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.

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. However, this approach does not inform outlier removal with the estimation task, leaving room for improvement. To address this limitation, we propose an automatic outlier rectification mechanism that integrates rectification and estimation within a joint optimization framework. We take the first step to utilize an optimal transport distance with a concave cost function to construct a rectification set in the space of probability distributions. Then, we select the best distribution within the rectification set to perform the estimation task. Notably, the concave cost function we introduced in this paper is the key to making our estimator effectively identify the outlier during the optimization process. We discuss the fundamental differences between our estimator and optimal transport-based distributionally robust optimization estimator. finally, we demonstrate the effectiveness and superiority of our approach over conventional approaches in extensive simulation and empirical analyses for mean estimation, least absolute regression, and the fitting of option implied volatility surfaces.

We introduce a new approach for generating sequences of implied volatility (IV) surfaces across multiple assets that is faithful to historical prices. We do so using a combination of functional data analysis and neural stochastic differential equations (SDEs) combined with a probability integral transform penalty to reduce model misspecification. We demonstrate that learning the joint dynamics of IV surfaces and prices produces market scenarios that are consistent with historical features and lie within the sub-manifold of surfaces that are essentially free of static arbitrage. Finally, we demonstrate that delta hedging using the simulated surfaces generates profit and loss (P&L) distributions that are consistent with realised P&Ls.