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.     

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     

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.     

Pricing VIX derivatives with infinite-activity jumps

In this paper, we investigate a two-factor VIX model with infinite-activity jumps, which is a more realistic way to reduce errors in pricing VIX derivatives, compared with Mencía and Sentana (2013), J Financ Econ, 108, 367–391. Our two-factor model features central tendency, stochastic volatility and infinite-activity pure jump Lévy processes which include the variance gamma (VG) and the normal inverse Gaussian (NIG) processes as special cases. We find empirical evidence that the model with infinite-activity jumps is superior to the models with finite-activity jumps, particularly in pricing VIX options. As a result, infinite-activity jumps should not be ignored in pricing VIX derivatives.     

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     

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     

On American VIX options under the generalized 3/2 and 1/2 models

In this paper, we extend the 3/2 model for VIX studied by Goard and Mazur and introduce the generalized 3/2 and 1/2 classes of volatility processes. Under these models, we study the pricing of European and American VIX options, and for the latter, we obtain an early exercise premium representation using a free‐boundary approach and local time‐space calculus. The optimal exercise boundary for the volatility is obtained as the unique solution to an integral equation of Volterra type. We also consider a model mixing these two classes and formulate the corresponding optimal stopping problem in terms of the observed factor process. The price of an American VIX call is then represented by an early exercise premium formula. We show the existence of a pair of optimal exercise boundaries for the factor process and characterize them as the unique solution to a system of integral equations.     

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.     

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.     

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 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.     

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).     

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     

The impact of net buying pressure on VIX option prices

This paper analyzes the impact of intraday trading activity on option prices in the Volatility Index (VIX) options market. Our results show that there is a temporal relationship between net buying pressure (NBP) and changes in implied volatility of VIX options. Moreover, an increase in NBPs lowers the next-day delta-hedged option returns. Using several measures proxying for limits to arbitrage, the average levels of the implied volatility curve rise when limits to arbitrage are severe. A trading strategy in the VIX futures market constructed by using the NBP generates an average annualized return of 10.09%.     

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.     

Valuation of VIX and target volatility options with affine GARCH models

In this paper we propose semiclosed-form solutions, subject to an inversion of the Fourier transform, for the price of VIX options and target volatility options under affine GARCH models based on Gaussian and Inverse Gaussian distributions. We illustrate the advantage of the proposed analytic expressions by comparing them with those obtained from benchmark Monte–Carlo simulations. The empirical performance of the two affine GARCH models is tested using different calibration exercises based on historical returns and market quotes on VIX and SPX options.     

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     

The information content of the S&P 500 index and VIX options on the dynamics of the S&P 500 index

Given that both S&P 500 index and VIX options essentially contain information about the future dynamics of the S&P 500 index, in this study, we set out to empirically investigate the informational roles played by these two option markets with regard to the prediction of returns, volatility, and density in the S&P 500 index. Our results reveal that the information content implied from these two option markets is not identical. In addition to the information extracted from the S&P 500 index options, all of the predictions for the S&P 500 index are significantly improved by the information recovered from the VIX options. Our findings are robust to various measures of realized volatility and methods of density evaluation. © 2011 Wiley Periodicals, Inc. Jrl Fut Mark     

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     

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