VIX Data Research Articles

Bayesian estimation of stochastic volatility jump diffusion model parameters using S&P 500 and VIX data

The Stochastic Volatility Jump Diffusion (SVJD) model is a common tool used for option pricing. Existing literature has employed many techniques in order to estimate the parameters of this model, including both Bayesian and frequentist approaches. In 2010, Duan and Yeh [Jump and volatility risk premiums implied by VIX. J Econ Dyn Control. 2010;34(11):2232–2244] used a frequentist approach with maximum likelihood estimation that included the VIX, the Chicago Board Options Exchange (CBOE) Volatility Index, which carries real time S&P 500 option price information in addition to the S&P 500 data. In this paper, we combine the previous Bayesian approach in Cape et al. [Estimating Heston's and Bates' models parameters using Markov Chain Monte Carlo simulation. J Stat Comput Simul. 2015;85(11):2295–2314] with the inclusion of VIX by Duan and Yeh. We provide the derivations of the conditional posterior distributions of the SVJD model's parameters and a guide to a Markov Chain Monte Carlo (MCMC) algorithm used to estimate the parameters. We apply the algorithm to the historical S&P 500 and VIX data and provide the subsequent parameter estimates. We offer these tools for others performing similar research with our R code available upon request.

Well-Conditioned Linear Minimum Mean Square Error Estimation

Linear minimum mean square error (LMMSE) estimation is often ill-conditioned, suggesting that unconstrained minimization of the mean square error is an inadequate approach to filter design. To address this, we first develop a unifying framework for studying constrained LMMSE estimation problems. Using this framework, we explore an important structural property of constrained LMMSE filters involving a certain prefilter. Optimality is invariant under invertible linear transformations of the prefilter. This parameterizes all optimal filters by equivalence classes of prefilters. We then clarify that merely constraining the rank of the filter does not suitably address the problem of ill-conditioning. Instead, we adopt a constraint that explicitly requires solutions to be well-conditioned in a certain specific sense. We introduce two well-conditioned filters and show that they converge to the unconstrained LMMSE filter as their truncation-power loss goes to zero, at the same rate as the low-rank Wiener filter. We also show extensions to the case of weighted trace and determinant of the error covariance as objective functions. Finally, our quantitative results with historical VIX data demonstrate that our two well-conditioned filters have stable performance while the standard LMMSE filter deteriorates with increasing condition number.

Fractional Barndorff-Nielsen and Shephard model: applications in variance and volatility swaps, and hedging

In this paper, we introduce and analyze the fractional Barndorff-Nielsen and Shephard (BN-S) stochastic volatility model. The proposed model is based upon two desirable properties of the long-term variance process suggested by the empirical data: long-term memory and jumps. The proposed model incorporates the long-term memory and positive autocorrelation properties of fractional Brownian motion with $H>1/2$, and the jump properties of the BN-S model. We find arbitrage-free prices for variance and volatility swaps for this new model. Because fractional Brownian motion is still a Gaussian process, we derive some new expressions for the distributions of integrals of continuous Gaussian processes as we work towards an analytic expression for the prices of these swaps. The model is analyzed in connection to the quadratic hedging problem and some related analytical results are developed. The amount of derivatives required to minimize a quadratic hedging error is obtained. Finally, we provide some numerical analysis based on the VIX data. Numerical results show the efficiency of the proposed model compared to the Heston model and the classical BN-S model.

Optimal investment strategy in the family of 4/2 stochastic volatility models

This paper derives optimal investment strategies for the 4/2 stochastic volatility model proposed in [Grasselli, M., The 4/2 stochastic volatility model: a unified approach for the Heston and the 3/2 model. Math. Finance, 2017, 27(4), 1013–1034] and the embedded 3/2 model [Heston, S.L., A simple new formula for options with stochastic volatility. 1997]. We maximize the expected utility of terminal wealth for a constant relative risk aversion (CRRA) investor, solving the corresponding Hamilton–Jacobi–Bellman (HJB) equations in closed form for both complete and incomplete markets. Conditions for the verification theorems are provided. Interestingly, the optimal investment strategy displays a very intuitive dependence on current volatility levels, a trend which has not been previously reported in the literature of stochastic volatility models. A full empirical analysis comparing four popular embedded models—i.e. the Merton (geometric Brownian motion), Heston (1/2), 3/2 and 4/2 models—is conducted using S&P 500 and VIX data. We find that the 1/2 model carries the larger weight in explaining the 4/2 behaviour, and optimal investments in the 1/2 and 4/2 models are similar, while investments in the 3/2 model are the most conservative in high-variance settings (20% of Merton's solution). Despite the similarities between the 1/2 and 4/2 models, wealth-equivalent losses due to deviations from the 4/2 model are largest for the1/2 and GBM models (40% over 10 years). Meanwhile, the wealth losses due to market incompleteness are harsher for the 1/2 model than for the 4/2 and 3/2 models (60% versus 40% and 30% respectively), highlighting the benefits of choosing the 4/2 or the 3/2 over the 1/2 model.

Stable Count Distribution for the Volatility Indices and Space-Time Generalized Stable Characteristic Function

This paper collects statistical and mathematical properties of the stable count distribution, with dispersed treatises on economic regime related to volatility. It is hypothesized that the stable count distribution is the marginal distribution of the volatility indices, such as CBOE VIX and VXTYN. Mathematically, its PDF is a special case of the Wright function. Analytical solutions for its moments and moment-generating function are carried out in terms of the Fox-Wright function. We show that it can be the kernel of a product distribution in fractional dynamics and quantitative finance. In the second half of the paper, the stable distribution family is generalized. A novel mathematical approach is taken by defining a space-time generalized stable characteristic function via the Mittag-Leffler function. The fractional PDF it produces is shown to be consistent with the fractional calculus of Riesz-Feller space derivative and Caputo-Mainardi time derivative. Small deviations in both time-fractional and space-fractional scenarios are studied to explain the ultra-high volatility tail observed in the VIX data during bear markets. The solutions match VIX data up to the last one percentile in the right tail, which corresponds to the largest market crashes in financial history.

Nature of VIX Jumps on Market Timing of Hedge Funds

The study indicates that Brownian motion, finite and infinite activity jumps are present in the ultra-high frequency VIX data. The total quadratic variation can be split into a continuous component of 29% and a jump component of 71%. Jump activities on ultra-high frequency VIX data are found informative in ex-ante identifying subgroups of hedge funds that deliver significant outperformance. In the months that follow large jumps, strategies exposing to long volatility and extreme risk tend to deliver positive performance in extreme market environments. In the months that follow small jumps, possibly as a result of trading illiquidity, most fund strategies exhibit losses in the jolting market environments. In the months that follow Brownian motion, strategies exposing to short volatility tend to deliver best performance. Hedge funds therefore deliver out-of-sample performance respective of types of jump activities on ultra-high frequency VIX.

STATISTICS OF VIX FUTURES AND APPLICATIONS TO TRADING VOLATILITY EXCHANGE-TRADED PRODUCTS

We study the dynamics of VIX futures and ETNs/ETFs. We find that contrary to classical commodities, VIX and VIX futures exhibit large volatility and skewness, consistent with the absence of cash-and-carry arbitrage. The constant-maturity futures (CMF) term-structure can be modeled as a stationary stochastic process in which the most likely state is contango with VIX [Formula: see text] and a long-term futures price [Formula: see text]. We analyze the behavior of ETFs and ETNs based on constant-maturity rolling futures strategies, such as VXX, XIV and VXZ, assuming stationarity and through a multi-factor model calibrated to historical data. We find that buy-and-hold strategies consisting of shorting ETNs that roll long futures, or buying ETNs that roll short futures, will produce theoretically-sure profits if it is assumed that CMFs are stationary and ergodic. To quantify further, we estimate a 2-factor lognormal model with mean-reverting factors to VIX and CMF historical data from 2011 to 2016. The results confirm the profitability of buy-and-hold strategies, but also indicate that the latter have modest Sharpe ratios, of the order of [Formula: see text] or less, and high variability over 1-year horizon simulations. This is due to the surges in VIX and CMF backwardations which are observed sporadically in the volatility futures market.

Reasonable evaluation of VIX options for the Taiwan stock index

This study aims to reasonably evaluate VXO (volatility index option) prices on the Taiwan stock index using four popular pricing models, namely, Black-Scholes (BS), square root (SQR), log-normal Ornstein-Uhlenbeck (LOU), and GARCH models. Since VXOs have not been traded yet in Taiwan options market, we directly employ the physical probability measure to evaluate the VXOs. This study examined the Taiwan and CBOE VIX data from December 2006 to January 2018 and found that the estimated option prices generally appear in the descending order as GARCH > BS > SQR > LOU. That is, when VIX has heteroscedasticity, the BS model could undervalue the VXOs. Additionally, when VIX has no mean-reverting property, the SQR and LOU models could seriously undervalue the VXOs, especially for the long-term maturity options. While existing studies have not sufficiently evaluated the Taiwan VXOs, the results of this study can provide future reference for investors and securities companies when evaluating the Taiwan VXOs.

Variance swaps valuation under non-affine GARCH models and their diffusion limits

In this article, we investigate the pricing and convergence of general non-affine non-Gaussian GARCH-based discretely sampled variance swaps. Explicit solutions for fair strike prices under two different sampling schemes are derived using the extended Girsanov principle as the pricing kernel candidate. Following standard assumptions on time-varying GARCH parameters, we show that these quantities converge respectively to fair strikes of discretely and continuously sampled variance swaps that are constructed based on the weak diffusion limit of the underlying GARCH model. An empirical study which relies on a joint estimation using both historical returns and VIX data indicates that an asymmetric heavier tailed distribution is more appropriate for modelling the GARCH innovations. Finally, we provide several numerical exercises to support our theoretical convergence results in which we further investigate the effect of the quadratic variation approximation for the realized variance, as well as the impact of discrete versus continuous-time modelling of asset returns.

Statistics of VIX Futures and Applications to Trading Volatility Exchange-Traded Products

We study the dynamics of VIX futures and ETNs/ETFs. We find that contrary to classical commodities, VIX and VIX futures exhibit large volatility and skewness, consistent with the absence of cash-and-carry arbitrage. The constant-maturity futures (CMF) term-structure can be modeled as a stationary stochastic process in which the most likely state is a contango with VIX ≈ 12% and a long-term futures price V∞ ≈ 20%. We analyze the behavior of ETFs and ETNs based on constant-maturity rolling futures strategies, such as VXX, XIV and VXZ, assuming stationarity and through a multi-factor model calibrated to historical data. We find that buy-and-hold strategies consisting of shorting ETNs that roll long futures, or buying ETNs that roll short futures, will produce theoretically-sure profits if it is assumed that CMFs are stationary and ergodic (see Proposition 3.1). To quantify further, we estimate a 2-factor lognormal model with mean-reverting factors to VIX and CMF historical data from 2011 to 2016. The results confirm the profitability of buy-and-hold strategies, but also indicate that the latter have modest Sharpe ratios, of the order of SR = 0.5 or less, and high variability over 1-year horizon simulations. This is due to the surges in VIX and CMF backwardations which are experienced sporadically, but also inevitably, in the volatility futures market.

A scaled version of the double-mean-reverting model for VIX derivatives

As the Heston model is not consistent with VIX data in real market well enough, alternative stochastic volatility models including the double-mean-reverting model of Gatheral (in: Bachelier Congress, 2008) have been developed to overcome its limitation. The double-mean-reverting model is a three factor model successfully reflecting the empirical dynamics of the variance but there is no closed form solution for VIX derivatives and SPX options and thus calibration using conventional techniques may be slow. In this paper, we propose a fast mean-reverting version of the double-mean-reverting model. We obtain a closed form approximation for VIX derivatives and show how it is effective by comparing it with the Heston model and the double-mean-reverting model.

Heston stochastic vol-of-vol model for joint calibration of VIX and S&P 500 options

A parsimonious generalization of the Heston model is proposed where the volatility-of-volatility is assumed to be stochastic. We follow the perturbation technique of Fouque et al [Multiscale Stochastic Volatility for Equity, Interest Rate, and Credit Derivatives, 2011, Cambridge University Press] to derive a first-order approximation of the price of options on a stock and its volatility index. This approximation is given by Heston’s quasi-closed formula and some of its Greeks. It can be efficiently calculated since it requires to compute only Fourier integrals and the solution of simple ODE systems. We exemplify the calibration of the model with S&P 500 and VIX data.

Statistics of VIX futures and their applications to trading volatility exchange-traded products

We study the dynamics of VIX futures and ETNs/ETFs. We find that contrary to classical commodities, VIX and VIX futures exhibit large volatility and skewness, consistent with the absence of cash-and-carry arbitrage. The constant-maturity futures (CMF) term-structure can be modeled as a stationary stochastic process in which the most likely state is a contango with VIX ≈ 12% and a long-term futures price V∞ ≈ 20%. We analyze the behavior of ETFs and ETNs based on constant-maturity rolling futures strategies, such as VXX, XIV and VXZ, assuming stationarity and through a multi-factor model calibrated to historical data. We find that buy-and-hold strategies consisting of shorting ETNs that roll long futures, or buying ETNs that roll short futures, will produce theoretically-sure profits if it is assumed that CMFs are stationary and ergodic (see Proposition 3.1). To quantify further, we estimate a 2-factor lognormal model with mean-reverting factors to VIX and CMF historical data from 2011 to 2016. The results confirm the profitability of buy-and-hold strategies, but also indicate that the latter have modest Sharpe ratios, of the order of SR = 0.5 or less, and high variability over 1-year horizon simulations. This is due to the surges in VIX and CMF backwardations which are experienced sporadically, but also inevitably, in the volatility futures market.

Oil and energy sector stock markets: An analysis of implied volatility indexes

The objective of our study is to assess the linkage between global oil and the US energy sector stock markets using their implied volatility indexes available from the Chicago Board of Options Exchange (CBOE). Our empirical analysis also includes the US VIX data in order to control for the effect of global equity market uncertainty. To investigate whether cointegration exists amongst the volatility series used, we consider applying the ARDL bound tests. The findings reveal that there exists a long-run relationship between oil and stock market implied volatility indexes. Besides, employing the Toda–Yamamoto version of the Granger causality test indicates short-run “lead-lag” associations between the implied volatilities of international oil and the US energy sector stock markets. The results carry important implications for investors and policymakers.

A Unified Valuation Framework for Variance Swaps under Non-Affine Stochastic Volatility Models

In this article, we investigate the pricing and convergence of general non-affine non-Gaussian GARCH-based variance swap prices. Explicit solutions for fair strike prices under two different sampling schemes are derived using the extended Girsanov principle as our pricing kernel candidate. Following standard assumptions on the time-varying GARCH parameters, we show that these quantities converge to discretely and continuously sampled variance swaps constructed based on the weak diffusion limit of the underlying GARCH model. An empirical study which relies on a joint estimation using both historical returns and VIX data indicates that an asymmetric heavier-tailed distribution is more appropriate for modelling the GARCH innovations. Finally, we provide several numerical exercises to support our theoretical convergence results in which we investigate the effect of the quadratic variation approximation for the realized variance, as well as the impact of discrete versus continuous-time modelling of asset returns.

Pure jump models for pricing and hedging VIX derivatives

Recent non-parametric statistical analysis of high-frequency VIX data (Todorov and Tauchen, 2011) reveals that VIX dynamics is a pure jump semimartingale with infinite jump activity and infinite variation. To our best knowledge, existing models in the literature for pricing and hedging VIX derivatives do not have these features. This paper fills this gap by developing a novel class of parsimonious pure jump models with such features for VIX based on the additive time change technique proposed in Li et al. (2016a, 2016b). We time change the 3/2 diffusion by a class of additive subordinators with infinite activity, yielding pure jump Markov semimartingales with infinite activity and infinite variation. These processes have time and state dependent jumps that are mean reverting and are able to capture stylized features of VIX. Our models take the initial term structure of VIX futures as input and are analytically tractable for pricing VIX futures and European options via eigenfunction expansions. Through calibration exercises, we show that our model is able to achieve excellent fit for the VIX implied volatility surface which typically exhibits very steep skews. Comparison to two other models in terms of calibration reveals that our model performs better both in-sample and out-of-sample. We explain the ability of our model to fit the volatility surface by evaluating the matching of moments implied from market VIX option prices. To hedge VIX options, we develop a dynamic strategy which minimizes instantaneous jump risk at each rebalancing time while controlling transaction cost. Its effectiveness is demonstrated through a simulation study on hedging Bermudan style VIX options.

Fourier Analysis of India's Implied Volatility Index

The objective of this study was to analyze and model periodic behavior observed in India’s Nifty VIX Index and to seek the origins of these previously unreported calendar variations. Implied volatility and its variations are important to understand as the pricing of many financial assets and derivatives depend upon this variable. A novel modeling approach to this task uses Fourier analysis for the first time in the literature of implied volatility and calendar effect modeling. For this purpose daily closing levels for the Nifty VIX Index were gathered covering trading days from 2010 through January 2014. Time series of VIX levels after removing a trendline were tested for normality, stationarity and autocorrelation. Nifty index and Nifty VIX data were also tested for two-way Granger Causality. Detrended Nifty VIX levels were Fourier analyzed to determine primary Fourier frequencies contributing to VIX periodic behavior. A Fourier model of VIX movements was constructed using four primary frequencies from the Fourier power spectrum. This model shows surprising accuracy in identifying the temporal location of VIX peaks and troughs. The probable origin of one recurring VIX calendar frequency is traced to India’s earnings release cycle.

Fourier Analysis of India's Implied Volatility Index

The objective of this study was to analyze and model periodic behavior observed in India's Nifty Implied Volatility (VIX) Index and to seek the origins of these previously unreported calendar variations. Implied volatility and its variations are important to understand as the pricing of many financial assets and derivatives depend upon this variable. A novel modeling approach to this task uses Fourier analysis for the first time in the literature of implied volatility and calendar effect modeling. For this purpose, daily closing levels for the Nifty VIX Index were gathered covering trading days from 2010 through January 2014. Time series of VIX levels after removing a trend line were tested for normality, stationarity, and autocorrelation. Nifty index and Nifty VIX data were also tested for two-way Granger causality. Detrended Nifty VIX levels were Fourier analyzed to determine primary Fourier frequencies contributing to VIX periodic behavior. A Fourier model of VIX movements was constructed using four primary frequencies from the Fourier power spectrum. This model showed surprising accuracy in identifying the temporal location of VIX peaks and troughs. The probable origin of one recurring VIX calendar frequency was traced to India's earnings release cycle.

Estimating and using GARCH models with VIX data for option valuation

This paper uses information on VIX to improve the empirical performance of GARCH models for pricing options on the S&P 500. In pricing multiple cross-sections of options, the models’ performance can clearly be improved by extracting daily spot volatilities from the series of VIX rather than by linking spot volatility with different dates by using the series of the underlying’s returns. Moreover, in contrast to traditional returns-based Maximum Likelihood Estimation (MLE), a joint MLE with returns and VIX improves option pricing performance, and for NGARCH, joint MLE can yield empirically almost the same out-of-sample option pricing performance as direct calibration does to in-sample options, but without costly computations. Finally, consistently with the existing research, this paper finds that non-affine models clearly outperform affine models.

A model for stock returns and volatility

We prove that Student’s t-distribution provides one of the better fits to returns of S&P component stocks and the generalized inverse gamma distribution best fits VIX and VXO volatility data. We further prove that stock returns are best fit by the product distribution of the generalized inverse gamma and normal distributions. We find Brown noise in VIX and VXO time series and explain the mean and the variance of the relaxation times on approach to the steady-state distribution.