Instantaneous squared VIX and VIX derivatives

In this paper, we propose a parsimonious and efficient model to price derivatives written on VIXs with different horizons. Our model is built on Luo and Zhang's (2012, J Futures Markets, 32, 1092–1123) concept of the instantaneous squared VIX (ISVIX) that is the sum of instantaneous diffusive and jump variances of the SPX return. Modeling the ISVIX as a mean-reverting jump-diffusion process with a stochastic long-term mean, we obtain analytical formulas for VIX options and futures. Estimation with VIX term structure and calibration with VIX options data show that our model performs well in matching both time series and cross-sectional VIX derivatives market prices.     

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     

Equity volatility,bond yields,and yield spreads

This study examines the impact of implied and contemporaneous equity market volatility on Treasury yields, corporate bond yields, and yield spreads over Treasuries. The CBOE VIX is the measure of implied volatility, and the measure of contemporaneous volatility is constructed using intraday squared S&P 500 returns. We find that bond yields and spreads respond to changes in equity market volatility in a manner consistent with a flight‐to‐quality effect. Both short‐ and long‐term Treasury yields fall in response to increases in implied volatility, and the yield curve flattens modestly. Yields on short‐term investment grade bonds fall in response to contemporaneous volatility shocks, while long‐term spreads on low‐quality issues widen. This indicates that investors “look ahead” in anticipation of changes in equity market volatility but respond more strongly to changes in contemporaneous market activity. © 2011 Wiley Periodicals, Inc. Jrl Fut Mark     

STOCHASTIC VOLATILITY MODELS AND THE PRICING OF VIX OPTIONS

In this paper, we examine and compare the performance of a variety of continuous‐time volatility models in their ability to capture the behavior of the VIX. The “3/2‐ model” with a diffusion structure which allows the volatility of volatility changes to be highly sensitive to the actual level of volatility is found to outperform all other popular models tested. Analytic solutions for option prices on the VIX under the 3/2‐model are developed and then used to calibrate at‐the‐money market option prices.     

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     

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     

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.     

Quantile Regression Analysis of the Asymmetric Return‐Volatility Relation

We use quantile regression to investigate the short‐term return‐volatility relation between stock index returns and changes in implied volatility index. Neither the leverage hypothesis nor the volatility feedback hypothesis effectively explains the asymmetric return‐volatility relation. Instead, behavioral explanations, such as the affect and representativeness heuristics, are supported by our results, particularly in the short‐term; the affect heuristic plays an important role. Moreover, in the context of an extreme volatility change distribution, the affect heuristic and time‐pressure dominate. Thus, we observe strong negative and asymmetric relations between each volatility index and its corresponding stock market index. The asymmetry increases monotonically from the median quantile to the uppermost quantile (i.e., 95%); therefore, ordinary least squares (OLS) regression underestimates this relation at upper quantiles. Additionally, the VIX presents the highest asymmetric return‐volatility relation, followed by the VSTOXX, VDAX, and VXN. Finally, the observed asymmetry is more pronounced with the new volatility index measure than with the old, at‐the‐money volatility index measure. © 2012 Wiley Periodicals, Inc. Jrl Fut Mark 33:235–265, 2013     

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.     

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     

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.     

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     

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     

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 futures pricing with conditional skewness

We develop a closed‐form VIX futures valuation formula based on the inverse Gaussian GARCH process by Christoffersen et al. that combines conditional skewness, conditional heteroskedasticity, and a leverage effect. The new model outperforms the benchmark in fitting the S&P 500 returns and the VIX futures prices. The fat‐tailed innovation underlying the model substantially reduced pricing errors during the 2008 financial crisis. The in‐ and out‐of‐sample pricing performance indicates that the new model should be a default modeling choice for pricing the medium‐ and long‐term VIX futures.     

A two‐mean reverting‐factor model of the term structure of interest rates

This article presents a two‐factor model of the term structure of interest rates. It is assumed that default‐free discount bond prices are determined by the time to maturity and two factors, the long‐term interest rate, and the spread (i.e., the difference) between the short‐term (instantaneous) risk‐free rate of interest and the long‐term rate. Assuming that both factors follow a joint Ornstein‐Uhlenbeck process, a general bond pricing equation is derived. Closed‐form expressions for prices of bonds and interest rate derivatives are obtained. The analytical formula for derivatives is applied to price European options on discount bonds and more complex types of options. Finally, empirical evidence of the model's performance in comparison with an alternative two‐factor (Vasicek‐CIR) model is presented. The findings show that both models exhibit a similar behavior for the shortest maturities. However, importantly, the results demonstrate that modeling the volatility in the long‐term rate process can help to fit the observed data, and can improve the prediction of the future movements in medium‐ and long‐term interest rates. So it is not so clear which is the best model to be used. © 2003 Wiley Periodicals, Inc. Jrl Fut Mark 23: 1075–1105, 2003     

When and How Does Supplier Opportunism Matter for Small Retailers' Channel Relationships with the Suppliers?

This study examines the moderating influences of power‐dependence structure and environmental volatility on the relationships between supplier opportunism and its outcomes. Data were collected from 102 small retailers using a mail survey. The results of multiple regressions indicate that supplier opportunism decreases small retailer credibility/benevolence and long‐term orientation. Further, the findings show that the lower the small retailer dependence, the stronger the negative influences of supplier opportunism on small retailer long‐term orientation. Finally, the lower the environmental volatility, the stronger the negative influences of supplier opportunism on small retailer credibility/benevolence and its long‐term orientation. Marketing implications and recommendations are provided.     

Pricing VIX options with volatility clustering

We investigate the valuation of volatility index (VIX) options by developing a model with a self-exciting Hawkes process that allows for clustering in the VIX. In the proposed framework, we find semianalytical expressions for the characteristic function and forward characteristic function, and then we solve the pricing problem of standard-start and forward-start options via the fast Fourier transform. The empirical results provide evidence to support the significance of accounting for volatility clustering when pricing VIX options.     

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.