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.     

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.     

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.     

PRICING COUPON-BOND OPTIONS AND SWAPTIONS IN AFFINE TERM STRUCTURE MODELS

This paper provides a numerically accurate and computationally fast approximation to the prices of European options on coupon-bearing instruments that is applicable to the entire family of affine term structure models. Exploiting the typical shapes of the conditional distributions of the risk factors in affine diffusions, we show that one can reliably compute the relevant probabilities needed for pricing options on coupon-bearing instruments by the same Fourier inversion methods used in the pricing of options on zero-coupon bonds. We apply our theoretical results to the pricing of options on coupon bonds and swaptions, and the calculation of "expected exposures" on swap books. As an empirical illustration, we compute the expected exposures implied by several affine term structure models fit to historical swap yields.     

SWAPTION PRICING IN AFFINE AND OTHER MODELS

This paper shows that Singleton and Umantsev's method for swaption pricing in affine models can be simplified and extended to other models. Two alternative methods for approximating the option exercise boundary are introduced: one based on the multivariate Taylor series expansion, and the other based on duration‐matched zero‐coupon bond approximation. Applied to affine models and quadratic‐Gaussian models, these methods are found to give accurate swaption prices.     

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.     

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.     

Inferring information from the S&P 500, CBOE VIX,and CBOE SKEW indices

This paper compares the information extracted from the S&P 500, CBOE VIX, and CBOE SKEW indices for the S&P 500 index option pricing. Based on our empirical analysis, VIX is a very informative index for option prices. Whether adding the SKEW or the VIX term structure can improve the option pricing performance depends on the model we choose. Roughly speaking, the VIX term structure is informative for some models, while the SKEW is very noisy and does not contain much important information for option prices. This paper also extends Zhang et al. (2017, J Futures Markets, 37, 211–237) into three typical affine models.     

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%.     

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.     

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     

ON THE CONSISTENCY OF REGRESSION‐BASED MONTE CARLO METHODS FOR PRICING BERMUDAN OPTIONS IN CASE OF ESTIMATED FINANCIAL MODELS

In many applications of regression‐based Monte Carlo methods for pricing, American options in discrete time parameters of the underlying financial model have to be estimated from observed data. In this paper suitably defined nonparametric regression‐based Monte Carlo methods are applied to paths of financial models where the parameters converge toward true values of the parameters. For various Black–Scholes, GARCH, and Levy models it is shown that in this case the price estimated from the approximate model converges to the true price.     

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.     

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 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     

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     

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.     

Volatility as an asset class: Holding VIX in a portfolio

Hedging market downturns without sacrificing upside has long been sought by investors. If VIX was directly investable, adding it as a hedge to the S&P 500 would result in significantly improved performance over the equity only portfolio. However, tradable VIX products do not provide the hedge or returns investors seek over long-term horizons. Alternatively, deconstructing VIX to find the key S&P 500 options which drive VIX movements leads to a synthetic VIX portfolio that provides a more effective hedge. Using these options captures correlations and returns similar to VIX, and combined with the S&P 500, outperforms the buy-and-hold index portfolio.