arXiv subjects

Eric Benhamou

Publications and source records attributed to Eric Benhamou.

35 records · Page 2Linked to original sources

arXiv2019

Connecting Sharpe ratio and Student t-statistic, and beyond

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.

q-fin.ST↗

arXiv2019

NGO-GM: Natural Gradient Optimization for Graphical Models

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.

cs.LG↗

arXiv2019

Similarities between policy gradient methods (PGM) in Reinforcement learning (RL) and supervised learning (SL)

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 influence the next state as opposed to SL (and USL) where, the next state remains the same, regardless of the decisions taken, either in batch or online learning. Although this difference is fundamental between SL and RL, there are connections that have been overlooked. In particular, we prove in this paper that gradient policy method can be cast as a supervised learning problem where true label are replaced with discounted rewards. We provide a new proof of policy gradient methods (PGM) that emphasizes the tight link with the cross entropy and supervised learning. We provide a simple experiment where we interchange label and pseudo rewards. We conclude that other relationships with SL could be made if we modify the reward functions wisely.

cs.LG↗

arXiv2019

BCMA-ES II: revisiting Bayesian CMA-ES

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.

cs.LG↗

arXiv2019

BCMA-ES: A Bayesian approach to CMA-ES

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.

cs.LG↗

arXiv2019

A discrete version of CMA-ES

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.

cs.LG↗

arXiv2019

Operator norm upper bound for sub-Gaussian tailed random matrices

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 $O(\sqrt n )$ 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.

math.PR↗

arXiv2018

Incremental Sharpe and other performance ratios

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.

q-fin.PM↗

arXiv2018

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

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.

q-fin.ST↗

arXiv2018

Trade Selection with Supervised Learning and OCA

In recent years, state-of-the-art methods for supervised learning have exploited increasingly gradient boosting techniques, with mainstream efficient implementations such as xgboost or lightgbm. One of the key points in generating proficient methods is Feature Selection (FS). It consists in selecting the right valuable effective features. When facing hundreds of these features, it becomes critical to select best features. While filter and wrappers methods have come to some maturity, embedded methods are truly necessary to find the best features set as they are hybrid methods combining features filtering and wrapping. In this work, we tackle the problem of finding through machine learning best a priori trades from an algorithmic strategy. We derive this new method using coordinate ascent optimization and using block variables. We compare our method to Recursive Feature Elimination (RFE) and Binary Coordinate Ascent (BCA). We show on a real life example the capacity of this method to select good trades a priori. Not only this method outperforms the initial trading strategy as it avoids taking loosing trades, it also surpasses other method, having the smallest feature set and the highest score at the same time. The interest of this method goes beyond this simple trade classification problem as it is a very general method to determine the optimal feature set using some information about features relationship as well as using coordinate ascent optimization.

cs.LG↗

arXiv2018

Feature selection with optimal coordinate ascent (OCA)

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.

stat.ML↗

arXiv2018

Gram Charlier and Edgeworth expansion for sample variance

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).

math.ST↗

arXiv2018

A few properties of sample variance

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.

math.ST↗

arXiv2018

T-statistic for Autoregressive process

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.

math.ST↗

arXiv2018

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

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

math.ST↗

arXiv2018

Trend without hiccups: a Kalman filter approach

Have you ever felt miserable because of a sudden whipsaw in the price that triggered an unfortunate trade? In an attempt to remove this noise, technical analysts have used various types of moving averages (simple, exponential, adaptive one or using Nyquist criterion). These tools may have performed decently but we show in this paper that this can be improved dramatically thanks to the optimal filtering theory of Kalman filters (KF). We explain the basic concepts of KF and its optimum criterion. We provide a pseudo code for this new technical indicator that demystifies its complexity. We show that this new smoothing device can be used to better forecast price moves as lag is reduced. We provide 4 Kalman filter models and their performance on the SP500 mini-future contract. Results are quite illustrative of the efficiency of KF models with better net performance achieved by the KF model combining smoothing and extremum position.

q-fin.TR↗

arXiv2008

Smart expansion and fast calibration for jump diffusion

Using Malliavin calculus techniques, we derive an analytical formula for the price of European options, for any model including local volatility and Poisson jump process. We show that the accuracy of the formula depends on the smoothness of the payoff function. Our approach relies on an asymptotic expansion related to small diffusion and small jump frequency/size. Our formula has excellent accuracy (the error on implied Black-Scholes volatilities for call option is smaller than 2 bp for various strikes and maturities). Additionally, model calibration becomes very rapid.

q-fin.PR↗