Stochastic Volatility Research Articles

Valuation of American options using machine learning: beyond Longstaff–Schwartz and hybrid models

This work explores the potential of different machine learning (ML) algorithms in the valuation of American options (Aos), contrasting them with the Longstaff–Schwartz (L–S) model. To carry out this research, the algorithms K-Nearest Neighbors (KNN), Random Forest (RF), Multi-Layer Perceptron (MLP) and Convolutional Neural Network (CNN) are employed using RStudio. The project specifically targets the prediction of the price of Apple’s put Aos through regression, utilizing data extracted from Bloomberg as a case study. To evaluate the model’s performance in a multidimensional context, we use both historical and stochastic volatility. The results show that these ML algorithms achieve a notable improvement in the performance and accuracy of Aos price predictions compared to the L-S model. RMSE values are very similar using historical and stochastic volatility, the most notable difference appearing in the L-S model. Prior trends in the literature show the development of hybrid models, which combine traditional techniques with the predictive capabilities of ML algorithms in the valuation of Aos in a more efficient and accurate way than the L-S model. However, our paper determines whether the supervised training of ML algorithms exclusively with historical financial market data, without relying on traditional methods, achieves better results than those of the L-S model. This ML “stand-alone” approach to price Aos faces the inability to derive hedging strategies and optimal exercise conditions. Future efforts could explore integrating deep reinforcement learning to identify optimal exercise policies or developing hybrid models that combine ML with traditional frameworks.

Observations concerning the estimation of Heston’s stochastic volatility model using HF data

This paper presents a comprehensive simulation study on estimating parameters for the popular Heston stochastic volatility model. Leveraging high-frequency data, we explore, in a data-science type exercise, various spot-volatility estimation and sampling techniques, improving existing methods to enhance parameter accuracy. Through extensive simulations, we report difficulties in generating correct parameter estimates for realistic parameter settings where the volatility dynamic does not satisfy the Feller condition. This study contributes valuable insights into the practical implementation of the Heston model and its applicability to high-frequency data. We find that the scheme of Azencott et al. (2020) with uniform kernel weighting provides reliable and efficient parameter estimates. It is advised to also apply a Jackknife estimation to corroborate the findings.

Real‐Time Forecasting Using Mixed‐Frequency VARs With Time‐Varying Parameters

ABSTRACTThis paper provides a detailed assessment of the real‐time forecast accuracy of a wide range of vector autoregressive models that allow for both structural change and indicators sampled at different frequencies. We extend the literature by evaluating a mixed‐frequency time‐varying parameter vector autoregressive model with stochastic volatility. Monte Carlo simulation shows that the novel model is well‐suited to estimate missing monthly observations in an environment that is subject to parameter instability. In a real‐time forecast exercise, the model delivers accurate now‐ and forecasts and, on average, outperforms its competitors. Particularly, inflation and unemployment rate forecasts are more precise.

Dimension reduction in vector autoregressive models for macroeconomic applications

The vector autoregressive (VAR) model is instrumental in analyzing multivariate time series applications. However, as the number of parameters grows with the dimension of the VAR response vector, dimension reduction techniques are often necessary to improve model efficiency. The envelope approach [Cook RD, Li B, Chiaromonte F. Envelope models for parsimonious and efficient multivariate linear regression. Stat Sin. 2010;927–960] is a novel technique that enhances estimation efficiency and prediction accuracy in multivariate analysis by incorporating dimension reduction. In this paper, we introduce a novel class of envelope VAR (EVAR) models featuring elliptically contoured and stochastic volatility error innovations, aimed at enhancing the analysis of macroeconomic applications. Moreover, the asymptotic properties of the EVAR models without Gaussian error innovations are established. The simulation experiments and empirical analysis using macroeconomic data from the Federal Reserve Economic Data (FRED) database, demonstrate that the envelope VAR models with these error innovations outperform the standard VAR (OLSVAR) models in both estimation efficiency and forecast accuracy, especially when the data includes irrelevant or immaterial information.

Optimal calculations for the space-time fractional derivative option pricing models with stochastic liquidity risk and volatility using a combination neural network.

This work incorporates a stochastic liquidity risk, stochastic volatility, and Caputo-type fractional derivatives into European option pricing and establishes two novel space-time fractional hybrid models to capture the nonlinearity and non-stationarity of price evolution processes. The combination neural network algorithm with the defined nonlinear test solutions is designed to solve the fractional derivative models with the initial condition, Dirichlet and Robin boundary conditions. When the influence of a liquidity risk is removed, the studied models are reduced to the space-time fractional Heston models, and the pricing results are compared with the analytical formula of the classical Heston model. In the presence of a liquidity risk, the pricing models under the Caputo and Caputo-Fabrizio fractional derivatives are tested based on the market data. The applications and comparison results prove that the dynamical models demonstrated in the work have small prediction errors and can highly fit market data. The designed combination neural network can effectively handle the mixed problems of the high-dimensional fractional derivative equations and derive the optimal approximations under a stochastic liquidity risk and volatility.

Exact simulation scheme for the Ornstein–Uhlenbeck driven stochastic volatility model with the Karhunen–Loève expansions

Exact simulation scheme for the Ornstein–Uhlenbeck driven stochastic volatility model with the Karhunen–Loève expansions

Option price computation under binary control regime switching triple-factor stochastic volatility model

This study presents an efficient pricing framework for European call options under a binary control regime that switches to a triple-factor stochastic volatility model, tailored for recessionary and stable market phases. The model captures regime transitions via binary controls and incorporates triple volatility sources. We derive the characteristic function and implement a semi-analytical pricing formula using trapezoidal and Gauss-Laguerre quadrature in MATLAB. The economic recovery process is influenced by the control parameter α, while the impacts θ₃ are considered secondary to other factors driving recovery. The results show that the option prices under recessionary conditions were lower compared to the recession-free regime, thereby validating the model’s sensitivity to macroeconomic uncertainty. It further confirms that the binary control regime switching triple-factor stochastic volatility model offers greater accuracy and adaptability across economic states, making it a promising tool for option pricing in dynamic financial environments.

The pricing of forward start options under jump diffusion model with stochastic interest rate and stochastic volatility

This article addresses the pricing problem of forward start options within the framework of a jump diffusion model, incorporating Vasicek stochastic interest rate, Heston stochastic volatility, and arbitrary jump size distributions. The study begins by applying a measure transformation to reformulate the expectation of the discounted payoff into the difference between the probabilities of two random events under new probability measures. Subsequently, the characteristic function is derived using the properties of Ito’s integral, compound Poisson processes, and the Feynman-Kac theorem. By leveraging the relationship between the characteristic function and the distribution function, the pricing formula for forward start options is established. The rationality and practicality of the proposed model are thoroughly demonstrated. Numerical experiments are conducted to analyze the accuracy and computational efficiency of the pricing formula. Additionally, sensitivity analyses are performed on key parameters to evaluate their influence on the option pricing outcomes.

Tail Dependence of Liquidity and Volatility in Carbon Futures Market: Evidence From EU ETS

ABSTRACTThis study constructs liquidity and volatility indicators based on the four phases of EU ETS and analyses tail dependence using Copula models. The results indicate strong tail dependence between liquidity and volatility in the fourth phase. The Amihud illiquidity ratio combined with the stochastic volatility model identifies high volatility risks during liquidity scarcity, while the Gibbs measure combined with the stochastic volatility model identifies low volatility risks. The robustness of the results is tested by classifying different periods based on structural breaks and assessing tail dependence, and by applying machine learning algorithms to remove outliers before measuring tail dependence.

Bayesian semiparametric inference for TVP-SVAR models with asymmetry and fat tails

Time-varying parameter (TVP) structural vector autoregressive models with stochastic volatility (SVAR-SV) usually assume Gaussian innovations and a smooth or discrete path for the coefficients. To account for possible skewness and fat tails, this work introduces a semiparametric mixture of multivariate restricted skew- t innovation distributions, also permitting the inference of clusters of asymmetry across data series. Moreover, a dynamic shrinkage prior is designed for the coefficients of the contemporaneous and lagged variables to model the path of the parameters flexibly. Inference in high-dimensional settings is performed via a Markov chain Monte Carlo algorithm that leverages the stochastic representation of the skew- t distribution for obtaining a conditional linear Gaussian state-space model. Then, the algorithm alternates between the centred and non-centred parametrizations to improve the mixing and samples from the joint smoothed distribution without loops. The proposed semiparametric approach is combined with a sparsification method to extract time-varying Granger-causal networks in different applications regarding the COVID-19 pandemic across Europe and financial contagion transmission in Europe and the world.

Sectoral Uncertainty: A Hierarchical-Volatility Approach

We propose a new empirical framework to estimate sectoral uncertainty from data-rich environments. We jointly decompose the conditional variance of economic time series into a common, a sector-specific, and an idiosyncratic component. By specifying a hierarchical-factor structure to stochastic volatility modeling, our framework combines both dimension reduction and flexibility. To estimate the model, we develop an efficient Markov chain Monte Carlo algorithm based on precision sampling techniques. We apply our framework to a large dataset of disaggregated industrial production series for the U.S. economy. Our findings suggest that: (i) uncertainty is heterogeneous at a sectoral level; and (ii) durable goods uncertainty may drive some business cycle effects typically attributed to aggregate uncertainty.

On deep calibration of (rough) stochastic volatility models

On deep calibration of (rough) stochastic volatility models

Flexible Bayesian MIDAS: Time-Variation, Group-Shrinkage and Sparsity

We propose a mixed-frequency regression prediction approach that models a time-varying trend and stochastic volatility in the trend and in the variable of interest. The coefficients of high-frequency indicators are regularized via a shrinkage prior that accounts for the grouping structure and within-group correlation among lags. A new sparsification algorithm on the posterior motivated by Bayesian decision theory derives inclusion probabilities over lag groups, thus making the results easy to communicate without imposing sparsity a priori. An empirical application on nowcasting UK GDP growth suggests that group-shrinkage improves nowcasting performance by relying on signals from a meaningful sub-set of predictors that include “hard” real activity indicators and, early in the data release, cycle additionally a number of surveys. Over the Covid-19 pandemic, a few additional indicators for the service and housing sectors are exploited that capture the disruptions from economic lockdowns. Accounting for a trend and stochastic volatility helps to stabilize the sparse nature of the variable selection during periods of large shocks, while accounting for uncertainty, especially early in the data release cycle.

Fuzzy Stochastic Modeling in Financial Risk Assessment and Economic Predictions

The current studies are concerned with introducing fuzzy stochastic models in predicting economics and analyzing financial risk. One class of models that has found notable traction in this domain are fuzzy stochastic models (which combine the merits of fuzzy logic and stochastic processes) as flexible representations for the uncertainty (fuzziness) and randomness (stochasticity) characteristic of economic variables (such as GDP growth, inflation, unemployment rates, or financial asset prices). In the study, introduced are fuzzy drift and volatility parameters to normalize uncertainty and stochastic volatility to portray said features through fuzzy stochastic models to predict macroeconomic indicators and stock prices for a defined duration. The results of the simulation suggest that fuzzy stochastic approach produces more dynamic and multilayered predictions, going beyond classical deterministic models and providing a more integrated perspective on how economic systems respond to uncertainty. Furthermore, a fuzzy Value at Risk (VaR) is applied to evaluate financial risk which also highlights the approach's opportunity to capture both the expected returns price fluctuation. To confirm the effect of altering fuzzy parameters and stochastic volatility on the model performance, a sensitivity analysis is conducted. In addition to suggesting avenues of future research that take advantage of machine learning and real-time data methods for better predictions, the study emphasizes the use of fuzzy stochastic models for financial risk assessment and for economic forecasting as they are well-placed to model risk and uncertainty in hyper-red, volatile non-linear systems.

State-Dependent and Time-Varying Effects of Monetary Policy

ABSTRACT Nonlinear models with Bayesian inference represent econometric techniques used to assess the monetary policy transmission mechanism. This paper provides two different approaches of vector autoregressive models to outline potential regime-dependent and time-varying effects: a two-state Markov-Switching model with time-invariant transition probabilities and a time-varying vector autoregressive model with stochastic volatility. The empirical results for the Romanian economy indicate the existence of asymmetric regime-dependent responses to monetary policy shock. Regarding the time-varying model, the first part of the period reveals a more cautious central bank behavior, with relatively low responses to shock, while recently, higher responses indicate improvements in the transmission of shocks.

Periodicity In Bitcoin Returns: A Time-Varying Volatility Approach

We examine if the day-of-the-week effect is present in Bitcoin return series. The model specification in use accounts for conditional heteroscedasticity, which is captured in the form of a stochastic volatility process that allows for periodic time-varying parameters. We find periodicity in Bitcoin returns, which is evidence against the market efficiency of Bitcoin.

How is artificial intelligence technology transforming energy security? New evidence from global supply chains

Research background: Ensuring energy security (ESR) is critical for national stability and sustainable growth. However, the role of artificial intelligence technology (AIT) in enhancing ESR, particularly through global supply chain (GSC) transmission channels, remains underexplored. Purpose of the article: This study investigates the dynamic relationship between AIT and ESR in the context of the United States, examining whether AIT ensures energy security via GSC optimization and how this interplay evolves over time. Methods: A time-varying parameter vector autoregressive model with stochastic volatility (TVP-VAR-SV) is developed to analyze the dynamic links between AIT, GSC, and ESR and capture time-varying effects and extreme-period impacts. Findings & value added: The results indicate that the impact of AIT on ESR is initially modest in the short term, but increases substantially in the medium to long term by optimizing energy production, distribution, and management. GSC stability and efficiency serve as key transmission channels, and post-2010 advancements amplify the influence of AIT. During crises, AIT ensures the continuity of energy security through intelligent GSC adjustments in the US. This study highlights the critical role of AIT-driven supply chain strategies in energy security and offers policy-makers actionable insights to leverage AIT and GSC optimization for sustainable energy systems.

Moment Generating Function of the Averaged Log-returns in the Heston's Stochastic Volatility Model

The aim of this short note is to investigate the domain of the moment generating function of the log-returns in the Heston's stochastic volatility model averaging the initial volatility value through its stationary distribution. This way we deal with the problem that arises from the fact that the volatility is a hidden market object and it is hard to be extracted.

ON THE IMPLIED VOLATILITY OF EUROPEAN AND ASIAN CALL OPTIONS UNDER THE STOCHASTIC VOLATILITY BACHELIER MODEL

In this paper, we study the short-time behavior of the at-the-money implied volatility for European and arithmetic Asian call options with fixed strike price. The asset price is assumed to follow the Bachelier model with a general stochastic volatility process. Using techniques of the Malliavin calculus such as the anticipating Itô’s formula, we first compute the level of the implied volatility when the maturity converges to zero. Then, we find a short-maturity asymptotic formula for the skew of the implied volatility that depends on the roughness of the volatility model. We apply our general results to the stochastic alpha–beta–rho (SABR), fractional Bergomi and local volatility models, and provide some numerical simulations that confirm the accurateness of the asymptotic formula for the skew.

Optimal Retirement Wealth Allocation under Volatile Interest Rates: A GARCH-Based Analysis

Managing post-retirement wealth effectively is crucial for ensuring financial security in uncertain market conditions. Traditional pension investment models assume constant interest rates, which fail to capture real-world financial volatility. This study develops an optimal investment strategy for post-retirement wealth management under stochastic interest rates, modeled using EGARCH and GJR-GARCH frameworks. By leveraging GARCH-type models, we estimate volatility dynamics and optimize asset allocation strategies. The Hamilton-Jacobi-Bellman (HJB) equation is applied within a stochastic control framework to derive the optimal investment policy. Sensitivity analysis is conducted to assess the impact of different risk aversion levels on portfolio allocation. The results demonstrate that accounting for stochastic interest rate volatility improves wealth sustainability in the post-retirement phase.