Natixis — Technical Presentation (October 2025)
34 slides from Natixis's technical presentation, Oct 14, 2025
Slide 1: Solving The Dynamic Volatility Fitting Problem
Solving The Dynamic Volatility Fitting Problem: A Deep Reinforcement Learning Approach in Continuous State Action Spaces Flow Equity Derivatives. Emmanuel Gnabeyeu1,2 and Omar Karkar 2 October 14, 2025 1 Ecole Polytechnique, Paris, France. 2 EMEA Head of EQA, Citigroup Global Markets Limited, London
Slide 2: Table of Contents
Table of Contents 1 General Overview of the Volatility Fitting Problem 2 A Reinforcement Learning Framework and Algorithm 3 Numerical Results 4 Conclusion and Prospects: E. Gnabeyeu, Ecole Polytechnique. Solving The Dynamic Volatility Fitting Problem: October 14, 2025 2 / 18
Slide 3: General Overview of the Volatility Fitting Problem
General Overview of the Volatility Fitting Problem Volatility Fitting: Problem Statement 1 General Overview of the Volatility Fitting Problem Volatility Fitting: Problem Statement Traditional approach for the Volatility Fitting Problem 2 A Reinforcement Learning Framework and Algorithm 3 Numerical R
Slide 4: Volatility Fitting Problem and why?
General Overview of the Volatility Fitting Problem Volatility Fitting: Problem Statement Volatility Fitting Problem and why ? n o ϕModel n o ϕ Ask Bid Ask Bid Mark-to-market quoted price P: PCall/Put , PCall/Put −−−→ σCall/Put , σCall/Put → σ Ask , σ Bid − for Model different moneyness (κ1 , ..., κn
Slide 5: Volatility Fitting Problem and why?
General Overview of the Volatility Fitting Problem Volatility Fitting: Problem Statement Volatility Fitting Problem and why ? n o ϕModel n o ϕ Ask Bid Ask Bid Mark-to-market quoted price P: PCall/Put , PCall/Put −−−→ σCall/Put , σCall/Put → σ Ask , σ Bid − for Model different moneyness (κ1 , ..., κn
Slide 6: Encoding the market data
General Overview of the Volatility Fitting Problem Volatility Fitting: Problem Statement Encoding the market data: Many methods in the industry to encode the implied volatility information 3 Non-Parametric grid of impl. Vol then interpolation via simple spline. Modelling the implied density of the a
Slide 7: Encoding the market data
General Overview of the Volatility Fitting Problem Volatility Fitting: Problem Statement Encoding the market data: Many methods in the industry to encode the implied volatility information 3 Non-Parametric grid of impl. Vol then interpolation via simple spline. Modelling the implied density of the a
Slide 8: Traditional approach for the Volatility Fitting Problem
General Overview of the Volatility Fitting Problem Traditional approach for the Volatility Fitting Problem 1 General Overview of the Volatility Fitting Problem Volatility Fitting: Problem Statement Traditional approach for the Volatility Fitting Problem 2 A Reinforcement Learning Framework and Algor
Slide 9: Traditional approach to volatility fitting with Parametrisation
General Overview of the Volatility Fitting Problem Traditional approach for the Volatility Fitting Problem Traditional approach to volatility fitting with Parametrisation: ⇒ Choice of the target objective function:5 : Among Others, the squared-error loss σ Mid (ti , κj ) − Ψvol s Pn 2 ⃗ (κj ) j=1 wj
Slide 10: Background on Reinforcement Learning
A Reinforcement Learning Framework and Algorithm Background on Reinforcement Learning 1 General Overview of the Volatility Fitting Problem 2 A Reinforcement Learning Framework and Algorithm Background on Reinforcement Learning Problem Statement in Reinforcement Learning Market environment Deep Deter
Slide 11: Towards a Novel Framework: Background on Reinforcement Learning
A Reinforcement Learning Framework and Algorithm Background on Reinforcement Learning Towards a Novel Framework: Background on Reinforcement Learning. An agent reinforced with awards based on its interaction with the environment would learn to operate optimally in that environment to maximize the re
Slide 12: Towards a Novel Framework: Background on Reinforcement Learning
A Reinforcement Learning Framework and Algorithm Background on Reinforcement Learning Towards a Novel Framework: Background on Reinforcement Learning. An agent reinforced with awards based on its interaction with the environment would learn to operate optimally in that environment to maximize the re
Slide 13: Problem Statement in Reinforcement Learning
A Reinforcement Learning Framework and Algorithm Problem Statement in Reinforcement Learning 1 General Overview of the Volatility Fitting Problem 2 A Reinforcement Learning Framework and Algorithm Background on Reinforcement Learning Problem Statement in Reinforcement Learning Market environment Dee
Slide 14: Problem Statement in Reinforcement Learning
A Reinforcement Learning Framework and Algorithm Problem Statement in Reinforcement Learning Problem Statement in Reinforcement Learning: (a) (b) Fixed expiry T, and observe the o n market at time ti , i.e. a collection Ask Bid σCall/Put (ti , κj ), σCall/Put (ti , κj ) of markets quotes for differe
Slide 15: RL Framework for Fitting
A Reinforcement Learning Framework and Algorithm Problem Statement in Reinforcement Learning RL Framework for Fitting: Starting from a prior parametrized volatility surface Ψvol ⃗ ( i.e. the latest fit is the prior input). θ ti E. Gnabeyeu, Ecole Polytechnique. Solving The Dynamic Volatility Fitting
Slide 16: Goodness of Fit in RL-Based Volatility Fitting
A Reinforcement Learning Framework and Algorithm Problem Statement in Reinforcement Learning Goodness of Fit in RL-Based Volatility Fitting Mean squared error (MSE) 6 defined by : n X 2 ξ(θ⃗ti ) := σ Mid (ti , κj ) − Ψvol ⃗ (ki ) θ (3) ti j=1 The problem is to find the shifted vector ∆θ⃗ti = ∆θt1i ,
Slide 17: Market environment
1 General Overview of the Volatility Fitting Problem 2 A Reinforcement Learning Framework and Algorithm Background on Reinforcement Learning Problem Statement in Reinforcement Learning Market environment Deep Deterministic Policy Gradient Algorithm 3 Numerical Results 4 Conclusion and Prospects:
Slide 18: Market environment and Non Linear functional approximation by FCNNs
A Reinforcement Learning Framework and Algorithm Market environment Market environment and Non Linear functional approximation by FCNNs: 1 States are the collection of observable market quotes: sti+1 = σ Bid (ti+1 , κj ), σ Ask (ti+1 , κj ), θt1i , ..., θtKi j 2 Actions are the bumps to apply in ord
Slide 19: Deep Deterministic Policy Gradient Algorithm
A Reinforcement Learning Framework and Algorithm Deep Deterministic Policy Gradient Algorithm 1 General Overview of the Volatility Fitting Problem 2 A Reinforcement Learning Framework and Algorithm Background on Reinforcement Learning Problem Statement in Reinforcement Learning Market environment De
Slide 20: Actor Critic: Deep Deterministic Policy Gradient
A Reinforcement Learning Framework and Algorithm Deep Deterministic Policy Gradient Algorithm Actor Critic: Deep Deterministic Policy Gradient.(Lillicrap et al., 2015) off-policy actor-critic method that learns a deterministic policy in continuous domain. E. Gnabeyeu, Ecole Polytechnique. Solving Th
Slide 21: Actor Critic: Deep Deterministic Policy Gradient
A Reinforcement Learning Framework and Algorithm Deep Deterministic Policy Gradient Algorithm Actor Critic: Deep Deterministic Policy Gradient.(Lillicrap et al., 2015) off-policy actor-critic method that learns a deterministic policy in continuous domain. Action network: DPG= Deep Policy gradient n
Slide 22: Actor Critic: Deep Deterministic Policy Gradient
A Reinforcement Learning Framework and Algorithm Deep Deterministic Policy Gradient Algorithm Actor Critic: Deep Deterministic Policy Gradient.(Lillicrap et al., 2015) off-policy actor-critic method that learns a deterministic policy in continuous domain. Action network: DPG= Deep Policy gradient n
Slide 23: Numerical Results: Static Market, MSE Reward
Numerical Results Static Market, MSE Reward 1 General Overview of the Volatility Fitting Problem 2 A Reinforcement Learning Framework and Algorithm 3 Numerical Results Static Market, MSE Reward Quasi Dynamic Scenario, MSE Reward 4 Conclusion and Prospects: E. Gnabeyeu, Ecole Polytechnique. Solving T
Slide 24: Numerical results: Convergence of DDPG Algorithm
Numerical Results Static Market, MSE Reward Numerical results: Convergence of DDPG Algorithm. To assess the performance of the RL algorithms for the volatility fitting, we consider toy problems of increasing complexity and perform a Monte Carlo on 5 differents random seeds: 1 Static scenario: we exp
Slide 25: Static Market, DDPG algorithm, MSE Reward
Numerical Results Static Market, MSE Reward Static Market, DDPG algorithm, MSE Reward: Below, best response of the agent amongst the last 1000 episodes (blue) and the mean of the last 1000 volatilities slices ( red) (a) High Smile (b) Skew (c) High Skew (d) Inverse Smile. (a) (b) E. Gnabeyeu, Ecole
Slide 26: DDPG: Convergence and Consistency
Numerical Results Static Market, MSE Reward DDPG: Convergence and Consistency. Performance of the RL algorithms: Monte Carlo on 5 differents random seeds: 1 Static scenario: RL agents perform as-good as traditional approaches in fitting volatility curve. Table: DDPG MSE Rewards type Skew High Smile
Slide 27: Quasi Dynamic Scenario, MSE Reward
Numerical Results Quasi Dynamic Scenario, MSE Reward 1 General Overview of the Volatility Fitting Problem 2 A Reinforcement Learning Framework and Algorithm 3 Numerical Results Static Market, MSE Reward Quasi Dynamic Scenario, MSE Reward 4 Conclusion and Prospects: E. Gnabeyeu, Ecole Polytechnique.
Slide 28: Quasi Dynamic Scenario: The market evolve freely
Numerical Results Quasi Dynamic Scenario, MSE Reward Quasi Dynamic Scenario: The market evolve freely. Figure: Training and Consistency over random seeds. E. Gnabeyeu, Ecole Polytechnique. Solving The Dynamic Volatility Fitting Problem: October 14, 2025 15 / 18
Slide 29: Quasi Dynamic Scenario, MSE Reward
Numerical Results Quasi Dynamic Scenario, MSE Reward Quasi Dynamic Scenario, MSE Reward 2.Validation Phase: We compare the performance of different successful agents under different seeds. Figure: Evolution of mean episodic rewards(several Figure: Evolution of mean episodic rewards(several random se
Slide 30: Quasi Dynamic Scenario, MSE Reward, Testing Phase
Numerical Results Quasi Dynamic Scenario, MSE Reward Quasi Dynamic Scenario, MSE Reward, Testing Phase 3.Testing Phase: we assess the performance of the best agent in a test episode. Play Video Vol fitting animation E. Gnabeyeu, Ecole Polytechnique. Solving The Dynamic Volatility Fitting Problem: Oc
Slide 31: Conclusion and Prospects
Conclusion and Prospects: Conclusion and Prospects 1 Sum up, A wide range of problems can be tackled with Deep RL Actor-Critic methods while neural networks help to approximate the solution function.7 We showed that DRL algorithms are natively tailored for the volatility fitting problem: native expl
Slide 32: Conclusion and Prospects
Conclusion and Prospects: Conclusion and Prospects 1 Sum up, A wide range of problems can be tackled with Deep RL Actor-Critic methods while neural networks help to approximate the solution function.7 We showed that DRL algorithms are natively tailored for the volatility fitting problem: native expl
Slide 33: Thanks for your attention!
Conclusion and Prospects: Thanks for your attention! E. Gnabeyeu, Ecole Polytechnique. Solving The Dynamic Volatility Fitting Problem: October 14, 2025 18 / 18
Slide 34: References
Conclusion and Prospects: References. 1 B. Hambly, R. Xu, and H. Yang, Recent Advances in Reinforcement Learning in Finance. 2 J. Gatheral, A. Jacquier. Arbitrage-Free SVI Volatility Surfaces 3 V. Zetocha, Sculpting implied volatility surfaces of illiquid assets 4 Sutton, R. S. and Barto, A. G, Rein