ABSTRACT We propose a novel hybrid framework that integrates stochastic volatility modeling with deep learning. The main contribution is a dual neural network architecture combining convolutional and LSTM networks (CNN-LSTM). This architecture learns and corrects pricing errors from the Heston model, and can optionally be enriched with GARCH-based time-varying volatility forecasts. The methodology involves three stages. First, the Heston model is calibrated using a genetic algorithm. Second, a GARCH model captures volatility dynamics over time. Third, a hybrid parametric model is built by integrating the CNN-LSTM network with the Heston framework (with or without GARCH predicted volatility). Results show that the hybrid Heston – CNN-LSTM model significantly lowers pricing errors and corrects systematic bias. Adding GARCH further enhances performance, especially for deep out-of-the-money options and under turbulent market conditions.

In energy markets, joint historical and implied calibration is of paramount importance for practitioners, yet notoriously challenging due to the need to align historical correlations of futures contracts with implied volatility smiles from the option market. We address this crucial problem with a multiplicative multi-factor Heath–Jarrow–Morton (HJM) model for forward curves, combined with a stochastic volatility factor coming from the lifted Heston model. We develop a sequential fast calibration procedure leveraging the Kemna–Vorst approximation of futures contracts: (i) historical correlations and the Variance Swap (VS) volatility term structure are captured through Level, Slope, and Curvature factors, (ii) the VS volatility term structure can then be corrected for a perfect match via a fixed-point algorithm, (iii) implied volatility smiles are calibrated using Fourier-based techniques. The main advantage of the proposed calibration framework is the decoupling of the calibration steps: each step tackles a simpler calibration subproblem and guaranties that the previously optimized parameters remain unchanged. Our model displays remarkable joint historical and implied calibration fits on the German power market and enables realistic interpolation within the implied volatility hypercube.

A development of a new measure for renewable energy uncertainty in Vietnam by using natural language processing and textual analysis

ABSTRACT Building upon factor decomposition to overcome the curse of dimensionality inherent in multivariate volatility processes, we develop a factor model-based multivariate stochastic volatility (fMSV) framework. We propose a two-stage estimation procedure for the fMSV model: in the first stage, estimators of the factor model are obtained, and in the second stage, the MSV component is estimated using the estimated common factor variables. We derive the asymptotic properties of the estimators, taking into account the estimation of the factor variables. Simulation experiments indicate that the fMSV model provides accurate measurements of the true underlying variance–covariance matrix, while empirical applications to portfolio allocation suggest superior forecasting performance compared to standard multivariate volatility models.

Motivated by empirical evidence from the joint behavior of realized volatility time series, we propose to model the joint dynamics of log-volatilities using a multivariate fractional Ornstein-Uhlenbeck process. This model is a multivariate version of the Rough Fractional Stochastic Volatility model introduced in Gatheral, Jaisson, and Rosenbaum, Quant. Finance, 2018. It allows for different Hurst exponents in the different marginal components and non trivial interdependencies. We discuss the main features of the model and propose a Generalized Method of Moments estimator that jointly identifies its parameters. We derive the asymptotic theory of the estimator and perform a simulation study that confirms the asymptotic theory in finite sample. We conduct an extensive empirical investigation of all realized-volatility time series covering the entire span of about two decades in the Oxford-Man realized library, and of a small spot-volatility system. Our analysis shows that these time series are strongly correlated and can exhibit asymmetries in their empirical cross-covariance function, accurately captured by our model. These asymmetries lead to spillover effects, which we derive analytically within our model and compute based on empirical estimates of model parameters. Moreover, in accordance with the existing literature, we observe behaviors close to non-stationarity and rough trajectories.

This paper develops an option pricing framework under a regime-switching doubly exponential co-jump model with stochastic volatility. The co-jump intensity between asset prices and volatility is governed by a continuous-time Markov chain to reflect varying frequencies of market shocks across economic states. We derive an analytical option pricing formula within an affine framework and utilize the Fast Fourier Transform for calculation. A procedure based on a particle filter combined with the expectation-maximization algorithm is proposed for parameter estimation. Monte Carlo simulations confirm the accuracy of the pricing formula and the estimation algorithm. An empirical analysis using S&P 500 index options demonstrates that this model yields lower in-sample and out-of-sample pricing errors compared to standard stochastic volatility and jump-diffusion models. This framework provides a tool for pricing derivatives and managing risk in markets with structural changes.

The integration of high-penetration renewable energy sources (RES) into global power systems necessitates advanced scheduling strategies to ensure supply-demand balance. Virtual Power Plants (VPPs) serve as critical aggregators for distributed resources; however, coordinating VPPs across multiple regions is hindered by the curse of dimensionality, partial observability, and stochastic volatility. Conventional centralized optimization lacks scalability for real-time applications, while single-agent approaches fail to effectively address complex collaborative dynamics. To overcome these limitations, this paper proposes a collaborative scheduling framework based on Multi-Agent Reinforcement Learning (MARL). We model the global system as a multi-regional environment where heterogeneous agents operate under a Centralized Training with Decentralized Execution (CTDE) architecture. A composite reward function is designed to balance economic efficiency with RES absorption, utilizing an attention-based mechanism to exploit time-zone complementarity. Simulation results demonstrate that the proposed method significantly outperforms baseline strategies. Specifically, it achieves a global RES accommodation rate of 94.2% and maintains a minimal tie-line violation rate of 0.8%, compared to only 76.5% accommodation with rule-based heuristics. Furthermore, the approach exhibits superior robustness in extreme-volatility scenarios where standard methods degrade. This study validates the efficacy of distributed intelligence in solving large-scale energy dispatch problems, offering a scalable and privacy-preserving pathway for managing the Global Energy Interconnection.

This paper investigates the robust optimal investment for an ambiguity averse member of a defined contribution (DC) pension plan in a fully-fledged, time consistent mean-variance modeling framework. In particular, the paper extends the literature on defined contribution pension plans in three directions: (1) We relax its assumption of purely continuous stock and/or contribution processes, which allows to introduce the effects of news, job loss, macroeconomic conditions, etc., into the model; (2) Unlike most studies in DC pension plans, we allow for ambiguity about both the mean arrival rate and jump size distribution of the stock returns and contribution rate processes of the member; (3) Ambiguity in our setting is time-varying. The model thus features stochastic stock volatility, stochastic interest rate, stochastic contribution rate, jumps in both stock and contribution rate processes, and time-varying ambiguity about diffusion parameters. Welfare analysis indicates that ignoring ambiguity can be very costly to the member. The framework proposed in this paper is general and adds significant realism to existing models in the literature.

ABSTRACT The literature on trade uncertainty (TU) primarily relies on low-frequency measures, with limited efforts to construct high-frequency TU or examine its effects. This study estimates daily TU using a mixed-frequency stochastic volatility model and applies cross-quantilogram and time-varying parameter local projection approaches to assess the high-frequency effects of China’s TU on U.S. business conditions (BC). Results show that China’s TU is event-driven and linked to exchange rate movements, predicts U.S. BC mainly at low and medium quantiles, and has time-varying effects on U.S. BC that reach their maximum magnitude during 2016–2020, particularly during the COVID-19 outbreak. These findings underscore the importance of easing trade frictions and maintaining bilateral trade stability for U.S. BC and provide implications for monitoring high-frequency TU and its time-varying spillovers.

A particle-mesh operator splitting framework for American option pricing under stochastic volatility

Analytically pricing vulnerable options under the stochastic volatility model with stochastic long-term mean and stochastic liquidity

ABSTRACT Economic policy uncertainty (EPU) is a critical yet often neglected factor in derivatives pricing, leading to systematic biases in existing valuation frameworks. This study proposes an advanced pricing model that explicitly incorporates a stochastic EPU factor with a regime‐switching long‐term mean into the stochastic volatility framework. The principal theoretical contribution of this research is a closed‐form analytical pricing formula for European options, which successfully overcomes the challenges of multiple stochastic factors and achieves substantial computational advantages over traditional numerical methods such as the Monte‐Carlo simulation. A preliminary empirical study using SSE 50 ETF option data demonstrates the model's superior pricing performance compared to other benchmarks.

This paper develops two neural-network-based frameworks for option pricing that incorporate financial option pricing PDEs while accommodating deviations from strict riskneutral valuation. The first approach, termed QINN, extends physics-informed neural networks (PINNs) by introducing a regularization parameter α that balances empirical data fitting with PDE-consistency. This enables QINN both to approximate PDE solutions directly and to infer latent model parameters, offering an alternative to conventional calibration techniques. The second approach, QINN 2 , removes the need for a prespecified model by embedding the volatility parameter of the Black–Scholes PDE into a separate neural network. This model-free formulation flexibly adapts to option data generated from different local and stochastic volatility models within a single framework. Numerical experiments across Black–Scholes, CEV, Heston, and 3/2 models demonstrate that QINN 2 matches or surpasses the accuracy of model-based QINN, especially for more complex dynamics. Together, these results highlight QINN and QINN 2 as practical and robust neural approaches to option pricing, bridging data-driven learning with financial model structure.

This study investigates the notion of the varying unpredictable volatility in the prices of green and traditional cryptocurrencies, and its symbiotic relationship with investor sentiment. Accordingly, the structural time varying parameter vector autoregression (TVP-VAR) method is employed for the 1 January 2018–23 September 2022 period. The findings revealed variable relationships of the sentiment index (SI) with cryptocurrencies amid various economic situations. The stochastic volatility of Bitcoin and Ethereum peaked around January 2018, trending lower since then, and rising back from 2021 to 2022, particularly in early 2021 and early 2022. During the Russia–Ukraine conflict, the stochastic volatility of Cardano was lower than those of Bitcoin and Ethereum. The simultaneous relationship of SI to the Bitcoin, IOTA, Cardano, and Ethereum shock vary between positive and negative in 2021 before turning negative in 2022. Bitcoin’s initial reactions to a positive investor mood shock are insignificantly different from zero. These results of SI to positive Bitcoin and IOTA shock decrease during our sample except in 2021. The impulse responses of Bitcoin to positive Ethereum and Cardano shocks decrease in 2018 and fluctuate during 2021–2022 and these results of SI to these negative shocks are significant due to the COVID-19 duration. In summary, the findings provide a novel and comprehensive analysis of the interdependencies between investor sentiment, and green and nongreen cryptocurrencies. In particular, the stochastic volatility of the cryptocurrency classes highlights spikes of investor sentiment during the Black Swan events (the COVID-19, the Russia–Ukraine conflict). The research offers valuable insights and cautionary implications regarding the transmission of uncertainty, and policy implications for regulators and investors.

When looking at financial returns, we consistently see phenomena like volatilityclustering and the leverage effect. This effect simply means negative returns pushfuture volatility higher than positive ones do. Stochastic Volatility (SV) modelsare great for capturing these dynamics, but finding their parameters is notoriouslytough. The main hurdle is that the hidden volatility states make the likelihood function intractable. To get around this problem, I apply Particle Markov Chain MonteCarlo (PMCMC) and specifically lean on the Particle Marginal Metropolis-Hastings(PMMH) approach to estimate an SV model that includes a leverage component.Running this on a simulated dataset (N = 2000) gave us a 34.22% acceptance rateand accurately pinned down the true parameters. Ultimately, the results prove PMCMC is a solid and reliable choice for estimating the correlation between price andvariance shocks without falling back on biased linear approximations.

ABSTRACT As a prominent tool for tail risk measurement, expectile‐based value at risk (EVaR) has attracted growing interest due to its sensitivity to extreme risks. Existing approaches face two principal challenges: the inefficiency of conventional models under finite samples and the tendency of single machine learning models toward overfitting or underfitting. This paper introduces a nonlinear expectile regression model based on a blending ensemble framework, integrating neural networks, support vector machines, XGBoost, LightGBM, and random forests as base learners, with an expectile regression forest as the metalearner. Monte Carlo simulations confirm the method's robustness in finite samples. Applied to Chinese stock indices, the model outperforms both traditional linear specifications and each individual machine learning model in EVaR forecasting. Performance gains are statistically significant under stochastic volatility and TGARCH settings, as verified by Diebold–Mariano and Giacomini–White tests. SHAP analysis further shows that XGBoost and LightGBM contribute most to prediction, enhancing interpretability and offering insight into ensemble decision mechanisms.

ABSTRACT This paper studies an investment‐reinsurance contract between an insurer and a reinsurer with asymmetric bargaining power. We assume that the surplus of the insurer follows a jump‐diffusion process. To reduce the risk of claims, the insurer can purchase proportional reinsurance, with the reinsurance premium calculated based on the expected value principle. The surplus of the insurer and the reinsurer can be allocated to a financial market consisting of a risk‐free asset and a risky asset, respectively. The price processes of the insurer's and reinsurer's risky assets satisfy different square root factor processes. To consider the benefits of both the insurer and the reinsurer, the optimization problem is formulated as an asymmetric Nash bargaining game. To maximize the weighted product of the expected exponential utility of the terminal wealth of both parties, explicit expressions for the Pareto‐optimal strategy and the corresponding value function are derived by employing stochastic control techniques and the Hamilton‐Jacobi‐Bellman (HJB) equation. In addition, we provide equilibrium strategies under several special cases, including cases where only the insurer or the reinsurer is considered, as well as models under the CEV (Constant Elasticity of Variance) model and the Heston stochastic volatility model. Finally, numerical examples illustrate the impact of bargaining power on optimal strategy.

Short Communication: Martingale Expansion for Stochastic Volatility

Parameter estimation of a distribution can be performed through two main approaches: the classical method and the Bayesian method. The Bayesian method integrates the sample distribution with the prior distribution, where random sampling is conducted via simulation techniques such as Markov Chain Monte Carlo (MCMC) with the Gibbs Sampling algorithm. This algorithm works by constructing a Markov Chain through recursive sampling from the full conditional posterior distribution for each parameter until convergence is reached. This study applies the Bayesian method with MCMC using the Gibbs Sampling algorithm to estimate the parameters of the Stochastic Volatility model, which allows asset price volatility to vary over time. The obtained Stochastic Volatility model is then used to predict the stock returns of PT. Aneka Tambang Tbk. (ANTM.JK), where the prediction results show good conformity with actual data. The resulting prediction values can be utilized by investors as a reference in making optimal investment portfolio decisions.

Valuation of American put options under a modified 4/2 stochastic volatility model