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.     

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.     

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.     

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.     

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.     

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.     

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.     

Portfolio choice with skewness preference and wealth-dependent risk aversion

This paper studies a dynamic portfolio choice problem for an investor with both wealth-dependent risk aversion and wealth-dependent skewness preferences. In a general economic setting, the solution is characterized in terms of a system of extended Hamilton-Jacobi-Bellman (EHJB) equations and the solution is given in closed form in some special cases. We demonstrate the effects of higher order risk preferences and state-dependent risk aversion on the optimal asset allocation decisions. We find that wealth-dependent risk aversion facilitates risk taking and the skewness preference leads to a more positively skewed portfolio in certain circumstances.     

Realized volatility and transactions

This paper re-examines the impact of number of trades, trade size and order imbalance on daily stock returns volatility. In contrast to prior studies, we estimate daily volatility using realized volatility obtained by summing up intraday squared returns. Consistent with the theory of quadratic variation, realized volatility estimates are shown to be less noisy than standard volatility measures such as absolute returns used in previous studies. In general, our results confirm [Jones, C.M., Kaul, G., Lipson, M.L., 1994. Transactions, volume, and volatility. Review of Financial Studies 7, 631–651] that number of trades is the dominant factor behind the volume–volatility relation. Neither trade size nor order imbalance adds significantly more explanatory power to realized volatility beyond number of trades. This finding is robust to different time periods, firm sizes and regression specifications. The implications of our results for microstructure theory are discussed.     

Conditional volatility persistence and volatility spillovers in the foreign exchange market

I investigate the magnitudes and determinants of volatility spillovers in the foreign exchange (FX) market, using realized measures of volatility and heterogeneous autoregressive (HAR) models. I confirm both meteor shower effects (i.e., inter-regional volatility spillovers) and heat wave effects (i.e., intra-regional volatility spillovers) in the FX market. Furthermore, I find that conditional volatility persistence is the dominant channel linking the changing market states of each region to future volatility and its spillovers. Market state variables contribute to more than half of the explanatory power in predicting conditional volatility persistence, with the model that calibrates volatility persistence and spillovers conditionally on market states performing statistically and economically better. The utilization of market state variables significantly extends our understanding of the economic mechanisms of volatility persistence and spillovers and sheds new light on econometric techniques for volatility modeling and forecasting.     

Margin regulation and volatility

An infinite-horizon asset-pricing model with heterogeneous agents and collateral constraints can explain why adjustments in stock market margins under US Regulation T had an economically insignificant impact on market volatility. In the model, raising the margin requirement for one asset class may barely affect its volatility if investors have access to another, unregulated class of collateralizable assets. Through spillovers, however, the volatility of the other asset class may substantially decrease. A very strong dampening effect on all assets׳ return volatilities can be achieved by a countercyclical regulation of all markets.     

Herding and market volatility

In this paper, we explore the impact of investor herding behavior on stock market volatility. We adopt a direct herding measure based on the variation of cross-sectional stock betas. The measure can be readily separated into positive and adverse components, whereby investors herd towards and away from the market portfolio, respectively. Using A-shares listed in the Chinese equity market from August 2005 to March 2021, we show that the market volatility is Granger caused by the measure, and that there exists an asymmetric effect between positive and adverse herding on volatility. Furthermore, we provide robust evidence that the information contained in the herding measure helps generate significantly improved volatility forecasts and add economic value to investors. Our paper not only contributes to the volatility forecasting literature but also advances our understanding of herding in the equity market.