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.     

Implied volatility and future portfolio returns

Prior studies find that the CBOE volatility index (VIX) predicts returns on stock market indices, suggesting implied volatilities measured by VIX are a risk factor affecting security returns or an indicator of market inefficiency. We extend prior work in three important ways. First, we investigate the relationship between future returns and current implied volatility levels and innovations. Second, we examine portfolios sorted on book-to-market equity, size, and beta. Third, we control for the four Fama and French [Fama, E., French, K., 1993. Common risk factors in the returns on stocks and bonds. Journal of Financial Economics 33, 3–56.] and Carhart [Carhart, M., 1997. On persistence in mutual fund performance. Journal of Finance, 52, 57–82.] factors. We find that VIX-related variables have strong predictive ability.     

Implied and realized volatility: empirical model selection

The paper studies the nonparametric connection between realized and implied volatilities. No-arbitrage identities and comparison inequalities are found. We formulate the multi-factor trading system on the volatility scale. To empirically determine the number of factors, we develop a high frequency analysis for sequential F-testing. We also design a cross validated estimate of quadratic variation.     

Implied volatility linkages among major European currencies

This paper examines linkages in expected future volatilities among major European currencies. For that purpose, volatility expectations implied by currency options on the euro, British pound, and Swiss franc quoted against the U.S. dollar are analysed. Vector autoregressive modelling is applied to ascertain the dynamics of the implied volatilities across currencies. The results show that the market expectations of future exchange rate volatilities are closely linked among major European currencies. Furthermore, it is found that the implied volatility of the euro significantly affects the volatility expectations of the British pound and the Swiss franc.     

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.     

Implied volatility index for the Norwegian equity market

We introduce and evaluate the NOVIX - an implied volatility index for the Norwegian equity index OBX. NOVIX is created according to the VIX methodology. We compare the NOVIX to the German VDAX-NEW and the U.S. VIX and find that NOVIX has similar properties as these two indices. We also evaluate the VIX, VDAX-NEW and NOVIX in terms of volatility forecasting. As a benchmark model we use a precise HAR model of Corsi (2009) based on high-frequency data. All three implied volatility indices significantly improve daily, weekly and monthly forecasts of volatility of their underlying equity indices. This improvement is largest for the VIX, followed by VDAX-NEW and NOVIX.     

Implied volatility sentiment: a tale of two tails

We propose a sentiment measure jointly derived from out-of-the-money index puts and single stock calls: implied volatility (IV-) sentiment. In contrast to implied correlations, our measure uses information from the tails of the risk-neutral densities from these two markets rather than across their entire moneyness structures. We find that IV-sentiment measure adds value over and above traditional factors in predicting the equity risk premium out-of-sample. Forecasting results are superior when constrained ensemble models are used vis-à-vis unregularized machine learning techniques. In a mean-reversion strategy, our IV-sentiment measure delivers economically significant results, with limited exposure to a set of cross-sectional equity factors, including Fama and French's five factors, the momentum factor and the low-volatility factor, and seems valuable in preventing momentum crashes. Our novel measure reflects overweight of tail events, which we interpret as a behavioral bias. However, we cannot rule out a risk-compensation rationale.     

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.     

Liquidity skewness

Bid–ask spreads in equities have declined on average but have become increasingly right-skewed. This finding holds across exchanges as well as size, price, and volume quartiles. Higher right-skewness is consistent with more competition among market makers; which may reduce cross-subsidization across periods of high and low asymmetric information, unlike a monopolistic regime that can maintain a relatively constant spread. Confirming this intuition, proportional differences in spreads between earnings announcements and normal periods have increased considerably even as trading costs have declined on average. Skewness also is cross-sectionally related to information proxies such as institutional holdings and analyst following.     

Risk aversion and skewness preference

Empirically, co-skewness of asset returns seems to explain a substantial part of the cross-sectional variation of mean return not explained by beta. This finding is typically interpreted in terms of a risk averse representative investor with a cubic utility function. This paper questions this interpretation. We show that the empirical tests fail to impose risk aversion and the implied utility function takes an inverse S-shape. Unfortunately, the first-order conditions are not sufficient to guarantee that the market portfolio is the global maximum for this utility function, and our results suggest that the market portfolio is more likely to represent the global minimum. In addition, if we do impose risk aversion, then co-skewness has minimal explanatory power.     

Measuring skewness premia

We provide a new methodology to empirically investigate the respective roles of systematic and idiosyncratic skewness in explaining expected stock returns. Using a large number of predictors, we forecast the cross-sectional ranks of systematic and idiosyncratic skewness, which are easier to predict than their actual values. Compared to other measures of ex ante systematic skewness, our forecasts create a significant spread in ex post systematic skewness. A predicted systematic skewness risk factor carries a significant and robust risk premium that ranges from 6% to 12% per year. In contrast, the role of idiosyncratic skewness in pricing stocks is less robust.     

A semiparametric factor model for implied volatility surface dynamics

We propose a semiparametric factor model, which approximatesthe implied volatility surface (IVS) in a finite dimensionalfunction space. Unlike standard principal component approachestypically used to reduce complexity, our approach is tailoredto the degenerated design of IVS data. In particular, we onlyfit in the local neighborhood of the design points by exploitingthe expiry effect present in option data. Using DAX index optiondata, we estimate the nonparametric components and a low-dimensionaltime series of latent factors. The modeling approach is completedby studying vector autoregressive models fitted to the latentfactors.     

Corporate insider trading and return skewness

Corporate insider trades predict idiosyncratic return skewness. CEO purchases are followed by an increase and CEO sales by a decrease in idiosyncratic skewness. The evidence suggests that this effect is driven by personal preferences rather than behavioral biases such as overconfidence. Our findings are consistent with the interpretation that CEOs, who are generally considered to be underdiversified, optimize their holdings by taking their preference for positive return skewness into account. We observe particularly robust results for CEO sales, which substantiates the less common notion that insider sales can be informative for investors.     

Implied integrated variance and hedging

If the volatility is stochastic, stock price returns and European option prices depend on the time average of the variance, i.e. the integrated variance, not on the path of the volatility. Applying a Bayesian statistical approach, we compute a forward-looking estimate of this variance, an option-implied integrated variance. Simultaneously, we obtain estimates of the correlation coefficient between stock price and volatility shocks, and of the parameters of the volatility process. Due to the convexity of the Black–Scholes formula with respect to the volatility, pricing and hedging with Black–Scholes-type formulas and the implied volatility often lead to inaccuracies if the volatility is stochastic. Theoretically, this problem can be avoided by using Hull–White-type option pricing and hedging formulas and the integrated variance. We use the implied integrated variance and Hull–White-type formulas to hedge European options and certain volatility derivatives.     

Implied Binomial Trees

This article develops a new method for inferring risk-neutral probabilities (or state-contingent prices) from the simultaneously observed prices of European options. These probabilities are then used to infer a unique fully specified recombining binomial tree that is consistent with these probabilities (and, hence, consistent with all the observed option prices). A simple backwards recursive procedure solves for the entire tree. From the standpoint of the standard binomial option pricing model, which implies a limiting risk-neutral lognormal distribution for the underlying asset, the approach here provides the natural (and probably the simplest) way to generalize to arbitrary ending risk-neutral probability distributions.     

Volume and skewness in international equity markets

We examine the relation between trading volume and skewness in 11 international stock markets using daily and monthly data from January 1980 to August 2004. We construct single equation and VAR models of the relation between the first three moments of market returns and trading volumes. Our results show hitherto unrecognised channels of influence, and support the investor heterogeneity approach to explaining return asymmetries.     

Implied Default Probability and Credit Derivatives

Recently many kinds of credit derivatives are traded in the market. The default probability implied in the market becomes important to price some credit derivatives. Also it is useful for managing the credit risk because it includes the market information. In this paper we show how to calculate the implied default probability in the default swap market or the defaultable bond market.This paper is developed from author’s master thesis (Matsumoto, 2000), Graduate School of Systems Management, the University of Tsukuba.     

Macroeconomic determinants of stock volatility and volatility premiums

How does stock market volatility relate to the business cycle? We develop, and estimate, a no-arbitrage model, and find that (i) the level and fluctuations of stock volatility are largely explained by business cycle factors and (ii) some unobserved factor contributes to nearly 20% to the overall variation in volatility, although not to its ups and downs. Instead, this “volatility of volatility” relates to the business cycle. Finally, volatility risk-premiums are strongly countercyclical, even more than stock volatility, and partially explain the large swings of the VIX index during the 2007–2009 subprime crisis, which our model captures in out-of-sample experiments.     

Growth and volatility

Growth and volatility correlate negatively across countries, but positively across sectors. Analytically, whether or not sectoral growth and volatility are correlated positively is irrelevant in the aggregate. Cross-country estimates identify the detrimental effects of macroeconomic volatility on growth, but they cannot be used to dismiss theories implying a positive growth-volatility coefficient, which appear to hold in sectoral data. In particular, volatile sectors command high investment rates, as they would in a mean-variance framework.