🎓 Information

I am Professor in Applied Mathematics at Ecole Polytechnique in the Mathematical Finance group of the CMAP department.   I defended my Habilitation à Diriger des Recherches in 2024 and my PhD Thesis in 2018.

Research interests:  Mathematical finance, path-signatures, Volterra processes, volatility modeling, machine learning.

Teaching: Stochastic modelling and derivatives and Volterra Processes and Path-Signatures in Finance (M2 Probabilité et Finance, Ecole Polytechnique-Sorbonne Université).

I co-organize the Bachelier Seminar and I am responsible of the 3rd-year engineering cycle internships in Mathematical Finance at Ecole Polytechnique. 

Mail: eduardo.abi-jaber@polytechnique.edu 

🌟 Awards

• AMIES PhD Award, best PhD in applied mathematics in collaboration with industry, 2019.

• Bachelier Finance Society Junior Scholar Award, most outstanding paper, 2018.

• DIM Math Innov, Postdoctoral Fellowship, 2018.

👊 Research Group

PhD Students

• Dimitri Sotnikov, co-supervised with Charles Bertucci (Cifre funding ENGIE), 2024-  (🌟 Best EU Quant Finance Master's Thesis 2024)

• Elie Attal, co-supervised with Mathieu Rosenbaum, 2024- 

• Maxime Guellil, co-supervised with Stéphane Crépey (Cifre funding CACIB), 2024- 

• Louis-Amand Gérard, co-supervised with Olivier Guéant (Cifre funding GEFIP), 2022-
  
  

• Nathan De Carvalho, co-supervised with Huyên Pham (Cifre funding ENGIE), 2021-2025. Lifting Energy Markets: from Volatility Modeling to Optimal Trading.  (🌟BFS Junior Scholar Award 2024)  

• Shaun Li, co-supervised with Bernard de Meyer (Cifre funding AXA-IM), 2021-2024. The Quintic Volatility Model for SPX and VIX: A 360 Tour.

Postdocs

• Francesca Primavera, 2025-
  
  

• Alessandro Bondi, co-supervised with Sergio Pulido, 2023-2024. (Currently Assistant Professor, Luiss University)

Master's Students

• Ronan Pécheul

📚 Publications 

[39] Hedging with memory: shallow and deep learning with signatures, with Louis-Amand Gérard, 2025.

[38] Exponentially Fading Memory Signature, with Dimitri Sotnikov, 2025.

[37] Weak solutions of Stochastic Volterra Equations in convex domains with general kernels, with Aurélien Alfonsi and Guillaume Szulda, 2025.

[36] Simulating integrated Volterra square-root processes and Volterra Heston models via Inverse Gaussian, with Elie Attal, 2025.

[35] Martingale property and moment explosions in signature volatility models, with Paul Gassiat and Dimitri Sotnikov, 2025.

[34] From Hyper Roughness to Jumps as H -> -1/2, with Elie Attal and Mathieu Rosenbaum, 2025.

[33] Capturing Smile Dynamics with the Quintic Volatility Model: SPX, Skew-Stickiness Ratio and VIX, with Shaun Li, 2025. (Jupyter notebook)

[32] Fredholm Approach to Nonlinear Propagator Models, with Alessandro Bondi, Nathan De Carvalho, Eyal Neuman and Sturmius Tuschmann, 2025. 

[31] Complex discontinuities of the square root of Fredholm determinants in the Volterra Stein-Stein model, with Maxime Guellil, 2025. (Jupyter notebook)

[30] The Volterra Stein-Stein model with stochastic interest rates, with Donatien Hainaut and  Edouard Motte, 2025.

[29] Heath-Jarrow-Morton meet lifted Heston in energy markets for joint historical and implied calibration, with Soukaïna Bruneau, Nathan De Carvalho, Dimitri Sotnikov and Laurent Tur, 2025. 

[28] State spaces of multifactor approximations of nonnegative Volterra processes, with Christian Bayer and Simon Breneis, 2024.

[27] Simulation of square-root processes made simple: applications to the Heston model, Risk Magazine (Cutting Edge Section), June 2025.

[26] Trading with propagators and constraints: applications to optimal execution and battery storage, with Nathan De Carvalho and Huyên Pham, 2024.

[25] Path-dependent processes from signatures, with Louis-Amand Gérard and Yuxing Huang, 2024. 

[24] Fourier-Laplace transforms in polynomial Ornstein-Uhlenbeck volatility models, with Shaun Li and Xuyang Lin, Finance and Stochastics, to appear, 2025. (Jupyter notebook) (🌟CFA Quant Awards 2024)

[23] Polynomial Volterra processes, with Christa Cuchiero,  Luca Pelizzari, Sergio Pulido and Sara Svaluto-Ferro, Electronic Journal of Probability, 29, 1-37, 2024.  . 

[22] Optimal Portfolio Choice with Cross-Impact Propagators, with Eyal Neuman and Sturmius Tuschmann, 2024. 

[21] Signature volatility models: pricing and hedging with Fourier, with Louis-Amand Gérard, SIAM Journal on Financial Mathematics, 16:2, 606-642, 2025. 

[20] Volatility models in practice: Rough, Path-dependent or Markovian?, with Shaun Li, Mathematical Finance, 35, no. 4: 796–817, 2025.

[19] Equilibrium in Functional Stochastic Games with Mean-Field Interaction, with Eyal Neuman and Moritz Voss, 2023. 

[18] Reconciling rough volatility with jumps, with Nathan De Carvalho, SIAM Journal on Financial Mathematics, 15(3), 785-823, 2024. (🌟BFS Junior Scholar Award 2024)

[17] The quintic Ornstein-Uhlenbeck volatility model that jointly calibrates SPX & VIX smiles, with Camille Illand and Shaun Li, Risk Magazine (Cutting Edge section), June 2023. (Jupyter notebook)

[16] Joint SPX-VIX calibration with Gaussian polynomial volatility models: deep pricing with quantization hints, with Camille Illand and Shaun Li, Mathematical Finance, 35, no. 2: 470-519, 2025. 

[15] Optimal Liquidation with Signals: the General Propagator Case, with Eyal Neuman, Mathematical Finance, 35, no. 4: 841–866,  2025.  (Jupyter notebook)       

[14] Gaussian Agency problems with memory and Linear Contracts, with Stéphane Villeneuve, Finance & Stochastics, 29, no. 1: 143-176, 2025.          

[13] The characteristic function of Gaussian stochastic volatility models: an analytic expression, Finance & Stochastics, 26, 733–769, 2022.                

[12] Markowitz portfolio selection for multivariate affine and quadratic Volterra models, with Enzo Miller and Huyên Pham, SIAM Journal on Financial Mathematics,  12(1), 369-409, 2021.  (Jupyter notebook).

[11] Weak existence and uniqueness for affine stochastic Volterra equations with L1-kernels, Bernoulli, 27(3), 1583-1615, 2021.

[10] The Laplace transform of the integrated Volterra Wishart process, Mathematical Finance, 32(1), 309-348, 2022.       

[9] Integral operator Riccati equations arising in stochastic Volterra control problems, with Enzo Miller and Huyên Pham, SIAM Journal on Control and Optimization, 59(2),  1581-1603, 2021.

[8] Linear-Quadratic control for a class of stochastic Volterra equations: solvability and approximation, with Enzo Miller and Huyên Pham, Annals of Applied Probability, 31(5), 2244-2274, 2021.     

[7] A weak solution theory for stochastic Volterra equations of convolution type, with Christa Cuchiero, Martin Larsson and Sergio Pulido, Annals of Applied Probability, 31(6), 2924-2952, 2021.

[6] Lifting the Heston model, Quantitative Finance, 1-19, 2019.   

[5] Markovian structure of the Volterra Heston model, with Omar El Euch, Statistics & Probability Letters, Volume 149, Pages 63-72, 2019.

[4] Multifactor approximation of rough volatility models, with Omar El Euch, SIAM Journal on Financial Mathematics, 10(2), 309-349, 2019. (🌟BFS Junior Scholar Award 2018)

[3] Affine Volterra processes, with Martin Larsson and Sergio Pulido, Annals of Applied Probability, Volume 29, Number 5, 3155-3200, 2019.        

[2] Stochastic invariance of closed sets for jump-diffusions with non-Lipschitz coefficients, Electronic Communications in Probability, 22, paper no. 53, 15 pp, 2017. 

[1] Stochastic invariance of closed sets with non-Lipschitz coefficients, with Bruno Bouchard and Camille Illand, Stochastic Processes and their Applications, Volume 129, Issue 5, Pages 1726-1748, 2019.      

   Thesis

• Volterra Processes in Finance, Habilitation, 2024. 

• Stochastic invariance and stochastic Volterra equations, PhD thesis, 2018. (🌟AMIES PhD Award)

🗣️ Talks

      Lectures

• Research in Options: RiO 2024,  Rio, December 4-8, 2024. Minicourse on Volterra processes in Quantitative Finance. (Invited)

• Stochastic Methods in Finance and Physics, Creta, Greece, 17-21 July 2023. Minicourse on Volterra processes in Finance. (Invited)

• Stochastic Numerics and Statistical Learning: Theory and Applications Workshop,  KAUST University,  Saudi Arabia, May 28-31, 2023. Minicourse on 'Non-Markovian models in Finance' with Christian Bayer.  (Invited)

     Conferences

• Stochastics & Computational Finance, Lisbon, September 2-5, 2025. Mini-symposium on Memory in Quantitative Finance.  

• SIAM Conference on Financial Mathematics and Engineering,  Miami, July 15-18, 2025. Mini-symposium on Memory in Quantitative Finance (opening talk). 

• Vienna Congress on Mathematical Finance,  WU Vienna, July 9-11, 2025. Volatility dynamics and memory: from the Quintic model to Signatures. (Plenary)

• AMaMeF Conference, 12th Edition, Verona, Italy, June 23-27, 2025. Heston: struggling with simulation?

• Signatures in 1 and 2 dimensions, Oslo, June 16-20, 2025. Chasing Stationarity with the Fading-Memory Signature. (Invited)

• Stochastic Numerics and Statistical Learning: Theory and Applications Workshop,  KAUST University,  Saudi Arabia, May 18-22, 2025. Path-Signatures: Memory and Stationarity. (Invited)

• XXVI Workshop on Quantitative Finance, Palermo, Italy, April 15-17, 2025.  Simulation of square-root processes made simple: applications to the Heston model.

• Stochastic & Rough Analysis, Berlin, Germany, April 7-11, 2025. Path-Signatures: Memory and Stationarity. (Invited)

• Financial Risks International Forum, Institut Louis Bachelier, Paris, March 18-19, 2025, Struggling with Heston? A New Perspective for Efficient Simulation.

• Research in Options: RiO 2024,  Rio, December 4-8, 2024. From the Quintic OU model to signature volatility models. (Invited)

• PGMO Days,  Palaiseau, November 19-20, 2024. Equilibrium in Functional Stochastic Games with Mean-Field Interaction. (Invited)

• Directions in rough analysis,  Oberwolfach, November 3-8, 2024. Some path-dependent processes from signatures.  (Invited)

• Stochastic control and Games for Risk and Regulation,  Hammamet, October 28-31, 2024. Invariance for stochastic Volterra equations. (Invited)

• 12th Bachelier World Congress of the Bachelier Finance Society,  Rio, July 8-12, 2024. Equilibrium in Functional Stochastic Games with Mean-Field Interaction. (Invited)

• Spring Colloquium on Probability and Finance,  Padova, April 19, 2024. From the Quintic model to signature volatility models: fast pricing and hedging with Fourier. (Invited)

• Recent Developments in the Mathematics of Machine Learning,  Wuppertal, March 26, 2024. Deep pricing with quantization hints: joint SPX-VIX calibration with Gaussian Polynomial Models. (Invited)

• Advances in Stochastic Analysis for Handling Risks in Finance and Insurance,  CIRM Luminy, France, September 11-15, 2023. Equilibrium in Functional Stochastic Games with Mean-Field Interaction (Invited)

• Workshop on rough volatility, Isle of Skye, Scotland, May 22-26, 2023. Rough volatility, too smooth? (Invited)

• Control & Optimisation Pisa 2023, Pisa, Italy, May 8-10, 2023.  A direct approach to a class of finite-players and mean-field games. (Invited)

• Workshop Stochastic Control and Risk, Hammamet, Tunisia, April 24-27, 2023.   A direct approach to a class of finite-players and mean-field games. (Invited)

• London-Paris Bachelier Workshop on Mathematical Finance, Paris, France, September 15-16, 2022, Gaussian Principal-Agent problems with memory.  (Invited)

• Modeling of Markets with Complex and Rough Regimes, Online, August 31, 2022. Quadratic Gaussian models: analytic expressions for pricing and portfolio allocation. (Invited)

• General AMaMeF Conference, 10th Edition, Online, June 22-25, 2021. Quadratic Gaussian models: analytic expressions for pricing and portfolio allocation. (Invited)

• Stochastic Analysis and Hermite Sobolev Spaces Workshop, Online, June 21-26, 2021. Linear-quadratic control of stochastic Volterra equations. (Invited)

• ILB Rising Talents in Finance and Insurance, Online, March 26, 2021. The characteristic function of Gaussian stochastic volatility models. (Invited)

• QuantMinds International, Online, November 2-6, 2020. Reconciling rough volatility with jumps.  (Invited)

• Advances in Financial Mathematics, Paris, France, January 14-17, 2020. A weak solution theory for stochastic Volterra equations. (Invited)

• Journées du Centre d'Economie de la Sorbonne, Université Paris 1, Paris, France, November 14, 2019. Rough volatility  modeling and stochastic Volterra equations. (Invited)

• Advances in Stochastic Analysis for Handling Risks in Finance and Insurance, CIRM, Luminy, France, October 21-25, 2019. Reconciling rough volatility with jumps. (Invited)

• Forum emploi maths, Paris, France, Otcober 15, 2019. The modeling puzzle in the presence of randomness. (AMIES Prize presentation)

• Vienna Congress on Mathematical Finance, Vienna, Austria, September 9-13, 2019. Reconciling hyper-rough volatility with jumps.

• General AMaMeF Conference, 9th Edition, Paris, France, June 11-14, 2019. Quadratic Volterra processes and multivariate rough volatility modeling. (Invited)

• Non linear methods for risk management, Paris, France, October 9, 2018. Lifting the Heston model. (Invited)

• Workshop on Finance, Insurance, Probability and Statistics, London, England, September 10-11, 2018. Lifting the Heston model (poster).

• European Summer School in Financial Mathematics, 11th Edition, Palaiseau, France, August 27-31, 2018. Lifting the Heston model.

• Bachelier Finance Society, 10th world congress, Dublin, Ireland, July 16-20, 2018. Markovian structure of the Volterra Heston model.

• IWAP, 9th International Workshop on Applied Probability, Budapest, Hungary, June 18-21, 2018. The Volterra square-root process: Markovian structure and approximation.

• Workshop "Long-range fractional processes in Finance", Ecole Polytechnique, Palaiseau, France, June 11, 2018. Lifting the Heston model. (Invited)

• Workshop "Robust Finance", Freiburg, Germany, May 16-18, 2018. Lifting the Heston model. (Invited)

• Berlin-Paris Young researchers Workshop, 2nd Edition, Stochastic Analysis with applications in Biology and Finance, Paris, France, May 2-4, 2018. Affine Volterra processes. (Invited)

• German Probability and Statistics Days 2018, 13th Edition, Freiburg, Germany, February 27 - March 2, 2018. Affine Volterra processes.

• Workshop on Quantitative Finance, XIX Edition, Roma, Italy, January 24-26, 2018. Affine Volterra processes.

• Bachelier Colloquium, Twelfth Colloquium on Mathematical Finance and Stochastic Calculus, Métabief, France, January 15-20, 2018. Affine Volterra processes.

• International Conference on Computational Finance, 2nd Edition, Lisbon, Portugal, September 4 - 8, 2017. Stochastic invariance and applications. (Invited)

• European Summer School in Financial Mathematics, 10th Edition, Dresden, Germany, August 28 - September 1, 2017. Affine Volterra processes. 

• Stochastic Processes and their Applications, 39th Conference, Moscow, Russia, July 24-28, 2017. Stochastic invariance and applications. 

• General AMaMeF Conference, 8th Edition, Amsterdam, Netherlands, June 19-23, 2017. Stochastic invariance and applications. 

• Workshop on Quantitative Finance, XVIII Edition, Milano, Italy, January 25-27, 2017. Stochastic invariance of closed sets with non-Lipschitz coefficients. [slides]

• Market Microstructure Confronting Many viewpoints, #4, Paris, France, December 6-9, 2016, Stochastic invariance of closed sets with non-Lipschitz coefficients. [slides] 

• Berlin-Paris Young Researchers Workshop, Stochastic Analysis with applications in Biology and Finance, Berlin, Germany, November 2-4, 2016. Stochastic invariance of closed sets with non-Lipschitz coefficients. [poster] 

• London-Paris Bachelier Workshop on Mathematical Finance, Paris, France, September 29-30, 2016. Stochastic invariance of closed sets with non-Lipschitz coefficients. [poster]

    Seminars

• Seminar Credit Agricole, Paris, June 19, 2025. Volatility dynamics and memory: from the Quintic model to Signatures.  

• Seminar BNP Paribas, Paris, May 28, 2025. Volatility dynamics and memory: from the Quintic model to Signatures. 

• Seminar in Mathematical Finance, BI Norwegian Business School, April 24, 2025. From the Quintic model to signatures volatility models. 

• Groupe de Travail MathsFi, Ecole Polytechnique, February 12, 2025. A new perspective for simulating square-root processes and Heston models.

• Rough Path Interest Group, Online, February 5, 2025. Some path-dependent processes from signatures.

• Stochastic and Finance seminar, University of Warwick, Warwick, January 15, 2025. Stochastic Fredholm equations: a passe-partout for propagator models and market impact.  

• Finance and Stochastics seminar, Imperial College, London, October 22, 2024.   Stochastic Fredholm equations: a passe-partout for propagator models and market impact. 

• Mathematical Finance Seminar Series,  Columbia University, New York, October 3, 2024, from the Quintic model to signature volatility models.

• FS Stochastische Analysis,  TU Berlin, June 13, 2024. Stochastic Fredholm equations: a passe-partout for propagator models with cross-impact, constraints and mean-field interactions. 

• Berlin Probability Colloquium,  Berlin, June 12, 2024. Some path-dependent processes from signatures.

• Séminaire de Probabilités et Statistique,  Le Mans Université, May 14, 2024. Signature volatility models: pricing and hedging with Fourier.

• Stochastic Analysis & Mathematical Finance Seminars,  Oxford, April 22, 2024. From the Quintic model to signature volatility models: fast pricing and hedging with Fourier.

• Risk & Stochastics and Financial Mathematics Seminar,  London School of Economics, February 7, 2024. Signature volatility models: pricing and hedging with Fourier.

• Groupe de Travail Mathématiques financières et probabilités, LPSM, November 23, 2023. From the Quintic OU model that jointly calibrates SPX-VIX smiles to signature volatility models. 

• Séminaire de Probabilités, Ecole Polytechnique, October 24, 2023.

• Seminar of the Department of Economics and Finance, Luiss University, Rome, September 21, 2023.  Equilibrium in Functional Stochastic Games with Mean-Field Interaction.  

• Seminar on Finance & Modeling, Université Paris 1, France, April 17, 2023. The Quintic model that jointly calibrates SPX/VIX.

• Finance and Stochastics seminar, Imperial College, London, England, March 28, 2023.  From the Quintic model that jointly calibrates SPX/VIX to Sig-Volatility models.

• Séminaire FDD-FIME, Institut Henri Poincaré, December 16, 2022, Optimal execution with transient impact and signals. 

• STAR seminar, University of Oslo, Online, May 6, 2022. Quadratic Gaussian models: analytic expressions for pricing and portfolio allocation. 

• Talks in Financial and Insurance Mathematics, ETH-Zürich, Switzerland, April 28, 2022. Quadratic Gaussian models: analytic expressions for pricing and portfolio allocation. 

• Séminaire Bachelier, January 14, 2021, Deep pricing with quantization hints.

• Osaka Webinar on Mathematical Finance, Online, November 25, 2021, Quadratic Gaussian models: analytic expressions for pricing and portfolio allocation. 

• Berlin Probability Colloquium of the International Research Training Group (IRTG),  June 2, 2021, Stochastic Volterra equations : theory, numerics and control. 

• Séminaire Bachelier, February 4, 2021, Quadratic Gaussian models: analytic expressions for pricing and portfolio allocation.

• Groupe de Travail: Finance mathématique, probabilités numériques et statistique des processus, Université Paris Diderot, January 21, 2021, Quadratic Gaussian models: analytic expressions for pricing and portfolio allocation.

• Groupe de Travail Probabilités et Mathématiques financières, Université d'Evry, Evry,  France, November 12, 2020. Linear-quadratic control of stochastic Volterra equations.

• Séminaire du SAMM, Université Paris 1, Paris, France, October 23, 2020. Linear-quadratic control of stochastic Volterra equations.

• Finance and Stochastics seminar, Imperial College, London, England, March 11, 2020. Linear-quadratic control of stochastic Volterra equations.

• Risk & Stochastics and Financial Mathematics Seminar, London School of Economics, London, England, March 12, 2020. Reconciling rough volatility with jumps.

• Vienna Seminar in Mathematical Finance and Probability, Vienna University, Austria, January 30, 2020. Linear-quadratic control of stochastic Volterra equations.

• Séminaire Bachelier, Paris, France, December 13, 2019. Linear-quadratic control of stochastic Volterra equations.

• Talks in Financial and Insurance Mathematics, ETH-Zürich, Switzerland, October 29, 2019. Linear quadratic control of stochastic Volterra equations.

• Seminar on Finance & Modeling, Université Paris 1, France, October 16, 2019. Invariance and viability: from finite dimensional ODEs to infinite dimensional SDEs.

• Séminaire du SAMM, Université Paris 1, Paris, France, October 11, 2019. Stochastic Volterra equations.

• Groupe de Travail Méthodes Stochastiques et Finance, Ecole des Ponts, INRIA & Université Paris Est, Noisy, France, April 18, 2019. Affine and quadratic Volterra processes and applications.

• Matinée scientifique du Ceremade, Université Dauphine, Paris, France, January 15, 2019. Stochastic Volterra equations.

• Stochastic Analysis and Stochastics of Financial Markets, Humboldt/TU Universities, Berlin, Germany, November 8, 2018. Lifting the Heston model.

• Groupe de Travail Probabilités et Mathématiques financières, Université d'Evry, Evry, France, February 1, 2018. Affine Volterra processes.

• Groupe de Travail Méthodes Stochastiques et Finance, Ecole des Ponts, INRIA & Université Paris Est, Noisy, France, October 19, 2017. Affine Volterra processes.

• Post/Doctoral Seminar in Mathematical Finance, ETH-Zürich, Switzerland, October 10, 2017. The Volterra square-root process.

• Séminaires de l’équipe de Probabilités et Statistiques, Université Paris 13, Paris, France, May 17, 2017. Stochastic invariance and applications. [slides]

• Séminaire des Jeunes Chercheurs du Ceremade, Université Paris Dauphine, Paris, France, February 23, 2017, Invariance of closed sets, from regular ODEs to irregular SDEs. [slides]

• PhD Students' Day - CEREMADE, Université Paris Dauphine, Paris, France, October 20, 2016. Invariance of closed sets, an intuitive overview. [slides] 

• Groupe de Travail Méthodes Stochastiques et Finance, Ecole des Ponts, INRIA & Université Paris Est, Noisy, France, October 13, 2016. Stochastic invariance of closed sets with non-Lipschitz coefficients. [slides] 

🧑‍🏫 Teaching

Ongoing at Ecole Polytechnique 

• Stochastic modelling and derivatives, M2 Probabilité et Finance, X-Sorbonne Université (21h)

• Volterra processes and Path-Signatures in Finance, M2 Probabilité et Finance, X-Sorbonne Université (15h)

Previous

Lectures

• 2022-2025: Deep Learning in Finance (MsC Data Science in Finance, Ecole Polytechnique-HEC)

• 2021-2022 at ENSAE: Numerical Methods in Financial Engineering (3A, 12h),  Introduction to mathematical finance (2A, 18h)

• 2019-2022 at Université Paris 1 Panthéon-Sorbonne: Market Risk Measures (M2, 18h), Calibration in Quantitative Finance (M2, 18h), Topics in Machine Learning (M2, 18h).

Tutorial classes

• 2022 at Ecole Polytechnique : Markov chains and martingales (MAP432, 20h)

• 2019-2022 at Université Paris 1 Panthéon-Sorbonne: Mathematics of insurance and risks (M2, 16h), Mathematical finance (M1, 24h), Integration and Probability (L3, 42h), Analysis (L3, 42h), Linear Algebra (L2, 30h).

• 2016-2018 at Université Paris Dauphine: Jump processes, Valuation and No Arbitrage (M2 MASEF).