regimes and to mean-revert to a normal level, underpinning Nobel prize-winning work on Generalized Autoregressive Heteroskedastic (GARCH) models. From a Reinforcement Learning (RL) point of view, this process can be interpreted as a model-based RL approach where the goal of the models is twofold: first, to represent the volatility dynamics and forecast its term structure and second, to compute a resulting allocation to match a given target volatility: hence the name ”volatility targeting method for risk-based portfolios”. However, the resulting volatility model-based RL approaches are hard to distinguish as each model results in similar performance without a clear dominant one. We therefore present an innovative approach with an additional supervised learning step to predict the best model(s), based on historical performance ordering of RL models. Our contribution shows that adding a supervised learning overlay to decide which model(s) to use provides improvement over a naive benchmark consisting in averaging all RL models. A salient ingredient in this supervised learning task is to adaptively select features based on their significance, thanks to minimum importance filtering. This work extends our previous work on combining model-free and model-based RL. It mixes different types of learning procedures, namely model-based RL and supervised learning opening new doors to combine different machine learning approaches.   Read more

Explainable AI (XAI) Models Applied to Planning in Financial Markets

Submitted: 8 June, 2021

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Authors: Eric Benhamou, Jean Jacques Ohana, David Saltiel, Beatrice Guez, and Steve Ohana

Abstract: Regime changes planning in financial markets is well known to be hard to explain and interpret. Can an asset manager ex-plain clearly the intuition of his regime changes prediction on equity market ? To answer this question, we consider a gradi-ent boosting decision trees (GBDT) approach to plan regime changes on S&P 500 from a set of 150 technical, fundamen-tal and macroeconomic features. We report an improved ac-curacy of GBDT over other machine learning (ML) methods on the S&P 500 futures prices. We show that retaining fewer and carefully selected features provides improvements across all ML approaches. Shapley values have recently been intro-duced from game theory to the field of ML. This approach allows a robust identification of the most important variables planning stock market crises, and of a local explanation of the crisis probability at each date, through a consistent features attribution. We apply this methodology to analyse in detail the March 2020 financial meltdown, for which the model of-fered a timely out of sample prediction. This analysis unveils in particular the contrarian predictive role of the tech equity sector before and after the crash.   Read more

Adaptive learning for financial markets mixing model-based and model-free RL for volatility targeting

Submitted: 22 April, 2021

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Authors: Eric Benhamou, David Saltiel, Serge Tabachnik, Sui Kai Wong and François Chareyron

Abstract: Model-Free Reinforcement Learning has achieved meaningful results in stable environments but, to this day, it remains problematic in regime changing environments like financial markets. In contrast, model-based RL is able to capture some fundamental and dynamical concepts of the environment but suffer from cognitive bias. In this work, we propose to combine the best of the two techniques by select selecting various model-based approaches thanks to Model-Free Deep Reinforcement Learning. Using not only past performance and volatility, we include additional contextual information such as macro and risk appetite signals to account for implicit regime changes. We also adapt traditional RL methods to real-life situations by considering only past data for the training sets. Hence, we cannot use future information in our training data set as implied by K-fold cross validation. Building on traditional statistical methods, we use the traditional "walk-forward analysis", which is defined by successive training and testing based on expanding periods, to assert the robustness of the resulting agent. Finally, we present the concept of statistical difference's significance based on a two-tailed T-test, to highlight the ways in which our models differ from more traditional ones. Our experimental results show that our approach outperforms traditional financial baseline portfolio models such as the Markowitz model in almost all evaluation metrics commonly used in financial mathematics, namely net performance, Sharpe and Sortino ratios, maximum drawdown, maximum drawdown over volatility.   Read more

From Forecast to Decisions in Graphical Models: A Natural Gradient Optimization Approach

Submitted: 26 March, 2021

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Authors: Eric Benhamou, David Saltiel, Beatrice Guez, Jamal Atif and Rida Laraki

Abstract: Graphical models and in particular Hidden Markov Models or their continuous space equivalent, the so called Kalman filter model, are a powerful tool to make some inference that can be used in decision making contexts. The estimation of their parameters is usually based on the Expectation Maximization approach as this is a natural statistical way to train them. When used for decision making, it may be more relevant to find parameters that are relevant to our decisions rather than just try to fit the model from a statistical point of view. Hence, we can reformulate the determination of graphical model as an inference problem where the true concern is the quality of the decisions from the forecast given by the model. We show that the resulting optimization problem can be reformulated as an information geometric optimization problem and introduce a natural gradient descent strategy that incorporates additional meta parameters. Graphical models and in particular Hidden Markov Models or their continuous space equivalent, the so called Kalman filter model, are a powerful tool to make some inference that can be used in decision making contexts. The estimation of their parameters is usually based on the Expectation Maximization approach as this is a natural statistical way to train them. When used for decision making, it may be more relevant to find parameters that are relevant to our decisions rather than just try to fit the model from a statistical point of view. Hence, we can reformulate the determination of graphical model as an inference problem where the true concern is the quality of the decisions from the forecast given by the model. We show that the resulting optimization problem can be reformulated as an information geometric optimization problem and introduce a natural gradient descent strategy that incorporates additional meta parameters.   Read more

Combining Model-Based and Model-Free RL for Financial Markets

Submitted: 25 March, 2021

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Authors: Eric Benhamou, David Saltiel, Serge Tabachnik, Sui Kai Wong and François Chareyron

Abstract: Model Free Reinforcement Learning has achieved great results in stable environments but has not been able sofar to generalize well in regime changing environments like financial markets. In contrast, model based RL are able to capture some fundamental and dynamical concepts of the environment but suffer from cognitive bias. In this work, we propose to combine the best of the two approaches by selecting thanks to Model free Deep Reinforcement Learning various model based approaches. Using not only past performance and volatility, we include additional contextual information to account for implicit regime changes like macro and risk appetite signals . We also adapt traditional RL methods to take into account that in real life training takes always place in the past. Hence we cannot use future information in our training data set as implied by K-fold cross validation. Building on traditional statistical methods, we introduce "walk-forward analysis", which is defined by successive training and testing based on expanding periods, to assert the robustness of the resulting agent. Last but not least, we present the concept of statistical difference significance based on a two-tailed T-test, to highlight the ways in which our models differ from more traditional ones. Our experimental results show that our approach outperforms traditional financial baselines portfolio models like Markowitz in almost all evaluation metrics commonly used in financial mathematics, namely net performance, Sharpe ratio, Sortino, maximum drawdown, maximum drawdown over volatility.   Read more

Explainable AI Models of Stock Crashes: A Machine-Learning Explanation of the Covid March 2020 Equity Meltdown

Submitted: 21 March, 2021

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Authors: Jean Jacques Ohana, Steve Ohana, Eric Benhamou, David Saltiel and Beatrice Guez

Abstract: We consider a gradient boosting decision trees (GBDT) approach to predict large S&P 500 price drops from a set of 150 technical, fundamental and macroeconomic features. We report an improved accuracy of GBDT over other machine learning (ML) methods on the S&P 500 futures prices. We show that retaining fewer and carefully selected features provides improvements across all ML approaches. Shapley values have recently been introduced from game theory to the field of ML. They allow for a robust identification of the most important variables predicting stock market crises, and of a local explanation of the crisis probability at each date, through a consistent features attribution. We apply this methodology to analyze in detail the March 2020 financial meltdown, for which the model offered a timely out of sample prediction. This analysis unveils in particular the contrarian predictive role of the tech equity sector before and after the crash.   Read more

Knowledge discovery with Deep RL for selecting financial hedges

Submitted: 9 February, 2021

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Authors: Eric Benhamou, David Saltiel, Sandrine Ungari, Abhishek Mukhopadhyay, Jamal Atif and Rida Laraki

Abstract: Can an asset manager gain knowledge from different data sources to select the right hedging strategy for his portfolio? We use Deep Reinforcement Learning (Deep RL or DRL) to extract information from not only past performances of the hedging strategies but also additional contextual information like risk aversion, correlation data, credit information and estimated earnings per shares. Our contributions are threefold: (i) the use of contextual information also referred to as augmented state in DRL, (ii) the impact of a one period lag between observations and actions that is more realistic for an asset management environment, (iii) the implementation of a new repetitive train test method called walk forward analysis, similar in spirit to cross validation for time series. Although our experiment is on trading bots, it can easily be translated to other bot environments that operate in sequential environment with regime changes and noisy data. Our experiment for an augmented asset manager interested in finding the best portfolio for hedging strategies achieves superior returns and lower risk.   Read more

Time your hedge with Deep Reinforcement Learning

Submitted: 9 November, 2020

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Authors: Eric Benhamou, David Saltiel, Sandrine Ungari and Abhishek Mukhopadhyay

Abstract: Can an asset manager plan the optimal timing for her/his hedging strategies given market conditions? The standard approach based on Markowitz or other financial rules aims to find the best portfolio allocation but fails to relate market conditions to hedging decisions. Deep Reinforcement Learning (DRL) creates a dynamic dependency between market information and hedging strategies allocation decisions. We present a realistic DRL framework using contextual information, delayed actions, anchored walk-forward training and leverage control. Results show superior returns and lower risk.   Read more

Detecting and adapting to crisis pattern with context based Deep Reinforcement Learning

Submitted: 9 November, 2020

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Authors: Eric Benhamou, David Saltiel, Jean Jacques Ohana and Jamal Atif

Abstract: Deep reinforcement learning (DRL) has reached super-human levels in complex tasks but remains challenging in finance. We present a DRL framework with dual subnetworks incorporating contextual indicators such as risk aversion and correlations. Using convolutional layers, LSTMs and adversarial training, the model outperforms traditional methods like Markowitz and successfully anticipates crisis patterns such as the Covid crash.   Read more

Bridging the gap between Markowitz planning and deep reinforcement learning

Submitted: 30 September, 2020

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Authors: Eric Benhamou, David Saltiel, Sandrine Ungari and Abhishek Mukhopadhyay

Abstract: Traditional portfolio planning relies on static optimization techniques. This paper bridges financial planning and Deep Reinforcement Learning (DRL), casting portfolio allocation as a continuous optimal control problem with delayed rewards. DRL adapts naturally to changing environments, incorporates multiple data sources, and removes traditional variance-based assumptions. Experiments using convolutional networks show promising results.   Read more

Time your hedge with Deep Reinforcement Learning

Submitted: 9 November, 2020

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Authors: Eric Benhamou, David Saltiel, Sandrine Ungari and Abhishek Mukhopadhyay

Abstract: Can an asset manager plan the optimal timing for her/his hedging strategies given market conditions? The standard approach based on Markowitz or other financial rules aims to find the best portfolio allocation but fails to relate market conditions to hedging decisions. Deep Reinforcement Learning (DRL) creates a dynamic dependency between market information and hedging strategies allocation decisions. We present a realistic DRL framework using contextual information, delayed actions, anchored walk-forward training and leverage control. Results show superior returns and lower risk.   Read more

Detecting and adapting to crisis pattern with context based Deep Reinforcement Learning

Submitted: 9 November, 2020

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Authors: Eric Benhamou, David Saltiel, Jean Jacques Ohana and Jamal Atif

Abstract: Deep reinforcement learning (DRL) has reached super-human levels in complex tasks but remains challenging in finance. We present a DRL framework with dual subnetworks incorporating contextual indicators such as risk aversion and correlations. Using convolutional layers, LSTMs and adversarial training, the model outperforms traditional methods like Markowitz and successfully anticipates crisis patterns such as the Covid crash.   Read more

Bridging the gap between Markowitz planning and deep reinforcement learning

Submitted: 30 September, 2020

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Authors: Eric Benhamou, David Saltiel, Sandrine Ungari and Abhishek Mukhopadhyay

Abstract: While researchers in the asset management industry have mostly focused on Markowitz-style planning, another community has advanced deep reinforcement learning for sequential decision-making. This paper bridges the gap by casting portfolio allocation as an optimal control problem with delayed rewards. DRL maps market conditions to actions, adapts to changing environments, relaxes classical risk assumptions, and can incorporate additional data. We present encouraging results using convolution networks.   Read more

Trade Selection with Supervised Learning and Optimal Coordinate Ascent (OCA)

Submitted: 15 September, 2020

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Authors: David Saltiel, Eric Benhamou, Rida Laraki and Jamal Atif

Abstract: Can we dynamically extract information and strong relationships between financial features to select trades over time? Despite end-to-end deep learning, feature selection remains key. This paper introduces Optimal Coordinate Ascent (OCA), an embedded feature selection method formulated as a coordinate-ascent optimization with block-variable dependencies. With few iterations it is efficient for gradient boosting methods (XGBoost). For convex smooth functions we prove a polynomial convergence rate in the dimension. We compare to RFE and Binary Coordinate Ascent and show competitive results for selecting financial trades.   Read more

AAMDRL: Augmented Asset Management with Deep Reinforcement Learning

Submitted: 29 September, 2020

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Authors: Eric Benhamou, David Saltiel, Sandrine Ungari, Abhishek Mukhopadhyay and Jamal Atif

Abstract: Can an agent learn efficiently in a noisy and self adapting environment with sequential, non-stationary observations? Through trading bots, we illustrate how Deep Reinforcement Learning (DRL) can tackle this challenge. Our contributions are: (i) contextual information (augmented state), (ii) one-period lag between observations and actions, and (iii) a repetitive train-test method called walk-forward analysis. The approach can translate to other sequential environments with regime changes and noisy data. Results show superior returns and lower risk.   Read more

Deep Reinforcement Learning for Portfolio Selection

Submitted: 8 September, 2020

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Authors: Eric Benhamou, David Saltiel, Jean Jacques Ohana, Jamal Atif and Rida Laraki

Abstract: Deep reinforcement learning (DRL) has reached an unprecedented level on complex tasks, but applications to real financial assets remain largely unexplored. We showcase state-of-the-art DRL methods for selecting portfolios according to financial environments, using a network that concatenates three subnetworks with convolutional layers to reduce complexity. The multi-input design captures dependencies from risk aversion, surprise indices, portfolio features and previous allocations. Results show the approach can outperform traditional portfolio optimization methods, with results available at our demo website.   Read more

Three remarkable properties of the Normal distribution

Submitted: 11 July, 2020

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Authors: Eric Benhamou, Beatrice Guez and Nicolas Paris

Abstract: In this paper, we present three remarkable properties of the normal distribution. First, if two independent variables' sum is normally distributed, then each random variable is normal (Lévy–Cramér theorem). Second, if two independent symmetric variables with finite variance have their sum and difference independent, then each is standard normal. Third, the normal distribution is the only one for which sample mean and variance are independent (Geary theorem). We provide new, quicker or self-contained proofs of these results.   Read more

Omega and Sharpe ratio

Submitted: 15 October, 2019

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Authors: Eric Benhamou, Beatrice Guez and Nicolas Paris

Abstract: Omega ratio has been advocated as a better performance indicator than Sharpe/Sortino because it uses the full return distribution. We compute Omega for the normal distribution and show that under symmetry assumptions Omega is oversold: it provides no additional information compared to Sharpe. For elliptic return distributions, the Omega-optimal portfolio matches the Sharpe-optimal one. Since elliptic distributions include many fat-tail families, this reduces the potential interest of Omega.   Read more

Variance Reduction in Actor Critic Methods (ACM)

Submitted: 23 July, 2019

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Authors: Eric Benhamou

Abstract: After presenting Actor Critic Methods (ACM), we show ACM are control variate estimators. Using the projection theorem, we prove that Q and Advantage Actor Critic (A2C) methods are optimal in L2 norm for the control variates spanned by functions conditioned by the current state and action. This gives a theoretical justification of A2C performance and enables a new formulation for Advantage Actor Critic with lower variance than traditional A2C.   Read more

Testing Sharpe ratio: luck or skill?

Submitted: 21 May, 2019

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Authors: Eric Benhamou, David Saltiel, Beatrice Guez and Nicolas Paris

Abstract: Sharpe ratio is widely used in asset management but relies on unknown expected returns and volatilities that must be estimated. This makes reported Sharpe ratios error-prone. We provide tests to measure Sharpe quality and whether a manager is lucky or skillful via statistical significance. We also compute a modified Sharpe insensitive to used capital, illustrate required trades for significance, and analyze autocorrelation effects with reference tables for minimum Sharpe levels and skill percentages.   Read more

Connecting Sharpe ratio and Student t-statistic, and beyond

Submitted: 14 May, 2019

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Authors: Eric Benhamou

Abstract: Sharpe ratio is widely used but subject to statistical estimation error. Lo (2002) and Mertens (2002) derive asymptotic distributions under i.i.d. normal returns. We provide the exact distribution for independent normal returns: up to a rescaling factor, it is a non-centered Student distribution. We also study AR(1) assumptions, compounding periods, and give formulas under heteroscedasticity and autocorrelation.   Read more

NGO-GM: Natural Gradient Optimization for Graphical Models

Submitted: 14 May, 2019

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Authors: Eric Benhamou, Jamal Atif, Rida Laraki and David Saltiel

Abstract: We reformulate parameter estimation in graphical models as an information geometric optimization problem. We introduce a natural gradient descent strategy with meta-parameters as an alternative to EM. It learns optimal parameters for the final objective without fitting an artificial distribution. We support theory with trend detection in financial markets, showing improved performance and reduced overfitting versus practitioner methods.   Read more

Similarities between policy gradient methods in reinforcement and supervised learning

Submitted: 2 May, 2019

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Authors: Eric Benhamou and David Saltiel

Abstract: Reinforcement learning (RL) is traditionally opposed to supervised learning (SL), but the gap is smaller than it seems. We show that gradient policy methods can be cast as an SL problem where true labels are replaced with discounted rewards. We provide a simple experiment interchanging labels and pseudo-rewards to demonstrate how SL techniques can translate into RL methods.   Read more

BCMA-ES II: revisiting Bayesian CMA-ES

Submitted: 9 April, 2019

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Authors: Eric Benhamou, David Saltiel, Beatrice Guez and Nicolas Paris

Abstract: We revisit Bayesian CMA-ES and provide updates for normal Wishart. We emphasize the difference between a normal and normal inverse Wishart prior. We show the key difference lies in expected covariance, which should be lower in the normal Wishart prior due to convexity of the inverse. We present a mixture model generalizing both priors, and provide numerical experiments comparing the approaches.   Read more

BCMA-ES: A Bayesian approach to CMA-ES

Submitted: 2 April, 2019

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Authors: Eric Benhamou, David Saltiel, Sebastien Verel and Fabien Teytaud

Abstract: We introduce a theoretically sound Bayesian approach for the CMA-ES algorithm. Assuming a conjugate prior for the multivariate normal parameters, we derive optimal updates at each iteration. This provides a justification for CMA-ES updates and yields two new versions depending on whether the likelihood is parametrized by covariance or precision (normal-Wishart vs normal-inverse-Wishart). Numerical experiments show faster convergence.   Read more

A discrete version of CMA-ES

Submitted: 11 February, 2019

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Authors: Eric Benhamou, Jamal Atif and Rida Laraki

Abstract: Evolution Strategies culminated in CMA-ES, which relies on a multivariate normal distribution and performs well for continuous optimization, but is not optimal for discrete variables. We extend CMA-ES to multivariate binomial correlated distributions. We show similarities with the multivariate normal (independence and correlation), discuss the exponential family, and prove the model can estimate pairwise and higher-order interactions. This yields a CMA-ES variant that accommodates discrete variables efficiently, with the corresponding algorithm.   Read more

Operator norm upper bound for sub-Gaussian tailed random matrices

Submitted: 19 January, 2019

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Authors: Eric Benhamou, Jamal Atif and Rida Laraki

Abstract: We investigate an upper bound of the operator norm for sub-Gaussian tailed random matrices. Beyond the classic case of uniformly bounded i.i.d. coefficients, we study matrices with unequal variances and dependence. We show a weaker condition suffices: independent rows that are uniformly sub-Gaussian. This does not require independent coefficients, only independent rows.   Read more

Kalman filter demystified: from intuition to probabilistic graphical model to real case in financial markets

Submitted: 13 December, 2018

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Authors: Eric Benhamou

Abstract: We revisit Kalman filter theory: intuition, underlying maths, and a graphical-model perspective. Presenting Kalman filters as graphical models connects them to Hidden Markov Models (HMM). We provide new inference algorithms for extended Kalman filters, propose CMA-ES as an alternative to EM for parameter estimation, and examine applications to financial markets with dynamics assumptions and tests. We connect the approach to trend-following technical analysis and show superior performance for trend detection.   Read more

Feature selection with optimal coordinate ascent (OCA)

Submitted: 3 December, 2018

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Authors: Eric Benhamou and David Saltiel

Abstract: Feature Selection (FS) is crucial for efficient machine learning algorithms. We present Optimal Coordinate Ascent (OCA) to select features among blocks and individual features for gradient boosting score maximization. OCA uses dependencies between variables forming blocks and reduces an NP-hard search to a polynomial one. Compared to earlier coordinate ascent approaches, OCA introduces block + individual selection (not binary), a k-best group initialization, and faster stopping criteria. We compare to prior methods and RFE, showing OCA yields the most compact dataset with top performance.   Read more

Gram Charlier and Edgeworth expansion for sample variance

Submitted: 18 September, 2018

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Authors: Eric Benhamou

Abstract: We derive a valid Edgeworth expansion for the Bessel-corrected empirical variance when data are generated by a strongly mixing process. We assume only a sufficiently fast tail decrease to ensure convergence. The results apply to strongly mixing normal processes and provide an alternative to Moschopoulos (1985) and Mathai (1982).   Read more

A few properties of sample variance

Submitted: 11 September, 2018

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Authors: Eric Benhamou

Abstract: A basic result is that the sample variance for i.i.d. observations is an unbiased estimator of the variance of the underlying distribution. But what happens if observations are neither independent nor identically distributed? Can we compute explicitly the first two moments of the sample mean and generalize Tukey (1957a, 1957b) formulae for the first two moments of the sample variance? We also know sample mean and variance are independent for i.i.d. normal data (used to derive Student’s distribution). Does this hold for other distributions? This paper answers these questions and extends Cho, Cho, and Eltinge (2004). We derive general formulae for the first two moments and variance of the sample variance under no specific assumption, and provide a faster proof of Lukacs (1942) using the log characteristic function of the unbiased sample variance estimator.   Read more

T-statistic for Autoregressive process

Submitted: 11 September, 2018

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Authors: Eric Benhamou

Abstract: We discuss the distribution of the t-statistic under a normal autoregressive assumption for the underlying discrete-time process. This generalizes the classical t-distribution case (uncorrelated normal observations). For AR(1), correlation breaks traditional results: the resulting t-statistic is a new distribution converging asymptotically to a normal. We give an explicit formula as a ratio of two dependent distributions (a normal and the norm of another independent normal). We also provide a modified statistic that follows a non-central t-distribution, derived from an orthogonal basis of the initial circulant Toeplitz covariance matrix. This has applications, including deriving the exact Sharpe ratio distribution under normal AR(1) assumptions.   Read more

Seven proofs of the Pearson Chi-squared independence test and its graphical interpretation

Submitted: 3 September, 2018

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Authors: Eric Benhamou and Valentin Melot

Abstract: This paper revisits the Pearson Chi-squared independence test, presenting the underlying theory with modern notations and new derivations. We also introduce an innovative and intuitive graphical presentation of the test, enabling visual interpretation and helping measure how close or far we are from accepting or rejecting the null hypothesis of independence.   Read more

Incremental Sharpe and other performance ratios

Submitted: 26 August, 2018

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Authors: Eric Benhamou and Beatrice Guez

Abstract: We present a methodology to compute incremental contributions for performance ratios such as Sharpe, Treynor, Calmar or Sterling. Using Euler's homogeneous function theorem, we decompose these ratios as a linear combination of individual modified performance ratios, enabling interpretation of drivers and a condition for a new asset to improve portfolio performance. We provide numerical examples.   Read more