Volatility Models Research Articles

On the Guyon–Lekeufack volatility model

On the Guyon–Lekeufack volatility model

Robustness of Hilbert space-valued stochastic volatility models

In this paper, we show that Hilbert space-valued stochastic models are robust with respect to perturbations, due to measurement or approximation errors, in the underlying volatility process. Within the class of stochastic-volatility-modulated Ornstein–Uhlenbeck processes, we quantify the error induced by the volatility in terms of perturbations in the parameters of the volatility process. We moreover study the robustness of the volatility process itself with respect to finite-dimensional approximations of the driving compound Poisson process and semigroup generator, respectively, when considering operator-valued Barndorff-Nielsen and Shephard stochastic volatility models. We also give results on square root approximations. In all cases, we provide explicit bounds for the induced error in terms of the approximation of the underlying parameter. We discuss some applications to robustness of prices of options on forwards and volatility.

A Barndorff-Nielsen and Shephard model with leverage in Hilbert space for commodity forward markets

We propose an extension of the model introduced by Barndorff-Nielsen and Shephard, based on stochastic processes of Ornstein–Uhlenbeck type taking values in Hilbert spaces and including the leverage effect. We compute explicitly the characteristic function of the log-return and the volatility processes. By introducing a measure change of Esscher type, we provide a relation between the dynamics described with respect to the historical and the risk-neutral measures. We discuss in detail the application of the proposed model to describe the commodity forward curve dynamics in a Heath–Jarrow–Morton framework, including the modelling of forwards with delivery period occurring in energy markets and the pricing of options. For the latter, we show that a Fourier approach can be applied in this infinite-dimensional setting, relying on the attractive property of conditional Gaussianity of our stochastic volatility model. In our analysis, we study both arithmetic and geometric models of forward prices and provide appropriate martingale conditions in order to ensure arbitrage-free dynamics.

Investment volatility during red sea crisis: Study in Asean country

The objective of this research is to investigate the volatility of investment instruments, with the aim of providing the best model for predicting investment value. In this topic, the main concepts and theories that underpin the research are presented. Efficient market hypothesis and rational expectations theory stand out, providing a solid basis for understanding the context of the investigation. The methodology adopted for this research GARCH modelling to analysis investment instrument and the impact of its volatility. Data collection was carried out through yahoo finance website and investing website during red sea war event. The research identified optimal GARCH models for different indexes and assets: GARCH (1,1) for JCI, STI, SET, and Bitcoin, E-GARCH (1,1) for KLSE, PSE, and VSE, and GJR-GARCH (1,1) for Gold. It discussed volatility persistence and leverage effects during the Red Sea War, considering model sensitivities, geopolitical impacts, and research limitations. This research's practical and theoretical implications inform financial risk management, highlighting the importance of robust volatility modeling. The findings can enhance portfolio optimization and investment strategies during geopolitical instability. These insights underscore the need for sophisticated approaches to manage financial risks effectively. This research uses advanced GARCH models to analyze volatility during the Red Sea Crisis, revealing asset-specific behaviors under geopolitical stress. The findings improve financial risk management and investment strategies in unstable conditions, enhancing volatility predictions and professional practices.

Conditional volatility targeting strategy considering jump effects: Evidence from sustainable ESG equity index

This paper considers the sustainable ESG equity index in the proposed conditional volatility targeting strategy. The research first detects jump risk using a jump test and then extends Bongaerts et al. (2020) by addressing jump risk and employing different volatility models to project volatilities under the conditional volatility targeting strategy. To capture consideration of the fact that index return dynamics, we propose an ARMA-GARCH jump model that can capture the characteristics of jump persistence, autocorrelation, and volatility clustering according to the return of the sustainable equity index. Our numerical analyses reveal that the portfolio allocation using a sustainable equity index to predict volatility, combined with a conditional volatility target strategy, can achieve higher performance. Furthermore, the proposed ARMA-GARCH jump model can enhance the performance with conditional volatility targeting strategy.

A generalized approach for pricing american options under a regime-switching model

Abstract Regime-switching models gained their popularity over the past decade because of their distinctive advantage of modelling different financial market statuses in a discrete manner rather than a continuous manner as in stochastic volatility models. When they are used in option pricing, the clear advantage is that they enable a larger parameter space for models to be calibrated for a specific market dynamics and thus allow a better quantitative risk management in terms of utilizing financial derivatives. However, when they are used to price American-style financial derivatives, a large number of economic statuses result in a demand for improved computational efficiency. This paper provides a new algorithm of high computational efficiency supplemented with a theorem that pre-analyzes the associated matrices and their eigenvalues, without concern about the possibility of duplicated roots being mistakenly identified as simple roots due to rounding errors or the presence of two extremely closely positioned simple roots within the original system.

Option Pricing in Sandwiched Volterra Volatility Model

Option Pricing in Sandwiched Volterra Volatility Model

Pricing exchange options under hybrid stochastic volatility and interest rate models

This paper investigates the pricing of exchange options under hybrid models integrating stochastic volatility and stochastic interest rates. It aims to achieve two primary objectives. First, we derive a closed-form pricing formula for exchange options under a two-factor Heston–Hull–White hybrid model, which accounts for long-term volatility and exhibits relatively broad correlations among the dynamics of asset prices, volatilities, and interest rates. Second, we explore the Heston model’s integration with a generalized single-factor stochastic interest rate model, illustrating that the price is not dependent on the specific form of the interest rate process. A closed-form pricing formula for exchange options under this framework is also derived. Our numerical experiments support the proposed formulas and elucidate the effects of various parameters on option prices.

Is Reinforcement Learning Good at American Option Valuation?

This paper investigates algorithms for identifying the optimal policy for pricing American Options. The American Option pricing is reformulated as a Sequential Decision-Making problem with two binary actions (Exercise or Continue), transforming it into an optimal stopping time problem. Both the least square Monte Carlo simulation method (LSM) and Reinforcement Learning (RL)-based methods were utilized to find the optimal policy and, hence, the fair value of the American Put Option. Both Classical Geometric Brownian Motion (GBM) and calibrated Stochastic Volatility models served as the underlying uncertain assets. The novelty of this work lies in two aspects: (1) Applying LSM- and RL-based methods to determine option prices, with a specific focus on analyzing the dynamics of “Decisions” made by each method and comparing final decisions chosen by the LSM and RL methods. (2) Assess how the RL method updates “Decisions” at each batch, revealing the evolution of the decisions during the learning process to achieve optimal policy.

Quantification of evaporative loss of volatile metals from planetary cores and metal-rich planetesimals

The processes responsible for the isotopic compositions and abundances of volatile elements in the early solar system remain highly debated. Orders of magnitude variation of (highly) volatile elements exist between different magmatic iron meteorite groups, but it is unclear to what extent their depletions can be explained by evaporation from metal melts during parent body accretion and/or subsequent break up. To this end, we present 86 new evaporation experiments with the aim of constraining the volatility of most volatile metals from metallic melts. The results confirm the previously proposed important effects of S in metal melt on the volatility of the elements of interest governed by their S-loving or S-phobic behavior. Nominally S-loving elements In, Sn, Te, Pb and Bi are significantly more volatile in Fe melt relative to FeS liquid, whereas nominally S-avoiding elements Ga and Sb are more volatile in FeS liquid relative to Fe melt, at a given pressure and temperature. The newly derived volatility sequences for S-free/poor and S-rich metallic melts were also compared with commonly used volatility models based on condensation temperatures. The results indicate significant differences between the latter, including the much more volatile behavior of Te, relative to Se, in both explored bulk compositions, which are traditionally assumed to be equally volatile. The (minimum) degree of volatile element depletion due to evaporation was quantified using the new experimental results and models. A comparison between the volatile element depletions in magmatic iron meteorites and the predicted depletions appropriate for evaporation from Fe melts shows that the latter depletions can be easily reconciled with (an) evaporation event(s). Altogether, the new data and models will provide an important framework when more accurate and precise estimates of magmatic iron meteorite bulk volatile element contents are available.

Extreme ATM skew in a local volatility model with discontinuity: joint density approach

This paper concerns a local volatility model in which the volatility takes two possible values, and the specific value depends on whether the underlying price is above or below a given threshold. The model is known, and a number of results have been obtained for it. In particular, option pricing formulas and a power-law behaviour of the implied volatility skew have been established in the case when the threshold is taken at the money. In this paper, we derive an alternative representation of option pricing formulas. In addition, we obtain an approximation of option prices by the corresponding Black–Scholes prices. Using this approximation streamlines obtaining the aforementioned behaviour of the skew. Our approach is based on the natural relationship of the model with skew Brownian motion and consists of the systematic use of the joint distribution of this stochastic process and some of its functionals.

Performance comparison of alternative volatility models applied to economic indexes: the role of asymmetric effects

ABSTRACT This article examines the empirical relationship between economic indexes – namely the industrial production and the consumer confidence – and their volatility by using the United States data over the periods 1919–2023 and 1960–2023, respectively. To address this topic, several GARCH-family models including volatility feedback and the asymmetric effects are estimated. We find that (i) the industrial production index exhibits asymmetric effects, with significant fitting performance over the standard GARCH model; (ii) the inclusion of t -distributed innovations in the error term improves the fitting performance of the competing models against their counterparts with a normal distribution assumption; (iii) both models tested for the consumer confidence index exhibit similar performance to fit with the data; (iv) by excluding the months during and after the COVID-19 pandemics and the Russia–Ukraine conflict, it leads to the same results and conclusions for both economic indexes. Conditional heteroscedasticity, economic growth, GARCH models, asymmetric effects, volatility feedback

Applications of GARCH Models in Forecasting Financial Market Volatility: Insights from Leading Global Stock Indexes

This study investigates the volatility dynamics of major global stock indexes, including the FTSE 100, Hang Seng Index, NIKKEI 225, and S&P 500, using a range of Generalized Autoregressive Conditional Heteroskedasticity (GARCH) models. The analysis spans a comprehensive 20-year period from January 1, 2004, to December 31, 2023, encompassing diverse market conditions such as bull and bear markets, the 2008 financial crisis, and the COVID-19 pandemic. The methodology includes preprocessing steps such as calculating daily log returns, performing descriptive statistics, and conducting stationarity and ARCH effect tests to ensure data suitability for volatility modelling. The study evaluates several GARCH models, including GARCH, EGARCH, NGARCH, APARCH, GJR-GARCH, and TGARCH, to forecast volatility and address both symmetric and asymmetric effects. The TGARCH model exhibits strong performance in capturing leverage effects and asymmetries, particularly for the FTSE 100 and Hang Seng Index. The APARCH model performs well for the S&P 500, demonstrating sensitivity to past shocks. Overall, the findings underscore the importance of advanced GARCH models in accurately predicting volatility in global financial markets, highlighting the TGARCH model's effectiveness in addressing asymmetries and providing insights into selecting appropriate models for enhanced financial analysis and risk management.

Option pricing in a stochastic delay volatility model

This work introduces a new stochastic volatility model with delay parameters in the volatility process, extending the Barndorff–Nielsen and Shephard model. It establishes an analytical expression for the log price characteristic function, which can be applied to price European options. Empirical analysis on S&P500 European call options shows that adding delay parameters reduces mean squared error. This is the first instance of providing an analytical formula for the log price characteristic function in a stochastic volatility model with multiple delay parameters. We also provide a Monte Carlo scheme that can be used to simulate the model.

Time-Varying Deterministic Volatility Model for Options on Wheat Futures

This study introduces a robust model that captures wheat futures’ volatility dynamics, influenced by seasonality, time to maturity, and storage dynamics, with minimal calibratable parameters. Our approach reduces error-proneness and enhances plausibility checks, offering a reliable alternative to models that are difficult to calibrate. Transferring estimated parameters from liquid to illiquid markets is feasible, which is challenging for models with numerous parameters. This is of practical importance as it improves the modeling of volatility in illiquid markets, where price discovery is less efficient. In liquid markets, on the other hand, where speculative activity is high, we find that implied volatility is usually the best measure. Additionally, the introduced volatility model is suitable for pricing options on wheat futures as a risk-neutral measure.

Convergence of the Two Point Flux Approximation method and the fitted Two Point Flux Approximation method for options pricing with local volatility function

In this paper, we deal with numerical approximations for solving the Black–Scholes Partial Differential Equation (PDE) for European and American options pricing with local volatility. This PDE is well-known to be degenerated. Local volatility model is a model where the volatility depends locally of both stock price and time. In contrast to constant volatility or time-dependent volatility models for which analytical representations of the exact solution is known for European Call options, there is no analytical solution for local volatility. The space discretization is performed using the classical finite volume method with Two-Point Flux Approximation (TPFA) and a novel scheme called Fitted Two-Point Flux Approximation (FTPFA). The Fitted Two-Point Flux Approximation (FTPFA) combines the fitted finite volume method and the standard TPFA method. More precisely the fitted finite volume method is used when the stock price approaches zero with the goal to handle the degeneracy of the PDE while the TPFA method is used on the rest of space domain. This combination yields our fitted TPFA scheme. The Euler method is used for the time discretization. We provide the rigorous convergence proofs of the two fully discretized schemes. Numerical experiments to support theoretical results are provided.

Persistent and transient variance components in option pricing models with variance-dependent Kernel

This paper examines theoretically and empirically a variance-dependent pricing kernel in the continuous-time two-factor stochastic volatility (SV) model. We investigate the relevance of such a kernel in the joint modeling of index returns and option prices. We contrast the pricing performance of this model in capturing the term structure effects and smile/smirk patterns to discrete-time GARCH models with similar variance-dependent kernels. We find negative and significant risk premium for both volatility factors, implying that investors are willing to pay for insurance against increases in volatility risk, even if it has little persistence. In-sample, the component GARCH model exhibits a slightly better fit overall and across all maturity buckets than the two-factor SV model. However, the two-factor SV model reduces strike price bias, giving rise to the model’s ability in reconciling the physical and risk-neutral distribution. Out-of-sample, the two-factor SV model has better fit to data.

A local volatility correction to mean-reverting stochastic volatility model for pricing derivatives

Generally, in the real market, empirical findings suggest that either local volatility (LV) or stochastic volatility (SV) models have a limit to capture the full dynamics and geometry of the implied volatilities of the given equity options. In this study, to overcome the disadvantage of such LV and SV models, we propose a special type of hybrid stochastic-local volatility (SLV∗) model in which the volatility is given by the squared logarithmic function of the underlying asset price added to a function of a fast mean-reverting process. By making use of asymptotic analysis and Mellin transform, we derive analytic pricing formulas for European derivatives with both smooth and non-smooth payoffs under the SLV∗ model. We run numerical experiments to verify the accuracy of the pricing formulas using a Monte-Carlo simulation method and to display that the proposed new model fits the geometry of the market implied volatility more closely than other models such as the Heston model, the stochastic elasticity of variance (SEV) model, the hybrid stochastic and CEV type local volatility (SVCEV) model and the multiscale stochastic volatility (MSV) model, especially for short time-to-maturity options.

Heterogeneous Volatility Information Content for the Realized GARCH Modeling and Forecasting Volatility

Abstract As the demand for accuracy in volatility modeling and forecasting increases, the literature tends to incorporate different volatility measures with heterogeneous information content to construct the hybrid volatility model. This study focuses on one of the popular hybrid volatility models: the Realized Generalized Autoregressive Heteroskedasticity (Realized GARCH) and embeds various volatility measures, including the CBOE VIX, VIX1D, Realized Volatility, and Daily Range to examine their heterogeneous impact on the conditional volatility estimation and forecasting. To evaluate the impact of the volatility measures, we first construct a volatility response function. This involves calculating the difference in one-step-ahead conditional volatility forecasts that incorporate information from both return and volatility measures against the forecasts based on return innovations only. Subsequently, the variance share is calculated to evaluate its role in explaining future variations in the Realized GARCH. Our results show that among these four volatility measures, VIX is the most informative volatility. Although VIX1D is overemphasized by the literature, its significance in volatility forecasting remains substantial, confirming that risk-neutral volatility measures are generally more informative than physical measures. Finally, we also find that incorporating multiple risk-neutral volatility measures does not improve forecasting performance compared to using a single measure due to overlapping information.

Approximation Rates for Deep Calibration of (Rough) Stochastic Volatility Models

Approximation Rates for Deep Calibration of (Rough) Stochastic Volatility Models