Implied Volatility Surface

Forecasting implied volatility surface with generative diffusion models ArXiv ID: 2511.07571 “View on arXiv” Authors: Chen Jin, Ankush Agarwal Abstract We introduce a conditional Denoising Diffusion Probabilistic Model (DDPM) for generating arbitrage-free implied volatility (IV) surfaces, offering a more stable and accurate alternative to existing GAN-based approaches. To capture the path-dependent nature of volatility dynamics, our model is conditioned on a rich set of market variables, including exponential weighted moving averages (EWMAs) of historical surfaces, returns and squared returns of underlying asset, and scalar risk indicators like VIX. Empirical results demonstrate our model significantly outperforms leading GAN-based models in capturing the stylized facts of IV dynamics. A key challenge is that historical data often contains small arbitrage opportunities in the earlier dataset for training, which conflicts with the goal of generating arbitrage-free surfaces. We address this by incorporating a standard arbitrage penalty into the loss function, but apply it using a novel, parameter-free weighting scheme based on the signal-to-noise ratio (SNR) that dynamically adjusts the penalty’s strength across the diffusion process. We also show a formal analysis of this trade-off and provide a proof of convergence showing that the penalty introduces a small, controllable bias that steers the model toward the manifold of arbitrage-free surfaces while ensuring the generated distribution remains close to the real-world data. ...

Risk-Sensitive Option Market Making with Arbitrage-Free eSSVI Surfaces: A Constrained RL and Stochastic Control Bridge ArXiv ID: 2510.04569 “View on arXiv” Authors: Jian’an Zhang Abstract We formulate option market making as a constrained, risk-sensitive control problem that unifies execution, hedging, and arbitrage-free implied-volatility surfaces inside a single learning loop. A fully differentiable eSSVI layer enforces static no-arbitrage conditions (butterfly and calendar) while the policy controls half-spreads, hedge intensity, and structured surface deformations (state-dependent rho-shift and psi-scale). Executions are intensity-driven and respond monotonically to spreads and relative mispricing; tail risk is shaped with a differentiable CVaR objective via the Rockafellar–Uryasev program. We provide theory for (i) grid-consistency and rates for butterfly/calendar surrogates, (ii) a primal–dual grounding of a learnable dual action acting as a state-dependent Lagrange multiplier, (iii) differentiable CVaR estimators with mixed pathwise and likelihood-ratio gradients and epi-convergence to the nonsmooth objective, (iv) an eSSVI wing-growth bound aligned with Lee’s moment constraints, and (v) policy-gradient validity under smooth surrogates. In simulation (Heston fallback; ABIDES-ready), the agent attains positive adjusted P&L on most intraday segments while keeping calendar violations at numerical zero and butterfly violations at the numerical floor; ex-post tails remain realistic and can be tuned through the CVaR weight. The five control heads admit clear economic semantics and analytic sensitivities, yielding a white-box learner that unifies pricing consistency and execution control in a reproducible pipeline. ...

Meta-Learning Neural Process for Implied Volatility Surfaces with SABR-induced Priors ArXiv ID: 2509.11928 “View on arXiv” Authors: Jirong Zhuang, Xuan Wu Abstract We treat implied volatility surface (IVS) reconstruction as a learning problem guided by two principles. First, we adopt a meta-learning view that trains across trading days to learn a procedure that maps sparse option quotes to a full IVS via conditional prediction, avoiding per-day calibration at test time. Second, we impose a structural prior via transfer learning: pre-train on SABR-generated dataset to encode geometric prior, then fine-tune on historical market dataset to align with empirical patterns. We implement both principles in a single attention-based Neural Process (Volatility Neural Process, VolNP) that produces a complete IVS from a sparse context set in one forward pass. On SPX options, the VolNP outperforms SABR, SSVI, and Gaussian process. Relative to an ablation trained only on market data, the SABR-induced prior reduces RMSE by about 40% and suppresses large errors, with pronounced gains at long maturities where quotes are sparse. The resulting model is fast (single pass), stable (no daily recalibration), and practical for deployment at scale. ...

Controllable Generation of Implied Volatility Surfaces with Variational Autoencoders ArXiv ID: 2509.01743 “View on arXiv” Authors: Jing Wang, Shuaiqiang Liu, Cornelis Vuik Abstract This paper presents a deep generative modeling framework for controllably synthesizing implied volatility surfaces (IVSs) using a variational autoencoder (VAE). Unlike conventional data-driven models, our approach provides explicit control over meaningful shape features (e.g., volatility level, slope, curvature, term-structure) to generate IVSs with desired characteristics. In our framework, financially interpretable shape features are disentangled from residual latent factors. The target features are embedded into the VAE architecture as controllable latent variables, while the residual latent variables capture additional structure to preserve IVS shape diversity. To enable this control, IVS feature values are quantified via regression at an anchor point and incorporated into the decoder to steer generation. Numerical experiments demonstrate that the generative model enables rapid generation of realistic IVSs with desired features rather than arbitrary patterns, and achieves high accuracy across both single- and multi-feature control settings. For market validity, an optional post-generation latent-space repair algorithm adjusts only the residual latent variables to remove occasional violations of static no-arbitrage conditions without altering the specified features. Compared with black-box generators, the framework combines interpretability, controllability, and flexibility for synthetic IVS generation and scenario design. ...

Implied Probabilities and Volatility in Credit Risk: A Merton-Based Approach with Binomial Trees ArXiv ID: 2506.12694 “View on arXiv” Authors: Jagdish Gnawali, Abootaleb Shirvani, Svetlozar T. Rachev Abstract We explore credit risk pricing by modeling equity as a call option and debt as the difference between the firm’s asset value and a put option, following the structural framework of the Merton model. Our approach proceeds in two stages: first, we calibrate the asset volatility using the Black-Scholes-Merton (BSM) formula; second, we recover implied mean return and probability surfaces under the physical measure. To achieve this, we construct a recombining binomial tree under the real-world (natural) measure, assuming a fixed initial asset value. The volatility input is taken from a specific region of the implied volatility surface - based on moneyness and maturity - which then informs the calibration of drift and probability. A novel mapping is established between risk-neutral and physical parameters, enabling construction of implied surfaces that reflect the market’s credit expectations and offer practical tools for stress testing and credit risk analysis. ...

Correct implied volatility shapes and reliable pricing in the rough Heston model ArXiv ID: 2412.16067 “View on arXiv” Authors: Unknown Abstract We use modifications of the Adams method and very fast and accurate sinh-acceleration method of the Fourier inversion (iFT) (S.Boyarchenko and Levendorskiĭ, IJTAF 2019, v.22) to evaluate prices of vanilla options; for options of moderate and long maturities and strikes not very far from the spot, thousands of prices can be calculated in several msec. with relative errors of the order of 0.5% and smaller running Matlab on a Mac with moderate characteristics. We demonstrate that for the calibrated set of parameters in Euch and Rosenbaum, Math. Finance 2019, v. 29, the correct implied volatility surface is significantly flatter and fits the data very poorly, hence, the calibration results in op.cit. is an example of the {"\em ghost calibration"} (M.Boyarchenko and Levendorkiĭ, Quantitative Finance 2015, v. 15): the errors of the model and numerical method almost cancel one another. We explain how calibration errors of this sort are generated by each of popular versions of numerical realizations of iFT (Carr-Madan, Lipton-Lewis and COS methods) with prefixed parameters of a numerical method, resulting in spurious volatility smiles and skews. We suggest a general {"\em Conformal Bootstrap principle"} which allows one to avoid ghost calibration errors. We outline schemes of application of Conformal Bootstrap principle and the method of the paper to the design of accurate and fast calibration procedures. ...

Solving The Dynamic Volatility Fitting Problem: A Deep Reinforcement Learning Approach ArXiv ID: 2410.11789 “View on arXiv” Authors: Unknown Abstract The volatility fitting is one of the core problems in the equity derivatives business. Through a set of deterministic rules, the degrees of freedom in the implied volatility surface encoding (parametrization, density, diffusion) are defined. Whilst very effective, this approach widespread in the industry is not natively tailored to learn from shifts in market regimes and discover unsuspected optimal behaviors. In this paper, we change the classical paradigm and apply the latest advances in Deep Reinforcement Learning(DRL) to solve the fitting problem. In particular, we show that variants of Deep Deterministic Policy Gradient (DDPG) and Soft Actor Critic (SAC) can achieve at least as good as standard fitting algorithms. Furthermore, we explain why the reinforcement learning framework is appropriate to handle complex objective functions and is natively adapted for online learning. ...

Robust financial calibration: a Bayesian approach for neural SDEs ArXiv ID: 2409.06551 “View on arXiv” Authors: Unknown Abstract The paper presents a Bayesian framework for the calibration of financial models using neural stochastic differential equations (neural SDEs), for which we also formulate a global universal approximation theorem based on Barron-type estimates. The method is based on the specification of a prior distribution on the neural network weights and an adequately chosen likelihood function. The resulting posterior distribution can be seen as a mixture of different classical neural SDE models yielding robust bounds on the implied volatility surface. Both, historical financial time series data and option price data are taken into consideration, which necessitates a methodology to learn the change of measure between the risk-neutral and the historical measure. The key ingredient for a robust numerical optimization of the neural networks is to apply a Langevin-type algorithm, commonly used in the Bayesian approaches to draw posterior samples. ...

Neural Term Structure of Additive Process for Option Pricing ArXiv ID: 2408.01642 “View on arXiv” Authors: Unknown Abstract The additive process generalizes the Lévy process by relaxing its assumption of time-homogeneous increments and hence covers a larger family of stochastic processes. Recent research in option pricing shows that modeling the underlying log price with an additive process has advantages in easier construction of the risk-neural measure, an explicit option pricing formula and characteristic function, and more flexibility to fit the implied volatility surface. Still, the challenge of calibrating an additive model arises from its time-dependent parameterization, for which one has to prescribe parametric functions for the term structure. For this, we propose the neural term structure model to utilize feedforward neural networks to represent the term structure, which alleviates the difficulty of designing parametric functions and thus attenuates the misspecification risk. Numerical studies with S&P 500 option data are conducted to evaluate the performance of the neural term structure. ...

The implied volatility surface (also) is path-dependent ArXiv ID: 2312.15950 “View on arXiv” Authors: Unknown Abstract We propose a new model for the forecasting of both the implied volatility surfaces and the underlying asset price. In the spirit of Guyon and Lekeufack (2023) who are interested in the dependence of volatility indices (e.g. the VIX) on the paths of the associated equity indices (e.g. the S&P 500), we first study how vanilla options implied volatility can be predicted using the past trajectory of the underlying asset price. Our empirical study reveals that a large part of the movements of the at-the-money-forward implied volatility for up to two years time-to-maturities can be explained using the past returns and their squares. Moreover, we show that this feedback effect gets weaker when the time-to-maturity increases. Building on this new stylized fact, we fit to historical data a parsimonious version of the SSVI parameterization (Gatheral and Jacquier, 2014) of the implied volatility surface relying on only four parameters and show that the two parameters ruling the at-the-money-forward implied volatility as a function of the time-to-maturity exhibit a path-dependent behavior with respect to the underlying asset price. Finally, we propose a model for the joint dynamics of the implied volatility surface and the underlying asset price. The latter is modelled using a variant of the path-dependent volatility model of Guyon and Lekeufack and the former is obtained by adding a feedback effect of the underlying asset price onto the two parameters ruling the at-the-money-forward implied volatility in the parsimonious SSVI parameterization and by specifying Ornstein-Uhlenbeck processes for the residuals of these two parameters and Jacobi processes for the two other parameters. Thanks to this model, we are able to simulate highly realistic paths of implied volatility surfaces that are free from static arbitrage. ...