Volatility Jumps Research Articles

Joint pricing of VIX and SPX options with stochastic volatility and jump models

Purpose – This paper studies the performance of commonly employed stochastic volatility and jump models in the consistent pricing of The CBOE Volatility Index (VIX) and The S&P 500 Index (SPX) options. With the existence of active markets for volatility derivatives and options on the underlying instrument, the need for models that are able to price these markets consistently has increased. Although pricing formulas for VIX and vanilla options are now available for commonly used models exhibiting stochastic volatility and/or jumps, it remains to be shown whether these are able to price both markets consistently. This paper fills this vacuum. Design/methodology/approach – In particular, the Heston model, the Heston model with jumps in returns and the Heston model with simultaneous jumps in returns and variance (SVJJ) are jointly calibrated to market quotes on SPX and VIX options together with VIX futures. Findings – The full flexibility of having jumps in both returns and volatility added to a stochastic volatility model is essential. Moreover, we find that the SVJJ model with the Feller condition imposed and calibrated jointly to SPX and VIX options fits both markets poorly. Relaxing the Feller condition in the calibration improves the performance considerably. Still, the fit is not satisfactory, and we conclude that one needs more flexibility in the model to jointly fit both option markets. Originality/value – Compared to existing literature, we derive numerically simpler VIX option and futures pricing formulas in the case of the SVJ model. Moreover, the paper is the first to study the pricing performance of three widely used models to SPX options and VIX derivatives.

A common jump factor stochastic volatility model

We introduce a new multivariate stochastic volatility model, based on the presence of a latent common factor with random jumps. The common factor is parameterized as a permanent component using a compound binomial process. This model can capture common jumps in the latent volatilities between markets, with particular relevance in the presence of crises and contagion in emerging markets.

Double-Jump Stochastic Volatility Model for VIX: Evidence from VVIX

This paper studies the continuous-time dynamics of VIX with stochastic volatility and jumps in VIX and volatility. Built on the general parametric affine model with stochastic volatility and jumps in the logarithm of VIX, we derive a linear relationship between the stochastic volatility factor and the VVIX index. We detect the existence of a co-jump of VIX and VVIX and put forward a double-jump stochastic volatility model for VIX through its joint property with VVIX. Using the VVIX index as a proxy for stochastic volatility, we use the MCMC method to estimate the dynamics of VIX. Comparing nested models of VIX, we show that the jump in VIX and the volatility factor are statistically significant. The jump intensity is also stochastic. We analyze the impact of the jump factor on VIX dynamics.

Superposition of COGARCH processes

We suggest three superpositions of COGARCH (sup-CO-GARCH) volatility processes driven by Lévy processes or Lévy bases. We investigate second-order properties, jump behaviour, and prove that they exhibit Pareto-like tails. Corresponding price processes are defined and studied. We find that the sup-CO-GARCH models allow for more flexible autocovariance structures than the COGARCH. Moreover, in contrast to most financial volatility models, the sup-CO-GARCH processes do not exhibit a deterministic relationship between price and volatility jumps. Furthermore, in one sup-CO-GARCH model not all volatility jumps entail a price jump, while in another sup-CO-GARCH model not all price jumps necessarily lead to volatility jumps.

Bayesian Modeling and Forecasting of 24-Hour High-Frequency Volatility

This article estimates models of high-frequency index futures returns using “around-the-clock” 5-min returns that incorporate the following key features: multiple persistent stochastic volatility factors, jumps in prices and volatilities, seasonal components capturing time of the day patterns, correlations between return and volatility shocks, and announcement effects. We develop an integrated MCMC approach to estimate interday and intraday parameters and states using high-frequency data without resorting to various aggregation measures like realized volatility. We provide a case study using financial crisis data from 2007 to 2009, and use particle filters to construct likelihood functions for model comparison and out-of-sample forecasting from 2009 to 2012. We show that our approach improves realized volatility forecasts by up to 50% over existing benchmarks and is also useful for risk management and trading applications. Supplementary materials for this article are available online.

Volatility Jumps and Their Economic Determinants

The volatility of financial returns is characterized by rapid and large increments. We propose an extension of the Heterogeneous Autoregressive model to incorporate jumps into the dynamics of the ex post volatility measures. Using the realized range measures of 36 NYSE stocks, we show that there is a positive probability of jumps in volatility. A common factor in the volatility jumps is shown to be related to a set of financial covariates (such as variance risk premium (VRP), S&P500 volume, credit default swap (CDS), and federal fund rates). The CDS on U.S. banks and VRP have predictive power on expected jump moves, thus confirming the common interpretation that sudden and large increases in equity volatility can be anticipated by credit deterioration of the U.S. bank sector as well as changes in the market expectations of future risks. Finally, the model is extended to incorporate the CDS and the VRP in the dynamics of the jump size and intensity.

The stochastic-volatility, jump-diffusion optimal portfolio problem with jumps in returns and volatility

This paper treats the risk-averse optimal portfolio problem with consumption in continuous time for a stochastic-jump-volatility, jump-diffusion (SJVJD) model of the underlying risky asset and the volatility. The new developments are the use of the SJVJD model with logtruncated-double-exponential jump-amplitude distribution in returns and exponential jumpamplitude distribution in volatility for the optimal portfolio problem. Although unlimited borrowing and short-selling play an important role in pure diffusion models, it is shown that borrowing and short-selling are unrealistically constrained for infinite-range jump-amplitudes. Finite-range jump-amplitude models can allow constraints to be very large in contrast to infinite range models which severely restrict the optimal instantaneous stock-fraction to [0,1]. The reasonable constraints in the optimal stock-fraction due to jumps in the wealth argument for stochastic dynamic programming jump integrals remove a singularity in the stock-fraction due to vanishing volatility. Main modifications for the usual constant relative risk aversion (CRRA) power utility model are for handling the partial integro-differential equation (PIDE) resulting from the additional variance independent variable, instead of the ordinary integro-differential equation (OIDE) found for the pure jump-diffusion model of the wealth process. In addition to natural constraints due to jumps when enforcing the positivity of wealth condition, other constraints are considered for all practical purposes under finite market conditions. Computational results are presented for optimal portfolio values, stock fraction and consumption policies.

Volatility Around the Clock: Bayesian Modeling and Forecasting of Intraday Volatility in the Financial Crisis

This paper estimates models of high frequency index futures returns using 'around the clock' 5-minute returns that incorporate the following key features: multiple persistent stochastic volatility factors, jumps in prices and volatilities, seasonal components capturing time of the day patterns, correlations between return and volatility shocks, and announcement effects. We develop an integrated MCMC approach to estimate interday and intraday parameters and states using high-frequency data without resorting to various aggregation measures like realized volatility. We provide a case study using financial crisis data from 2007 to 2009, and use particle filters to construct likelihood functions for model comparison and out-of-sample forecasting from 2009 to 2012. We show that our approach improves realized volatility forecasts by up to 50% over existing benchmarks.

The Distribution of Uncertainty: Evidence from the VIX Options Market

AbstractThis article investigates the informational content implied in the risk‐neutral distribution of the VIX, a leading barometer of economic uncertainty. We extract the risk‐neutral distribution from VIX option prices over the sample period from 2006 to 2011 using a non‐parametric approach. We analyze the time‐series behavior of the option‐implied moments and assess whether the information implied in the risk‐neutral distribution has predictive power. The risk‐neutral distribution considerably changed shape during the financial crisis. Furthermore, risk‐neutral moments contain useful information with respect to the likelihood of upward jumps in volatility. Consistent with investors disliking high levels of economic uncertainty, we find that the overall shape of the estimated volatility pricing kernel is increasing. For certain periods, there is a puzzling U‐shape. The behavior of the volatility pricing kernel over time reveals that the financial crisis has affected investors’ attitudes towards risk. © 2014 Wiley Periodicals, Inc. Jrl Fut Mark 35:597–624, 2015

The Distribution of Uncertainty: Evidence from the VIX Options Market

This paper investigates the informational content implied in the risk-neutral distribution of the VIX, a leading barometer of economic uncertainty. We extract the risk-neutral distribution from VIX option prices over the sample period from 2006 to 2011 using a non-parametric approach. We analyze the time-series behavior of the option-implied moments and assess whether the information implied in the risk-neutral distribution has predictive power. The risk-neutral distribution considerably changed shape during the financial crisis. Furthermore, risk-neutral moments contain useful information with respect to the likelihood of upward jumps in volatility. Consistent with investors disliking high levels of economic uncertainty, we find that the overall shape of the estimated volatility pricing kernel is increasing. For certain periods, there is a puzzling U-shape. The behavior of the volatility pricing kernel over time reveals that the financial crisis has affected investors' attitudes towards risk.

Hedging Industrial Metals With Stochastic Volatility Models

The financialization of commodities documented in [Tang and Xiong (2012) Financial Analyst Journal, 68:54–74] has led commodity prices to exhibit not only time‐varying volatility, but also price and volatility jumps. Using the class of stochastic volatility (SV) models, we incorporate such extreme price movements to generate out‐of‐sample hedge ratios. In‐sample estimation on China's copper (CU) and aluminum (AL) spot and futures markets confirms the presence of price jumps and price‐volatility jump correlations. Out‐of‐sample hedge ratios from the [Bates (1996) Review of Financial Studies, 9:69–107] SV with price jumps model deliver the greatest risk reduction on the unhedged positions at 59.55% for CU and 49.85% for AL. But it is the [Duffie, Pan, and Singleton (2000) Econometrica, 68:1343–1376] SV model with correlated price and volatility jumps that produces hedge ratios which yield the largest Sharpe Ratios of 0.644 for CU and 0.886 for AL. © 2014 Wiley Periodicals, Inc. Jrl Fut Mark 34:704–730, 2014

Fast Exponential Time Integration for Pricing Options in Stochastic Volatility Jump Diffusion Models

AbstractThe stochastic volatility jump diffusion model with jumps in both return and volatility leads to a two-dimensional partial integro-differential equation (PIDE). We exploit a fast exponential time integration scheme to solve this PIDE. After spatial discretization and temporal integration, the solution of the PIDE can be formulated as the action of an exponential of a block Toeplitz matrix on a vector. The shift-invert Arnoldi method is employed to approximate this product. To reduce the computational cost, matrix splitting is combined with the multigrid method to deal with the shift-invert matrix-vector product in each inner iteration. Numerical results show that our proposed scheme is more robust and efficient than the existing high accurate implicit-explicit Euler-based extrapolation scheme.

Intraday Serial Correlation,Volatility, and Jump: Evidence from China's Stock Market

In this article, we investigate the relationships among intraday serial correlation, jump-robust volatility, positive and negative jumps based on Shanghai composite index high frequency data. We implement variance ratio test to quantify intraday serial correlation. We also measure the continuous part of realized volatility using jump-robust MedRV estimator and disentangle positive and negative jumps using Realized Downside Risk Measure and Realized Upside Potential Measure proposed by Bi et al., (2013). We find that intraday serial correlation are positively correlated with jump-robust volatility and negatively correlated with negative jumps which confirm the LeBaron effect.

Nonparametric Estimation of Periodicity of Power Volatility and Discontinuous Daily Jump and Intraday Jump

Nonparametric Estimation of Periodicity of Power Volatility and Discontinuous Daily Jump and Intraday Jump

Resolution of Policy Uncertainty and Sudden Declines in Volatility

We introduce downward volatility jumps into a general non-affine modeling framework of the term structure of variance. With variance swaps and S&P 500 returns, we find that downward volatility jumps are associated with a resolution of policy uncertainty, in particular through statements from Federal Open Market Committee meetings and speeches of the Federal Reserve chairman. We also find that such jumps are priced with positive risk premia, which reflect the price of the put protection offered by the Federal Reserve. Ignoring downward volatility jumps may lead to an exaggeration of the negative total variance risk premia, hence a biased-interpretation of the price of tail events. We also find variance risk premia tend to be insignificant or even positive at the inception of crises. On the modeling side, we explore the structural differences and relative goodness-of-fits of factor specifications, and find that the log-volatility model with two Ornstein-Uhlenbeck factors and double-sided jumps are superior in capturing volatility dynamics and pricing variance swaps.

Volatility of Volatility and Tail Risk Premiums

This paper reports on tail risk premiums in two tail risk hedging strategies: the S&P 500 puts and the VIX calls. As a new measure of tail risk, we suggest using a model-free, risk-neutral measure of the volatility of volatility implied by a cross section of the VIX options, which we call the VVIX index. The tail risk measured by the VVIX index has forecasting power for future tail risk hedge returns. Specifically, consistent with the literature on rare disasters, an increase in the VVIX index raises the current prices of tail risk hedges and thus lowers their subsequent returns over the next three to four weeks. Furthermore, we find that volatility of volatility risk and its associated risk premium both significantly contribute to the forecasting power of the VVIX index, and that the predictability largely results from the integrated volatility of volatility rather than volatility jumps.

Volatility activity: Specification and estimation

The paper examines volatility activity and its asymmetry and undertakes further specification analysis of volatility models based on it. We develop new nonparametric statistics using high-frequency option-based VIX data to test for asymmetry in volatility jumps. We also develop methods for estimating and evaluating, using price data alone, a general encompassing model for volatility dynamics where volatility activity is unrestricted. The nonparametric application to VIX data, along with model estimation for S&P index returns, suggests that volatility moves are best captured by an infinite variation pure-jump martingale with a symmetric jump compensator around zero. The latter provides a parsimonious generalization of the jump-diffusions commonly used for volatility modeling.

Understanding Jumps in the High-Frequency VIX

I conduct a comprehensive nonparametric study of volatility jumps and leverage effect by examining high-frequency data on the VIX and S&P 500 from 1992 to 2010. I argue that the VIX data prior to 1998 are too noisy to provide a reliable inference. After 1999, the dataset is cleaner but still controversial. More specifically, the high-frequency dynamics of the VIX jumps challenges the assumptions of commonly used stochastic volatility jump-diffusion models. I explain this phenomenon by hypothesizing that most jump-like movements in the VIX are pseudo-jumps i.e., these jumps are large but temporary deviations from fundamental values.

Fast Fourier transform option pricing with stochastic interest rate, stochastic volatility and double jumps

In this paper, we focus on pricing European options in a double exponential jump diffusion model with stochastic volatility and stochastic interest rate. Firstly, using fast Fourier transform (FFT) technique, we obtain numerical solutions for option prices. Then, we analyze several effects on option prices under the proposed model, including correlation between stock returns and volatility, stochastic interest rate. Simulations show that FFT is fast and efficient, stock returns are negatively correlated with volatility and the effect of stochastic interest rate over longer time horizons is significant.

Stochastic Volatility Jump‐Diffusions for European Equity Index Dynamics

AbstractMajor research on equity index dynamics has investigated only US indices (usually the S&P 500) and has provided contradictory results. In this paper a clarification and extension of that previous research is given. We find that European equity indices have quite different dynamics from the S&P 500. Each of the European indices considered may be satisfactorily modelled using either an affine model with price and volatility jumps or a GARCH volatility process without jumps. The S&P 500 dynamics are much more difficult to capture in a jump‐diffusion framework.