An approximation formula for normal implied volatility under general local stochastic volatility models

We approximate normal implied volatilities by means of an asymptotic expansion method. The contribution of this paper is twofold: to our knowledge, this paper is the first to provide a unified approximation method for the normal implied volatility under general local stochastic volatility models. Second, we applied our framework to polynomial local stochastic volatility models with various degrees and could replicate the swaptions market data accurately. In addition we examined the accuracy of the results by comparison with the Monte‐Carlo simulations.     

The implied volatility smirk of commodity options

This paper studies the implied volatility (IV) smirks in four commodity markets by adopting Zhang and Xiang's methodology. First, we document the term structure and dynamics of IV smirks. Overall, the commodity IV curves are negatively skewed with a positive curvature. Then we analyze the commodity and S&P 500 returns' predictability based on in‐sample and out‐of‐sample tests and find that the information embedded in IV smirks can significantly predict monthly commodity and S&P 500 returns. For example, the risk‐neutral fourth cumulant (FC) from the crude oil market outperforms all of the standard predictors in predicting the S&P 500 returns.     

Properties and the predictive power of implied volatility in the New Zealand dairy market

This study develops a dairy implied volatility index (DVIX), derived from New Zealand Exchange traded options on whole milk powder (WMP) futures. We document an inverse return–volatility relation which is asymmetric, where increases in WMP futures prices are associated with larger absolute changes in the DVIX than decreases. In sample, the results strongly suggest that the DVIX has a high information content regarding conditional variance and that the inclusion of historical information further improves the predictive power. Out of sample, we find that the DVIX provides substantial information about future realized volatility. We also document that a combination of historical volatility and the DVIX provides the best out-of-sample forecasts.     

The impacts of public news announcements on intraday implied volatility dynamics

We examine the responses of intraday option-implied volatilities to scheduled announcements of macroeconomic indicators. The increase in implied volatility around macroeconomic news announcements is more pronounced for puts than for calls and is stronger for announcements made during trading hours than for those made during nontrading hours. These effects are also more pronounced in the crisis and postcrisis periods than in the precrisis period. Monetary policy announcements have a more substantial impact on volatility than other announcements have, even after controlling for news surprise components. The impact appears to be greater for policy rate hikes than for policy rate cuts.     

Informed options trading on the implied volatility surface: A cross-sectional approach

This study investigates the cross-sectional implication of informed options trading across different strikes and maturities. We explore the term structure perspective of the one-way information transmission from options markets to stock markets by adopting well-known option-implied volatility measures to examine stock return predictability. Using equity options data for U.S. listed stocks spanning 2000–2013, we find that the shape of the long-term implied volatility curve exhibits extra predictive power for stock returns of subsequent months even after orthogonalizing the short-term components. Our findings indicate that the inter-market information asymmetry rapidly disappears before the expiration of long-term option contracts.     

Uncertainty and the volatility forecasting power of option-implied volatility

This study investigates the impact of uncertainty on the volatility forecasting power of option-implied volatility. Option-implied volatility is a powerful predictor of future volatility, particularly during periods of high uncertainty. This is consistent with option-implied volatility being largely determined by volatility-informed traders (rather than directional traders) when uncertainty is high. New volatility forecasting models that incorporate such interaction outperform benchmark models, both in- and out-of-sample. The new models also better predict future volatility during the 2008 global financial crisis, for which benchmark models perform poorly. The results are robust to alternative choices of benchmark models, loss functions, and estimation windows.     

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.     

THE MOMENT FORMULA FOR IMPLIED VOLATILITY AT EXTREME STRIKES

Consider options on a nonnegative underlying random variable with arbitrary distribution. In the absence of arbitrage, we show that at any maturity T , the large-strike tail of the Black-Scholes implied volatility skew is bounded by the square root of  2| x |/ T   , where x is log-moneyness. The smallest coefficient that can replace the 2 depends only on the number of finite moments in the underlying distribution. We prove the moment formula , which expresses explicitly this model-independent relationship. We prove also the reciprocal moment formula for the small-strike tail, and we exhibit the symmetry between the formulas. The moment formula, which evaluates readily in many cases of practical interest, has applications to skew extrapolation and model calibration.     

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.     

Forecasting bitcoin volatility: Evidence from the options market

This paper studies a large number of bitcoin (BTC) options traded on the options exchange Deribit. We use the trades to calculate implied volatility (IV) and analyze if volatility forecasts can be improved using such information. IV is less accurate than AutoRegressive–Moving-Average or Heterogeneous Auto-Regressive model forecasts in predicting short-term BTC volatility (1 day ahead), but superior in predicting long-term volatility (7, 10, 15 days ahead). Furthermore, a combination of IV and model-based forecasts provides the highest accuracy for all forecasting horizons revealing that the BTC options market contains unique information.     

A CLOSED‐FORM EXACT SOLUTION FOR PRICING VARIANCE SWAPS WITH STOCHASTIC VOLATILITY

In this paper, we present a highly efficient approach to price variance swaps with discrete sampling times. We have found a closed‐form exact solution for the partial differential equation (PDE) system based on the Heston's two‐factor stochastic volatility model embedded in the framework proposed by Little and Pant. In comparison with the previous approximation models based on the assumption of continuous sampling time, the current research of working out a closed‐form exact solution for variance swaps with discrete sampling times at least serves for two major purposes: (i) to verify the degree of validity of using a continuous‐sampling‐time approximation for variance swaps of relatively short sampling period; (ii) to demonstrate that significant errors can result from still adopting such an assumption for a variance swap with small sampling frequencies or long tenor. Other key features of our new solution approach include the following: (1) with the newly found analytic solution, all the hedging ratios of a variance swap can also be analytically derived; (2) numerical values can be very efficiently computed from the newly found analytic formula.     

MOMENT EXPLOSIONS AND LONG‐TERM BEHAVIOR OF AFFINE STOCHASTIC VOLATILITY MODELS

We consider a class of asset pricing models, where the risk‐neutral joint process of log‐price and its stochastic variance is an affine process in the sense of Duffie, Filipovic, and Schachermayer. First we obtain conditions for the price process to be conservative and a martingale. Then we present some results on the long‐term behavior of the model, including an expression for the invariant distribution of the stochastic variance process. We study moment explosions of the price process, and provide explicit expressions for the time at which a moment of given order becomes infinite. We discuss applications of these results, in particular to the asymptotics of the implied volatility smile, and conclude with some calculations for the Heston model, a model of Bates and the Barndorff‐Nielsen–Shephard model.     

The Fed's policy decisions and implied volatility

This study examines how the Fed's monetary policy decisions affect the implied volatility of the S&P 500 index. The results show that stock market uncertainty is significantly affected by the Fed's policy decisions. In particular, we find that implied volatility generally decreases after FOMC meetings, while the relationship between target rate surprises and market uncertainty appears positive. However, our results also suggest that the apparent positive relationship between policy surprises and implied volatility is mostly driven by the volatility‐reducing effects of negative surprises. We further document that implied volatility is affected by both scheduled and unscheduled policy actions, with the scheduled path surprises having the strongest impact on volatility. Finally, our findings indicate that the impact of monetary policy decisions on implied volatility is more pronounced during periods of expansive policy. © 2010 Wiley Periodicals, Inc. Jrl Fut Mark     

CLOSED FORM PRICING FORMULAS FOR DISCRETELY SAMPLED GENERALIZED VARIANCE SWAPS

Most of the existing pricing models of variance derivative products assume continuous sampling of the realized variance processes, though actual contractual specifications compute the realized variance based on sampling at discrete times. We present a general analytic approach for pricing discretely sampled generalized variance swaps under the stochastic volatility models with simultaneous jumps in the asset price and variance processes. The resulting pricing formula of the gamma swap is in closed form while those of the corridor variance swaps and conditional variance swaps take the form of one‐dimensional Fourier integrals. We also verify through analytic calculations the convergence of the asymptotic limit of the pricing formulas of the discretely sampled generalized variance swaps under vanishing sampling interval to the analytic pricing formulas of the continuously sampled counterparts. The proposed methodology can be applied to any affine model and other higher moments swaps as well. We examine the exposure to convexity (volatility of variance) and skew (correlation between the equity returns and variance process) of these discretely sampled generalized variance swaps. We explore the impact on the fair strike prices of these exotic variance swaps with respect to different sets of parameter values, like varying sampling frequencies, jump intensity, and width of the monitoring corridor.     

IMPLIED VOLATILITY IN THE HULL–WHITE MODEL

We study the implied volatility K ↦ I ( K ) in the Hull–White model of option pricing, and obtain asymptotic formulas for this function as the strike price K tends to infinity or zero. We also prove that the function I is convex near zero and concave near infinity, and characterize the behavior of the first two derivatives of this function.     

Multifactor implied volatility functions for HJM models

This study evaluates two one‐factor, two two‐factor, and two three‐factor implied volatility functions in the HJM class, with the use of eurodollar futures options across both strike prices and maturities. The primary contributions of this article are (a) to propose and test three implied volatility multifactor functions not considered by K. I. Amin and A. J. Morton (1994), (b) to evaluate models using the AIC criteria as well as other standard criteria neglected by S. Y. M. Zeto (2002), and (c) to .nd that multifactor models incorporating the exponential decaying implied volatility functions generally outperform other models in .tting and prediction, in sharp contrast to K. I. Amin and A. J. Morton, who find the constantvolatility model superior. Correctly specified and calibrated simple constant and square‐root factor models may be superior to inappropriate multifactor models in option trading and hedging strategies. © 2006 Wiley Periodicals, Inc. Jrl Fut Mark 26:809–833, 2006     

THE NORMALIZING TRANSFORMATION OF THE IMPLIED VOLATILITY SMILE

We study specific nonlinear transformations of the Black–Scholes implied volatility to show remarkable properties of the volatility surface. No arbitrage bounds on the implied volatility skew are given. Pricing formulas for European payoffs are given in terms of the implied volatility smile.     

The forecast quality of CBOE implied volatility indexes

We examine the forecast quality of Chicago Board Options Exchange (CBOE) implied volatility indexes based on the Nasdaq 100 and Standard and Poor's 100 and 500 stock indexes. We find that the forecast quality of CBOE implied volatilities for the S&P 100 (VXO) and S&P 500 (VIX) has improved since 1995. Implied volatilities for the Nasdaq 100 (VXN) appear to provide even higher quality forecasts of future volatility. We further find that attenuation biases induced by the econometric problem of errors in variables appear to have largely disappeared from CBOE volatility index data since 1995. © 2005 Wiley Periodicals, Inc. Jrl Fut Mark 25:339–373, 2005     

LEVERAGED ETF IMPLIED VOLATILITIES FROM ETF DYNAMICS

The growth of the exchange‐traded fund (ETF) industry has given rise to the trading of options written on ETFs and their leveraged counterparts (LETFs). We study the relationship between the ETF and LETF implied volatility surfaces when the underlying ETF is modeled by a general class of local‐stochastic volatility models. A closed‐form approximation for prices is derived for European‐style options whose payoffs depend on the terminal value of the ETF and/or LETF. Rigorous error bounds for this pricing approximation are established. A closed‐form approximation for implied volatilities is also derived. We also discuss a scaling procedure for comparing implied volatilities across leverage ratios. The implied volatility expansions and scalings are tested in three settings: Heston, limited constant elasticity of variance (CEV), and limited SABR; the last two are regularized versions of the well‐known CEV and SABR models.     

Panel quantile regressions for estimating and predicting the value-at-risk of commodities

Using a flexible panel quantile regression framework, we show how the future conditional quantiles of commodities returns depend on both ex post and ex ante uncertainty. Empirical analysis of the most liquid commodities covering main sectors, including energy, food, agriculture, and precious and industrial metals, reveal several important stylized facts. We document common patterns of the dependence between future quantile returns and ex post as well as ex ante volatilities. We further show that the conditional returns distribution is platykurtic. The approach can serve as a useful risk management tool for investors interested in commodity futures contracts.