VIX derivatives: Valuation models and empirical evidence (original) (raw)
Related papers
2010
Abstract: We conduct an extensive empirical analysis of VIX derivative valuation models before, during and after the recent financial crisis. Since the restrictive assumptions about mean reversion and heteroskedasticity of existing models yield large distortions during the ...
Volatility Model Specification: Evidence from the Pricing of VIX Derivatives
2013
This study examines whether a jump component or an additional factor better supports volatility modeling by investigating the pricing of VIX derivatives. To reduce the computational burdens for the empirical estimation significantly, we propose an efficient and easily implemented numerical approximation for the pricing of VIX derivatives. In terms of the term structure of VIX futures, we show that the additional volatility factor can replicate the common empirical patterns and explain the changes in the term structure, but the jump component cannot. In terms of the pricing of VIX options, we find that the two-factor volatility models significantly outperform the jump volatility models and that adding jumps in volatility only provides a minor improvement. Therefore, our general findings support the merit of the two-factor volatility specification.
Modeling VIX Futures and Pricing VIX Options in the Jump Di usion Modeling
2014
In this thesis, a closed-form solution for the price of options on VIX futures is derived by developing a term-structure model for VIX futures. We analyze the VIX futures by the Merton Jump Diffusion model and allow for stochastic interest rates in the model. The performance of the model is investigated based on the daily VIX futures prices from the Chicago Board Option Exchange (CBOE) data. Also, the model parameters are estimated and option prices are calculated based on the estimated values. The results imply that this model is appropriate for the analysis of VIX futures and is able to capture the empirical features of the VIX futures returns such as positive skewness, excess kurtosis and decreasing volatility for long-term expiration. ∗Postal address: Mathematical Statistics, Stockholm University, SE-106 91, Sweden. E-mail: fatemeharamian@yahoo.com. Supervisor: Mia Hinnerich.
An analytical formula for VIX futures and its applications
Journal of Futures Markets, 2011
In this paper 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.
SSRN Electronic Journal, 2017
We consider a modeling setup where the VIX index 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 obtained expansions 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 been previously 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.
Simple Heuristics for Pricing VIX Options
SSRN Electronic Journal, 2000
The article presents a simple parameterization of the volatility surface for options on the S&P 500 volatility index, VIX. Specifically, we document the following features of VIX implied volatility: (i) VIX at-the-money (ATM) implied volatility correlates strongly with the volatility skew in S&P 500 options; (ii) VIX ATM implied volatility declines exponentially with options' time to expiry; (iii) a SABR-type model can be used to model the smile observed in VIX options. These observations lead to simple heuristics for quoting prices (in terms of implied volatility) of VIX options with almost arbitrary strike and expiry, obtaining values that are reasonably close to market levels.
The performance of VIX option pricing models: Empirical evidence beyond simulation
Journal of Futures Markets, 2011
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 contention that "simpler is better" is supported by the empirical evidence using actual VIX option market data. Our tests employ three representative models for VIX options: Whaley (1993), , and Carr and Lee . We also compare our results to Lin and Chang , 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. In general, Whaley's Black-like option model produces the best overall results, supporting the "simpler is better" contention. 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.
SSRN Electronic Journal, 2013
A double gamma model is proposed for the VIX. The VIX is modeled as gamma distributed with a mean and variance that respond to a gamma-distributed realized variance over the preceeding month. Conditional on VIX and the realized variance, the logarithm of the stock is variance gamma distributed with affine conditional drift and quadratic variation. The joint density for the triple realized variance, VIX, and the SPX is in closed form. Maximum likelihood estimation on time series data addresses model adequacy. A joint calibration of the model to SPX and VIX options is employed to illustrate a risk management application hedging realized volatility options.
A remark on Lin and Chang's paper ‘Consistent modeling of S&P 500 and VIX derivatives’
Journal of Economic Dynamics and Control, 2012
Lin and Chang (2009, 2010) establish a VIX futures and option pricing theory when modeling S&P 500 index by using a stochastic volatility process with asset return and volatility jumps. In this note, we prove that Lin and Chang's formula is not an exact solution of their pricing equation. More generally, we show that the characteristic function of their pricing equation cannot be exponentially affine, as proposed by them. Furthermore, their formula cannot serve as a reasonable approximation. Using the (Heston, 1993) model as a special case, we demonstrate that Lin and Chang formula misprices VIX futures and options in general and the error can become substantially large.
Willow Tree Algorithms for Pricing VIX Derivatives Under Stochastic Volatility Models
SSRN Electronic Journal, 2019
VIX futures and options are the most popular contracts traded in the Chicago Board Options Exchange. The bid-ask spreads of traded VIX derivatives remain to be wide, possibly due to lack of reliable pricing models. In this paper, we consider pricing VIX derivatives under the consistent model approach, which considers joint modeling of the dynamics of the S&P index and its instantaneous variance. Under the affine jump-diffusion formulation with stochastic volatility, analytic integral formulas can be derived to price VIX futures and options. However, these integral formulas invariably involve Fourier inversion integrals with cumbersome hyper-geometric functions, thus posing various challenges in numerical evaluation. We propose a unified numerical approach based on the willow tree algorithms to price VIX derivatives under various common types of joint process of the S&P index and its instantaneous variance. Given the analytic form of the characteristic function of the instantaneous variance of the S&P index process in the Fourier domain, we apply the fast Fourier transform algorithm to obtain the transition density function numerically in the real domain. We then construct the willow tree that approximates the dynamics of the instantaneous variance process up to the fourth order moment. Our comprehensive numerical tests performed on the willow tree algorithms demonstrate high level of numerical accuracy, runtime efficiency and reliability for pricing VIX futures and both European and American options under the affine model and 3/2-model. We also examine the implied volatility smirks and the term structures of the implied skewness of VIX options.