Volatility Jumps Research Articles

Bayesian estimation of stochastic volatility jump diffusion model parameters using S&P 500 and VIX data

The Stochastic Volatility Jump Diffusion (SVJD) model is a common tool used for option pricing. Existing literature has employed many techniques in order to estimate the parameters of this model, including both Bayesian and frequentist approaches. In 2010, Duan and Yeh [Jump and volatility risk premiums implied by VIX. J Econ Dyn Control. 2010;34(11):2232–2244] used a frequentist approach with maximum likelihood estimation that included the VIX, the Chicago Board Options Exchange (CBOE) Volatility Index, which carries real time S&P 500 option price information in addition to the S&P 500 data. In this paper, we combine the previous Bayesian approach in Cape et al. [Estimating Heston's and Bates' models parameters using Markov Chain Monte Carlo simulation. J Stat Comput Simul. 2015;85(11):2295–2314] with the inclusion of VIX by Duan and Yeh. We provide the derivations of the conditional posterior distributions of the SVJD model's parameters and a guide to a Markov Chain Monte Carlo (MCMC) algorithm used to estimate the parameters. We apply the algorithm to the historical S&P 500 and VIX data and provide the subsequent parameter estimates. We offer these tools for others performing similar research with our R code available upon request.

Heterogeneity, Jumps and Co-Movements in Transmission of Volatility Spillovers Among Cryptocurrencies

Abstract We analyze properties identified in the price volatility of Bitcoin and some of the leading cryptocurrencies namely Litecoin, Ripple, and Ethereum. We employ Heterogeneous Autoregressive models (HAR) in both a univariate and multivariate level of analysis. First, the significance of heterogeneity and jumps is examined, considering the ability of several univariate HAR models, to predict realized volatility of cryptocurrencies. Second, we examine the relevance of realized volatility jumps and covariances in the transmission of volatility spillovers among cryptocurrencies. We perform a comparative spillover analysis of the multivariate HAR models in two versions, considering variances only and covariances as well. Our results indicate that covariances and jumps inclusion lead to an increase in spillovers. The time-varying spillover analysis indicates higher dependency between Bitcoin and the other cryptocurrencies mostly at short frequencies.

Empirical analysis of crude oil dynamics using affine vs. non-affine jump-diffusion models

This paper investigates the dynamics of the United States oil (USO) exchange traded fund (ETF). Daily USO returns are modelled using stochastic volatility (SV) frameworks derived from three different model classes: SV models with contemporaneous jumps in returns and volatility (SVCJ); SV model with jumps in returns only (SVJ); and a pure SV model class without jumps. Six affine and non-affine models are considered within each model class that depend on specification of the drift and the diffusion terms in the variance process, resulting in a total of 18 models that are estimated using particle Markov Chain Monte Carlo (PMCMC) approach. Model evaluation is conducted using the Deviance Information Criterion (DIC), Bayes factors, probability plots, and deviation measures to assess the discrepancy between the estimated volatility and key benchmarks, the crude oil ETF volatility index (OVX) and the realised volatility (RV). Our analysis indicates that models incorporating jumps, particularly the SVCJ-PLY-0.5 and SVCJ-PLY-1.0, more accurately capture USO dynamics than standard SV models. The SVCJ-PLY-0.5 model ranks highest based on DIC statistics and Bayes factors, and both models excel in aligning their estimated volatility with the OVX and RV benchmarks. Overall, the statistical criteria employed in our comparison favour models with jumps over the standard SV model class, suggesting that models incorporating jumps in both return and variance processes (SVCJ) are superior to those with jumps solely in the return process (SVJ). The affine models SVJ-LIN-0.5 and SVCJ-LIN-0.5 with linear variance drift and square root diffusion that are particularly interesting for theoretical finance applications are highly ranked among considered frameworks, outperforming several non-affine alternatives. Our analysis of the regression model for volatility forecasting reveals a significant predictive accuracy in the evaluated models, demonstrating their effectiveness in anticipating future volatility trends.

Beyond GARCH in cryptocurrency volatility modelling: superiority of range-based estimators

ABSTRACT Cryptoassets are extremely volatile with possible volatility jumps and infrastructure noise, making the estimation of true volatility process challenging. When the high-frequency data are not available, the true volatility needs to be estimated to be further studied or forecasted. The GARCH-family models have become a norm in the field. Here, we examine the performance of 6 GARCH-type specifications with 4 distributional assumptions and compare them with 4 non-parametric range-based models built on the daily ‘candles’. Our study focuses on five popular cryptocurrencies (Bitcoin, Ethereum, BNB, XRP, and Dogecoin) between 1 July 2019 and 30 September 2022, utilizing Binance 5-minute data for realized measures as the high-frequency estimators of the true volatility process. The results reveal that the Garman-Klass estimator clearly outperforms the GARCH-family models in all studied settings, and the other range-based estimators remain competitive with the GARCH-family models. These results are crucial for studies on volatility in cryptoassets where using the GARCH-type models is a standard. When the high-frequency data are not available, the range-based estimators, and the Garman-Klass estimator in particular, should be preferred as proxies for the true volatility process over the GARCH-type models, be it in the in-sample, more qualitative studies, or the forecasting, out-of-sample exercises.

Price dynamics and volatility jumps in bitcoin options

In the FinTech era, we contribute to the literature by studying the pricing of Bitcoin options, which is timely and important given that both Nasdaq and the CME Group have started to launch a variety of Bitcoin derivatives. We find pricing errors in the presence of market smiles in Bitcoin options, especially for short-maturity ones. Long-maturity options display more of a “smirk” than a smile. Additionally, the ARJI-EGARCH model provides a better overall fit for the pricing of Bitcoin options than the other ARJI-GARCH type models. We also demonstrate that the ARJI-GARCH model can provide more precise pricing of Bitcoin and its options than the SVCJ model in term of the goodness-of-fit in forecasting. Allowing for jumps is crucial for modeling Bitcoin options as we find evidence of time-varying jumps. Our empirical results demonstrate that the realized jump variation can describe the volatility behavior and capture the jump risk dynamics in Bitcoin and its options.

Total variation regularization analysis for inverse volatility option pricing problem

We investigate in this paper an inverse problem of recovering the space-dependent volatility in option pricing. To enhance precision across the domain, we transform the original problem into an inverse source problem of a bounded degenerate parabolic equation, utilizing linearization and variable substitution. Unlike classical methods, we apply a total variation regularization combined with a novel generalized finite integration technique. This approach accommodates volatility jumps in an overnight rate scenario. Leveraging an optimal control framework, we demonstrate that the inverse problem can be reformulated as an optimal control problem whose existence, necessary conditions, local uniqueness and stability for the minimizer of control functional are obtained. For numerical verification, we derive the Euler equation and design a discretization algorithm with generalized finite integration technique. Numerical examples showcase the robustness of our approach, highlighting advantages in accuracy and effectiveness over other strategies.

The risks of trading on cryptocurrencies: A regime-switching approach based on volatility jumps and co-jumping behaviours

ABSTRACT Previous research has shown volatility jumps and co-jumping behaviours in cryptocurrency markets. Motivated by these findings, we employ the herding effect and financial contagion channel to outline a theoretical framework of volatility-state-dependent correlations in cryptocurrency markets. We show that digital currency markets are more strongly correlated when experiencing an identical volatility regime, which echoes co-jumping behaviours addressed by the literature. Moreover, the strong correlation that occurs when the paired cryptocurrencies simultaneously experience a high volatility regime results in the least effectiveness of diversification in terms of a minimum portfolio risk reduction. Last but not least, the proposed state-dependent approach in this study proves effective at the task of risk forecasting and risk reduction for cryptocurrency portfolios, beyond the bivariate GARCH-based models, which are a pure and simple time-dependent approach.

Strategic Asset Allocation of a Reserves’ Portfolio: Hedging Against Shocks

Central bank reserves function as a liquidity buffer to mitigate country exposure and vulnerability to external shocks. Emerging Market Economies are the countries most exposed to the volatility of capital flows and have usually preferred to build up large war-chests of international reserves as a self-insurance mechanism, as it is under their full discretion. Nevertheless, the standard practice of immobilizing large amounts of “cash” to insure against jumps in volatility and risk-aversion could be enhanced. The inclusion of hedging strategies in the strategic asset allocation decision can help to enhance the risk management of the national balance sheet, transferring funds to those scenarios when reserves are most needed. This paper presents a practical approach that we propose to enhance the analysis of the strategic asset allocation of a central bank, and to explore the benefits of including in the construction of the efficient portfolio the analysis of correlations between the reserves’ portfolio and the country’s main vulnerabilities to external shocks.

Using implied volatility jumps for realized volatility forecasting: Evidence from the Chinese market

This study examines the Chinese implied volatility index (iVIX) to determine whether jump information from the index is useful for volatility forecasting of the Shanghai Stock Exchange 50ETF. Specifically, we consider the jump sizes and intensities of the 50ETF and iVIX as well as cojumps. The findings show that both the jump size and intensity of the 50ETF can improve the forecasting accuracy of the 50ETF volatility. Moreover, we find that the jump size and intensity of the iVIX provide no significant predictive ability in any forecasting horizon. The cojump intensity of the 50ETF and iVIX is a powerful predictor for volatility forecasting of the 50ETF in all forecasting horizons, and the cojump size is helpful for forecasting in short forecasting horizon. In addition, for a one-day forecasting horizon, the iVIX jump size in the cojump is more predictive of future volatility than that of the 50ETF when simultaneous jumps occur. Our empirical results are robust and consistent. This work provides new insights into predicting asset volatility with greater accuracy.

Power‐type derivatives for rough volatility with jumps

AbstractThis paper proposes a novel analytical pricing–hedging framework for volatility derivatives which simultaneously takes into account rough volatility and volatility jumps. Directly targeting the instantaneous variance of a risky asset, our model consists of a generalized fractional Ornstein–Uhlenbeck process driven by a Lévy subordinator and an independent sinusoidal‐composite Lévy process, and allows the characteristic function of average forward variance to be obtainable in semiclosed form, without having to invoke any geometric‐mean approximations. Pricing–hedging formulae are proposed for a general class of power‐type derivatives, in the spirit of numerical Fourier transform. A comparative empirical study is conducted on two independent recent data sets on Volatility Index options, before and during the COVID‐19 pandemic, to demonstrate that the proposed framework is highly amenable to efficient model calibration under various choices of kernels. The price dynamics of the underlying asset can be readily considered and the possibility of studying rough volatility of volatility is given as well.

Leverage effect in cryptocurrency markets

In this article we study the leverage effect in cryptocurrency markets using a stochastic volatility model with simultaneous and correlated jumps in returns and volatility. We estimate the model using an efficient sequential learning algorithm with daily data on four actively traded cryptocurrencies including Bitcoin, Ethereum, Chainlink, and Litecoin. Doing so allows us to sequentially learn about the return-volatility relationships and the leverage effect in these cryptocurrencies when new data come in. We find that these relationships depend on both the diffusive and jump components of correlations between returns and volatility. Interestingly, the diffusive and jump components often have opposite signs for these currencies; that is, while the diffusive component may exhibit a negative return-volatility relationship (the “leverage effect”), the jump component may show a positive relationship (the “inverse leverage effect”). As a result, the total leverage effect can be quite different from the diffusive leverage effect, due to the presence of correlated jumps in returns and volatility. Overall, we provide evidence that these jumps matter greatly to the total leverage effect in cryptocurrency markets.

AMPLITUDE ADJUSTMENT WITH FIWASVJ MODEL

Andrianantenainarinoro\cite{maReference101} remarked that the price amplitudes of financial models may not correspond to the reality and we propose here a model in continuous time Fractionally Integrated WASC Stochastic Volatility Jump. To do this, we introduce a fractal index in the WASC Stochastic Volatility Jump model and we have two others characteristics: amplitude adjustment and memory of process. We present also several theories in stochastic calculus, algebraic, differential geometry, numerical method and estimating method which can use to financial such us: sense of a fractional integral, relationship between trace and determinant operator, Euler's approximation for an unresolved differential equation and convergence speed.

Outliers and Time-Varying Jumps in the Cryptocurrency Markets

We examine the presence of outliers and time-varying jumps in the returns of four major cryptocurrencies (Bitcoin, Ethereum, Ripple, Dogecoin, Litecoin), and a broad cryptocurrency index (CCI30). The results indicate that only Bitcoin returns are contaminated with outliers. Time-varying jumps are present in Bitcoin, Litecoin, Ripple, and the cryptocurrency index. Notably, the presence of jumps in Bitcoin is significant after correcting for outliers. The main findings point to a price instability in some major cryptocurrencies and thereby the importance of accounting for large shocks and time-varying jumps in modelling volatility in the debatable cryptocurrency markets.

Volatility Jump of Multi-Power and Maximum Outlyingness with Periodicity Filters in China yuan/US dollar Exchange Rates

The discrete daily and intraday jump probabilities of China yuan/US dollar returns from June 1, 2012 to April 30, 2021 is analyzed using five-minute returns considering periodicity filters of volatility. When we use Z-daily jump using either tripower quarticity and quadpower quarticity, the daily jump probability seems to be 66% and 70%, respectively, at α = 0.999. Thus, if the periodicity filters of volatility are not considered, the jump probabilities may be overestimated. When we use the intraday observations or periods, with the periodicity filters of volatility such as such as MAD and short half scale, the daily jump probability seems to be 39% and 40% at α = 0.999, respectively. When we consider the periodicity filters of volatility, the five-minute returns of China yuan/US dollar exchange rates have considerably lower daily jump probabilities than we use using max outlying daily jump statistics without MAD periodicity filters of volatility.

Modelling high frequency crude oil dynamics using affine and non-affine jump–diffusion models

This paper investigates the dynamics of high frequency crude oil markets proxied by the United States Oil (USO) exchange traded fund (ETF). USO returns are modelled using a stochastic framework with jumps and stochastic volatility (SVCJ). We consider three nested models within the affine SVCJ framework, and extend our analysis to three additional models within the non-affine framework. We compare six modelling frameworks (affine/non-affine and with/without jumps) based on their ability to capture high frequency USO dynamics. The modelling parameters are estimated using the Particle Markov Chain Monte Carlo (PMCMC) approach. We investigate whether jumps in USO returns and volatility are important, and whether it is necessary to leave the affine model class for the non-affine frameworks when jumps are included in the model. Using various statistical criteria and based on volatility forecasting performance, we document that (i) jumps in the returns and stochastic volatility are essential when modelling USO dynamics; (ii) the non-affine specifications are preferred to the affine specifications: The non-affine SVCJ-NL and SVJ-NL models stand out when modelling high frequency USO returns, followed by the affine SVCJ-L and SVJ-L models. The latter might be better suited for theoretical finance applications, such as derivative pricing, as closed form solutions for option prices are available under the affine frameworks.

Pricing VIX derivatives using a stochastic volatility model with a flexible jump structure

This paper proposes a novel stochastic volatility model with a flexible jump structure. This model allows both contemporaneous and independent arrival of jumps in return and volatility. Moreover, time-varying jump intensities are used to capture jump clustering. In the proposed framework, we provide a semi-analytical solution for the pricing problem of VIX futures and options. Through numerical experiments, we verify the accuracy of our pricing formula and explore the impact of the jump structure on the pricing of VIX derivatives. We find that the correct identification of the market jump structure is crucial for pricing VIX derivatives, and misspecified model setting can yield large errors in pricing.

Testing the volatility jumps based on the high frequency data

This article tests volatility jumps based on the high frequency data. Under the null hypothesis that the volatility process is a continuous semimartingale, our test statistic converges to a normal distribution, and under the alternative hypothesis where the volatility has jumps, the statistic diverges to infinity. Compared to the test statistic of Bibinger et al. (Bibinger et al. (2017). Annals of Statistics 45, 1542–1578), our proposed statistic diverges to infinity at a faster rate, and has a better power. Simulation studies confirm the theoretical results, and an empirical analysis shows that some real financial data possess volatility jumps.

Equity Risk and Return across Hidden Market Regimes

The key to understanding the dynamics of stock markets, particularly the mechanisms of their changes, is in the concept of the market regime. It is regarded as a regular transition from one state to another. Although the market agenda is never the same, its functioning regime allows us to reveal the logic of its development. The article employs the concept of financial turbulence to identify hidden market regimes. These are revealed through the ratio of the components, which describe single changes of correlated risks and volatility. The combinations of typical and atypical variates of correlational and magnitude components of financial turbulence allowed four hidden regimes to be revealed. These were arranged by the degree of financial turbulence, conceptually analyzed and assessed from the perspective of their duration. The empirical data demonstrated ETF day trading profits for S&P 500 sectors, covering the period of January 1998–August 2020, as well as day trade profits of the Russian blue chips within the period of October 2006–February 2021. The results show a significant difference in regard to the market performance and volatility, which depend on hidden regimes. Both sample data groups demonstrated similar contemporaneous and lagged effects, which allows the prediction of volatility jumps in the periods following atypical correlations.

Detecting Jump Risk and Jump-Diffusion Model for Bitcoin Options Pricing and Hedging

In this paper, we conduct a fast calibration in the jump-diffusion model to capture the Bitcoin price dynamics, as well as the behavior of some components affecting the price itself, such as the risk of pitfalls and its ambiguous effect on the evolution of Bitcoin’s price. In addition, in our study of the Bitcoin option pricing, we find that the inclusion of jumps in returns and volatilities are significant in the historical time series of Bitcoin prices. The benefits of incorporating these jumps flow over into option pricing, as well as adequately capture the volatility smile in option prices. To the best of our knowledge, this is the first work to analyze the phenomenon of price jump risk and to interpret Bitcoin option valuation as “exceptionally ambiguous”. Crucially, using hedging options for the Bitcoin market, we also prove some important properties: Bitcoin options follow a convex, but not strictly convex function. This property provides adequate risk assessment for convex risk measure.

Economic policy uncertainty and gold return dynamics: Evidence from high-frequency data

This paper examines the dynamic effect of economic policy uncertainty (EPU) on return and volatility in gold futures via the time varying parameter VAR (TVP-VAR) model with stochastic volatility applied to high-frequency data. We show that the impulse responses of gold returns and volatility to EPU shocks are time-varying and exhibit asymmetric patterns alternately over the sample period. While the effect of EPU on gold volatility is generally stronger in the intermediate to long horizons, the opposite is observed in the case of gold returns, underscoring the additive effects of uncertainty shocks on gold market volatility. The effect of EPU on gold volatility is found to be asymmetric with respect to the nature of shocks and stronger in times of crisis or major events, with such events implied by high EPU values associated with positive volatility jumps in the gold market. Our findings have significant implications for the valuation of gold derivatives and the effectiveness of gold as a potential hedge for financial investors and suggest that investors may risk mis-valuations and consequently, ineffective and costly hedges in response to uncertainty shocks if the time-variation in the response of gold return dynamics to uncertainty shocks is ignored.