VIX futures and its closed‐form pricing through an affine GARCH model with realized variance

This paper studies the forecasting of volatility index (VIX) and the pricing of its futures by a generalized affine realized volatility model proposed by Christoffersen et al. This model is a weighted average of a GARCH and a pure realized variance (RV) model that incorporates each volatility component into the new dynamics. We rewrite the VIX in terms of both volatility components and then derive closed‐form formulas for the VIX forecasting and its futures pricing. Our empirical studies find that a unification of the GARCH and the RV in the modeling substantially improves the forecasting of this index and the pricing of its futures.     

An analytical formula for VIX futures and its applications

In this study we present a closed‐form, exact solution for the pricing of VIX futures in a stochastic volatility model with simultaneous jumps in both the asset price and volatility processes. The newly derived formula is then used to show that the well‐known convexity correction approximations can sometimes lead to large errors. Utilizing the newly derived formula, we also conduct an empirical study, the results of which demonstrate that the Heston stochastic volatility model is a good candidate for the pricing of VIX futures. While incorporating jumps into the underlying price can further improve the pricing of VIX futures, adding jumps to the volatility process appears to contribute little improvement for pricing VIX futures. © 2011 Wiley Periodicals, Inc. Jrl Fut Mark     

A simplified pricing model for volatility futures

We develop a general model to price VIX futures contracts. The model is adapted to test both the constant elasticity of variance (CEV) and the Cox–Ingersoll–Ross formulations, with and without jumps. Empirical tests on VIX futures prices provide out‐of‐sample estimates within 2% of the actual futures price for almost all futures maturities. We show that although jumps are present in the data, the models with jumps do not typically outperform the others; in particular, we demonstrate the important benefits of the CEV feature in pricing futures contracts. We conclude by examining errors in the model relative to the VIX characteristics. © 2010 Wiley Periodicals, Inc. Jrl Fut Mark 31:307–339, 2011     

The new market for volatility trading

This study analyses the new market for trading volatility; VIX futures. We first use market data to establish the relationship between VIX futures prices and the index itself. We observe that VIX futures and VIX are highly correlated; the term structure of average VIX futures prices is upward sloping, whereas the term structure of VIX futures volatility is downward sloping. To establish a theoretical relationship between VIX futures and VIX, we model the instantaneous variance using a simple square root mean‐reverting process with a stochastic long‐term mean level. Using daily calibrated long‐term mean and VIX, the model gives good predictions of VIX futures prices under normal market situation. These parameter estimates could be used to price VIX options. © 2010 Wiley Periodicals, Inc. Jrl Fut Mark 30:809–833, 2010     

Modeling VXX under jump diffusion with stochastic long-term mean

We develop a model for the VXX, the most actively traded VIX futures exchange-traded note, using Duffie, Pan, and Singleton's affine jump diffusion framework, where the volatility process has jumps and a stochastic long-term mean. We calibrate the model parameters using the VIX term structure data and show that our model provides the theoretical link between the VIX, VIX futures, and the VXX. Our model can be used for pricing VIX futures, the VXX and other short-term VIX futures exchange-traded products (ETPs). Our model could be extended to price options on the VXX and other short-term VIX futures ETPs.     

A dimension-invariant cascade model for VIX futures

We propose a new stochastic volatility model by allowing for a cascading structure of volatility components. The model, under a minor assumption, allows us to add as many components as desired with no additional parameters, effectively defeating the curse of dimensionality often encountered in traditional models. We derive a semi-closed-form solution to the VIX futures price, and find that our six-factor model with only six parameters can closely fit spot VIX and VIX futures prices from 2004 to 2015 and produce out-of-sample pricing errors of magnitudes similar to those of in-sample errors.     

Pricing VIX futures: Evidence from integrated physical and risk‐neutral probability measures

This study derives closed‐form solutions to the fair value of VIX (volatility index) futures under alternate stochastic variance models with simultaneous jumps both in the asset price and variance processes. Model parameters are estimated using an integrated analysis of integrated volatility and VIX time series from April 21, 2004 to April 18, 2006. The stochastic volatility model with price jumps outperforms for the short‐dated futures, whereas additionally including a state‐dependent volatility jump can further reduce out‐of‐sample pricing errors for other futures maturities. Finally, adding volatility jumps enhances hedging performance except for the short‐dated futures on a daily‐rebalanced basis. © 2007 Wiley Periodicals, Inc. Jrl Fut Mark 27:1175–1217, 2007     

Causality in the VIX futures market

This study examines the price‐discovery function and information efficiency of a fast growing volatility futures market: the Chicago Board of Option Exchange VIX futures market. A linear Engle–Granger cointegration test with an error correction mechanism (ECM) shows that during the full sample period, VIX futures prices lead spot VIX index, which implies that the VIX futures market has some price‐discovery function. But a modified Baek and Brock nonlinear Granger test detects bi‐directional causality between VIX and VIX futures prices, suggesting that both spot and futures prices react simultaneously to new information. Quarter‐by‐quarter investigations show that, on average, the estimated parameters are not significantly different from zero, thus providing further evidence supporting information efficiency in the VIX futures market. © 2011 Wiley Periodicals, Inc. Jrl Fut Mark     

Volatility components: The term structure dynamics of VIX futures

In this study we empirically study the variance term structure using volatility index (VIX) futures market. We first derive a new pricing framework for VIX futures, which is convenient to study variance term structure dynamics. We construct five models and use Kalman filter and maximum likelihood method for model estimations and comparisons. We provide evidence that a third factor is statistically significant for variance term structure dynamics. We find that our parameter estimates are robust and helpful to shed light on economic significance of variance factor model. © 2009 Wiley Periodicals, Inc. Jrl Fut Mark 30:230–256, 2010     

Improving volatility prediction and option valuation using VIX information: A volatility spillover GARCH model

We develop a new generalized autoregressive conditional heteroskedasticity (GARCH) model that accounts for the information spillover between two markets. This model is used to detect the usefulness of the CBOE volatility index (VIX) for improving the performance of volatility forecasting and option pricing. We find the significant ability of VIX to predict stock volatility both in-sample and out-of-sample. VIX information also helps to greatly reduce the option pricing error. The proposed volatility spillover GARCH model performs better than the related approaches proposed by Kanniainen et al. (2014, J Bank Finance, 43, pp. 200-211) and P. Christoffersen et al. (2014, J Financ Quant Anal, 49, pp. 663–697).     

VIX futures

VIX futures are exchange‐traded contracts on a future volatility index (VIX) level derived from a basket of S&P 500 (SPX) stock index options. The authors posit a stochastic variance model of VIX time evolution, and develop an expression for VIX futures. Free parameters are estimated from market data over the past few years. It is found that the model with parameters estimated from the whole period from 1990 to 2005 overprices the futures contracts by 16–44%. But the discrepancy is dramatically reduced to 2–12% if the parameters are estimated from the most recent one‐year period. © 2006 Wiley Periodicals, Inc. Jrl Fut Mark 26:521–531, 2006     

Testing mean reversion in financial market volatility: Evidence from S&P 500 index futures

This article presents a comprehensive study of continuous time GARCH (generalized autoregressive conditional heteroskedastic) modeling with the thintailed normal and the fat‐tailed Student's‐t and generalized error distributions (GED). The study measures the degree of mean reversion in financial market volatility based on the relationship between discrete‐time GARCH and continuoustime diffusion models. The convergence results based on the aforementioned distribution functions are shown to have similar implications for testing mean reversion in stochastic volatility. Alternative models are compared in terms of their ability to capture mean‐reverting behavior of futures market volatility. The empirical evidence obtained from the S&P 500 index futures indicates that the conditional variance, log‐variance, and standard deviation of futures returns are pulled back to some long‐run average level over time. The study also compares the performance of alternative GARCH models with normal, Student's‐ t, and GED density in terms of their power to predict one‐day‐ahead realized volatility of index futures returns and provides some implications for pricing futures options. © 2008 Wiley Periodicals, Inc. Jrl Fut Mark 28:1–33, 2008     

VIX term structure and VIX futures pricing with realized volatility

Using an extended LHARG model proposed by Majewski et al. (2015, J Econ, 187, 521–531), we derive the closed-form pricing formulas for both the Chicago Board Options Exchange VIX term structure and VIX futures with different maturities. Our empirical results suggest that the quarterly and yearly components of lagged realized volatility should be added into the model to capture the long-term volatility dynamics. By using the realized volatility based on high-frequency data, the proposed model provides superior pricing performance compared with the classic Heston–Nandi GARCH model under a variance-dependent pricing kernel, both in-sample and out-of-sample. The improvement is more pronounced during high volatility periods.     

The performance of VIX option pricing models: Empirical evidence beyond simulation

We examine the pricing performance of VIX option models. Such models possess a wide‐range of underlying characteristics regarding the behavior of both the S&P500 index and the underlying VIX. Our tests employ three representative models for VIX options: Whaley ( 1993 ), Grunbichler and Longstaff ( 1996 ), Carr and Lee ( 2007 ), Lin and Chang ( 2009 ), who test four stochastic volatility models, as well as to previous simulation results of VIX option models. We find that no model has small pricing errors over the entire range of strike prices and times to expiration. In particular, out‐of‐the‐money VIX options are difficult to price, with Grunbichler and Longstaff's mean‐reverting model producing the smallest dollar errors in this category. Whaley's Black‐like option model produces the best results for in‐the‐money VIX options. However, the Whaley model does under/overprice out‐of‐the‐money call/put VIX options, which is opposite the behavior of stock index option pricing models. VIX options exhibit a volatility skew opposite the skew of index options. © 2010 Wiley Periodicals, Inc. Jrl Fut Mark31:251–281, 2011     

Conditional Volatility and the GARCH Option Pricing Model with Non‐Normal Innovations

On the basis of the theory of a wedge between the physical and risk‐neutral conditional volatilities in Christoffersen, P., Elkamhi, R., Feunou, B., & Jacobs, K. (2010), we develop a modification of the GARCH option pricing model with the filtered historical simulation proposed in Barone‐Adesi, G., Engle, R. F., & Mancini, L. (2008). The one‐day‐ahead conditional volatilities under physical and risk‐neutral measures are the same in the previous model, but should have been allowed to be different. Using extensive data on S&P 500 index options, our approach, which employs one‐day‐ahead risk‐neutral conditional volatility estimated from the cross‐section of the option prices (in contrast to the existing GARCH option pricing models), maintains theoretical consistency under conditional non‐normality, and improves the empirical performances. Remarkably, the risk‐neutral volatility dynamics are stable over time in this model. In addition, the comparison between the VIX index and the risk‐neutral integrated volatility economically validates our approach. © 2011 Wiley Periodicals, Inc. Jrl Fut Mark 33:1–28, 2013     

The short‐time behavior of VIX‐implied volatilities in a multifactor stochastic volatility framework

We consider a modeling setup where the volatility index (VIX) dynamics are explicitly computable as a smooth transformation of a purely diffusive, multidimensional Markov process. The framework is general enough to embed many popular stochastic volatility models. We develop closed‐form expansions and sharp error bounds for VIX futures, options, and implied volatilities. In particular, we derive exact asymptotic results for VIX‐implied volatilities, and their sensitivities, in the joint limit of short time‐to‐maturity and small log‐moneyness. The expansions obtained are explicit based on elementary functions and they neatly uncover how the VIX skew depends on the specific choice of the volatility and the vol‐of‐vol processes. Our results are based on perturbation techniques applied to the infinitesimal generator of the underlying process. This methodology has previously been adopted to derive approximations of equity (SPX) options. However, the generalizations needed to cover the case of VIX options are by no means straightforward as the dynamics of the underlying VIX futures are not explicitly known. To illustrate the accuracy of our technique, we provide numerical implementations for a selection of model specifications.     

A CONSISTENT PRICING MODEL FOR INDEX OPTIONS AND VOLATILITY DERIVATIVES

We propose a flexible framework for modeling the joint dynamics of an index and a set of forward variance swap rates written on this index. Our model reproduces various empirically observed properties of variance swap dynamics and enables volatility derivatives and options on the underlying index to be priced consistently, while allowing for jumps in volatility and returns. An affine specification using Lévy processes as building blocks leads to analytically tractable pricing formulas for volatility derivatives, such as VIX options, as well as efficient numerical methods for pricing of European options on the underlying asset. The model has the convenient feature of decoupling the vanilla skews from spot/volatility correlations and allowing for different conditional correlations in large and small spot/volatility moves. We show that our model can simultaneously fit prices of European options on S&P 500 across strikes and maturities as well as options on the VIX volatility index.     

VIX option pricing

Substantial progress has been made in developing more realistic option pricing models for S&P 500 index (SPX) options. Empirically, however, it is not known whether and by how much each generalization of SPX price dynamics improves VIX option pricing. This article fills this gap by first deriving a VIX option model that reconciles the most general price processes of the SPX in the literature. The relative empirical performance of several models of distinct interest is examined. Our results show that state‐dependent price jumps and volatility jumps are important for pricing VIX options. © 2009 Wiley Periodicals, Inc. Jrl Fut Mark 29:523–543, 2009     

The Term Structure of VIX

In this study, we extend the Chicago Board Options Exchange volatility index, VIX, from 30‐day to any arbitrary time‐to‐maturity, and study the term structure of VIX. We propose new concepts of instantaneous and long‐term squared VIXs as the limits at the short and long ends of the term structure respectively. Modeling the volatility process with instantaneous and long‐term squared VIXs, we establish a parsimonious approach to capture information contained in the term structure of VIX. Our study provides an efficient setup to further study the pricing of VIX derivatives and their relation with S&P 500 options.     

Information Role of U.S. Futures Trading in a Global Financial Market

Using a bivariate, asymmetric generalized autoregressive conditional heteroskedasticity model, we examine the patterns of information flows for three financial futures contracts that are dual‐listed on U.S. and Asian markets (i.e., Nikkei 225 Index, Eurodollar, and dollar–yen currency futures). The results indicate that the U.S. market plays a leading role in terms of pricing‐information transmission across markets. In terms of volatility spillover across markets, however, foreign markets seem to play a similar role (e.g., Nikkei Index futures) or even a more significant role than the United States (e.g., Eurodollar futures in Singapore and dollar–yen currency futures in Japan). © 2001 John Wiley & Sons, Inc. Jrl Fut Mark 21:1071–1090, 2001