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.     

Variance and skew risk premiums for the volatility market: The VIX evidence

We extract variance and skew risk premiums from volatility derivatives in a model-free way and analyze their relationships along with volatility index and equity index returns. These risk premiums can be synthesized through option trading strategies. Using a time series of option prices on the VIX, we find that variance swap excess return can be partially explained by volatility index and equity index excess returns while these latter variables carry little information for the skew swap excess return. The results sharply contrast with those obtained for the equity index option market underlining very specific characteristics of the volatility derivative market.     

FROM SMILE ASYMPTOTICS TO MARKET RISK MEASURES

The left tail of the implied volatility skew, coming from quotes on out‐of‐the‐money put options, can be thought to reflect the market's assessment of the risk of a huge drop in stock prices. We analyze how this market information can be integrated into the theoretical framework of convex monetary measures of risk. In particular, we make use of indifference pricing by dynamic convex risk measures, which are given as solutions of backward stochastic differential equations, to establish a link between these two approaches to risk measurement. We derive a characterization of the implied volatility in terms of the solution of a nonlinear partial differential equation and provide a small time‐to‐maturity expansion and numerical solutions. This procedure allows to choose convex risk measures in a conveniently parameterized class, distorted entropic dynamic risk measures, which we introduce here, such that the asymptotic volatility skew under indifference pricing can be matched with the market skew. We demonstrate this in a calibration exercise to market implied volatility data.     

Who knows more about future currency volatility?

We use four currency pairs from October 1, 2001 to September 29, 2006 to compare the predictive power of the implied volatility derived from currency option prices that are traded on the Philadelphia Stock Exchange (PHLX), Chicago Mercantile Exchange (CME), and over‐the‐counter market (OTC). Among the competing implied volatility forecasts, OTC‐implied volatility subsumes the information content of PHLX‐ and CME‐implied volatility. Consistent with extant studies our result also shows that the implied volatility provides more information about future volatility–regardless of whether it is from the OTC, PHLX, or CME markets–than time series based volatility. © 2009 Wiley Periodicals, Inc. Jrl Fut Mark 29:270–295, 2009     

The quality of volatility traded on the over‐the‐counter currency market: A multiple horizons study

Previous studies of the quality of market‐forecasted volatility have used the volatility that is implied by exchange‐traded option prices. The use of implied volatility in estimating the market view of future volatility has suffered from variable measurement errors, such as the non‐synchronization of option and underlying asset prices, the expiration‐day effect, and the volatility smile effect. This study circumvents these problems by using the quoted implied volatility from the over‐the‐counter (OTC) currency option market, in which traders quote prices in terms of volatility. Furthermore, the OTC currency options have daily quotes for standard maturities, which allows the study to look at the market's ability to forecast future volatility for different horizons. The study finds that quoted implied volatility subsumes the information content of historically based forecasts at shorter horizons, and the former is as good as the latter at longer horizons. These results are consistent with the argument that measurement errors have a substantial effect on the implied volatility estimator and the quality of the inferences that are based on it. © 2003 Wiley Periodicals, Inc. Jrl Fut Mark 23:261–285, 2003     

Implied volatility forecasts in the grains complex

This article finds that the implied volatilities of corn, soybean, and wheat futures options 4 weeks before option expiration have significant predictive power for the underlying futures contract return volatilities through option expiration from January 1988 through September 1999. These implied volatilities also encompass the information in out‐of‐sample seasonal Glosten, Jagannathan, and Runkle (GJR;1993) volatility forecasts. Evidence also demonstrates that when corn‐implied volatility rises relative to out‐of‐sample seasonal GJR volatility forecasts, implied volatility substantially overpredicts realized volatility. However, simulations of trading rules that involve selling corn option straddles when corn‐implied volatility is high relative to out‐of‐sample GJR volatility forecasts indicate that none of the trading rules would have been significantly profitable. This finding suggests that these options are not necessarily overpriced. © 2002 Wiley Periodicals, Inc. Jrl Fut Mark 22:959–981, 2002     

Do Chinese Retail Option Traders Know Anything about Market Volatility?

This paper examines the information content of implied volatility in the Chinese covered warrant market and finds that the implied volatility is consistently higher than the realized volatility for all warrants and across all maturities. The implied volatility has very little information content for future volatility in the Chinese warrant market which is dominated by retail investors. Possible explanations for the results are regulatory issues such as restrictions on the short-selling of warrants, differential trading rules for stocks and warrants, high leverage and low trading costs and a market dominated by retail investors.     

Stock return dynamics,option volume,and the information content of implied volatility

This article reports new empirical results on the information content of implied volatility, with respect to modeling and forecasting the volatility of individual firm returns. The 50 firms with the highest option volume on the Chicago Board Options Exchange between 1988 and 1995 are examined. First, the results indicate that the ability of implied volatility to subsume all relevant information about conditional variance depends on option trading volume. For the most active options in the sample, implied volatility reliably outperforms GARCH and subsumes all information in return shocks beyond the first lag. For these active options, implied volatility performs substantially better than indicated by the prior results of Lamoureux and Lastrapes ( 1993 ), despite significant methodological improvements in the time‐series volatility models in this study including the use of high‐frequency intraday return shocks. For the lower option‐volume firms in the sample, the performance of implied volatility deteriorates relative to time‐series volatility models. Finally, compared to a time‐series approach, the implied volatility of equity index options provides reliable incremental information about future firm‐level volatility. © 2003 Wiley Periodicals, Inc. Jrl Fut Mark 23:615–646, 2003     

The information content of implied volatility in agricultural commodity markets

In this article we compare the incremental information content of lagged implied volatility to GARCH models of conditional volatility for a collection of agricultural commodities traded on the New York Board of Trade. We also assess the relevance of the additional information provided by the implied volatility in a risk management framework. It is first shown that past squared returns only marginally improve the information content provided by the lagged implied volatility. Secondly, value‐at‐risk (VaR) models that rely exclusively on lagged implied volatility perform as well as VaR models where the conditional variance is modelled according to GARCH type processes. These results indicate that the implied volatility for options on futures contracts in agricultural commodity markets provides relevant volatility information that can be used as an input to VaR models. © 2003 Wiley Periodicals, Inc. Jrl Fut Mark 23:441–454, 2003     

Does global fear predict fear in BRICS stock markets? Evidence from a Bayesian Graphical Structural VAR model

We examine the predictive power of implied volatility in the commodity and major developed stock markets for the implied volatility in individual BRICS stock markets. We use daily data from March 2011 to October 2016 and employ the newly developed Bayesian Graphical Structural Vector Autoregressive (BGSVAR) model of Ahelegbey et al. (2016). Evidence suggests that the predictability of individual implied volatilities in BRICS is generally a function of both global and within the group stock market implied volatilities, and that the role of commodity market volatility is marginal, except for South Africa. Important implications for policy-makers and portfolio-managers are discussed.     

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     

Factors explaining movements in the implied volatility surface

This article explores the relationship of changes in the S&P 500 index implied volatility surface to economic state variables. Observable variables can explain some of the variation in implied volatility, with the majority of explanatory power from index returns. Although the contemporaneous return is most important for explaining changes in short dated volatility, the path of the index is important for explaining changes in long dated volatility. Other variables also display statistically significant relations to volatility changes. Shocks to the Nikkei 225, short‐term interest rates, and the corporate/government bond yield spread are correlated with small, systematic changes in implied volatility. The results suggest a multifactor model for market volatility, with factors other than index returns adding negligible explanatory ability. © 2002 Wiley Periodicals, Inc. Jrl Fut Mark 22:915–937, 2002     

SERIES EXPANSION OF THE SABR JOINT DENSITY

Under the SABR stochastic volatility model, pricing and hedging contracts that are sensitive to forward smile risk (e.g., forward starting options, barrier options) require the joint transition density. In this paper, we address this problem by providing closed‐form representations, asymptotically, of the joint transition density. Specifically, we construct an expansion of the joint density through a hierarchy of parabolic equations after applying total volatility‐of‐volatility scaling and a near‐Gaussian coordinate transformation. We then establish an existence result to characterize the truncation error and provide explicit joint density formulas for the first three orders. Our approach inherits the same spirit of a small total volatility‐of‐volatility assumption as in the original SABR analysis. Our results for the joint transition density serve as a basis for managing forward smile risk. Through numerical experiments, we illustrate the accuracy of our expansion in terms of joint density, marginal density, probability mass, and implied volatilities for European call options.     

The information content of implied volatility in light of the jump/continuous decomposition of realized volatility

In the framework of encompassing regressions, the information content of the jump/continuous components of historical volatility is assessed when implied volatility is included as an additional regressor. The authors' empirical application focuses on daily and intradaily data for the S&P100 and S&P500 indexes, and daily data for the associated VXO and VIX implied volatility indexes. The results show that the total explanatory power of the encompassing regressions barely changes when the jump/continuous components are included, although the weekly and monthly continuous components are usually significant. This evidence supports the view that implied volatility has very high information content, even when extended decompositions of past realized volatility are used. Moreover, adding GARCH‐type volatility forecasts in the regressions confirms these results. © 2007 Wiley Periodicals, Inc. Jrl Fut Mark 27:337–359, 2007     

EXPLICIT IMPLIED VOLATILITIES FOR MULTIFACTOR LOCAL‐STOCHASTIC VOLATILITY MODELS

We consider an asset whose risk‐neutral dynamics are described by a general class of local‐stochastic volatility models and derive a family of asymptotic expansions for European‐style option prices and implied volatilities. We also establish rigorous error estimates for these quantities. Our implied volatility expansions are explicit; they do not require any special functions nor do they require numerical integration. To illustrate the accuracy and versatility of our method, we implement it under four different model dynamics: constant elasticity of variance local volatility, Heston stochastic volatility, three‐halves stochastic volatility, and SABR local‐stochastic volatility.     

Bid and ask prices of index put options: Which predicts the underlying stock returns?

In this study, we separately estimate the implied volatility from the bid and ask prices of deep out-of-the-money put options on the S&P500 index. We find that the implied volatility of ask prices has stronger predictive power for stock returns than does the implied volatility of bid prices. We identify two sources of the better performance of the ask price implied volatility: one is its stronger predictive power during economic recessions and in the presence of increasing intermediary capital risk, and the other is its richer information about the future market variance risk premium.     

OPTION HEDGING AND IMPLIED VOLATILITIES IN A STOCHASTIC VOLATILITY MODEL1

In the stochastic volatility framework of Hull and White (1987), we characterize the so-called Black and Scholes implied volatility as a function of two arguments the ratio of the strike to the underlying asset price and the instantaneous value of the volatility By studying the variation m the first argument, we show that the usual hedging methods, through the Black and Scholes model, lead to an underhedged (resp. overhedged) position for in-the-money (resp out-of the-money) options, and a perfect partial hedged position for at the-money options These results are shown to be closely related to the smile effect, which is proved to be a natural consequence of the stochastic volatility feature the deterministic dependence of the implied volatility on the underlying volatility process suggests the use of implied volatility data for the estimation of the parameters of interest A statistical procedure of filtering (of the latent volatility process) and estimation (of its parameters) is shown to be strongly consistent and asymptotically normal.     

隐含波动率的信息含量及其在我国的应用

国内现有关于波动率的研究多集中于时间序列模型,忽略了另一类预测波动率的方法即隐含波动率法。文章在回顾、评述了国内关于波动率的研究后,对国外关于隐含波动率的研究进行了梳理,为在我国大陆地区发展股指期权市场、通过提高期权市场的效率,以运用隐含波动率法更好地预测波动率提供了理论基础和政策建议。     

Smiling less at LIFFE

This study investigates the structure of the implied volatility smile, using the prices of equity options traded on the LIFFE. First, the slope of the implied volatility curve is significantly negative for both individual stocks and index options, and the slope is less negative for longer‐term options. The implied volatility skew can be described by risk‐neutral skewness and kurtosis, with the former having the first‐order effect. Moreover, the implied volatility skew for individual stock options is less severe than for index options. Finally, the relationship between the real and risk‐neutral moments implied in option prices is significant. The results indicate that, for equity options traded on the LIFFE, the slope of the implied volatility skew is flatter than that on the Chicago Board of Exchange (CBOE). © 2008 Wiley Periodicals, Inc. Jrl Fut Mark 28:57–81, 2008     

CREDIT SPREADS, OPTIMAL CAPITAL STRUCTURE, AND IMPLIED VOLATILITY WITH ENDOGENOUS DEFAULT AND JUMP RISK

We propose a two-sided jump model for credit risk by extending the Leland–Toft endogenous default model based on the geometric Brownian motion. The model shows that jump risk and endogenous default can have significant impacts on credit spreads, optimal capital structure, and implied volatility of equity options: (1) Jumps and endogenous default can produce a variety of non-zero credit spreads, including upward, humped, and downward shapes; interesting enough, the model can even produce, consistent with empirical findings, upward credit spreads for speculative grade bonds. (2) The jump risk leads to much lower optimal debt/equity ratio; in fact, with jump risk, highly risky firms tend to have very little debt. (3) The two-sided jumps lead to a variety of shapes for the implied volatility of equity options, even for long maturity options; although in general credit spreads and implied volatility tend to move in the same direction under exogenous default models, this may not be true in presence of endogenous default and jumps. Pricing formulae of credit default swaps and equity default swaps are also given. In terms of mathematical contribution, we give a proof of a version of the "smooth fitting" principle under the jump model, justifying a conjecture first suggested by Leland and Toft under the Brownian model.