Volatility Model Specification: Evidence from the Pricing of VIX Derivatives (original) (raw)
Related papers
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.
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.
VIX derivatives: Valuation models and empirical evidence
Pacific-Basin Finance Journal, 2019
This study proposes an efficient approach for the pricing of VIX derivatives under the affine framework and investigates the respective value of two variance components and variance jumps in the pricing of VIX derivatives. Our numerical results show that our approach significantly reduce the computational burden. Our empirical findings provide support for the use of two-variance component models as the means of capturing the fickle term structure of VIX derivatives, and the use of variance jumps is vital when included in the long-run variance component.
We propose a class of discrete-time stochastic volatility models that, in a parsimonious way, captures the time-varying higher moments observed in financial series. We build this class of models in order to reach two desirable results. Firstly, we have a recursive procedure for the characteristic function of the log price at maturity that allows a semianalytical formula for option prices as in Heston and Nandi [2000]. Secondly, we try to reproduce some features of the Vix Index. We derive a simple formula for the Vix index and use it for option pricing purposes.
A simplified pricing model for volatility futures
Journal of Futures Markets, 2011
We develop a general model to price VIX futures contracts. The model is adapted to test both the constant elasticity of variance (CEV) and the Cox-Ingersoll-Ross formulations, with and without jumps. Empirical tests on VIX futures prices provide out-of-sample estimates within 2% of the actual futures price for almost all futures maturities. We show that although jumps are present in the data, the models with jumps do not typically outperform the others; in particular, we demonstrate the important benefits of the CEV feature in pricing futures contracts. We conclude by examining errors in the model relative to the VIX characteristics. 1 When the S&P 500 options approach expiration then the nearby maturity is dropped from the calculation when five days are left until expiration; at that time and the first and second deferred maturities are employed in the calculation.
Constructing a Class of Stochastic Volatility Models: Empirical Investigation with Vix Data
SSRN Electronic Journal, 2000
We propose a class of discrete-time stochastic volatility models that, in a parsimonious way, captures the time-varying higher moments observed in financial series. We build this class of models in order to reach two desirable results. Firstly, we have a recursive procedure for the characteristic function of the log price at maturity that allows a semianalytical formula for option prices as in Heston and Nandi [2000]. Secondly, we try to reproduce some features of the Vix Index. We derive a simple formula for the Vix index and use it for option pricing purposes.
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.
THE NELSON–SIEGEL MODEL OF THE TERM STRUCTURE OF OPTION IMPLIED VOLATILITY AND VOLATILITY COMPONENTS
We develop the Nelson-Siegel model in the context of option-implied volatility term structure and study the time series of volatility components. Three components, corresponding to the level, slope, and curvature of the volatility term structure, can be interpreted as the long-, medium-, and short-term volatilities. The long-term component is persistent and driven by macroeconomic variables, the medium-term by market default risk, and the short-term by financial market conditions. The three-factor Nelson-Siegel model has superior performance in forecasting the volatility term structure, with better out-of-sample forecasts than the popular deterministic implied volatility function and a restricted two-factor model, providing support to the literature of component volatility models. has reached a consensus that long-term volatility reacts differently to volatility shocks than short-term volatility. Therefore, a natural way to extend the one-factor volatility models is to decompose the volatility into long-term and short-term components. Recently, these component volatility models (CVM) (see have been shown to perform better than one-factor volatility models in modeling the implied volatility term structure.
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.