Hojjat Rakhshani

Lhassane Idoumghar

Julien Lepagnot

Mathieu Brévilliers
email: mathieu.brevilliers@uha.fr2e.c.keedwell@exeter.ac.uk

Edward Keedwell

Automatic hyperparameter selection in Autodock

Keywords: Surrogate-assisted computation, Autodock, docking, multi-objective optimization

Autodock is a widely used molecular modeling tool which predicts how small molecules bind to a receptor of known 3D structure. The current version of AutoDock uses metaheuristic algorithms in combination with local search methods for doing the conformation search. Appropriate settings of hyperparameters in these algorithms are important, particularly for novice users who often find it hard to identify the best configuration. In this work, we design a surrogate based multiobjective algorithm to help such users by automatically tuning hyperparameter settings. The proposed method iteratively uses a radial basis function model and non-dominated sorting to evaluate the sampled configurations during the search phase. Our experimental results using Autodock show that the introduced component is practical and effective.

I. INTRODUCTION

AutoDock is a protein-ligand docking tool which has now been distributed to more than 29000 users around the world [START_REF] Trott | Autodock vina: improving the speed and accuracy of docking with a new scoring function, efficient optimization, and multithreading[END_REF]. The application for this software arises from problems in computer-aided drug design. The ideal procedure would optimize the interaction energy between the substrate and the target protein. It employs a set of meta-heuristic and local search optimization methods to meet this demand. The Autodock puts protein-ligand docking at the disposal of nonexpert users how often do not know how to choose among the dozens of configurations for such algorithms. This motivated us to bring the benefits of hyperparameter tuning to users of AutoDock.

The global optimization problem arises in many real-world applications as well as molecular docking. A standard continuous optimization problem seeks a parameter vector x * to minimize an objective function f (x) : R D → R, i.e. f (x * ) < f (x) for all x ∈ Ω, where Ω = R D denotes the search domain (a maximization problem can be obtained by negating the f ). Over the years, optimization algorithms have been effectively applied to tackle this problem. These algorithms, however, may need to be fine-tuned which is a major challenge to their successful application. This is primarily due to the highly nonlinear and complex properties associated with the objective function. In this context, the advantages of algorithm configuration techniques become clear. *This work is accepted at 2018 IEEE International Conference on Bioinformatics and Biomedicine.

The hyperparameter tuning domain has been dominated by model-based optimization [START_REF] Hutter | Sequential modelbased optimization for general algorithm configuration[END_REF] that adopt probabilistic surrogate models to replace in part the original computationally expensive evaluations. They construct computationally cheap to evaluate surrogate models in order to provide a fast approximation of the expensive fitness evaluations during the search process. Surrogate based techniques try to model conditional probability p(y|ϕ) of a m-dimensional configuration ϕ given n observations S: (1) with the corresponding evaluation metrics y: y = y (1) , ..., y (n) T = y(x (1) ), ..., y(x

S = x (1) , ..., x (n) T ∈ R n×m , x = {x 1 , ..., x m } ∈ R m
(n) ) T ∈ R n (2)
The essential question arises in model based algorithms is which individuals should be chosen to be evaluated using the exact fitness function. This is most characterized by making a good balance between the exploration and exploitation capabilities. One of the earliest researches in this direction was performed by Jones et al. who adopted a Kriging surrogate to fit the data and make a balance between exploration and exploitation [START_REF] Jones | Efficient global optimization of expensive black-box functions[END_REF]. The exploration property of their proposed efficient global optimization (EGO) algorithm is enhanced by the fact that the expected improvement (EI) (an acquisition function) is conditioned on points with large uncertainty and low values of the surrogate. In another study, Booker et al. [START_REF] Booker | A rigorous framework for optimization of expensive functions by surrogates[END_REF] sought out a balanced search strategy by taking into account sample points with low surrogate predictions and high mean square error. Moreover, Wang et al. [START_REF] Wang | Mode-pursuing sampling method for global optimization on expensive black-box functions[END_REF] introduced the modepursuing approach which favors the trial points with low surrogate values according to a probability function. Regis and Shoemaker [START_REF] Regis | Constrained global optimization of expensive black box functions using radial basis functions[END_REF] also put forward an approach according to which the next candidate point is chosen to be the one that minimizes the surrogate value subject to distance from previously evaluated points. The distance starts from a high value (global search) and ends with a low value (local search). They also proposed Stochastic Response Surface (SRS) [START_REF] Regis | A stochastic radial basis function method for the global optimization of expensive functions[END_REF] algorithm that cycles from emphasis on the objective to emphasis on the distance using a weighting strategy. The SRS, however, mitigated some of the requirements for inner acquisition function optimization. For this reason, we focused on proposing a new algorithm configuration approach based on SRS model. The proposed algorithm, called as MO-SRS, modify the SRS so as to be able to handle multi-objective problems. More precisely, MO-SRS is equipped with the idea of multi-objective particle swarm optimization (PSO) [START_REF] Coello | Mopso: a proposal for multiple objective particle swarm optimization[END_REF]. We used the MO-SRS to make a balance between intermolecular energy and the Root Mean Square Deviation (RMSD) during the hyperparameter optimization in Autodock.

The rest of the paper is organized as follows. Section II provides us with a brief review on the related works. Section III gives a brief description for the docking problem. Section IV elaborates technical details of our proposed approach. In Section V, the performance of the introduced components is investigated by conducting a set of experiments. The last section summarizes the paper and draws conclusions.

II. RELATED WORKS

Sequential Model-based Algorithm Configuration (SMAC) [START_REF] Hutter | Sequential modelbased optimization for general algorithm configuration[END_REF], Hyperband [START_REF] Li | Hyperband: A novel bandit-based approach to hyperparameter optimization[END_REF], Spearmint [START_REF] Snoek | Practical bayesian optimization of machine learning algorithms[END_REF], F-race [START_REF] Birattari | F-race and iterated f-race: An overview[END_REF] and Treestructure Parzen Estimator (TPE) [START_REF] Bergstra | Algorithms for hyper-parameter optimization[END_REF] are examples of well known methods for hyperparameter optimization. SMAC adopted a random forests model and Expected Improvement (EI) to compute p(y|ϕ). Similarly, TPE et al. defined a configuration algorithm based on tree-structure Parzen estimator and EI. To tackle the curse of dimensionality, TPE assigns particular values of other elements to the hyperparameters which are known to be irrelevant. Ilievski et al. [START_REF] Ilievski | Efficient hyperparameter optimization for deep learning algorithms using deterministic rbf surrogates[END_REF] proposed a deterministic method which employs dynamic coordinate search and radial basis functions (RBFs) to find most promising configurations. By using the RBFs [START_REF] Park | Universal approximation using radial-basisfunction networks[END_REF] as surrogate model, they mitigated some of the requirements for inner acquisition function optimization. In another work [START_REF] Snoek | Scalable bayesian optimization using deep neural networks[END_REF], the authors put forward neural networks as an alternative to Gaussian process for modeling distributions over functions. Interestingly, Google introduced Google Vizier [16], an internal service for surrogate based optimization which incorporates Batched Gaussian Process Bandits along with the EI acquisition function.

Although the above-mentioned approaches have been proven successful, they are not able to handle several objective functions during the tuning process. Consequently, this paper presents a novel multi-objective algorithm which integrates the idea of a multi-objective meta-heuristics with the SRS method to find a subset of promising configurations.

III. MOLECULAR DOCKING

Molecular docking enables us to find an optimal conformation between a ligand L and receptor R. In another word, it predicts the preferred orientation of L to R when bound to each other in order to form a stable complex. A schematic example of this procedure is illustrated in Fig. 1. This problem can be formulated as a multi-objective problem consisting of minimizing the RMSD score and the binding energy E inter . In Autodock, energy function E inter is defined as follows [START_REF] Trott | Autodock vina: improving the speed and accuracy of docking with a new scoring function, efficient optimization, and multithreading[END_REF]:

E inter = Q R-L bound + Q R-L unbound (3)
Fig. 1: Schematic example of docking a ligand to a protein

Q = W vdw i,j A ij r 12 ij - B ij r 6 ij + W hbond i,j E(t) C ij r 12 ij - D ij r 10 ij + W elec i,j q i q j ε(r ij )r ij ) + W sol i,j (S j V i + S i V j )e (- r 2 ij 2σ 2 )
In ( 3), Q R-L bound and Q R-L unbound represent the states of ligandprotein complex in bound and unbound modes, respectively.

The pairwise energetic terms take into the account evaluations for repulsion (vdw), electrostatics (elec), desolvation (sol) and hydrogen bonding (hbond). The constant weights W vdw , W elec , W hbond and W sol belong to Van der Waals, electrostatic interactions, hydrogen bonds and desolvation, respectively. Moreover, r ij , A ij , B ij , C ij and D ij denotes Lennard-Jones parameters, while E(t) function is the angledependent directionality. Also, V is the volume of atoms that surround a given atom, weighted by a solvation parameter (S). For a detailed report of all the variables please refer to [START_REF] Morris | Autodock4 and autodocktools4: Automated docking with selective receptor flexibility[END_REF].

IV. THE PROPOSED METHOD

This section discusses in detail the main components of the proposed MO-SRS. We extend the idea of stochastic RBF to the multi-objective case to be suitable for the Autodock application. Compared to the evolutionary algorithms like genetic algorithm, MO-SRS need less computational time by virtue of surrogate modeling techniques. A workflow of the introduced approach is illustrated in Fig 2.

The MO-SRS starts the optimization procedure by generating a set of random configurations (i.e., initial population). During this phase, it might be possible to miss a considerable portion of the promising area due to the high dimensionality of the configuration space. It should be noticed that we have a small and a fix computational budget and increasing size of the initial population cannot remedy the issue. Furthermore, it is crucial for a model-based algorithm to efficiently explore the search space so as to approximate the nonlinear behavior of the objective function. The Design of Computer Experiments (DoCE) methods are often used to partially mitigate high dimensionality of the search space. Among them, MO-SRS Fig. 2: Workflow of the introduced automatic hyperparameter tuning approach for Autodock utilized the Latin Hypercube Sampling (LHS) [START_REF] Mckay | Comparison of three methods for selecting values of input variables in the analysis of output from a computer code[END_REF] which is a DoCE method for providing a uniform cover on the search space using a minimum number of population. The main advantage of LHS is that it does not require an increased initial population size for more dimensions.

As the next step, we evaluate all the generated configurations x i (i = 1, 2, • • • , n) using the Autodock to yield outcomes y

(1) i = f (1) (x i ) and y (2) i = f (2) (x i ).
Here, f 1 and f 2 denote the energy and RMSD function variables, respectively. Thereafter, we adopt surrogate model techniques to approximate the fitness function using this data set.

Surrogate models offer a set of mathematical tools for predicting the output of an expensive objective function f . Particularly, they try to predict the fitness function f for any unseen input vector x according to computed data points (x i , y i ). Given solution x and objective function f , a surrogate model f can be defined as in [START_REF] Booker | A rigorous framework for optimization of expensive functions by surrogates[END_REF], where is the approximation error.

f (x) = f (x) + (4)
Among different surrogate models, we utilized the Radial Basis Function (RBF) which is a good model for high dimensional problems [START_REF] Park | Universal approximation using radial-basisfunction networks[END_REF], [START_REF] Díaz-Manríquez | On the selection of surrogate models in evolutionary optimization algorithms[END_REF]. It belongs to an interpolation method for scattered multivariate data which considers all the sample points. In this light, it introduces linear combinations based on a RBF h(x) to approximate the desired response function f ; as presented in [START_REF] Wang | Mode-pursuing sampling method for global optimization on expensive black-box functions[END_REF].

f (x) = n i=1 w i h(|x -x i |) + b x + a (5)
In [START_REF] Wang | Mode-pursuing sampling method for global optimization on expensive black-box functions[END_REF], w i shows i th unknown weight coefficient, h determines a radial basis function, x is an unseen point and are independent errors with a variance σ 2 . A radial function h : R m → R has the property h(x) = h( x ). By having a suitable kernel h; the unknown parameters a, b and w could be obtained by solving the following system of linear equations:

Φ P P 0 w c = f ψ 0 (6)
Here, Φ is a n × n matrix with Φ i,j = h( x i -x i ), c = (b , a) and

P = x 1 x 2 • • • x n 1 1 • • • 1 (7)
The linear system (6) has a unique solution and it can be used to approximate function values.

As opposite to the standard SRS method, we have two main objective functions which should be optimized: the intermolecular energy and the RMSD. Hence, we repeat the above mentioned procedure for both the objective functions. More precisely, we will train two surrogate models f1 and f2 to approximate the energy and RMSD output variables, respectively. Now we are ready to use the trained surrogate models inside a pre-selection strategy. Accordingly, we generate a set of N candidates to be evaluated using f1 and f2 . In the single-objective SRS code, the following equation generates neighborhoods using the current best solution f2 , a randomly produced vector v ∈ R m and adaptive parameter γ:

σ j = x best + γ ⊗ v, j = 1...N (8)
Generally speaking, in multi-objective optimization the best solutions need to be chosen based on two objectives and there is no single best solution. So, we cannot directly apply the standard search operators of the SRS. Moreover, we have to update our surrogate models f1 and f2 based on a best obtained solution which is another problem for the MO-SRS. Having this in mind, we borrowed the idea of leader selection in multi-objective PSO [START_REF] Coello | Mopso: a proposal for multiple objective particle swarm optimization[END_REF] algorithm which handles the same problem (update equation of the PSO also depends on the global best solution and finding a leader particle is an important task). Inferred from [START_REF] Coello | Mopso: a proposal for multiple objective particle swarm optimization[END_REF], MO-SRS first builds an archive which contains the Pareto-optimal solutions. A solution vector is Pareto-optimal if there is not another solution that dominates it. In mathematical terms, solution x 1 is said to dominate another solution x 2 , if:

• f i (x 1 ) f i (x 2 ) for all i ∈ {1, 2, ..., k} and • f i (x 1 ) f i (x 2 ) for at least one i ∈ {1, 2, ...

, k}

The MO-SRS divides the objective space into hypercubes and assigns a fitness value to each hypercube based on the number of Pareto-optimal solutions that lie in it. Then, it employs the roulette-wheel to find the superior hypercube. A randomly selected solution from that hypercube is determined to be the best solution x best . All the N candidates are evaluated by the previously trained surrogates f1 and f2 . Again, we select the best solution according to the above mentioned strategy. This best solution is then used to update our surrogate models. All the aforementioned steps continue until some stopping criteria are met. To sum up, MO-SRS approach is illustrated in Fig. 3. The MO-SRS will be available soon on https://github.com/ML-MHs/Auto-Autodock.

Algorithm 1 MO-SRS Use LHS to initialize a population X ← {x 1 , • • • , x n0 } n ← n 0 y (1) i = f (1) (x i ) : i ← 1 • • • n y (2) i = f (2) (x i ) : i ← 1 • • • n repeat
Find Pareto-optimal P using y

(1) i and y

(2) i Set x best using P and described hypercube technique Fit surrogate f (1) by B n ← x i , y

: i ← 1 • • • n Fit surrogate f (2) by C n ← x i , y (1) i
: i ← 1 • • • n Generate new solutions σ j : j ← 1 • • • N as in (8) (2) i
Apply the bound constraints on each solution σ j ŷ(1)

j = f (1) (σ j ) : j ← 1 • • • N ŷ(2) j = f (2) (σ j ) : j ← 1 • • • N
Find Pareto-optimal P using ŷ(1) and ŷ(2) Set x n+1 using P and described hypercube technique y

(1)

n+1 = f (1) (x n+1 ) y (2) n+1 = f (2) (x n+1 ) B n+1 ← B n ∪ (x n+1 , y (1) n+1 ) C n+1 ← C n ∪ (x n+1 , y (2)
n+1 ) n ← n + 1 until stopping criteria are met Fig. 3. Pseudocode of the MO-SRS method

V. EXPERIMENTAL RESULTS

In this section, the performance of MO-SRS hyperparameter tuning method on Autodock task is investigated. We considered two scenarios from Autodock documentations which should be minimized: 1)1DWD and 2) HSG1. All simulations are performed using Matlab. We adopted the same implementation and parameter configuration for the SRS as suggested in [START_REF] Müller | Matsumoto: The matlab surrogate model toolbox for computationally expensive black-box global optimization problems[END_REF]. The Cubic function is used as the kernel in the RBF.

We applied the MO-SRS to tune the hyperparameters of Autodock for the genetic algorithm (GA) [START_REF] Morris | Automated docking using a lamarckian genetic algorithm and an empirical binding free energy function[END_REF], simulated annealing (SA) [START_REF] Kirkpatrick | Optimization by simulated annealing[END_REF], a local search (LS) called as pseudo-Solis-Wet [START_REF] Trott | Autodock vina: improving the speed and accuracy of docking with a new scoring function, efficient optimization, and multithreading[END_REF] and a hybrid method (HB) which combines the GA and LS. The GA is a well-known evolutionary optimization algorithm which is highly sensitive ot the initial value of its parameters. The performance of the GA is dependent on the structure of the problem at hand and do not scale well with complexity. Consequently, parameters of GA like crossover, population size and mutation should be chosen with care. A small population size will lead to the early convergence problem, while using a large population increases the computational cost. The configuration tuning of GA becomes even more challenging when there is a correlation between its hyperparameters. For example, mutation is more effective on smaller population sizes, while the crossover is likely to benefit from large populations. All the mentioned reasons make the GA a suitable algorithm for benchmarking the performance of MO-SRS. The same situation could happen for the considered LS and SA algorithms. From other points of view, the adopted algorithms are introduced to measure search performance of the MO-SRS under different dimensions. The LS and GA belong to low dimensional problems, SA is a medium dimension and HB algorithm is a high dimensional hyperparameter tuning problem. A detailed information for the optimized configurations for GA, SA and LS are presented in Tables I-III, respectively.

The obtained results are reported in Tables IV andV. The MO-SRS should obtain a reliable search performance using a limited computational budget and so the number of evaluations is set to 100. For each algorithm, the search bounds are reported in Tables I-III. To reduce the influence of stochastic error, experiments are repeated 10 times for each problem. In Tables IV-V, the tuned prefix denotes the optimized algorithms using the MO-SRS. The MO-SRS offers a set of final solutions for each of the cases and we used a Pareto-optimal graph to illustrate such solutions; as depicted in Fig. 4. However, it should be noticed that the results offered by the default methods in Autodock are based on single-objective algorithms. For this reason, we compared the obtained results of the MO-SRS according to each of those objectives. The subindex 1 in tables IV-V shows the performance of the compared algorithms for the binding energy and subindex 2 denotes the same for the RMSD criterion. The results are averaged over 10 runs. The energy unit is in kcal/mol and RMSD unit is in Angstroms.

The results of the first case study are presented in Table IV. As it can be seen, MO-SRS converges closer to the global optimum. In the case of LS method, we can see that the tuned algorithm yields a minimized energy -17.1530 and RMSD 0.0860. In docking domain, RMSD < 1.5 Angstrom is always wise to consider. Similarly, MO-SRS provides more accurate results for the second scenario according to the results in Table V.

In these tables, we can see how the introduced method tries to find more than one objective simultaneously. In this regard, there are two main points. The first one is that the diversity of the obtained solutions strongly depends on the performance of the algorithm at hand. For example, LS performs better than the GA and consequently we can see the Pareto-optimal for LS contains more solution. The second point is that all Paretooptimal solutions are obtained only after 100 evaluations. This confirms our hypothesis that MO-SRS can effectively be use to tune hyperparameters of the Autodock. The main components of MO-SRS are the adopted multi-objective and surrogate modeling techniques which can be further improved in the future works.

TABLE I :

 I Details for the optimized hyperparameters of GA

	Hyperparameter	Type	Range	Default
	seed	Integer	[0,10000000] Random
	ga pop size	Integer	[50,500]	150
	ga elitism	Binary	0,1	1
	ga mutation rate	Continuous [0.2,0.99]	0.02
	ga crossover rate Continuous [0.2,0.99]	0.80

VI. CONCLUSION A typical docking scenario in Autodock includes applying optimization algorithms. Following those steps, users should select a set of appropriate hyperparameters to maximize the performance of their final results. As it is often beyond the abilities of novice users, we proposed to develop a multiobjective approach for automatically tuning highly parametric algorithms of the Autodock. Automating end-to-end process of applying the proposed MO-SRS offers the advantages of more accurate solutions. The experimental results clearly show that the obtained results by MO-SRS outperform hand-tuned models. We hope that the introduced MO-SRS helps nonexpert users to more effectively apply Autodock to their applications.


Lourme Valentin

Analyse de la prédiction d'une Nappe de Volatilité par machine Learning

Keywords: Machine Learning, Volatilité, Nappe, Interpolation, Prédiction (5 maximum): Machine Learning, Volatility, Surface, Interpolation, Prediction

analysis of the predictor of a volatility surface by machine learning

Valentin Lourme

Introduction (thème de l'article) : Réalisation d'une étude qui compare les méthodes d'interpolation traditionnelles avec les algorithmes de Machine Learning pour prédire les données manquantes dans les nappes de volatilité. L'objectif est d'améliorer les estimations de volatilité et les méthodes de valorisation des produits financiers. Question de recherche : Les Méthodes de Machine Learning sont-elles adaptées à la construction d'une surface de volatilité qui ne soit pas arbitrable et qui permettent une estimation en accord avec les évènements présents sur le marché ? Méthode : Comparaison entre la prédiction du Machine Learning et l'interpolation par spline Cubique. Récupération des données de nappes de Volatilité sur les mois de Mars, Avril et Mai 2023. Entrainement du modèle de Machine Learning le plus adapté (XGBoost) sur les données historiques et comparaison des deux approches sur des points manquant d'une nappe de volatilité en date du 1 er juin 2023 Résultats : Obtention de 4 courbes de comparaison entre le modèle de Machine Learning et l'interpolation par spline cubiques. Ces courbes sont étudiées pour des valeurs de strikes autour de 100%. L'erreur entre chaque modèle et les données de volatilité observées permet de mesurer l'écart de performance entre les deux approches. Conclusion (synthèse de l'article) : Le modèle de machine learning (XGBoost) est recommandé pour des valeurs de maturités courtes et des strikes proches de 100%, tandis que l'interpolation par spline cubique est préférable pour des maturités plus longues. Cependant, les résultats dépendent des données d'entrée, et le modèle de machine learning peut souffrir de sur-apprentissage.

I. Introduction

L'avènement des nappes de volatilité dans le domaine de la finance de marché est étroitement lié à une avancée révolutionnaire dans la valorisation des produits dérivés : le modèle Black-Scholes. Développé par Fischer Black et Myron Scholes dans les années 1970, ce modèle a introduit une nouvelle approche mathématique pour évaluer les options et a apporté les bases de la gestion moderne des risques [START_REF] Black | The pricing of options and corporate liabilities[END_REF].

Le [START_REF] Breiman | Random Forests[END_REF].

1. Les Arbres de décisions À l'origine, les arbres de décision ont été conçus pour résoudre des problèmes de classification. Cependant, ils ont depuis été adaptés pour traiter des problèmes de régression. En utilisant un ensemble de données mesurables et normalisées, un algorithme basé sur les arbres de décision cherche à créer des noeuds de décision afin de diviser de manière optimale les données d'entrée et prédire une valeur 𝑦. Les arbres de régression cherchent à prédire une valriable y par rapport à un jeu de données d'entrée X. La moyenne 𝑦 ̅ est calculée sur l'ensemble des données y. Les données sont classifiées en fonction des noeuds ou critères de décision, puis un calcul de la moyenne pour chaque sous-catégorie est effectué. Ainsi, lorsqu'une nouvelle donnée est étudiée, elle est classifiée en fonction des différents noeuds de décision, et la valeur prédite est la moyenne de la sous-catégorie à laquelle cette donnée appartient [START_REF] Maimon | Data Mining and Knowledge Discovery Handbook Decision Trees[END_REF]. À chaque noeud de décision, un indice de pureté appelé MSE (Mean Squared Error) est calculé pour évaluer la précision de la régression pour ce noeud.

𝑀𝑆𝐸² = ∑ |𝑦 ̅ -𝑦 𝑖 𝑖∈𝑁𝑜𝑒𝑢𝑑𝑠 |²
La fiabilité de l'estimation des valeurs y par l'arbre de régression augmente à mesure que l'erreur moyenne de prédiction (MSE) diminue. Une fonction de coût est attribuée à chaque noeud, et en ajustant les conditions de chaque noeud, il est possible de réduire au minimum cette fonction de coût.

Cette fonction notée J a la forme suivante :

C. L'interpolation par spline cubique

Lorsque les nappes de volatilité ont pu être estimées à partir du modèle LSVi, il est possible d'interpoler les données de volatilité en certains points précis par la méthode du spline cubique. Cette approche consiste à construire une courbe lisse à partir des points de volatilité disponibles, en utilisant des polynômes de degré trois (cubiques) entre les points de données adjacents. L'interpolation par spline cubique offre plusieurs avantages. Tout d'abord, elle permet de combler les lacunes dans les données de volatilité en fournissant des estimations pour les points manquants. Cette méthode est particulièrement utile lorsque les données disponibles sont dispersées ou incomplètes. En utilisant des polynômes cubiques, l'interpolation par spline permet de capturer les variations complexes de la volatilité dans les nappes. Elle produit ainsi une courbe lisse qui s'ajuste mieux aux données disponibles, ce qui conduit à une estimation plus précise de la volatilité.

Lors de la résolution d'un problème d'interpolation, le but est de trouver une fonction S qui passe par n points donnés. Néanmoins, le phénomène de Runge se présente lorsque le nombre de points à interpoler augmente, notamment lors de l'application de l'interpolation de Lagrange (J.P. Demailly, 2006). En effet, dans certains cas les polynômes interpolateurs de Lagrange ne constituent pas une approche d'approximation efficace lorsque le nombre de points n augmente.

Pour éviter le phénomène de Runge, une approche alternative serait d'utiliser plusieurs polynômes de degré inférieur ou égal à trois sur chaque intervalle entre les points d'interpolation, puis de les assembler pour définir une fonction S sur l'intervalle complet. La fonction S doit satisfaire des critères de continuité et de dérivabilité spécifiques. Cela implique que le degré des polynômes interpolateurs augmente. L'objectif est de trouver un compromis optimal, et celui qui est généralement adopté est celui des splines cubiques.

Soient 𝐾 1 , . . . , 𝐾 𝑛 , 𝜎 1 , . . . , 𝜎 𝑛 des nombres réels. On appelle Spline cubique associé à la famille (𝐾 𝑖 , 𝜎 𝑖 ) toute fonction S de classe C² polynomiale de degré au plus trois sur chacun des intervalles [𝐾 𝑖 , 𝐾 𝑖+1 [ et telle que pour tout i dans {1, …, n} S(𝐾 𝑖 ) = 𝜎 𝑖 .

Une telle fonction S nécessite quatre coefficients à déterminer sur chacun des n-1 intervalles ce qui fait un total de 4n-4 coefficients à définir. Afin de déterminer ces coefficients, on applique des conditions aux limites aux points à interpoler sur chaque intervalle, ce qui donne lieu à 2n-2 équations. De plus, nous souhaitons que la dérivée et la dérivée seconde de S soient continues en 𝐾 𝑖 , pour i allant de 2 à n-1, ce qui ajoute 2n-4 conditions. Pour les deux dernières conditions, nous imposons une contrainte de minimisation d'énergie, à savoir que les dérivées seconde aux points 𝐾 1 et 𝐾 𝑛 soient nulles.

Dans le cadre de cette étude, l'algorithme XGBoost sera appliqué pour estimer les valeurs manquantes de volatilité. En comparant les résultats obtenus par l'algorithme XGBoost avec les estimations produites par l'interpolation par spline cubique, nous serons en mesure d'évaluer la performance et l'exactitude des deux approches. L'interpolation par spline cubique n'est pas remise en question lorsque moins de deux points consécutifs sont manquants à la surface de volatilité.

Il reste alors à définir un intervalle de strike, un intervalle de maturité et un nombre de points consécutifs pour lesquels la méthode d'interpolation par spline cubique pourrait éventuellement être remplacée par un algorithme de Machine Learning. Si cet algorithme est correctement entraîné, il pourrait capturer la dynamique de la surface de volatilité en ces points spécifiques.

B. Domaine de définition de l'étude D'après les analyses de prédictions par spline cubique, ce sont les valeurs de volatilité pour les options à la monnaie qui sont les plus écartées vis-à-vis des valeurs réelles. L'analyse se focalise donc pour des strike en pourcent, proche de l'ATM. Pour le choix de la maturité et du nombre de points à prendre pour l'étude, l'erreur moyenne agit comme indicateur de la prédiction par spline cubique en fonction du nombre de points et de la maturité (en jours) :

𝐸𝑟𝑟(𝑛, 𝑇𝑇𝑀) = 1 𝑛 ∑ |𝑦 𝑘,𝑇=𝑇𝑇𝑀 -𝑦 𝑘,𝑇=𝑇𝑇𝑀 ̂| 𝑛 𝑘 = 1
Avec n qui représente le nombre de points enlevés autour du pourcentage de strike 100% (option à la monnaie), TTM correspond à la maturité de l'option, 𝑦 𝑘,𝑇=𝑇𝑇𝑀 représente la valeur réelle pour la valeur de strike K et de maturité TTM et 𝑦 𝑘,𝑇=𝑇𝑇𝑀 ̂ représente la valeur prédite par le spline cubique pour un strike K et une maturité TTM (en jours)

Le graphique suivant montre l'évolution de l'erreur en fonction du nombre de points enlevés et du 'Time To Maturity'.

C1 -Public N atixis D. Comparaison des deux modèles sur le domaine de définition de l'étude A partir des résultats de prédiction de l'algorithme « XGBoost regression » et des résultats de l'interpolation par spline cubique sur des données de volatilité basé sur des options vanilles européennes de maturité 60 jours (2 mois) pour des valeurs de strike comprises entre 80% et 120% (à la monnaie), pour sept points supprimes à estimer, nous pouvons calculer l'erreur de l'estimation de chacun de ces deux modèles par rapport à la volatilité observée pour en déduire la performance de chacun de ces deux modèle. En ce qui concerne l'algorithme de XGBoost, ce dernier réalise des prédictions avec une erreur plutôt faible, cependant, étant donné que l'erreur mesuré sur le jeu d'entrainement est négligeable devant celle mesurée sur le jeu de test, nous pouvons considérer que le modèle XGBoost rencontre un problème de sur-apprentissage.

VII. Conclusion

Pour conclure, Le modèle de machine Learning est à privilégier pour des valeurs de maturités faibles, inférieures à 1 an, pour des valeurs de strike autour de 100% tandis que l'interpolation par spline cubique est plus avantageuse pour des maturités élevées qui sont supérieures à un an en raison de l'allure hyperbolique de la courbe de volatilité. Cependant, ces résultats dépendent des données fournies en entrée. En effet, Le modèle s'est entrainé sur des données des trois mois de mars, Avril et Mai. Cet entrainement sur ces données spécifiques est à remettre en question au regard des évènements financiers majeurs qu'il y a pu y avoir sur le marché (faillite de la Silicon Valley Bank) pendant cette période. De plus, les caractéristiques que nous avons fournies au modèle à savoir la moyenne, l'écart-type, la valeur minimale et la valeur maximale sont critiquables car d'autres caractéristiques descriptives auraient pu être prises à la place, comme la tendance sur 7 jours. De plus le modèle de machine Learning est sujet à une problématique de sur-apprentissage.

  à bien cette étude comparative, il est essentiel de définir les données à récupérer ainsi que la plage d'historique spécifiée. J'ai pu collecter quotidiennement les données des nappes de volatilité pour l'indice CAC 40 pendant les mois de mars, avril et mai, correspondant à la volatilité observée sur le marché à la clôture. À titre d'exemple, voici la structure générale d'une nappe de volatilité ainsi que l'affichage de la nappe en date du 2 mai : Le "Strike Percent" représente la valeur du strike en pourcentage par rapport au forward. Ce strike Percent prend des valeurs comprises dans l'intervalle [0.05, 5]. Le « Time To Maturity » est la maturité des options portant sur le CAC40 mesurée en jours. Elle est calculée par différence entre la « Valuation Date » et la « Maturity ». Le 'Time To Maturity' varie selon les jours d'observations et elle prend souvent ses valeurs dans l'intervalle [1, 7200]. « La valuation Date » correspond au jour d'observation de la nappe et enfin le Forward correspond au prix Forward du contrat. Ces nappes récupérées sont donc assez complètes en termes d'informations fournies. Nous allons donc nous concentrer uniquement sur les intervalles de 'Strike Percent' et de 'Time To Maturity' sur lesquelles l'interpolation par spline cubique s'avère moins performante que sur les autres intervalles. C1 -Public N atixis On remarque que pour les maturités inférieures à 360 jours, pour des valeurs d'options à la monnaie, ou le strike en pourcent est proche de 1, on observe un écart entre les valeurs interpolées et les valeurs réelles plus important que sur le reste de la courbe lorsque sept points successifs de la courbe sont supprimés. Nous décidons donc de focaliser notre étude sur des maturités comprises entre 15 jours et 1an (360 jours) et pour des valeurs de strike en pourcent comprises dans l'intervalle [0.5, 1.5]. Ainsi la nouvelle nappe de notre étude aura la forme suivante :Le jeu de données de volatilité considéré est celui en date du 1 er juin 2023. Ces valeurs mesurent l'écart de performance entre les prédictions de l'algorithme de Machine Learning XGBoost et l'interpolation du spline cubique.Pour filtrer les valeurs aberrantes dans les données récupérées, j'ai vérifié que les données historiques récupérées sur les mois de mars, Avril et Mai étaient non-arbitrables. Ainsi dans le cadre de cette étude, l'objectif était de vérifier si les valeurs de volatilité des options hors de la monnaie étaient plus élevées que celles des options à la monnaie. Si ce n'est pas le cas, une stratégie d'arbitrage peut s'appliquer, la stratégie Call-Spread. Nous avons aussi cherché à vérifier si, pour les options à la monnaie, la volatilité augmentait avec la maturité. Dans le cas inverse, une stratégie d'arbitrage peut s'appliquer, il s'agit du Calendar-Spread. Mes données récupérées vérifiaient les deux conditions de non-arbitrage et donc ne contenaient pas de valeurs aberrantes pour l'entrainement du modèle de machine Learning. Ainsi dans la suite de cette étude, nous allons supposer que les données d'étude sont non-arbitrables. marché sont considérées comme non-arbitrables et adaptées à l'entraînement d'un modèle de Machine Learning. Pour la comparaison entre le modèle de Machine Learning et l'interpolation par spline cubique, ce sont les données en date du 1er juin 2023 qui serviront de référence pour évaluer les performances des deux méthodes.A. Limitation de l'interpolation par spline cubique La méthode d'interpolation par spline cubique est utilisée pour compléter les données manquantes dans les nappes de volatilité. Cependant, certaines limitations sont constatées lorsque l'interpolation par spline cubique tente d'interpoler plus de trois points consécutifs en fonction du strike pour une valeur de maturité fixée. De plus, l'erreur entre les points d'interpolation et les points de volatilité réellement observés est plus prononcée pour les options à la monnaie avec des maturités inférieures à 1 an. Cette observation peut être expliquée par la forme spécifique des courbes 𝜎(𝐾 | 𝑇 = 𝑇 𝑗 ) avec 𝑇 𝑗 la maturité comprise entre une journée et une année. L'Hypothèse principale formulée est alors la suivante :

  suivantes, nous avons tracé la volatilité observée sur le marché, la volatilité prédite par le modèle XGBoost ainsi que la volatilité interpolée par spline cubique en fonction du strike pour quatre maturités différentes. Les maturités sont comprises dans l'ensemble {60 jours, 120 jours, 180 jours, 360 jours}. Les erreurs entre les modèles et les valeurs observées sont référencées dans le tableau suivant : Nous pouvons observer sur la première ligne du tableau la maturité en jours ainsi que les quatre maturités caractéristiques de notre étude {60, 120, 180, 360}. Sur la deuxième ligne apparait l'erreur entre le modèle XGBoost et les valeurs de volatilité observées sur le marché pour chacune des maturités. Enfin, sur la troisième ligne figure l'erreur entre l'interpolation par spline cubique et les valeurs de volatilité du marché. Nous pouvons observer directement le modèle qui approche le mieux les données de marché avec la nuance de couleur pour chaque maturité. Chacune des valeurs de ce tableau représente un écart de volatilité moyen. C1 -Public N atixis L'évolution de l'erreur pour chacun des deux modèles est obtenue à l'aide du graphique suivant : Par ailleurs, l'histogramme suivant donne la répartition en pourcentage de l'importance de chacune des caractéristiques. Nous observons que la volatilité au jour d'étude occupe une place prépondérante pour la prédiction C1 -Public N atixis

  modèle Black-Scholes repose sur l'hypothèse selon laquelle la volatilité du prix d'un actif financier est constante et connue. Cependant, dans la réalité, la volatilité est une variable dynamique et peut varier au fil du temps. La prise en compte de cette réalité a conduit à l'émergence des nappes de volatilité (M. Britten-Jones et[START_REF] Britten-Jones | Option prices, implied price processes, and stochastic volatility[END_REF].Les nappes de volatilité sont des outils graphiques qui illustrent la relation entre la volatilité implicite, les prix d'exercice et les échéances des options. Elles permettent aux professionnels de la finance d'appréhender la structure et la dynamique de la volatilité sur une gamme d'échéances et de niveaux de prix. Ainsi, les nappes de volatilité fournissent une vision plus complète et plus précise de la volatilité que la simple considération d'une valeur constante.

	II.	Revue de littérature
	Un ensemble de techniques de prédiction a été testé sur le jeu de données via une technique de
	la librairie automl (lazypredict), Les techniques basées sur les arbres de décisions ont eu une
	meilleure précision de prédiction, de ce fait, uniquement ces techniques sont détaillées dans le
	chapitre suivant, ainsi que les techniques de références, splines cubique et LSVi
	A. Le Machine Learning
	Le Machine Learning, ou apprentissage automatique, est un domaine de l'intelligence
	artificielle qui vise à développer des algorithmes capables d'apprendre à partir de données et
	d'effectuer des tâches sans être explicitement programmés. En 1986, le chercheur Ross Quinlan
	développa un modèle basé sur des arbres de décisions pour effectuer une classification
	supervisée des données. Ce modèle constitue la base de développement des algorithmes de
	Random Forest (L.
		nous
	espérons contribuer à l'amélioration des méthodes de valorisation des produits financiers et à la
	prise de décisions plus éclairées dans le domaine de la finance.
	C1 -Public N atixis

En reconnaissant l'importance de la volatilité comme facteur clé dans la détermination des prix des produits dérivés et dans la gestion des risques, les acteurs de la finance de marché ont commencé à utiliser les nappes de volatilité pour améliorer leurs modèles d'évaluation et leurs stratégies de négociation. Ces nappes offrent une perspective plus nuancée et plus réaliste de la volatilité, permettant ainsi une meilleure estimation des prix et des risques associés aux options d'achat et de vente (D.

[START_REF] Backus | Accounting for biases in Black-Scholes[END_REF]

.

Les nappes de volatilité peuvent cependant être incomplètes en raison d'un manque d'information. Dans ce cas, les méthodes d'interpolations spline cubique ou LSVi sont généralement utilisées pour prédire les données n'ayant pas pu être récupérées. Une étude doit être menée spécifiquement sur ces nappes qui peuvent être comblées et prédites, en comparant les approches traditionnelles d'interpolation avec les avancées récentes des algorithmes de Machine Learning. En explorant ces approches, nous cherchons à améliorer la précision des estimations de la volatilité.

Cette étude permet de mieux comprendre les avantages et les limites des deux approches, ainsi que leur pertinence dans le contexte financier actuel. En évaluant la capacité des algorithmes de Machine Learning à prédire les données manquantes dans les nappes de volatilité,

  Pour un noeud de décision d'indice k donné, en notant 𝜔 𝑇𝑟𝑢𝑒 , 𝜔 𝐹𝑎𝑙𝑠𝑒 , les proportions de population présentes respectivement dans la branche ou la condition du noeud k est vérifiée et dans celle ou la condition du noeud k n'est pas vérifiée, et 𝑀𝑆𝐸 𝑇𝑟𝑢𝑒 , 𝑀𝑆𝐸 𝐹𝑎𝑙𝑠𝑒 , les erreurs moyennes de prédiction respectives des branches ou la condition du noeud k est vérifiée et ou cette condition n'est pas vérifiée, on obtient et dans leur facilité d'apprentissage. Cependant, les arbres de décisions sont très sensibles à des petites variations dans le jeu de données d'entrée et ont un problème de sur-apprentissage si trop de branches sont créées. Afin de surmonter ces limites, l'algorithme de la forêt aléatoire est apparu pour combler les lacunes des arbres de décisions.2. L'algorithme de Random Forest L'algorithme de Random Forest utilise l'« ensemble Learning » pour effectuer une prédiction. Les méthodes d'« ensemble Learning » utilisent plusieurs algorithmes d'apprentissage et intègrent les résultats de ces modèles pour améliorer les performances prédictives par rapport aux modèles individuels. Les arbres de décision qui composent une forêt aléatoire doivent satisfaire une condition importante : le critère de diversité de la population, c'est-à-dire qu'ils doivent avoir des noeuds de décision différents pour que les prédictions varient d'un arbre à l'autre. L'algorithme de la Random Forest fonctionne en créant plusieurs arbres de décision avec un faible nombre de noeuds qui s'entraînent individuellement sur des partitions aléatoires de l'ensemble de données (M.Belgiu et L. Dragut, 2016). Cette sélection aléatoire vise à maximiser la diversité des données. La prédiction de la forêt aléatoire est obtenue en prenant la moyenne des prédictions des arbres de décision. La limitation majeure de l'algorithme de la Random Forest réside dans sa partition aléatoire des données. Un nouvel algorithme va alors permettre de combler cette lacune : Le gradient Boosting Tree. Dans ce modèle, la volatilité est considérée comme une variable aléatoire qui peut changer au cours du temps. Le modèle 'Local Stochastic Volatility implied' suppose que la volatilité suit un processus stochastique (M.[START_REF] Lorig | Explicit implied volatilities for multifactor local stochastic volatility models[END_REF]. Ce modèle nécessite plusieurs paramètres pour être calibrer correctement. Actuellement, les experts utilisent ce modèle pour estimer la volatilité implicite des options par rapport au marché.

𝐽(𝑘) = 𝜔 𝑇𝑟𝑢𝑒 * 𝑀𝑆𝐸 𝑇𝑟𝑢𝑒 + 𝜔 𝐹𝑎𝑙𝑠𝑒 * 𝑀𝑆𝐸 𝐹𝑎𝑙𝑠𝑒 En ayant plusieurs règles de décision et en calculant le cout de chacune des règles, on cherche celle qui minimise la fonction J. L'avantage majeur des arbres de décisions réside dans leur C1 -Public N atixis simplicité d'utilisation 3. Le gradient Boosting Tree et le XGBoost

Le XGBoost est une implémentation de l'algorithme de gradient boosting tree. Cet algorithme commence par calculer la moyenne des données notée 𝑦 ̅. À partir de cette moyenne, il calcule l'erreur entre la moyenne et chaque valeur y de la série. Ensuite, il cherche à prédire l'erreur (également appelée résidu). En utilisant à la fois la moyenne et la prédiction de l'erreur, il arrive à une nouvelle prédiction de la donnée y (T.

Chen et C. Guestrin, 2016)

. Ce processus est répété jusqu'à ce que la prédiction de l'erreur fournisse une erreur globale acceptable. Le XGBoost présente tous les avantages des arbres de décision et du random forest en rajoutant le fait que le modèle pondère chacun des arbres de décision. Il apparait ainsi comme un modèle robuste de machine Learning que nous nous proposons d'utiliser dans le cadre de l'étude de prédiction des données manquantes sur une nappe de volatilité. L'objectif de cette étude consiste à utiliser l'algorithme XGBoost pour déterminer les données de volatilité manquantes, afin de comparer les prédictions générées par l'apprentissage automatique avec la méthode actuelle d'interpolation par spline cubique. B. Calibration de la nappe de volatilité par le modèle LSVi Le modèle 'Local Stochastic Volatility implied' (LSVi) est un modèle qui estime la volatilité d'un actif sous-jacent dans le temps. Lorsque les nappes de volatilité sont calculées par le modèle LSVi, il est alors possible d'interpoler certains points de volatilité par la méthode du spline cubique. C1 -Public N atixis

  𝜎 𝑘,𝑇=𝑇𝑇𝑀 est la valeur de volatilité observée sur le marché et 𝜎 𝑘,𝑇=𝑇𝑇𝑀 ̂ est la valeur estimée par le modèle XGBoost. La même formule s'applique pour l'interpolation par spline cubique en modifiant la valeur 𝜎 𝑘,𝑇=𝑇𝑇𝑀 ̂ par celle estimée par l'interpolation par spline cubique.

	L'erreur entre le modèle XGBoost et les données de volatilité observée est calculée selon la
	formule suivante :
	𝐸𝑟𝑟(𝑛, 𝑇𝑇𝑀) =	1 𝑛	𝑛 ∑ |𝜎 𝑘,𝑇=𝑇𝑇𝑀 -𝜎 𝑘,𝑇=𝑇𝑇𝑀 ̂| 𝑘 = 1
	Avec n=7, le strike K prenant ses valeurs dans l'ensemble {80%, 90%, 95%, 100%, 105%,
	110%, 120%}.

  VI. DiscussionsA partir des quatre courbes que nous observons, nous remarquons qu'à mesure que le temps à maturité augmente, l'erreur entre les données de volatilité observées sur le marché et l'interpolation par spline cubique diminue. Il en est de même pour l'erreur entre les prédictions XGBoost et les données observées. Nous remarquons que pour les maturités courtes {60 jours, 120 jours, 180 jours} l'algorithme de machine Learning XGBoost est plus performant comparé à l'interpolation par spline cubique. En revanche, lorsque la maturité dépasse 360 jours, l'interpolation par spline cubique a une valeur de MAE plus faible. L'algorithme XGBoost arrive à capturer les tendances du marché pour toutes les courbes de maturités différentes, en particulier celles ou les strikes sont proches de 100%. Dans cette zone, la volatilité ne suit pas de tendance particulière et c'est pourquoi le spline cubique rencontre des difficultés pour interpoler les données en ces points particulier. Er revanche, pour des maturités longues (supérieures à 1 an) la volatilité décrit une courbe hyperbolique que le spline cubique arrive à bien capturer, c'est pourquoi sur ces valeurs de maturités, l'erreur MAE du spline cubique est plus faible que l'erreur MAE du XGBoost. Par ailleurs, l'histogramme représentant l'importance des caractéristiques fournies en entrées pour l'entrainement du XGBoost montre que la volatilité au jour J représente près de 60% d'importance pour la prédiction des données manquantes, ensuite, la moyenne et la valeur maximale à long terme (42 jours) de volatilité représentent chacune 10% d'importance pour la calibration du modèle.

Remerciements

Pour ce travail, je tiens à remercier mon professeur Jean-Yves Dantan, ainsi que mes collègues Mezouar Nassim, Le Razer Yannick, Brik Réda, Huang Men, Perelman Paul et Garcin Vincent, qui m'ont aidé et guidé dans la réalisation du plan de cette revue et qui m'ont apporté des pistes d'étude pour étudier ce sujet. Je tiens également à remercier l'ensemble du corps enseignant du master qui m'a apporté une initiation au monde de la finance ainsi qu'à l'ensemble des membres de l'équipe IPV Equity de Natixis pour leur temps et toutes les explications sur le monde de la finance de Marché.


Emilie Walezak

Angela Carter

A S Byatt

Sarah Hall

Rose Tremain

Jeanette Winterson

Women and Myths in A. S. Byatt's Anthology, Medusa's Ankles

This article investigates Byatt's use of myths in her short fiction. It explores the various interpretations of myths through the reading of three stories. "Medusa's Ankles" exemplifies Byatt's feminist demystification of the Freudian castration complex. "Heavenly Bodies" contrasts the ancient myth of the Virgin and the modern myths of female pop stars, thus mobilising Barthes's definition of myths as cultural phenomena. Finally, "A Stone Woman" constitues Byatt's own creation of a female myth that also talks to current environmental concerns. Short fiction allows Byatt to tackle the female body-mind dilemma charactersitic of her work in uncompromising ways that focus on the language of the stories rather than the psychology of characters.

Cet article explore l'usage que fait Byatt du mythe dans ses nouvelles. À travers la lecture de trois nouvelles, il s'agit de parcourir les différentes interprétations de la notion de mythe. La nouvelle "Medusa's Ankles" permet ainsi à Byatt de démystifier le complexe de castration freudien sur un mode comique. "Heavenly Bodies" met en contraste le mythe ancien de la Vierge et les mythes modernes au sens de Barthes comme phénomènes culturels. Enfin, la nouvelle "A Stone Woman" permet à Byatt de créer sa propre mythologie du féminin en racontant une histoire qui s'empare également des enjeux actuels concernant l'environnement. Alors que le dilemme corps-esprit parcourt l'oeuvre de Byatt, la forme spécifique de la nouvelle lui permet de le mettre en fiction de manière audacieuse, en se focalisant sur les jeux de la langue plutôt que sur la psychologie des personnages.

In May 2021, Chatto & Windus published a selection of A. S. Byatt's short stories named after "Medusa's Ankles," the introductory story in the 1993 collection The Matisse Stories. The new volume gathers together, in chronological order of publication, a selection of thirteen stories from Byatt's five previous collections-Sugar and Other Stories, The Matisse Stories, The Djinn in the Nightingale's Eye: Five Fairy Stories, Elementals: Stories of Fire and Ice, Little Black Book of Stories-as well as one story, "Dolls' Eyes," originally published in Sarah Eyre and Ra Page's 2009 The New Uncanny: Tales of Unease and four uncollected stories published in the press: "Heavenly Bodies," published in The Sunday Times in December 1998, "The Narrow Jet," published in the Paris Review in spring 2005, "The Lucid Dreamer," published in the New Statesman in June 2011, and "Sea Story," published in The Guardian in March 2013. As David Mitchell notes in his introduction to Medusa's Ankles, the selection demonstrates the great variety of Byatt's short fiction: it features ghost stories -"The July Ghost"-realist stories-"Racine and the Tablecloth"-fantastic stories-"The Djinn in the Nightingale's Eye"-ekphrastic stories-"Christ in the House of Mary and Martha"-fairy stories -"Cold"-and metafictional stories-"Raw Material."

In accordance with the multiplicity of genres displayed in these texts, Byatt has always included side narratives in her novels to offer insights into other worlds than the realist one: ekphrases in the Quartet, gothic stories and fairy tales in Possession, metafictional fairy stories in The Children's Book. She has explained that "the story within the story always represents a different kind of grip on the world, a different kind of reality, and also it's always of course about a reading experience" (" Entretien" 20). Interestingly enough, her short stories themselves usually embed other stories as she is not a follower of the modernist creed and has stated that she "hate[s] the word epiphany" ("World Literature" 18:16), considering instead that "short stories can go round corners and sideways" ("World Literature" 19:02). However, whereas the embedded stories in her novels are designed to shed light on the characters' world views and to augment the world of the book itself, the short stories proper do not care about people: "I don't write my short stories for the people, I write them for the language" ("World Literature" 17:10). Thus Byatt sees short fiction as a thinking form quite distinct from her novels: "And so I started seeing things in this very condensed clear way, as images, not necessarily to be strung together in a long narrative, but to be thought out from" ("Entretien" 26). The indifference to characters along with Byatt's use of short fiction "to accommodate the strange" (13) can be paralleled with her recourse to myth. In the final metafictional chapter of her latest novel, Ragnarok, Byatt exposes her "Thoughts on Myth." She distinguishes between mythological figures characterised by their attributes and novelistic figures characterised by their personality. Myths do not deal with psychology, Byatt writes, they are strange, "more and less real" (159), they "haunt the mind" (161) with their strangeness.

In addition to the Norse gods, the other mythological figure that impacted Byatt in her childhood was Cassandra.1 She has said in an interview that Cassandra stood in between the mythical world and the fictional world. Her prophetic features make her mythical while her refusal to have sex with Apollo makes her human. This relates to Byatt's own exploration of femininity as a conflict between body and mind:

There was the idea of knowing everything and refusing to be a sexual being and therefore nobody believing everything she said, so she was mad. . . . I felt this was an image of me, this person who knew what was going on and nobody believed a word of it. You have to compromise yourself with the world if you want people to believe you. (Byatt, "Ragnarok") While Byatt's novels-whose "threads of myth . . . are an essential part of the thought and the form of the books, and of the way the characters take in the world" (Byatt, Ragnarok 160)-might be her compromise with the world, her short stories, this contribution means to suggest, may read as Cassandra-like attempts not to compromise. This article aims to investigate myth by focusing on the choice of title for the selection, Medusa's Ankles, as the mythological echoes of Medusa's name recall Byatt's latest venture in rewriting the Norse myth of the end of the world. Taking the example of three of the collected stories, it seeks to explore the different meanings of myth and Byatt's own thinking out of the body/mind conundrum with the short form. The title story, "Medusa's Ankles," corresponds to Byatt's comic take on the psychoanalytical use of myth and fits with a feminist agenda of demystification. "Heavenly Bodies" also reads as a feminist endeavor to debunk modern myths, as Roland Barthes defined them, by calling on older mythological forms. Finally, with "A Stone Woman," Byatt creates a new female myth.

Demystifying Psychoanalysis

Upon reading the title, Medusa's Ankles, one cannot but think of Hélène Cixous's "Laugh of the Medusa" and of the 1970s feminist contestation of the Freudian myth of Oedipus. Instead of descrying universal structuring taboos and fears in myths, feminists meant to unveil the latent phallocentric content in psychoanalytical readings of them. Before "Medusa's Ankles" was first published in 1993, Byatt had already attacked the Oedipus complex in the 1992 novella "Morpho Eugenia," which upended the taboo of incest with its focus on insects, the sphinx being turned into a moth. The novella is built up to disclose not so much incest itself as William Adamson's entrapment in a nineteenth-century patriarchal worldview or, as Margaret Pearce has demonstrated, to upset his male gaze. Standing half-way between short and long fiction, the novella, that pivots around such games as scrambled words and riddles, is indeed an exercise in reading that focuses on language and embeds the allegorical fairy story by Matty Compton, "Things Are Not What They Seem." Writing on analogies in "Morpho Eugenia," Sue Sorensen has reproached Byatt with "smother[ing]" her characters in ideas (186). However, if one considers the novella to be more on the side of short fiction, it then corresponds to Byatt's avowed intent of thinking things out from the language itself, aside from the psychology of characters.

The originality of "Medusa's Ankles" lies with its comic aspect, which precisely stems from the characters' psychology while the language itself turns them into signs designed to mock Freudian mythology. Susannah is a middle-aged woman who has reached a peak in her career and is scheduled to appear on TV. She goes to her hairdresser only to find out that the comforting pink boudoir displaying Matisse's Rosy Nude has been refitted into a cold and unforgiving environment, and the hairdresser Lucian has left his wife-whose ankles had grown too fat-for a younger model. Upon seeing her middle-aged hairstyle, Susannah flies into a rage and breaks everything in the salon. When she goes back home, her husband looks at her in an unusual way and compliments her hairdo.

The story, told in free indirect speech from the point of view of Susannah, is one of humiliation. The cruel comic arises from Susannah's willful submission to the hairdresser's mistreatment. Her passive attitude is heralded from the beginning of the story, when she remarks: "A woman's relation with her hairdresser is anatomically odd. Her head meets his belt . . ." (Medusa 115). The implicit sexual content of bowing to a fellatio was already suggested a few sentences earlier when Susannah recalled her first time at Lucian's: "He had done her himself, the owner" (115). Throughout the text, he keeps bending her head authoritatively while criticising her: "He pressed close, he pushed at the nape of her neck, her nose was near his discreet zip. 'You've been washing it without conditioner,' he said" (118); "She bent her head submissively, and he scraped the base of her skull. 'You could have highlights,' he said in a tone of no enthusiasm" (119); "She kept still as a mouse, her head bowed under his repressing palm" (123); "He had her by the short hairs at the nape of her neck" (125). Even though Susannah is smart enough to recognise his negligence and self-centeredness-he arrives late, ignores her requests-her insecurity in the face of ageing makes her tolerate his abuse and trust his professional competence. She goes mad when he leaves her in the hands of a female assistant. While "Morpho Eugenia" detracts the male gaze, "Medusa's Ankles" comically summons the female gaze as internalised self-deprecation and sexual objectification: at the hands of her hairdresser, Susannah is only able to see herself as a fat middle-aged woman instead of a professional achiever.

However, if the diegesis itself focuses on the characters' relations, the language of the story calls for an iconographic and mythographic reading, such as pursued by Laurence Petit when she analyses the multiplication of frames in the story. Matisse's Pink Nude, which was complemented by a reproduction of La Chevelure as the story's frontispiece in the original collection, signals the story's connection to art history. As a consequence, Susannah's very name cannot but call to mind the iconography of Susannah and the Elders, whose story of the libidinous male gaze invading the privacy of a young girl is ironically twisted around by Byatt. Susannah has grown old and is no longer the object of male desire. It is no longer simply the character that matters but the idea of the female beauty canon in painting which is called into question. Medusa herself is the subject of a long iconographic tradition to which Byatt's insistent depictions of the "rolls of hair" (116), "curls and waves," "bubbles and wisps" (117), "[s]ausages and snail-shells, grape-clusters and twining coils" (125-26) or "fatal coils" (128) pay homage. Thus Susannah the character migrates from a fictional signifier to an ekphrastic signified of youth preyed upon by old age while her inter-relation to the nude tradition emblematic of Matisse turns her into a sign mediating between pictorial representations and canonical concepts. This migration process characterizes the whole story: the pink colour, about which Byatt said she liked "la couleur comique de la rose" ("Rose"), migrates from the adjectival rosy, which first comforts Susannah, to a destructive verb when she vents her fury in the salon: "Rage rose in her" (125). The very title of the story points to a transference process from the traditional canonical metonymic sign characterising Medusa-the snake-like hair-to the character's ankles, which symbolise the redundancy of her middle-aged identity. The mundane tale of an old woman petrifying the customers of a hairdresser's with her frenzy mocks the Freudian mythologizing of the fear of castration.

The character of Lucian is also a displaced figure of the psychoanalyst, his first name bringing to mind the painter and grandson of Sigmund, Lucian Freud, also famous for his fleshy, unforgiving nudes. Lucian's deafness to Susannah's words along with his self-centered chattering away reads as a dig at Freud's talking cure: "He created his own psychiatrist and guru from his captive hearer" (117). Furthermore, Lucian is the one repeatedly threatening Susannah, brandishing his scissors, "wav[ing] the scissors dangerously near her temples" (120), scraping the nape of her neck, which reads as an inverted version of Dalila's story.

Through metonymic displacements then, Byatt upends at one and the same time the iconographic myth of Susannah and the Elders and the psychoanalytical myth of Medusa by portraying an ageing woman enraged by her thinning hair because it makes her feel she has become invisible in the eyes of a male-dominated society that values youthful attractive women. Byatt's insistence on bringing down to earth the mythic, her use of metonymies that never reach towards the metaphoric, aim at demystifying the canon of art history and psychoanalysis. Like Cassandra, Susannah stands halfway between the fictional and the mythical, she is a middle-aged woman in crisis while her first name fractures into a variety of connotations: it references her biblical namesake, contains the end letters of Medusa; the ending of her name may read as a reference to Anna Freud, while the repetition of the letter S can be interpreted in relation to Samson. Byatt invites her readers to a game of anagrams, the same way she did in "Morpho Eugenia" in which the signifier Eugenia reads as the first name of the fictional character, a reference to the butterfly named after the French empress Eugénie, as well as an allusion to Francis Galton's eugenics. The use of metonymic displacements can furthermore be interpreted, in relation to Freudian dream theory, as a detraction of the psychoanalytical myth of the paternal metaphor as governing the symbolic order of language. In her own way, Byatt follows Cixous' agenda of writing a sext 2 by jointly fictionalizing a female gaze struggling with internalised oppression and by altering the myth of Medusa. The comic impulse of the story contributes to its process of demystification by showing a quotidian version of petrifying fear: the mirror does not defeat Susannah, she smashes it to pieces. The comic, however, does not break the rules, but rather makes apparent the limits of transgression. Therefore, the diegesis itself remains on the side of compromise and the story's ending portrays the character as reintegrating the patriarchal order. The language, though, allows for endless recombinations, replacing the comic with the ludic. Byatt is playful in her short stories, much more so than in her novels. By making the reader a partner of her language games, she invites a shared thinking through of myths and their potential revisions.

Modern Myths and Old Myths

"Heavenly Bodies" precisely revises old myths by contrasting them with modern ones, in the sense of Barthes, as cultural phenomena. It tells the story of a publicity stunt in the form of a glittering skywoman, sent sailing around the earth, whom incarnations of the zodiac come to dismantle. The skywoman can be interpreted as a modern myth as she is the representation of a pop star, Lucy Furnix, with the text referencing the Beatles' song "Lucy in the Sky with Diamonds" (324). She is the lover of a media tycoon who indulges his fancy while making money out of it, selling "patented . . . Lucy tubes" (324) to viewers around the world. By adopting Brad's point of view in free indirect speech, the beginning of the text reproduces the arguments of the marketer invoking ethical branding when advertising Lucy's social usefulness in lighting dark street corners, and her environmental-friendly usage in helping crops with her light. Just like "Sea Story" tackles the question of ocean pollution, "Heavenly Bodies" deals with light pollution and space junk as Brad Macmamman is "one of the few people powerful and rich enough to assemble a skywoman without fear of international complaints about light pollution, or advertising controls" (323). Soon enough, after the first amazement at the technical feat, people start to loathe Lucy, going from Lucy craze to "Lucy craziness" (325), making her responsible for all sorts of ills "from milk-souring to muggings" (325).

The commodification of the sky parallels that of Lucy herself portrayed as a caricature of desirable female traits: "full-breasted, narrow-waisted, with a cloud of shimmering hair and shapely legs" (323), with "a pouting mouth, partly open to show sparkling teeth, with a plump upper lip" (324); "her eyes were huge. . . . Her nose was pert and her cheekbones pronounced" (324). She displays all the standards of the modern beauty myth as outlined by Naomi Wolf in The Beauty Myth. While Wolf does not call on Barthes, she nevertheless chooses the word "myth" to highlight the latent political content of beauty as a new social fiction designed to oppress women liberated from the feminine mystique of domesticity. According to Barthes, modern myths naturalise such concepts as beauty so that they are never questioned. In this story, Byatt parallels the unquestioned appropriation of celestial bodies through space conquest with that of female bodies through marketing. Lucy Furnix is a signifier sailing in the sky displaying a variety of advertising signifieds: she holds a mirror that "streamed with white rays like the Columbia Motion Pictures totem" (323)-the word "totem" referencing once again the Freudian theory on myth-she wears a beauty pageant sash "like the guerdon of Miss World" displaying the tycoon's logo (324), and despite her blue eyes and blond lashes she also displays orientalist elements with her "harem pants" (324), further typifying her as a sign emblematic of the many while hinting at cultural appropriation. She is commonly known as "the Usherette," that quaint figure of the cinema industry kin to the stereotyped airplanes' stewardess. Brad calls her "The Lady with the Lamp" (324), degrading the myth of female professional competency represented by Florence Nightingale. Meanwhile, the sky is crowded with "tiny satellites, carrying light-emitting polymers and mirrors" (323). In "Sea Story" also, Byatt parallels the commodification of the sea and of women with the free indirect speech of the character-narrator, Harold, fantasising Laura, a student in oceanography, as a sea creature, while extolling his romantic views of the masterless ocean. The underlying irony of the narrative is designed to uncover his misguided perspective as Laura proves to be uninterested in a relationship with him, giving him a fake email address, and the Perrier bottle he sends sailing with a love letter in it contributes to augment the Pacific trash vortex.

Both "Sea Story" and "Heavenly Bodies" abruptly switch their focus from the human perspective to the non-human. However, while the former adopts a harshly realist view on the destruction wrought by the bottle at sea, in "Heavenly Bodies," old myths take over to dismantle the marketed skywoman. The zodiac signs fantastically materialise in the sky to destroy her piece by piece. The modern myth's objectification is highlighted by its metonymic dismemberment with the Scorpion "shear[ing] away her painted toes" (326), the Archer using her logos as a "target practice" (327), the Goat "carry[ing] away her legs on its horntips" (327), the Water-carrier "carry[ing] away the harem pants, the exiguous brassiere, and her well-coiffeured waves" (327), the Fish "eating away her buttocks and sex" (328), the Ram "transfix[ing] the mirror of her now dull lamp between its horns" (328), the Twins "slic[ing] off her arms" (329), the Crab "surgically remov[ing] her jutting breasts" (329), and the Lion decapitating her. All the while, the lifeless man-made skywoman sails on "with no diminution of her anodyne smile" (329). The violence of her rending as she is stripped, shaved and amputated by virile mythical figures sporting "rearing tail[s]," "muscular arms" (326), "wicked curved slender horns," "a silver beard" (327), "a magnificent pendulous sex" (328), "sword belts and plumed helmets" (328), reads as a deconstructive exposé of her female attributes marketed as desirable by men. It mirrors the public's switch from Lucy craze to Lucy craziness as her segmentation evokes a castration-based punishment. She is kin to the femme fatale, the object of the male gaze whose threat of castration is turned against her as it reduces her to fetishes (see Mulvey).

Byatt, however, exposes the process brutally, thus not endorsing the male gaze itself, but rather offering an alternative narrative that closes with the ascendancy of the Virgin. By contrast with Lucy Furnix's dismemberment, the non-human agents that "sip" and "nip" and "claw" at her undergo the reverse process: they first appear as myriads of small shapeshifting elements pertaining to all kindsbirds, insects, fish, plants, reptiles-that gradually coalesce into larger zodiac figures. Byatt constructs her own myth by contrast with the deconstruction of the publicity stunt. The figure of the Virgin that finally appears echoes Byatt's earlier recourse to Astraea in The Virgin in the Garden in which she builds on Elizabeth I's appropriation of the myth of the Golden Age. Elizabeth is like Cassandra who refuses to be a sexual being. The queen's motto, semper eadem, is interpreted by Byatt to refer not only to her constancy but to the suspension of her sexual condition: "Tudor rose, blood, flesh, marble, a spring shut up, a fountain sealed, ego flos campi, not to be cut off by the butcher. . . . Elizabeth would not bleed. She would neither be butchered nor marry. She would be a stone that did not bleed, a Princess, semper eadem and single" (Virgin 130). She becomes a myth by setting aside her humanity. While Lucy Furnix "sailed into the sky without annunciation" (Medusa 323), the Virgin, whose ascendancy is heralded for months by the zodiac, encompasses a variety of old myths. She is the Christian virgin, recalling Da Vinci's Renaissance painting The Virgin of the Rocks, "a figure who sat veiled on a starry rock, with a dark blue robe" (331), heralded by the Angel. Her flower cloak "a great field of flowers . . . which had its own valleys," "lily-white" (331), recalls the lover in the Song of Songs, "ego flos campi et lilium convalium" ("I am the flower of the field and the lily between the valleys"). She is also reminiscent of Demeter with her "sheaf of lovely grasses and ears of corn, oats and barley" (331), a figure which Byatt referenced in Possession by ekphrasising Lord Leighton's pre-Raphaelite painting, "The Return of Persephone." Finally she is Astraea who, according to Ovid, fled the earth during the Iron Age and ascended to heaven to become the Virgo constellation. She stands next to Libra and is also a figure of justice, thus "she held also a pair of scales" (331). Byatt is probably gesturing towards Milton's Paradise Lost when Satan is expelled from Eden by the angel Gabriel: In counterpoise, now ponders all events (Book I, lines 996-1001, 106) Byatt's own description of the celestial bodies is reminiscent of Milton's depiction of the heavenly struggle taking place amidst the zodiac signs, especially as she uses the same adjective "pendulous" when evoking the Ram's sex. In her version however, it is the Virgin who weighs all things. She brings back wonder on earth, for when Lucy's diamond is finally lost in the flower garment and everything turns back to night, the humans marvel at "the lovely lights visible from our small planet, the Milky Way, the galaxies, the constellations, the travelling planets and moons" (332). The supernatural mediates with the ordinariness of the night sky by reminding the inhabitants of the earth of how what they take for granted has been a source of fascination for thousands of years.

Byatt's reinterpretation of old myths in the light of current concerns about air rights, space law and the objectification of women in advertising talks to their malleability, as opposed to modern myths that mascarade as perpetual. The Virgin as "the stone that would not bleed" furthermore illustrates Byatt's career-long effort to devise a female myth that would escape the constraints of masculinist biology: "The frozen, stony women became my images of choosing the perfection of the work, rejecting . . . the imposed biological cycle, blood, kiss, roses, birth, death, and the hungry generations" (On Histories and Stories 164).

Creating New Myths: "A Stone Woman" "A Stone Woman," originally published in the 2003 collection Little Black Book of Stories, presents a revision of Ovid's myth of Pygmalion (see Lara Rallo, "Myth") by telling the reverse story of a woman turning to stones. The plural is important as instead of her expected death by petrification, Ines will become a new lively being. The short story is Byatt's uncompromising implementation of Maud's suggestion in Possession of replacing sexuality with geology: "The whole of our scholarship-the whole of our thought-we question everything except the centrality of sexuality-Unfortunately feminism can hardly avoid privileging such matters. I sometimes wish I had embarked on geology myself" (222). The tale of this new geological being (see Lara Rallo, "Geology") contrasts the expectations of a patriarchal world view with an unexpected fantastic transmogrification (see Coelsh-Foisner). Its fantastic logic is very similar to the strategy employed in "The Djinn in the Nightingale's Eye," that displays the fantastic makeover of Gillian who is granted wishes by the djinn. Like the narratologist Gillian, the lexicographer Ines feels redundant in her late middle age and the story partly reads as a menopausal tale. Both characters at first conform to phallocentric views of women's bodies. In "A Stone Woman," more particularly, the perspective is that of the scientific split between subject and object that Ines first adopts to observe her transformation, as if from a detached vantage point, researching the names of stones in the dictionary. Yet, the more the metamorphosis defies her expectations, the more she becomes involved once again in her own bodily experience. Her changing body, which at first terrifies and disgusts her, becomes something she marvels at. With the help of the Icelandic sculptor, Thorsteinn, Ines eventually accepts her metamorphosis and ultimately joins the dancing trolls in Iceland. This fantastic switch of perspective echoes the revision of Ovid's myth. While the myth tells of the power of the creator over his sculpture, whose fidelity to nature is so powerful that the statue comes to life, the short story emphasises the creative abundance of nature itself as the long lists of geological words illustrate, testifying to the plethora of stones sprouting on Ines's body. The story mirrors the shift from the disembodied gaze of the scientific recorder to Ines's embodied empirical experience (see Beltrami). It compares to Spinoza's distinction between "natura naturata" and "natura naturans" that Jane Bennett uses to differentiate "brute matter" from "generativity" (117), promoting a new perspective on matter not as inert but as an agent equal to humans. From that angle, the choice of stones is all the more significant as they are not living organisms. Still, Byatt gives them life through her walking metamorphosis, just as she elaborated on Michelet's conceit of the cathedral builders as "the masters of live stones" in Possession. That she provides a veritable taxonomy of stones with the lists of stone words displayed in the story, in addition to the growing stones on Ines's body, also contributes to include them in biological diversity. Byatt thus writes a feminist posthuman tale that seeks to overcome the body/mind divide by emphasising the agency of inert matter. Subverting the dormancy of stones parallels Ines's emancipation from passivity. Just as the stones become live stones, bursting forth, swelling and breeding over her body, Ines becomes not a petrified statue but a living volcano.

Byatt thereby signals that the current period needs to imagine new myths to come to terms with the changes in our relationship to nature wrought by climate change. Indeed, in addition to its treatment of ageing (see Matthews, and Lara-Rallo, "Time"), the geological short story may also read as an Anthropocene tale on the geological force that human beings have become akin to, a comparison popularised along with the word "Anthropocene" itself by atmospheric chemist Paul J. Crutzen. The story answers the current need "to scale up our imagination of the human-we are actually changing the face of the planet. . . . We can no longer separate the biological agency of humans from their geological agency" (Chakrabarty). Byatt's portrayal of a non-human character in "A Stone Woman" echoes both her depiction of the celestial bodies in "Heavenly Bodies" and her even more radical narrative of the journey of a plastic bottle in "Sea Story." In all three stories, environmental concerns about human impacts loom large and reiterate the worry expressed in Ragnarok about the extinction of species. The short form allows Byatt more leeway to reinterpret myths uncompromisingly by allowing for sudden shifts to the uncanny-"A Stone Woman"-or the marvelous-"Heavenly Bodies," with the fantastic3 helping break ontological boundaries between humans and things. Her more recent stories could thus be described as the "onto-stories" that Jane Bennett wishes to see emerge in order to address the current change of paradigm in our understanding of the agency of matter, whether animate or inanimate: "an . . . onto-tale enhances one's awareness of the vitality of the world with which one is enmeshed" (361). Even though Bennett is referring to thoughtexperiment in philosophical fiction, it is my contention that literary speculations are suited to the "scal[ing] up" of our imagination. The recourse to myths furthermore underlines not only the plasticity of myths but also their eschatological function: the world that Byatt knew as a child is coming to an end and her stories are tales of rebirth after cataclysmic endings. They literally embody a radical change of perspective on the world: Ines's gaze becomes non-human, "her new eyes could not quite bring the dancing black letters to have any more meaning than the spiders and ants which scurried round her feet" (Medusa 383). The readers are meant to experience this radical upset

  Th'Eternal to prevent such horrid fray Hung forth in Heav'n his golden Scales, yet seen Betwixt Astraea and the Scorpion sign, Wherein all things created first he weigh'd, The pendulous round Earth with balanc't Air

"Cassandra . . . was my other favourite figure beside the Norse Gods when I was a girl"(Byatt, "Ragnarok").

"They [the psychoanalysts] riveted us between two horrifying myths: between the Medusa and the abyss. That would be enough to set half the world laughing, except that it's still going on. For the phallocentric sublation is with us, and it's militant, regenerating the old patterns, anchored in the dogma of castration. They haven't changed a thing: they've theorized their desire for reality! Let the priests tremble, we're going to show them our sexts" (Cixous 885).

I am referring here to Tzvetan Todorov's theory. Incidentally, Todorov makes an appearance as a character in "The Djinn in the Nightingale's Eye."


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Censure, encadrement et régulation des images aux États-Unis, XXème-XXIème siècles.

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Date-limite d'envoi des propositions : 16 septembre 2023 Nom de la revue : Transatlantica (Revue en ligne de l'Association française d'études américaines [AFEA]) Adresse email de contact : adrienne.boutang@univ-fcomte.fr Le numéro interrogera les processus de régulation des images (peinture, photographie, bande-dessinée, cinéma et audiovisuel) aux États-Unis. L'objectif sera de mettre en lumière les normes, implicites ou explicites, qui régissent la production, la circulation et la réception d'images troublantes ou transgressives. Le dossier couvrira aussi bien les dispositifs de régulation explicites -Code de Production du cinéma hollywoodien, par exemple, ou bien tentatives étatiques de « façonner la citoyenneté visuelle » par la propagande (Roeder, 1993, 2) -que les mécanismes de contrôle ou de protestation plus diffus et l'utilisation de l'opinion publique (groupes de pression, presse, réseaux sociaux, stratégies de boycott et blacklisting, etc.) Qu'il s'agisse de dispositifs de domination étatiques et institutionnels visant à policer l'espace public, ou de protestations issues de la société civile et de groupes de pression, les dispositifs d'encadrement des images ne seront pas abordés uniquement sous l'angle de la limitation ou de l'interdiction mais aussi sous celui de la contrainte créatrice. Il s'agira aussi de rendre compte de l'inventivité suscitée par le contournement de certaines normes, la production de contre-discours ou d'images alternatives, de détournements, permettant d'interroger les stéréotypes culturels dominants et de révéler des mécanismes de domination.

La question de la régulation des images pourra être abordée dans une perspective esthétique (question de la censure productive, esthétique de la litote, bon goût, etc.) et sociale (évolution des normes morales, des moeurs, apparition de nouvelles normes liées à la prise en compte de processus de domination ou de subordination). Les enjeux de financement et de diffusion des images seront également inclus dans la réflexion (régulation/limitation du financement des oeuvres, diffusions dans des espaces publics de ces dernières ou de matériaux promotionnels comme des affiches, etc.).

Les axes couverts par le dossier pourront notamment être : Mécanismes de régulation, et de censure ou d'autocensure, leur fonctionnement et leur incidence : Censure d'État, enjeux juridiques Pratiques de régulation internes, allant de la censure effective à l'autocensure, l'édulcoration, et à la catégorisation par publics ou selon les contextes de diffusion (monde de l'art, de l'édition, industries créatives : cinéma, télévision et internet, notamment plateformes de distribution, résistances aux restrictions, par exemple via le lancement de la galerie virtuelle « Don't Delete Art » en 2020).

La catégorisation des publics pourra faire l'objet d'études spécifiques (publics fragiles déterminés par des mécanismes de panique morale, incidence de la mondialisation sur la prise en compte des susceptibilités, etc.). La régulation des paratextes (bandes-annonces, affiches d'exposition, etc.) pourra également être étudiée.

Incidences sur le financement des projets artistiques et leur diffusion, des polémiques suscitées par certains financements de la NEA aux débats et polémiques plus récents autour des représentations de groupes sociaux minoritaires.

Critères présidant à la régulation et à la réception des images :

La question de la réception, qui participe des dispositifs d'encadrement des images en circulation dans l'espace public, pourra aussi être abordée.

Critères esthétiques, idéologiques, relatifs aux bonnes moeurs, à la protection des publics. Là encore, il s'agira d'étudier non seulement les dispositifs d'interdiction mais aussi les résistances productives à ces derniers. Une attention plus particulière pourra être portée à la question de la distinction entre les contextes de diffusion et consommation des images, notamment entre espace public et espace privé.

Étude des discours accompagnant la réception des images, qu'il s'agisse de discours critiques officiels ou de débats dans l'espace public.

Des abstracts de 250 à 300 mots, rédigés en anglais ou en français, et une notice biographique de 150 mots, seront à envoyer à adrienne.boutang@univ-fcomte.fr pour le 16 Septembre 2023. Les auteur•rice•s recevront une réponse le 7 octobre 2023. Les articles devront être envoyés pour expertise pour le 5 février 2024. La feuille de style de la revue est disponible ici : https://journals.openedition.org/transatlantica/5220.

Bibliographie indicative

  ANDREWS, David. Soft in the MiddleThe Contemporary Softcore Feature in Its Contexts, Columbus : Ohio State University Press, 2006. BOURGET, Jean-Loup. Hollywood, la norme et la marge. Paris : Armand Colin Cinéma, (1998), 2005. BROUWER, Ceciel. « The bigger picture. Rethinking curatorial approaches to photographs of childhood. » Curating Under Pressure, International Perspectives on Negotiating Conflict and Upholding Integrity. Dir. Janet Marstine et Svetlana Mintcheva. Londres : Routledge, 2020.

Censoring, framing and regulating images in the Twentieth and Twenty-First Century United States

Deadline for submissions: September 16, 2023 full name / name of organization: Transatlantica (online journal of the French Association of American Studies [AFEA]) contact email: adrienne.boutang@univ-fcomte.fr This issue of Transatlantica focuses on the regulation and restriction of images and visual imagery (painting, photography, comics, film and audiovisual media) in the United States. It aims to shed light on the norms, either implicit or explicit, governing the production, circulation and reception of disturbing or controversial images. The issue will cover both explicit regulatory mechanisms -the Motion Picture Production Code, for example, or federal, state, or local governments attempts to "shape visual citizenship" through propaganda (Roeder, 1993, 2) -and more diffuse mechanisms of control or protest, including the use of public opinion (pressure groups, press, social networks, boycott and blacklisting strategies, etc.). The question of image regulation can be approached from an aesthetic perspective (censorship as a productive force, the aesthetics of litotes, good taste, etc.) and from a social perspective (the evolution of moral standards and mores, the emergence of new standards linked to processes of domination or subordination). Restrictions on the funding (state support or private sponsorships) and publication of images will also be considered (regulation/ limitation of public grants, regulation of promotional materials, such as posters, in the public space, etc.).

Questions authors might consider include the following:

Regulation, censorship and self-censorship mechanisms and their impact:

State censorship, legal issues

Internal regulatory practices, ranging from effective censorship to self-censorship, age-based categorization, restrictions based on specific distribution/exhibition contexts (the art world, publishing, creative industries: cinema, television and social media platforms, along with modes of resistance to these restrictions, as exemplified by the launching of the virtual gallery "Don't Delete Art" in 2020).

Audience classification (vulnerable audiences, determined by moral panic mechanisms, the impact of globalization and consideration of alternative cultural policies). Topics may also include the regulation of paratexts (trailers, exhibition posters, etc.)

Regulation through funding and distribution, from the controversies sparked by National Endowment for the Arts grants to recent debates regarding representations of minority group members.

Assessing images: criteria for the regulation and reception of images.

How does reception contribute to the restriction, framing and legitimation of images circulating in the public space? Aesthetic, ideological and moral criteria; justification of censorship through the assumption of vulnerable audiences. Suggested topics may cover not only prohibition mechanisms, but also productive resistances. Particular attention will be paid to the distinction between the contexts in which images are distributed and consumed, and between public and private spaces.

Studies into the reception of, and discourses surrounding, controversial or transgressive images.

Abstracts of 250-300 words, in English or in French, along with biographical statements of 150 words, should be submitted at adrienne.boutang@univ-fcomte.fr by September 16, 2023. Authors of selected proposals will be notified by October 7, 2023. They will be invited to submit their essays for peer-review by February 5, 2024. The journal stylesheet can be found at: https://journals.openedition.org/transatlantica/5220.