The information content of implied volatility, skewness and kurtosis: empirical evidence from long-term CAC 40 options

Implied standard deviation is widely believed to be the best available forecast of the volatility of returns over the remaining contract life (Jorion, 1995 ). In this paper, we take this result two steps further to the higher moments of the distribution (skewness and kurtosis) based on a Gram–Charlier series expansion of the normal distribution (Corrado and Su, 1996 ) using long-term CAC 40 option prices contract, named PXL. First, we found that implied first moments contain a substantial amount of information for future moments of CAC 40 returns although this amount decreases with respect to the moment's order. Secondly, we found that the different shapes of the volatility smile are consistent with different distribution of the underlying returns. Based on these results, we also observed that including other implied moments significantly improves the out-of-sample pricing performance of the Black–Scholes, (1973) model.     

SKEWNESS AND KURTOSIS IN S&P 500 INDEX RETURNS IMPLIED BY OPTION PRICES

The Black-Scholes (1973) model frequently misprices deep-in-the-money and deep-out-of-the-money options. Practitioners popularly refer to these strike price biases as volatility smiles. In this paper we examine a method to extend the Black-Scholes model to account for biases induced by nonnormal skewness and kurtosis in stock return distributions. The method adapts a Gram-Charlier series expansion of the normal density function to provide skewness and kurtosis adjustment terms for the Black-Scholes formula. Using this method, we estimate option-implied coefficients of skewness and kurtosis in S&P 500 stock index returns. We find significant nonnormal skewness and kurtosis implied by option prices.     

The information content of risk-neutral skewness for volatility forecasting

The paper investigates whether risk-neutral skewness has incremental explanatory power for future volatility in the S&P 500 index. While most of previous studies have investigated the usefulness of historical volatility and implied volatility for volatility forecasting, we study the information content of risk-neutral skewness in volatility forecasting model. In particular, we concentrate on Heterogeneous Autoregressive model of Realized Volatility and Implied Volatility (HAR-RV-IV). We find that risk-neutral skewness contains additional information for future volatility, relative to past realized volatilities and implied volatility. Out-of-sample analyses confirm that risk-neutral skewness improves significantly the accuracy of volatility forecasts for future volatility.     

The information content of implied skewness and kurtosis changes prior to earnings announcements for stock and option returns

We use option prices to examine whether changes in stock return skewness and kurtosis preceding earnings announcements provide information about subsequent stock and option returns. We demonstrate that changes in jump risk premiums can lead to changes in implied skewness and kurtosis and are also associated with the mean and variability of the stock price response to the earnings announcement. We find that changes in both moments have strong predictive power for future stock returns, even after controlling for implied volatility. Additionally, changes in both moments predict call returns, while put return predictability is primarily linked to changes in skewness.     

Alternative bankruptcy prediction models using option-pricing theory

We examine the empirical properties of the theoretical Black–Scholes–Merton (BSM) bankruptcy model. We evaluate the predictive ability of various existing modifications of the BSM model and extend prior studies by estimating volatility directly from market-observable returns on firm value. We show that parsimonious models using our direct market-observable volatility estimate perform better than alternative, more sophisticated, models. Our findings suggest the adoption of simpler modelling approaches relying on market data when implementing the BSM model.     

Implied volatility skews and stock return skewness and kurtosis implied by stock option prices

The Black-Scholes* option pricing model is commonly applied to value a wide range of option contracts. However, the model often inconsistently prices deep in-the-money and deep out-of-the-money options. Options professionals refer to this well-known phenomenon as a volatility ‘skew’ or ‘smile’. In this paper, we examine an extension of the Black-Scholes model developed by Corrado and Su that suggests skewness and kurtosis in the option-implied distributions of stock returns as the source of volatility skews. Adapting their methodology, we estimate option-implied coefficients of skewness and kurtosis for four actively traded stock options. We find significantly nonnormal skewness and kurtosis in the option-implied distributions of stock returns.     

Ex Ante Skewness and Expected Stock Returns

We use option prices to estimate ex ante higher moments of the underlying individual securities’ risk‐neutral returns distribution. We find that individual securities’ risk‐neutral volatility, skewness, and kurtosis are strongly related to future returns. Specifically, we find a negative (positive) relation between ex ante volatility (kurtosis) and subsequent returns in the cross‐section, and more ex ante negatively (positively) skewed returns yield subsequent higher (lower) returns. We analyze the extent to which these returns relations represent compensation for risk and find evidence that, even after controlling for differences in co‐moments, individual securities’ skewness matters.     

Information Flows Between the US and China’s Agricultural Commodity Futures Markets—Based on VAR–BEKK–Skew-t Model

The information flow in the volatility and the skewness of returns are two factors closely influences the hedging risks for cross-border transactions. This article adopts a VAR–BEKK–MGARCH model with multivariate skew-t error terms to investigate the mean and volatility spillovers, while accounting for the potential skewness. The model is applied to real returns of corn, wheat, and soybeans futures in United States and China. The empirical results indicate the major role of United States in information transmission, and the increasing volatility spillovers of China to United States in highly marketized commodities and after trading structure changes. The analysis of skewness provides evidences for market inefficiency and implication on the investment decision and trading strategies.     

Oil prices implied volatility or direction: Which matters more to financial markets?

We examine the impact of oil price uncertainty on US stock returns by industry using the US Oil Fund options implied volatility OVX index and a GJR-GARCH model. We test the effect of the implied volatility of oil on a wide array of domestic industries’ returns using daily data from 2007 to 2016, controlling for a variety of variables such as aggregate market returns, market volatility, exchange rates, interest rates, and inflation expectations. Our main finding is that the implied volatility of oil prices has a consistent and statistically significant negative impact on nine out of the ten industries defined in the Fama and French (J Financ Econ 43:153–193, 1997) 10-industry classification. Oil prices, on the other hand, yield mixed results, with only three industries showing a positive and significant effect, and two industries exhibiting a negative and significant effect. These findings are an indication that the volatility of oil has now surpassed oil prices themselves in terms of influence on financial markets. Furthermore, we show that both oil prices and their volatility have a positive and significant effect on corporate bond credit spreads. Overall, our results indicate that oil price uncertainty increases the risk of future cash flows for goods and services, resulting in negative stock market returns and higher corporate bond credit spreads.     

Calibration of a nonlinear feedback option pricing model

We consider the option pricing model proposed by Mancino and Ogawa, where the implementation of dynamic hedging strategies has a feedback impact on the price process of the underlying asset. We present numerical results showing that the smile and skewness patterns of implied volatility can actually be reproduced as a consequence of dynamical hedging. The simulations are performed using a suitable semi-implicit finite difference method. Moreover, we perform a calibration of the nonlinear model to market data and we compare it with more popular models, such as the Black–Scholes formula, the Jump-Diffusion model and Heston's model. In judging the alternative models, we consider the following issues: (i) the consistency of the implied structural parameters with the times-series data; (ii) out-of-sample pricing; and (iii) parameter uniformity across different moneyness and maturity classes. Overall, nonlinear feedback due to hedging strategies can, at least in part, contribute to the explanation from a theoretical and quantitative point of view of the strong pricing biases of the Black–Scholes formula, although stochastic volatility effects are more important in this regard.     

The cross-section of currency volatility premia

We identify a global risk factor in the cross-section of implied volatility returns in currency markets. A zero-cost strategy that buys forward volatility agreements with downward sloping implied volatility curves and sells those with upward slopes–a volatility carry strategy–generates significant excess returns. The covariation with volatility carry returns fully explains the cross-sectional variation of our slope-sorted portfolios. The lower the slope, the more the forward volatility agreement is exposed to volatility carry risk.     

The pricing of idiosyncratic risk: evidence from the implied volatility distribution

A recent strand in the literature has investigated the relationship between idiosyncratic risk and future stock returns. Although several authors have found significant predictive power of idiosyncratic volatility, the magnitude and direction of the dependence is still being debated. Using a sample of all S&P 100 constituents, we identify positive risk premia for option-implied idiosyncratic risk. Depending on the model used to identify unsystematic risk, we observe a statistically and economically significant average annual premium of 1.72 percent. To investigate whether this impact is driven by the definition of idiosyncratic risk, we extend the pricing kernel by implied skewness. Using a double-sorting procedure, we show that the compensation of unsystematic risk is mainly driven by firms with high positive implied skewness.     

Market skewness risk and the cross section of stock returns

The cross section of stock returns has substantial exposure to risk captured by higher moments of market returns. We estimate these moments from daily Standard & Poor's 500 index option data. The resulting time series of factors are genuinely conditional and forward-looking. Stocks with high exposure to innovations in implied market skewness exhibit low returns on average. The results are robust to various permutations of the empirical setup. The market skewness risk premium is statistically and economically significant and cannot be explained by other common risk factors such as the market excess return or the size, book-to-market, momentum, and market volatility factors, or by firm characteristics.     

Commonality in the time-variation of stock–stock and stock–bond return comovements

We jointly investigate time-varying comovements between stock returns across countries and between long-term government bond and stock returns within countries. Our focus is on how daily return comovements vary with stock uncertainty, as measured by the implied volatility (IV) from equity index options. Cross-country stock return comovements tend to be stronger (weaker) following high (low) IV days and on days with large (small) changes in IV. Stock–bond return comovements tend to be substantially positive (negative) following low (high) IV days and on days with small (large) changes in IV. A regime-switching analysis also indicates a striking temporal commonality in the stock–stock and stock–bond comovement variations. Our findings bear on understanding the influence of time-varying uncertainty on price formation and the diversification benefits of stock–bond and cross-country stock holdings.     

Stochastic volatility and option pricing

Through a simple Monte Carlo experiment, Dimitrios Gkamas documents the effects that stochastic volatility has on the distribution of returns and the inability of the normal distribution utilized by the Black–Scholes model to fit empirical returns. He goes on to investigate the implied volatility patterns that stochastic volatility models can generate and potentially explain.     

Model-driven statistical arbitrage on LETF option markets

In this paper, we study the statistical properties of the moneyness scaling transformation, which adjusts the moneyness coordinate of the implied volatility smile in an attempt to remove the discrepancy between the IV smiles for levered and unlevered ETF options. We construct bootstrap uniform confidence bands which indicate that the implied volatility smiles are statistically different after moneyness scaling has been performed. An empirical application shows that there are trading opportunities possible on the LETF market. A statistical arbitrage type strategy based on a dynamic semiparametric factor model is presented. This strategy presents a statistical decision algorithm which generates trade recommendations based on comparison of model and observed LETF implied volatility surface. It is shown to generate positive returns with a high probability. Extensive econometric analysis of the LETF implied volatility process is performed including out-of-sample forecasting based on a semiparametric factor model and a uniform confidence bands' study. These provide new insights into the latent dynamics of the implied volatility surface. We also incorporate Heston stochastic volatility into the moneyness scaling method for better tractability of the model.     

Do fear indices help predict stock returns?

This study investigates the forecasting power of implied volatility indices on forward looking returns. Prior studies document that negative innovations to returns are associated with increasing implied volatility of the underlying indices; thus, suggesting a possible relationship between extremely high levels of implied volatility and positive short term returns. We investigate this issue by examining the predictive power of three implied volatility indices, VIX, VXN and VDAX, on the underlying index returns. We extend previous research by also focusing on characterised selected stocks and examine the relationship between implied volatility indices and future returns across different sectors and classified portfolios. Our findings suggest that implied volatility indices are good predictors of 20-days and 60-days forward looking returns and illustrate insignificant predictive power for very short term (1-day and 5-days) returns.     

Modeling Conditional Skewness in Stock Returns

Abstract

In this paper, we propose a new GARCH-in-Mean (GARCH-M) model allowing for conditional skewness. The model is based on the so-called z distribution capable of modeling skewness and kurtosis of the size typically encountered in stock return series. The need to allow for skewness can also be readily tested. The model is consistent with the volatility feedback effect in that conditional skewness is dependent on conditional variance. Compared to previously presented GARCH models allowing for conditional skewness, the model is analytically tractable, parsimonious and facilitates straightforward interpretation.Our empirical results indicate the presence of conditional skewness in the monthly postwar US stock returns. Small positive news is also found to have a smaller impact on conditional variance than no news at all. Moreover, the symmetric GARCH-M model not allowing for conditional skewness is found to systematically overpredict conditional variance and average excess returns.     

Jump risk,stock returns,and slope of implied volatility smile

In the presence of jump risk, expected stock return is a function of the average jump size, which can be proxied by the slope of option implied volatility smile. This implies a negative predictive relation between the slope of implied volatility smile and stock return. For more than four thousand stocks ranked by slope during 1996–2005, the difference between the risk-adjusted average returns of the lowest and highest quintile portfolios is 1.9% per month. Although both the systematic and idiosyncratic components of slope are priced, the idiosyncratic component dominates the systematic component in explaining the return predictability of slope. The findings are robust after controlling for stock characteristics such as size, book-to-market, leverage, volatility, skewness, and volume. Furthermore, the results cannot be explained by alternative measures of steepness of implied volatility smile in previous studies.     

The skewness risk premium in equilibrium and stock return predictability

In this study, we investigate the skewness risk premium in the financial market under a general equilibrium setting. Extending the long-run risks (LRR) model proposed by Bansal and Yaron (J Financ 59:1481–1509, 2004) by introducing a stochastic jump intensity for jumps in the LRR factor and the variance of consumption growth rate, we provide an explicit representation for the skewness risk premium, as well as the volatility risk premium, in equilibrium. On the basis of the representation for the skewness risk premium, we propose a possible reason for the empirical facts of time-varying and negative risk-neutral skewness. Moreover, we also provide an equity risk premium representation of a linear factor pricing model with the variance and skewness risk premiums. The empirical results imply that the skewness risk premium, as well as the variance risk premium, has superior predictive power for future aggregate stock market index returns, which are consistent with the theoretical implication derived by our model. Compared with the variance risk premium, the results show that the skewness risk premium plays an independent and essential role for predicting the market index returns.