This paper proposes a hybrid methodology to improve the approximation of SABR (Stochastic Alpha Beta Rho) implied volatility by combining analytical structure with machine learning. The approach augments the neural-network input representation with geometric features derived from the stochastic differential equations of the SABR model. Unlike approaches that fully replace analytical formulas with black-box models, the proposed framework preserves the analytical backbone of the model. The hybridization operates along two complementary dimensions. First, geometry-aware variables reflecting intrinsic properties of the SABR dynamics are used as structured inputs to the network. Second, the neural network is trained to learn the residual error relative to Hagan's closed-form approximation rather than implied volatility directly. The resulting model acts as a structured residual correction to the analytical formula, retaining interpretability while capturing higher-order effects that are not included in the asymptotic expansion. Numerical experiments conducted over realistic parameter domains, as well as stressed environments, show that the method improves accuracy and robustness compared with both analytical approximations and standard neural-network approaches. Because the correction remains lightweight and structurally consistent with the underlying model, the framework is well suited for real-time pricing and calibration in practical trading environments.

Option prices encode the market's collective outlook through implied density and implied volatility. An explicit link between implied density and implied volatility translates the risk-neutrality of the former into conditions on the latter to rule out static arbitrage. Despite earlier recognition of their parity, the two had been studied in isolation for decades until the recent demand in implied volatility modeling rejuvenated such parity. This paper provides a systematic approach to build neural representations of option implied information. As a preliminary, we first revisit the explicit link between implied density and implied volatility through an alternative and minimalist lens, where implied volatility is viewed not as volatility but as a pointwise corrector mapping the Black-Scholes quasi-density into the implied risk-neutral density. Building on this perspective, we propose the neural representation that incorporates arbitrage constraints through the differentiable corrector. With an additive logistic model as the synthetic benchmark, extensive experiments reveal that deeper or wider network structures do not necessarily improve the model performance due to the nonlinearity of both arbitrage constraints and neural derivatives. By contrast, a shallow feedforward network with a single hidden layer and a specific activation effectively approximates implied density and implied volatility.

Recent work in financial machine learning has shown the virtue of complexity: the phenomenon by which deep learning methods capable of learning highly nonlinear relationships outperform simpler approaches in financial forecasting. While transformer architectures like Informer have shown promise for financial time series forecasting, the application of transformer models for options data remains largely unexplored. We conduct preliminary studies towards the development of a transformer model for options data by training the Vision Transformer (ViT) architecture, typically used in modern image recognition and classification systems, to predict the realized volatility of an asset over the next 30 days from its implied volatility surface (augmented with date information) for a single day. We show that the ViT can learn seasonal patterns and nonlinear features from the IV surface, suggesting a promising direction for model development.

In this paper, we propose a novel conceptual framework to detect outliers using optimal transport with a concave cost function. Conventional outlier detection approaches typically use a two-stage procedure: first, outliers are detected and removed, and then estimation is performed on the cleaned data.

Atypically to standard NN applications, financial industry practitioners use such models equally to replicate market prices and to value other financial instruments. In other words, low training losses are as important as generalization capabilities.

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.

Providing an arbitrage-free valuation formula and specifying risk-neutral dynamics are essentially two sides of the same coin in option pricing. Yet, the modeling methodology has been leaning towards the latter for decades. That is, the invention of an option pricing model typically starts with proposing a stochastic process that is a martingale for the underlying asset, so that the corresponding risk-neural measure is constructed, and henceforth the arbitrage-free option valuation can be determined either analytically or numerically. Such a methodology was established through the pioneering work of Bachelier [4] and Black and Scholes [9], and since then, almost all of the prevailing models have been invented along this paradigm. The list includes but is not limited to local volatility models by Dupire [17], Cox [14], stochastic volatility models by Heston [20], Hagan et al. [18], Bates [8], jump-diffusion models by Merton [28], Kou [24], and other models built upon Lévy processes by Madan et al. [26], Barndorff-Nielsen [7]. Nonetheless, the reverse approach, which first provides an arbitrage-free valuation formula as in Carr and Madan [11], Davis and Hobson [15] and then finds the underlying martingale supporting the formula, is still possible, as noted in [21, 27]. In recent work, Carr and Torricelli [12] starts with one particular pricing formula that yields logistically distributed marginals. Although there is no underlying Lévy process that produces such marginals, by allowing the increment to be nonstationary, an additive logistic process can be constructed to support that pricing formula.

As such, indicators in the options market, such as options prices, implied volatility, and the Greeks, are seen as "smarter" compared to indicators in the securities market. Numerous studies have confirmed this perspective and have explored the discovery function of options implied volatility on securities prices. For instance, Ni et al. (2020) found that the degree of skewness in implied volatility smiles has a significant predictive ability for stock market returns, while Han and Li (2021) discovered that the difference between call and put implied volatility has significant predictive power for stock market returns. Additionally, there is more research on the predictive ability of options implied volatility on realized volatility, dating back to Latane and Rendleman (1976-05) reverse use of the BS formula to derive the implied standard deviation of options and constructing a weighted implied standard deviation (WISD) using delta-neutral weighting, which was found to predict actual volatility significantly better than methods based on historical volatility. In recent years, numerous studies have incorporated the VIX index and the HAR method proposed by Corsi (2009), achieving notable results in predicting stock market volatility Byun and Kim (2013); Zhang (2020); Wan and Tian (2023). Preprint submitted to Elsarticle May 18, 2024 However, indicators in the options market should not be treated as the gold standard.