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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.
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Submitted: June 6, 2022

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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.
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Submitted: September 15, 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.
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Submitted: June 8, 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.
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Submitted: 22 April, 2021

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Authors: __Eric Benhamou__, __David Saltiel__, __Beatrice Guez__, __Jamal Atif__ and __Rida Laraki__

Submitted: 26 March, 2021

Collaboration with
**LOMBARD ODIER **
team

Access to the video and to the SSRN paper

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.
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Submitted: March 25, 2021

Collaboration with
**HOMA CAPITAL**
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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.
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Submitted: March 21, 2021

Collaboration with
**SOCIETE GENERALE**
team

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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.
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Submitted: 9 February, 2021

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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 more or less sophisticated financial
rules aims to find the best portfolio allocation thanks to
forecasted expected returns and risk but fails to fully relate
market conditions to hedging strategies decision. In contrast,
Deep Reinforcement Learning (DRL) can tackle this challenge by
creating a dynamic dependency between market information and
hedging strategies allocation decisions.
In this paper, we present
a realistic and augmented DRL framework that: (i) uses additional
contextual information to decide an action, (ii) has a one period
lag between observations and actions to account for one day lag
turnover of common asset managers to rebalance their hedge, (iii)
is fully tested in terms of stability and robustness thanks to a
repetitive train test method called anchored walk forward
training, similar in spirit to k fold cross validation for time
series and (iv) allows managing leverage of our hedging strategy.
Our experiment for an augmented asset manager interested in sizing
and timing his hedges shows that our approach achieves superior
returns and lower risk.
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Submitted: 9 November, 2020

Collaboration with
**HOMA CAPITAL**
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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 like game solving (Go, StarCraft
II), and autonomous driving. However, it remains an open question
whether DRL can reach human level in applications to financial
problems
and in particular in detecting pattern crisis and
consequently dis-investing. In this paper, we present an
innovative DRL framework consisting in two subnetworks fed
respectively with portfolio strategies past performances and
standard deviations as well as additional contextual features.
The second sub network plays an important role as it captures
dependencies with common financial indicators features like risk
aversion, economic surprise index and correlations between assets
that allows taking into account context based information. We
compare different network architectures either using layers of
convolutions to reduce network’s complexity or LSTM block to
capture time dependency and whether previous allocations is
important in the modeling. We also use adversarial training to
make the final model more robust. Results on test set show this
approach substantially over-performs traditional portfolio
optimization methods like Markovitz and is able to detect and
anticipate crisis like the current Covid one.
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Submitted: 9 November, 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 techniques based on financial and risk planning
techniques like Markowitz efficient frontier, minimum variance,
maximum diversification or equal risk parity, in parallel, another
community in machine learning has started working on reinforcement
learning and more particularly deep reinforcement learning to
solve other decision making problems for challenging tasks
like autonomous driving, robot learning, and on a more conceptual side
games solving like Go. This paper aims to bridge the gap between
these two approaches by showing Deep Reinforcement Learning (DRL)
techniques can shed new lights on portfolio allocation thanks to a
more general optimization setting that casts portfolio allocation
as an optimal control problem that is not just a one-step
optimization, but rather a continuous control optimization with a
delayed reward.
The advantages are numerous: (i) DRL maps directly
market conditions to actions by design and hence should adapt to
changing environment, (ii) DRL does not rely on any traditional
financial risk assumptions like that risk is represented by
variance, (iii) DRL can incorporate additional data and be a multi
inputs method as opposed to more traditional optimization methods.
We present on an experiment some encouraging results using
convolution networks.
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Submitted: 30 September, 2020

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Authors: __David Saltiel__, __Eric Benhamou__, __Rida Laraki__ and __Jamal Atif__

Abstract: Can we dynamically extract some information and strong
relationship between some financial features in order to select
some financial trades over time?
Despite the advent of
representation learning and end-to-end approaches, mainly through
deep learning, feature se- lection remains a key point in many
machine learning scenarios. This paper introduces a new
theoretically motivated method for feature se- lection. The
approach thatfits within the family of embedded methods, casts the
feature selection conundrum as a coordinate ascent optimiza- tion
with variables dependencies materialized by block variables.
Thanks to a limited number of iterations, it proves eficiency for
gradient boost- ing methods, implemented with XGBoost. In case of
convex and smooth functions, we are able to prove that the
convergence rate is polynomial in terms of the dimension of the
full features set. We provide comparisons with state of the art
methods, Recursive Feature Elimination and Bi- nary Coordinate
Ascent and show that this method is competitive when selecting
some financial trades.
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Submitted: 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 and non-homogeneous
observations? Through trading bots, we illustrate how Deep Reinforcement Learning (DRL) can
tackle this challenge. 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 shows that AAMDRL achieves superior returns and lower risk.
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Submitted: 29 September, 2020

Collaboration with
**HOMA CAPITAL**
team

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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
unprecedent level on complex tasks like game solving (Go,
StarCraft II), and autonomous driving. However, applications to
real Financial assets are still largely unexplored
and it remains
an open question whether DRL can reach super human level. In this
demo, we showcase state-of-the-art DRL methods for selecting
portfolios according to financial environment, with a final
network concatenating three individual networks using lay- ers of
convolutions to reduce network's complexity.
The multi entries of our network enables capturing dependencies from common financial
indicators features like risk aversion, citigroup index surprise,
portfolio specific features and previous portfolio allocations.
Results on test set show this approach can overperform traditional
portfolio optimization methods with results available at our demo
website.
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Submitted: September, 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 that if two independent
variables's sum is normally distributed, then each random variable follows a normal
distribution (which is referred to as the Levy Cramer theorem), second a variation
of the Levy Cramer theorem that states that if two independent symmetric random variables
with finite variance have their
sum and their difference independent, then each random variable follows a standard normal
distribution, and third that the normal distribution is characterized by the fact that it is
the only distribution for which the sample mean and variance are independent (which is a central
property for deriving the Student distribution and referred as the Geary theorem). The novelty of
this paper is to provide new, quicker or self contained proofs of theses theorems.
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Submitted: 11 July, 2020

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Authors: __Eric Benhamou__, __Beatrice Guez__ and __Nicolas Paris__

Abstract: Omega ratio, defined as the
probability-weighted ratio of gains over losses at a given level of expected
return, has been advocated as a better performance indicator compared to Sharpe
and Sortino ratio as it depends on the full return distribution and hence encapsulates
all information about risk and return. We compute Omega ratio for the normal distribution and show that under some distribution
symmetry assumptions, the Omega ratio is oversold as it does not provide any additional
information compared to Sharpe ratio. Indeed, for returns that have elliptic distributions,
we prove that the optimal portfolio according to Omega ratio is the same as the optimal portfolio
according to Sharpe ratio. As elliptic distributions are a weak form of symmetric distributions that
generalized Gaussian distributions and encompass many fat tail distributions, this reduces tremendously
the potential interest for the Omega ratio.
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Submitted: 15 October, 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
the Q and Advantage Actor Critic (A2C) methods are optimal in the sense of the L2 norm for the
control variate estimators spanned by functions conditioned by the current state and action. This
straightforward application of Pythagoras theorem provides a
theoretical justification of the strong performance of QAC and AAC most often referred
to as A2C methods in deep policy gradient methods. This enables us to derive a new formulation
for Advantage Actor Critic methods that has lower variance and improves the traditional A2C method.
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Submitted: 23 July, 2019

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Authors: __Eric Benhamou__, __David Saltiel__, __Beatrice Guez__ and __Nicolas Paris__

Abstract: Sharpe ratio (sometimes also
referred to as information ratio) is widely used in asset management to compare
and benchmark funds and asset managers. It computes the ratio of the (excess) net
return over the strategy standard deviation. However, the elements to compute the
Sharpe ratio, namely, the expected returns and the volatilities are unknown numbers and
need to be estimated statistically.
This means that the Sharpe ratio used by funds is likely to be error prone because
of statistical estimation errors. In this paper, we provide various tests to measure
the quality of the Sharpe ratios. By quality, we are aiming at measuring whether a manager
was indeed lucky of skillful. The test assesses this through the statistical significance of
the Sharpe ratio. We not only look at the traditional Sharpe ratio but also compute a modified Sharpe
insensitive to used Capital. We provide various statistical tests that can be used to precisely quantify
the fact that the Sharpe is statistically significant. We illustrate in particular the number of trades for
a given Sharpe level that provides statistical significance as well as the impact of auto-correlation by providing
reference tables that provides the minimum required Sharpe ratio for a given time period and correlation. We also provide
for a Sharpe ratio of 0.5, 1.0, 1.5 and 2.0 the skill percentage given the auto-correlation level.
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Submitted: 21 May, 2019

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Authors: __Eric Benhamou__

Abstract: Sharpe ratio is widely
used in asset management to compare and benchmark funds and asset managers.
It computes the ratio of the excess return over the strategy standard deviation.
However, the elements to compute the Sharpe ratio, namely, the expected returns and
the volatilities are unknown numbers and need to be estimated statistically. This means that the Sharpe ratio used by funds is subject
to be error prone because of statistical estimation error. Lo (2002), Mertens (2002)
derive explicit expressions for the statistical distribution of the Sharpe ratio using standard
asymptotic theory under several sets of assumptions (independent normally distributed - and identically
distributed returns). In this paper, we provide the exact distribution of the Sharpe ratio for independent
normally distributed return. In this case, the Sharpe ratio statistic is up to a rescaling factor a non centered
Student distribution whose characteristics have been widely studied by statisticians. The asymptotic behavior of our
distribution provide the result of Lo (2002). We also illustrate the fact that the empirical Sharpe ratio is asymptotically
optimal in the sense that it achieves the Cramer Rao bound. We then study the empirical SR under AR(1) assumptions and investigate
the effect of compounding period on the Sharpe (computing the annual Sharpe with monthly data for instance). We finally provide general
formula in this case of heteroscedasticity and autocorrelation.
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Submitted: 14 May, 2019

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Authors: __Eric Benhamou__, __Jamal Atif__, __Rida Laraki__ and __David Saltiel__

Abstract: This paper deals with estimating model parameters
in graphical models. We reformulate it as an information geometric optimization problem and introduce
a natural gradient descent strategy that incorporates additional meta parameters. We show that our approach
is a strong alternative to the celebrated EM approach for learning in graphical models. Actually, our natural
gradient based strategy leads
to learning optimal parameters for the final objective function without artificially trying to fit a
distribution that may not correspond to the real one. We support our theoretical findings with the question
of trend detection in financial markets and show that the learned model performs better than traditional practitioner
methods and is less prone to overfitting.
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Submitted: 14 May, 2019

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Authors: __Eric Benhamou__ and __David Saltiel__

Abstract: Reinforcement learning (RL) is about sequential decision
making and is traditionally opposed to supervised learning (SL)
and unsupervised learning (USL).
In RL, given the current state,
the agent makes a decision that may in uence the next state as
opposed to SL where the next state remains the same, regardless of
decisions taken. Although this difference is fundamental, SL and
RL are not so different.
In particular, we emphasize in this paper
that gradient policy methods can be cast as a SL problem where
true label are replaced with discounted rewards. We pro- vide a
simple experiment where we interchange label and pseudo rewards to
show that SL techniques can be directly translated into RL
methods.
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Submitted: 2 May, 2019

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Authors: __Eric Benhamou__, __David Saltiel__, __Beatrice Guez__ and __Nicolas Paris__

Abstract: This paper revisits the Bayesian CMA-ES and provides updates
for normal Wishart. It emphasizes the difference between a normal and normal inverse Wishart prior. After some
computation, we prove that the only difference relies surprisingly in the expected covariance. We prove that the
expected covariance should be lower in the normal Wishart prior model because of the convexity of the inverse. We
present a mixture model that generalizes both normal Wishart and normal inverse Wishart model. We finally present
various numerical experiments to compare both methods as well as the generalized method.
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Submitted: 9 April, 2019

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Authors: __Eric Benhamou__, __David Saltiel__, __Sebastien Verel__ and __Fabien Teytaud__

Abstract: This paper introduces a novel theoretically
sound approach for the celebrated CMA-ES algorithm. Assuming the parameters of the multi variate
normal distribution for the minimum follow a conjugate prior distribution, we derive their optimal update
at each iteration step. Not only provides this Bayesian framework a justification for the update of the CMA-ES
algorithm but it also gives two new
versions of CMA-ES either assuming normal-Wishart or normal-Inverse Wishart priors, depending whether we parametrize
the likelihood by its covariance or precision matrix. We support our theoretical findings by numerical experiments that
show fast convergence of these modified versions of CMA-ES.
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Submitted: 2 April, 2019

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Authors: __Eric Benhamou__, __Jamal Atif__ and __Rida Laraki__

Abstract: Modern machine learning uses
more and more advanced optimization techniques to find optimal hyper parameters.
Whenever the objective function is non-convex, non continuous and with potentially
multiple local minima, standard gradient descent optimization methods fail. A last resource
and very different method is to assume that the optimum(s), not necessarily unique, is/are distributed according to
a distribution and iteratively to adapt the distribution according to tested points.
These strategies originated in the early 1960s, named Evolution Strategy (ES) have culminated
with the CMA-ES (Covariance Matrix Adaptation) ES. It relies on a multi variate normal distribution
and is supposed to be state of the art for general optimization program. However, it is far from being
optimal for discrete variables. In this paper, we extend the method to multivariate binomial correlated
distributions. For such a distribution, we show that it shares similar features to the multi variate normal:
independence and correlation is equivalent and correlation is efficiently modeled by interaction between different
variables. We discuss this distribution in the framework of the exponential family. We prove that the model can estimate
not only pairwise interactions among the two variables but also is capable of modeling higher order interactions. This allows
creating a version of CMA ES that can accommodate efficiently discrete variables. We provide the corresponding algorithm and conclude.
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Submitted: 11 February, 2019

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Authors: __Eric Benhamou__, __Jamal Atif__ and __Rida Laraki__

Abstract: This paper investigates an upper
bound of the operator norm for sub-Gaussian tailed random matrices. A lot of attention
has been put on uniformly bounded sub-Gaussian tailed random matrices with independent coefficients.
However, little has been done for sub-Gaussian tailed random matrices whose matrix coefficients variance
are not equal or for matrix for which coefficients are not independent.
This is precisely the subject of this paper. After proving that random matrices with uniform sub-Gaussian
tailed independent coefficients satisfy the Tracy Widom bound, that is, their matrix operator norm remains bounded
by O(n−−√) with overwhelming probability, we prove that a less stringent condition is that the matrix rows are independent
and uniformly sub-Gaussian. This does not impose in particular that all matrix coefficients are independent, but only their rows,
which is a weaker condition.
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Submitted: 19 January, 2019

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Authors: __Eric Benhamou__

Abstract: In this paper, we revisit the Kalman filter
theory. After giving the intuition on a simplified financial markets example, we revisit the
maths underlying it. We then show that Kalman filter can be presented in a very different fashion
using graphical models. This enables us to establish the connection between Kalman filter and Hidden
Markov Models. We then look at their application in financial
markets and provide various intuitions in terms of their applicability for
complex systems such as financial markets. Although this paper has been written
more like a self contained work connecting Kalman filter to Hidden Markov Models
and hence revisiting well known and establish results, it contains new results and
brings additional contributions to the field. First, leveraging on the link between
Kalman filter and HMM, it gives new algorithms for inference for extended Kalman filters.
Second, it presents an alternative to the traditional estimation of parameters using EM algorithm
thanks to the usage of CMA-ES optimization. Third, it examines the application of Kalman filter and
its Hidden Markov models version to financial markets, providing various dynamics assumptions and tests.
We conclude by connecting Kalman filter approach to trend following technical analysis system and showing
their superior performances for trend following detection.
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Submitted: 13 December, 2018

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Authors: __Eric Benhamou__ and __David Saltiel__

Abstract: In machine learning, Feature Selection
(FS) is a major part of efficient algorithm. It fuels the algorithm and is the starting
block for our prediction. In this paper, we present a new method, called Optimal Coordinate
Ascent (OCA) that allows us selecting features among block and individual features. OCA relies
on coordinate ascent to find an optimal solution for gradient boosting methods score
(number of correctly classified samples). OCA takes into account the notion of dependencies
between variables forming blocks in our optimization. The coordinate ascent optimization solves
the issue of the NP hard original problem where the number of combinations rapidly explode making a
grid search unfeasible. It reduces considerably the number of iterations changing this NP hard problem
into a polynomial search one. OCA brings substantial differences and improvements compared to previous
coordinate ascent feature selection method: we group variables into block and individual variables instead
of a binary selection. Our initial guess is based on the k-best group variables making our initial point more
robust. We also introduced new stopping criteria making our optimization faster. We compare these two methods
on our data set. We found that our method outperforms the initial one. We also compare our method to the Recursive
Feature Elimination (RFE) method and find that OCA leads to the minimum feature set with the highest score. This is
a nice byproduct of our method as it provides empirically the most compact data set with optimal performance.
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Submitted: 3 December, 2018

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Authors: __Eric Benhamou__

Abstract: In this paper, we derive a valid Edgeworth expansions
for the Bessel corrected empirical variance when data are generated by a strongly mixing process whose
distribution can be arbitrarily. The constraint of strongly mixing process makes the problem not easy.
Indeed, even for a strongly mixing normal process, the distribution is unknown. Here, we do not assume any other assumption than a
sufficiently fast decrease of the underlying distribution to make the Edgeworth expansion
convergent. This results can obviously apply to strongly mixing normal process and provide
an alternative to the work of Moschopoulos (1985) and Mathai (1982).
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Submitted: 18 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 (see for instance Casella and Berger (2002)). But what happens if the observations are neither
independent nor identically distributed. What can we say? Can we in particular compute explicitly the first two
moments of the sample mean and hence generalize
formulae provided in Tukey (1957a), Tukey (1957b) for the first two moments of the sample variance?
We also know that the sample mean and variance are independent if they are computed on an i.i.d. normal distribution.
This is one of the underlying assumption to derive the Student distribution Student alias W. S. Gosset (1908). But does
this result hold for any other underlying distribution? Can we still have independent sample mean and variance if the distribution
is not normal? This paper precisely answers these questions and extends previous work of Cho, Cho, and Eltinge (2004). We are able to
derive a general formula for the first two moments and variance of the sample variance under no specific assumption. We also provide a
faster proof of a seminal result of Lukacs (1942) by using the log characteristic function of the unbiased sample variance estimator.
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Submitted: 11 September, 2018

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Authors: __Eric Benhamou__

Abstract: In this paper, we discuss the
distribution of the t-statistic under the assumption of normal autoregressive
distribution for the underlying discrete time process. This result generalizes the classical
result of the traditional t-distribution where the underlying discrete time process follows an uncorrelated normal
distribution. However, for AR(1), the underlying process is correlated.
All traditional results break down and the resulting t-statistic is a new distribution
that converges asymptotically to a normal. We give an explicit formula for this new distribution
obtained as the ratio of two dependent distribution (a normal and the distribution of the norm of another
independent normal distribution). We also provide a modified statistic that follows a non central t-distribution.
Its derivation comes from finding an orthogonal basis for the the initial circulant Toeplitz covariance matrix.
Our findings are consistent with the asymptotic distribution for the t-statistic derived for the asympotic case of large
number of observations or zero correlation. This exact finding of this distribution has applications in multiple fields and
in particular provides a way to derive the exact distribution of the Sharpe ratio under normal AR(1) assumptions.
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Submitted: 11 September, 2018

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Authors: __Eric Benhamou__ and __Valentin Melot__

Abstract: This paper revisits the Pearson
Chi-squared independence test. After presenting the underlying theory with modern notations
and showing new way of deriving the proof, we describe an innovative and intuitive graphical presentation
of this test. This enables not only interpreting visually the test but also measuring how close or far we are
from accepting or rejecting the null hypothesis of non
independence.
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Submitted: 3 September, 2018

Access to the SSRN paper

Authors: __Eric Benhamou__ and __Beatrice Guez__

Abstract: We present a new methodology of computing
incremental contribution for performance ratios for portfolio like Sharpe, Treynor, Calmar
or Sterling ratios. Using Euler's homogeneous function theorem, we are able to decompose these
performance ratios as a linear combination of individual modified performance ratios. This allows
understanding the drivers of these performance ratios as well as deriving
a condition for a new asset to provide incremental performance for the portfolio. We provide various
numerical examples of this performance ratio decomposition.
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Submitted: 26 August, 2018

Access to the SSRN paper

Authors: __Eric Benhamou__, __David Saltiel__ and __Sebastien Verel__

Abstract: This paper introduces a novel theoretically sound
approach for the celebrated CMA-ES algorithm. Assuming the parameters of the multi variate normal
distribution for the minimum follow a conjugate prior distribution, we derive their optimal update
at each iteration step. Not only provides this Bayesian framework a justification for the update of
the CMA-ES algorithm but it also gives two new
versions of CMA-ES either assuming normal-Wishart or normal-Inverse Wishart
priors, depending whether we parametrize the likelihood by its covariance or
precision matrix. We support our theoretical findings by numerical experiments
that show fast convergence of these modified versions of CMA-ES.
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Submitted: 2 April, 2019