ABSTRACT A collection of time series are “related” if they follow similar stochastic processes and/or they are statistically dependent. This paper proposes a related time series (RTS) forecasting model that exploits these relationships. The model's foundation is a set of univariate Gaussian autoregressions, one for each series, which are then augmented to incorporate stochastic volatility, heavy‐tailed innovations, additive outliers, time‐varying parameters and common factors. The model is estimated and forecasts are computed using Bayesian methods with hierarchical priors that pool information across series. Computationally efficient MCMC methods are proposed. The RTS model is applied to three datasets and yields encouraging pseudo‐out‐of‐sample forecasting results.

We argue that negative skew and positive mean of the distribution of stock returns are largely due to the broken symmetry of stochastic volatility governing gains and losses. Starting with stochastic differential equations for stock returns and for stochastic volatility, we argue that the distribution of stock returns can be effectively split in two—for gains and losses—assuming difference in parameters of their respective stochastic volatilities. A modified Jones–Faddy skew t-distribution utilized here allows to reflect this in a single organic distribution which tends to meaningfully capture this asymmetry. We illustrate its application on distribution of daily S&P500 returns and analyze its tails.

Modern financial systems do not exist in isolation but form part of a complex global network of interconnected financial systems. This globalization of financial systems significantly increases the risk of contagion in financial markets, impacting asset prices and other important economic factors, including interest rates and market volatility. This phenomenon informs not only investors’ investment strategies but also the prices of contingent claims. In this article, we present a derivative pricing model in an incomplete and globalized financial market. To appreciate the dynamics and impact of some important market factors, particularly default risks due to contagion, we consider two different financial markets with defaultable assets: in one market, we consider a stock whose price process follows a Heston stochastic volatility model, and in the other, a stock that follows a Hawkes-type jump diffusion model whose intensity is subjected to external systemic shocks. In both markets, we derive an indifference price for a contingent claim that is subject to the risk of default and show the impacts the investor’s risk aversion and external shocks on the price of the contingent claim.

This paper develops a unified analytical framework for pricing discretely sampled volatility-average swaps under the 4/2 stochastic volatility model. The model accommodates a broad range of volatility dynamics by combining affine and inverse-affine components in the instantaneous volatility specification, thereby unifying and extending the structural features of the classical Heston and 3/2 stochastic volatility models. Closed-form expressions for the conditional complex moments of the asset price are derived and serve as the fundamental building blocks for obtaining explicit analytical pricing formulas for volatility-average swaps under discrete sampling. The validity of the proposed pricing formulas is rigorously established within the admissible parameter space of the model. Extensive numerical experiments verify the accuracy and computational efficiency of the analytical results when compared with Monte Carlo simulations. The numerical analysis further reveals that discretely sampled volatility swap prices converge to their continuous-time counterparts in a manner that may be monotonic or non-monotonic, depending on the interaction between the volatility and inverse-volatility components of the 4/2 model, thereby emphasizing the importance of sampling effects in volatility derivative valuation. A detailed sensitivity analysis demonstrates how variations in the parameters governing the volatility and inverse-volatility components influence the fair strike prices, underscoring the structural flexibility of the 4/2 stochastic volatility model. Overall, the proposed framework provides an analytically tractable and computationally efficient approach for pricing volatility-linked derivatives under discrete sampling, offering valuable insights for both theoretical research and practical applications in volatility markets.

Analytic approximations for pricing perpetual American strangle options under constant elasticity of variance model with stochastic volatility

Variance and volatility swap valuation with stochastic liquidity and regime switching stochastic volatility

Vulnerable options under a Hawkes jump-diffusion model with two-factor stochastic volatility

This paper proposes a stochastic volatility model in which leverage evolves through a heterogeneous autoregressive (HAR) structure that combines daily and weekly return innovations. The resulting HARA specification can be viewed as a parsimonious restriction of multi-lag leverage drivers used in the stochastic volatility literature, delivering an interpretable multiscale decomposition of leverage dynamics and a closed-form leverage-propagation function. Parameters are estimated using data cloning, providing stable likelihood-based inference. Monte Carlo experiments confirm accurate recovery of the daily/weekly leverage components under both Gaussian and heavy-tailed innovations. Empirically, estimates for four international equity indices reveal substantial cross-asset heterogeneity in the horizon and persistence of leverage propagation. Benchmark comparisons are reported in an appendix as validation.

We present a new approach to commodity pricing that enhances accuracy by integrating four distinct risk factors: the spot price, stochastic volatility, convenience yield, and stochastic interest rates. We build on Yan [Valuation of commodity derivatives in a new multi-factor model. Rev. Deriv. Res., 2002, 5, 251–271], the only model to our knowledge that incorporates all four sources of risk, and extend it by adding a more flexible correlation structure that captures state-dependent co-movements and time-varying risk premia. A further contribution is the explicit inclusion of the stochastic interest-rate factor within a unified Kalman-filter framework, which allows us to jointly filter the state variables and estimate model parameters using both commodity and bond market data. An empirical analysis of crude-oil futures shows that our four-factor model captures the complex dynamics of the futures term structure and consistently outperforms existing benchmarks.

In recent years, major international economic frictions and regional security tensions have intensified, contributing to a sustained rise in global geopolitical risk (GPR) and exerting profound influences on commodity markets characterized by both physical and financial attributes. Natural gas, aluminum, and copper, as key commodities in the energy and industrial sectors, exhibit price fluctuations that are closely associated with industrial chain stability and macroeconomic performance. This study focuses on the natural gas, aluminum, and copper markets across three major economic regions-China, the United States, and the European Union-by constructing a time-varying parameter structural vector autoregression model with stochastic volatility (TVP-SVAR-SV) and employing Markov Chain Monte Carlo (MCMC) methods for parameter estimation. Combined with impulse response analysis and comparative examination of major events, this paper systematically investigates the time-varying transmission patterns of geopolitical risk across nine segmented commodity markets. The empirical results indicate that: first, the transmission of geopolitical risk to commodity markets demonstrates a clear unidirectional characteristic, with geopolitical risk functioning as the primary external driver of market volatility, while feedback effects from commodity market fluctuations to geopolitical risk remain limited; second, transmission effects exhibit pronounced horizon dependence and short-term persistence, as strong short-term shocks gradually converge over longer horizons; third, significant heterogeneity exists across commodity categories and regions, with energy commodities displaying higher overall sensitivity than metal commodities, the European market experiencing comparatively stronger impacts, and the Chinese market showing relatively greater resilience supported by policy coordination and well-integrated industrial systems; fourth, major global public health emergencies and large-scale international economic frictions amplify transmission mechanisms, with shock intensity increasing in line with the severity of external disturbances. This study not only enriches the theoretical understanding of the dynamic relationship between geopolitical risk and commodity markets but also provides valuable empirical evidence for policymakers designing differentiated regulatory frameworks and for enterprises seeking to manage exposure to commodity price volatility.

Testing of Heston’s Stochastic Volatility Models in China’s Emerging Index Options Market

Active lock-in options are a class of complex derivatives characterized by pronounced path dependence and optimal decision making features, and they possess significant application value in the design of structured financial products and risk management. This paper investigates the pricing of active lock-in call options under a stochastic volatility framework. The lock-in decision is formulated as an optimal stopping problem and is further reformulated as a partial differential equation with obstacle constraints. By introducing a linear complementarity problem formulation, the structural properties of the option value function and the optimal lock-in boundary are systematically characterized. From a numerical perspective, an IMEX time discretization scheme is employed to transform the continuous problem into a sequence of time-layered discrete complementarity systems. These systems are efficiently solved using the projected successive over relaxation (PSOR) algorithm. Numerical experiments are conducted to analyze the structural features and economic interpretations of the value function and the associated free boundary surface.

This paper examines the complex and time-varying linkages between inflation, EPU, and economic resilience in emerging market economies. By applying a TVP-SV-VAR model on quarterly data from 2000 to 2024, we explore the dynamic interaction of these macroeconomic variables over time. Our results indicate that inflation and EPU exert a constant negative impact on economic resilience, with these impacts heterogeneous across different time periods and economic cycles. The analysis also shows that uncertainty and inflation demonstrate more pronounced negative effects during crisis times, implying that policy credibility and macroeconomic stability are accentuated when economies are under external shocks. The estimated stochastic volatility showed significant heteroskedasticity in all series, advocating the importance of considering time-varying volatility in macroeconomic modeling. Our results carry important implications for policymakers in emerging economies, highlighting the importance of credible monetary policy frameworks, transparent communication, and strong institutional arrangements necessary to build economic resilience.

ABSTRACT This study provides a new perspective on the well-known risk-return relationship across 18 cryptocurrencies, which together account for approximately 86% of the total cryptocurrency market capitalization. It adopts a novel stochastic volatility in mean model with time-varying parameters, offering significant improvements over commonly adopted model specifications in the literature. The findings reveal that the risk-return relationship in cryptocurrency markets displays time-varying behaviour. Furthermore, the Bitcoin, Ethereum, and Chainlink markets are simultaneously characterized by a reverse leverage effect and a time-varying volatility feedback effect, which is a novel empirical finding. By contrast, only the reverse leverage effect holds for the Binance Coin, Tron, Polkadot, Mantra Dao, and Avalanche markets, suggesting a positive intertemporal link between the returns and volatility processes of the Bitcoin, Ethereum, and Chainlink markets. Moreover, instead of the traditional static models widely used in the literature, more flexible and appropriate models are needed to capture the time-varying, nonlinear, and complex risk-return interactions in cryptocurrency markets. Finally, in most cases, volatility in cryptocurrency markets, including Bitcoin and Ethereum, tends to decline.

Spatial Stochastic Volatility Models: A Bayesian Approach with Spatial Moving Average Process

ABSTRACT The paper proposes a new integrated realized stochastic volatility–mixed data sampling–geopolitical risk (RSV–MIDAS–GPR) model to model and forecast crude oil futures volatility. The model jointly models returns and the realized measure of volatility, leverages contemporaneous volatility information, and captures the effects of GPR on crude oil futures volatility. The empirical results demonstrate a significant positive correlation between GPR and crude oil futures volatility. Meanwhile, the RSV–MIDAS–GPR model, which incorporates both GPR and realized volatility, exhibits a synergistic effect, leading to a substantial improvement in out‐of‐sample forecasting performance. Furthermore, the model demonstrates notable capability in identifying high‐volatility states and achieves higher forecasting accuracy than competing models during market turmoil. Finally, economic value tests confirm that the inclusion of GPR provides valuable guidance for investor decision‐making. These findings offer both methodological and empirical contributions to the related research field.

This article presents a model for pricing an exchange option considering stochastic volatility and liquidity risk. The impact of liquidity risk on an asset price is considered by utilizing a liquidity discount process that is influenced by both market and asset-specific liquidity. Girsanov’s theorem is applied to transform from the real-world probability measure to equivalent probability measures, such as the risk-neutral probability measure. The Feynman–Kac theorem is applied to transform the exchange option pricing formula into the vanilla option pricing formula. The analytical expression is derived through the characteristic function approach. The accuracy of the proposed formula is validated through comparisons with Monte Carlo simulation, where the relative error remains below 0.93% across different values of S(0) and τ. Furthermore, numerical experiments highlight that incorporating liquidity risk leads to higher option prices. As the maturity increases from 0.1 to 2.0, the percentage gap between the option prices increases from 1.65% to 20.2%. Finally, sensitivity analysis is conducted to examine the influence of various parameters and to demonstrate the impact of stochastic volatility and liquidity in exchange option valuation.

ABSTRACT We develop a new and powerful method to analyse time series to rigorously detect flares in the presence of an irregularly oscillatory baseline, and apply it to stellar light curves observed with the Transiting Exoplanet Survey Satellite. First, we remove the underlying non-stochastic trend using a time-varying amplitude harmonic model. We then model the stochastic component of the light curves in a manner analogous to financial time series, as an ARMA + GARCH process, allowing us to detect and characterize impulsive flares as large deviations inconsistent with the correlation structure in the light curve. We apply the method to exemplar light curves from TIC 13955147 (a G5V eruptive variable), TIC 269797536 (an M4 high-proper motion star), and TIC 441420236 (AU Mic, an active dMe flare star), detecting up to 145, 460, and 403 flares, respectively, at rates ranging from ${\approx }0.4~\mathrm{ to}~8.5$ d$^{-1}$ over different sectors and under different detection thresholds. We detect flares down to amplitudes of 0.03 per cent, 0.29 per cent, and 0.007 per cent of the bolometric luminosity for each star, respectively. We model the distributions of flare energies and peak fluxes as power laws, and find that the solar-like star exhibits values similar to those on the Sun ($\alpha _{E,P}\approx 1.85,2.36$), while for the less- and highly-active low-mass stars $\alpha _{E,P}\gt2$ and $\lt 2,$ respectively.

This study develops a generalized Newton method to address the nonlinear Kolmogorov forward equation (KFE) under the local stochastic volatility (LSV) framework. Analytical convergence conditions are derived via the geometric series theorem, and empirical validation is conducted using 15 years of monthly crude oil spot price data (2011–2025), with the parameters set using maximum likelihood estimation. Sensitivity analyses confirm the stable convergence of the iterative scheme under realistic scenarios while also identifying parameter ranges that may lead to divergence. These findings demonstrate that the proposed methodology provides a tractable and accurate approach to probability density estimation in commodity markets, with a clear potential for extension to multi-dimensional settings and richer datasets.

ABSTRACT This paper studies the effects of key underlying structural macroeconomic shocks on the U.S. trend inflation rate. To do so, we consider eight structural shocks that incorporate a broad set of information for the U.S. economy and that can be regarded as the main structural determinants of the latter. Using a Bayesian estimation procedure, we estimate the effects of these structural shocks on the trend inflation rate via an unobserved components model with stochastic volatility and structural shocks. We find that four structural shocks have significant and quantitatively important effects on trend inflation. Price markup and government policy shocks raise trend inflation, indicating long-run inflationary effects, while finance and productivity shocks lower trend inflation, implying long-run deflationary effects.