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.     

Invisible Parameters in Option Prices

This paper characterizes contingent claim formulas that are independent of parameters governing the probability distribution of asset returns. While these parameters may affect stock, bond, and option values, they are “invisible” because they do not appear in the option formulas. For example, the Black-Scholes ( 1973 ) formula is independent of the mean of the stock return. This paper presents a new formula based on the log-negative-binomial distribution. In analogy with Cox, Ross, and Rubinstein's ( 1979 ) log-binomial formula, the log-negative-binomial option price does not depend on the jump probability. This paper also presents a new formula based on the log-gamma distribution. In this formula, the option price does not depend on the scale of the stock return, but does depend on the mean of the stock return. This paper extends the log-gamma formula to continuous time by defining a gamma process. The gamma process is a jump process with independent increments that generalizes the Wiener process. Unlike the Poisson process, the gamma process can instantaneously jump to a continuum of values. Hence, it is fundamentally “unhedgeable.” If the gamma process jumps upward, then stock returns are positively skewed, and if the gamma process jumps downward, then stock returns are negatively skewed. The gamma process has one more parameter than a Wiener process; this parameter controls the jump intensity and skewness of the process. The skewness of the log-gamma process generates strike biases in options. In contrast to the results of diffusion models, these biases increase for short maturity options. Thus, the log-gamma model produces a parsimonious option-pricing formula that is consistent with empirical biases in the Black-Scholes formula.     

Skewness and Stock Option Prices

Abstract

In the classical Black-Scholes model, the logarithm of the stock price has a normal distribution, which excludes skewness. In this paper we consider models that allow for skewness. We propose an option-pricing formula that contains a linear adjustment to the Black-Scholes formula. This approximation is derived in the shifted Poisson model, which is a complete market model in which the exact option price has some undesirable features. The same formula is obtained in some incomplete market models in which it is assumed that the price of an option is defined by the Esscher method. For a European call option, the adjustment for skewness can be positive or negative, depending on the strike price.     

The valuation of compound options

This paper presents a theory for pricing options on options, or compound options. The method can be generalized to value many corporate liabilities. The compound call option formula derived herein considers a call option on stock which is itself an option on the assets of the firm. This perspective incorporates leverage effects into option pricing and consequently the variance of the rate of return on the stock is not constant as Black-Scholes assumed, but is instead a function of the level of the stock price. The Black-Scholes formula is shown to be a special case of the compound option formula. This new model for puts and calls corrects some important biases of the Black-Scholes model.     

The Variance Gamma Process and Option Pricing

A three parameter stochastic process, termed the variance gamma process, that generalizes Brownian motion is developed as a model for the dynamics of log stock prices. The process is obtained by evaluating Brownian motion with drift at a random time given by a gamma process. The two additional parameters are the drift of the Brownian motion and the volatility of the time change. These additional parameters provide control over the skewness and kurtosis of the return distribution. Closed forms are obtained for the return density and the prices of European options. The statistical and risk neutral densities are estimated for data on the S & P500 Index and the prices of options on this Index. It is observed that the statistical density is symmetric with some kurtosis, while the risk neutral density is negatively skewed with a larger kurtosis. The additional parameters also correct for pricing biases of the Black Scholes model that is a parametric special case of the option pricing model developed here.     

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.     

Monthly Pattern and Portfolio Effect on Higher Moments of Stock Returns: Empirical Evidence from Hong Kong

Using a direct test, this paper studies the month-of-the-year effect on the higher moments of six industrial stock indices of the Hong Kong market. We also examine the portfolio effect on skewness and kurtosis across month of the year to see if such an anomaly exists. The empirical results support a weak month-of-the-year effect in higher moments of stock returns. Using a complete sample of all possible combinations for each portfolio size, we show that portfolio effect varies across month of the year for both skewness and kurtosis. In particular, our results show that diversification does not necessarily provide benefits to rational investors when the stock return distribution is non-normal, even though portfolio formation can reduce standard deviation. In June, August and October, diversification across industrial sectors results in a more negatively skewed and leptokurtic return distribution, which is not preferred by investors with risk-aversion. Two (one) possible explanations for the portfolio effect on skewness (kurtosis) are also provided. Our empirical results add new evidence to the existence of anomalies in the Hong Kong stock market. This revised version was published online in August 2006 with corrections to the Cover Date.     

Properties of Multinomial Lattices with Cumulants for Option Pricing and Hedging

In this paper, we analyze properties of multinomial lattices that model general stochastic dynamics of the underlying stock by taking into account any given cumulants (or moments). First, we provide a parameterization of multinomial lattices, and demonstrate that mean, variance, skewness, and kurtosis of the underlying may be matched using five branches. Then, we investigate the convergence of the multinomial lattice when the basic time period approaches zero, and prove that the limiting process of the multinomial lattice that matches annualized mean, variance, skewness and kurtosis is given by a compound Poisson process. Finally, we illustrate the effect of higher order moments in the underlying asset process on the price of derivative securities through numerical experiments using the multinomial lattice, and provide a comparison with jump-diffusion models.     

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.     

A closed-form solution for options with stochastic volatility with applications to bond and currency options

I use a new technique to derive a closed-form solution for theprice of a European call option on an asset with stochasticvolatility. The model allows arbitrary correlation between volatilityand spot asset returns. I introduce stochastic interest ratesand show how to apply the model to bond options and foreigncurrency options. Simulations show that correlation betweenvolatility and the spot asset's price is important for explainingreturn skewness and strike-price biases in the Black-Scholes(1973) model. The solution technique is based on characteristicfunctions and can be applied to other problems     

On the robustness of higher-moment factors in explaining average expected returns: Evidence from Australia

This study tests the importance of systematic skewness and systematic kurtosis of Australian stock returns in the spirit of the higher-moment asset pricing model. We apply the Dagenais and Dagenais (1997) higher-moment estimators to correct for the errors-in-variables (EIVs) problems commonly found in the Fama and MacBeth (1973) two-pass regression methodology. After correcting for the EIVs problems, the two higher-moment factors, especially systematic skewness, are important in pricing Australian stocks. Systematic kurtosis appears to replace beta which plays a diminished role in the heavy-tailed return distribution.     

Robust estimation with flexible parametric distributions: estimation of utility stock betas

The distributions of stock returns and capital asset pricing model (CAPM) regression residuals are typically characterized by skewness and kurtosis. We apply four flexible probability density functions (pdfs) to model possible skewness and kurtosis in estimating the parameters of the CAPM and compare the corresponding estimates with ordinary least squares (OLS) and other symmetric distribution estimates. Estimation using the flexible pdfs provides more efficient results than OLS when the errors are non-normal and similar results when the errors are normal. Large estimation differences correspond to clear departures from normality. Our results show that OLS is not the best estimator of betas using this type of data. Our results suggest that the use of OLS CAPM betas may lead to erroneous estimates of the cost of capital for public utility stocks.     

Discretely adjusted option hedges

This paper analyses the distribution of returns on a hedged portfolio, consisting of a European call option and its associated stock, when the portfolio is rebalanced at discrete time intervals. Under the assumptions of the Black-Scholes model this distribution is particularly skew. In tests of the average return on a hedged portfolio this skewness leads to biased t-statistics. The paper explores the nature and extent of this bias and suggests procedures for overcoming it. Other aspects of discrete hedging are also discussed.     

International asset allocation under regime switching, skew, and kurtosis preferences

This paper investigates the international asset allocation effectsof time-variations in higher-order moments of stock returnssuch as skewness and kurtosis. In the context of a four-momentInternational Capital Asset Pricing Model (ICAPM) specificationthat relates stock returns in five regions to returns on a globalmarket portfolio and allows for time-varying prices of covariance,co-skewness, and co-kurtosis risk, we find evidence of distinctbull and bear regimes. Ignoring such regimes, an unhedged USinvestor's optimal portfolio is strongly diversified internationally.The presence of regimes in the return distribution leads toa substantial increase in the investor's optimal holdings ofUS stocks, as does the introduction of skewness and kurtosispreferences.     

Nonparametric Tests of Alternative Option Pricing Models Using All Reported Trades and Quotes on the 30 Most Active CBOE Option Classes from August 23, 1976 through August 31, 1978

The tests reported here differ in several ways from those of most other papers testing option pricing models: an extremely large sample of observations of both trades and bid-ask quotes is examined, careful consideration is given to discarding misleading records, nonparametric rather than parametric statistical tests are used, reported results are not sensitive to measurement of stock volatility, special care is taken to incorporate the effects of dividends and early exercise, a simple method is developed to test several option pricing formulas simultaneously, and the statistical significance and consistency across subsamples of the most important reported results are unusually high. The three key results are: (1) short-maturity out-of-the-money calls are priced significantly higher relative to other calls than the Black-Scholes model would predict, (2) striking price biases relative to the Black-Scholes model are also statistically significant but have reversed themselves after long periods of time, and (3) no single option pricing model currently developed seems likely to explain this reversal.     

Some extensions of the CAPM for individual assets

There is ample evidence that stock returns exhibit non-normal distributions with high skewness and excess kurtosis. Experimental evidence has shown that investors like positive skewness, dislike extreme losses and show high levels of prudence. This has motivated the introduction of the four-moment capital asset pricing model (CAPM). This extension, however, has not been able to successfully explain average returns. Our paper argues that a number of pitfalls may have contributed to the weak and conflicting empirical results found in the literature. We investigate whether conditional models, whether models that use individual stocks rather than portfolios and whether models that extend both the moment and factor dimension can improve on more traditional static, portfolio-based, mean–variance models. More importantly, we find that the use of a scaled coskewness measure in cross-section regression is likely to be spurious because of the possibility for the market skewness to be close to zero, at least for some periods. We provide a simple solution to this problem.     

Institutional Ownership and Distribution of Equity Returns

Although there is considerable evidence of the importance of skewness and kurtosis in equity returns, much less attention has been paid to their determinants. Recent theoretical and empirical advances in the literature suggest that the information structure and other market characteristics affect the nature of return distributions. One such characteristic is the degree of institutional ownership in the stock. This study hypothesizes and documents a significant inverse relationship between the degree of institutional ownership and the standard deviation, skewness, and kurtosis of equity returns.     

Approximate option valuation for arbitrary stochastic processes

We show how a given probability distribution can be approximated by an arbitrary distribution in terms of a series expansion involving second and higher moments. This theoretical development is specialized to the problem of option valuation where the underlying security distribution, if not lognormal, can be approximated by a lognormally distributed random variable. The resulting option price is expressed as the sum of a Black-Scholes price plus adjustment terms which depend on the second and higher moments of the underlying security stochastic process. This approach permits the impact on the option price of skewness and kurtosis of the underlying stock's distribution to be evaluated.     

On Jumps in Common Stock Prices and Their Impact on Call Option Pricing

The Black-Scholes call option pricing model exhibits systematic empirical biases. The Merton call option pricing model, which explicitly admits jumps in the underlying security return process, may potentially eliminate these biases. We provide statistical evidence consistent with the existence of lognormally distributed jumps in a majority of the daily returns of a sample of NYSE listed common stocks. However, we find no operationally significant differences between the Black-Scholes and Merton model prices of the call options written on the sampled common stocks.     

Interacting biases,non-normal return distributions and the performance of tests for long-horizon event studies

We report simulations of one-, three-, and five-year abnormal buy-and-hold stock return tests. Using benchmark portfolios purged of new-listings and rebalancing biases, we find severe misspecification of most tests, due in part to skewness. Control-firm matching also results in misspecification, particularly in large samples. We document a negative relation between skewness bias and sample size, and an overlapping-horizons bias. Both biases become more severe as the holding period lengthens. The biases interact such that tests can be well-specified in one situation but not another. A two groups test using winsorized abnormal returns yields correct specification and considerable power in many situations.