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.

We develop an arbitrage-free deep learning framework for yield curve and bond price forecasting based on the Heath-Jarrow-Morton (HJM) term-structure model and a dynamic Nelson-Siegel parameterization of forward rates. Our approach embeds a no-arbitrage drift restriction into a neural state-space architecture by combining Kalman, extended Kalman, and particle filters with recurrent neural networks (LSTM/CLSTM), and introduces an explicit arbitrage error regularization (AER) term during training. The model is applied to U.S. Treasury and corporate bond data, and its performance is evaluated for both yield-space and price-space predictions at 1-day and 5-day horizons. Empirically, arbitrage regularization leads to its strongest improvements at short maturities, particularly in 5-day-ahead forecasts, increasing market-consistency as measured by bid-ask hit rates and reducing dollar-denominated prediction errors.

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.

Group decision making is an important field with interesting applications in various disciplines, among which environmental economics. Group decision, often requires that all or the majority of agents in the group agree to a single proposal or opinion, i.e. consensus. This is particularly true in cases where there is no coercion involved in the implementation of the decision made, so that the implementation of the decision depends on the good will, or rather the acceptance of the common decision by all members of the group. To make the discussion more concrete we consider the following generic situation: Assume that a group of agents, G, has to reach a common decision concerning policies regarding a future contingency X. Policies may refer for instance to the cost of abatement measures for protection against X, which clearly require the acceptance of a commonly acceptable estimate for the value of X by every member of the group as well as the acceptance of a commonly acceptably discount factor. Typically, different member of the group will have different valuations for X, therefore report different costs for the adverse effects of X. Moreover, different members of the group will have different discount rates for calculating the present value of the future adverse effect X.

Most representative decision tree ensemble methods have been used to examine the variable importance of Treasury term spreads to predict US economic recessions with a balance of generating rules for US economic recession detection. A strategy is proposed for training the classifiers with Treasury term spreads data and the results are compared in order to select the best model for interpretability. We also discuss the use of SHapley Additive exPlanations (SHAP) framework to understand US recession forecasts by analyzing feature importance. Consistently with the existing literature we find the most relevant Treasury term spreads for predicting US economic recession and a methodology for detecting relevant rules for economic recession detection. In this case, the most relevant term spread found is 3 month to 6 month, which is proposed to be monitored by economic authorities. Finally, the methodology detected rules with high lift on predicting economic recession that can be used by these entities for this propose. This latter result stands in contrast to a growing body of literature demonstrating that machine learning methods are useful for interpretation comparing many alternative algorithms and we discuss the interpretation for our result and propose further research lines aligned with this work.

"Attention takes two sentences, turns them into a matrix where the words of one sentence form the columns, and the words of another sentence form the rows, and then it makes matches, identifying relevant context." Check out the graphic from the Attention is All You Need paper below. It's two sentences, in different languages (French and English), translated by a professional human translator. The attention mechanism can generate a heat map, showing what French words the model focused on to generate the translated English words in the output.

Consistent Recalibration models (CRC) have been introduced to capture in necessary generality the dynamic features of term structures of derivatives' prices. Several approaches have been suggested to tackle this problem, but all of them, including CRC models, suffered from numerical intractabilities mainly due to the presence of complicated drift terms or consistency conditions. We overcome this problem by machine learning techniques, which allow to store the crucial drift term's information in neural network type functions. This yields first time dynamic term structure models which can be efficiently simulated.

Quants are in the business of making helpful assumptions around modelling the characteristics of an asset. But when it comes to interest rates, many of those simplifying conventions tend to break down. That has prompted a growing number of quants to explore the use of machine-learning techniques to better predict the term structure of interest rates. For instance, it is not uncommon to model stock price returns by assuming they are driven by a normal distribution and a volatility that is independent of the level of the stock. But movements in interest rates have been shown to depend heavily on the absolute level of rates at a given point in time.

Yield curve forecasting is an important problem in finance. In this work we explore the use of Gaussian Processes in conjunction with a dynamic modeling strategy, much like the Kalman Filter, to model the yield curve. Gaussian Processes have been successfully applied to model functional data in a variety of applications. A Gaussian Process is used to model the yield curve. The hyper-parameters of the Gaussian Process model are updated as the algorithm receives yield curve data. Yield curve data is typically available as a time series with a frequency of one day. We compare existing methods to forecast the yield curve with the proposed method. The results of this study showed that while a competing method (a multivariate time series method) performed well in forecasting the yields at the short term structure region of the yield curve, Gaussian Processes perform well in the medium and long term structure regions of the yield curve. Accuracy in the long term structure region of the yield curve has important practical implications. The Gaussian Process framework yields uncertainty and probability estimates directly in contrast to other competing methods. Analysts are frequently interested in this information. In this study the proposed method has been applied to yield curve forecasting, however it can be applied to model high frequency time series data or data streams in other domains.

Inductive theorem provers often diverge. This paper describes a simple critic, a computer program which monitors the construction of inductive proofs attempting to identify diverging proof attempts. Divergence is recognized by means of a ``difference matching'' procedure. The critic then proposes lemmas and generalizations which ``ripple'' these differences away so that the proof can go through without divergence. The critic enables the theorem prover Spike to prove many theorems completely automatically from the definitions alone.