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.     

Option Prices as Predictors of Equilibrium Stock Prices

The Black-Scholes option pricing model, modified for dividend payments, is used to calculate jointly implied stock prices and implied standard deviations. A comparison of the implied stock prices with observed stock prices reveals that the implied prices contain information regarding equilibrium stock prices that is not fully reflected in observed stock prices. The implications of this finding are discussed.     

Dividend Surprises Inferred from Option and Stock Prices

This paper introduces a new method to measure the unexpected component of dividend announcements. While measures used previously were based on various arbitrary models of dividend expectations, our suggested method compares the reaction of stock and option prices to dividend announcements. Our measure is compared to commonly used model-based measures, to a Box-Jenkins time-series-based measure, and to a Value-Line Investor Survey-based measure of dividend surprises. The new measure is more highly correlated with the market's reaction to the announcements than are alternative measures of dividend surprises. The new measure is also shown to be insensitive to the extent to which the options used to identify unexpected dividend announcements are in- or out-of-the-money.     

The Information in Option Volume for Future Stock Prices

We present strong evidence that option trading volume containsinformation about future stock prices. Taking advantage of aunique data set, we construct put-call ratios from option volumeinitiated by buyers to open new positions. Stocks with low put-callratios outperform stocks with high put-call ratios by more than40 basis points on the next day and more than 1% over the nextweek. Partitioning our option signals into components that arepublicly and nonpublicly observable, we find that the economicsource of this predictability is nonpublic information possessedby option traders rather than market inefficiency. We also findgreater predictability for stocks with higher concentrationsof informed traders and from option contracts with greater leverage.     

Option Volume and Stock Prices: Evidence on Where Informed Traders Trade

This paper investigates the informational role of transactions volume in options markets. We develop an asymmetric information model in which informed traders may trade in option or equity markets. We show conditions under which informed traders trade options, and we investigate the implications of this for the linkage between markets. Our model predicts an important informational role for the volume of particular types of option trades. We empirically test our model's hypotheses with intraday option data. Our main empirical result is that negative and positive option volumes contain information about future stock prices.     

Do Stock Prices and Volatility Jump? Reconciling Evidence from Spot and Option Prices

This paper examines the empirical performance of jump diffusion models of stock price dynamics from joint options and stock markets data. The paper introduces a model with discontinuous correlated jumps in stock prices and stock price volatility, and with state-dependent arrival intensity. We discuss how to perform likelihood-based inference based upon joint options/returns data and present estimates of risk premiums for jump and volatility risks. The paper finds that while complex jump specifications add little explanatory power in fitting options data, these models fare better in fitting options and returns data simultaneously.     

Stochastic Properties of Time-Averaged Financial Data: Explanation and Empirical Demonstration Using Monthly Stock Prices

This article considers the potential statistical problems resulting from the use of averaged rather than end‐of‐period data in financial research. Averaged data are widely employed throughout the literature without explicit recognition that the use of such data results in biased estimates of the variance, covariance and autocorrelation of the first as well as higher order changes. We illustrate the magnitude of the biases, using the S&P 500 end‐of‐month series over the period March 1957 to February 2001. Results confirm the predictions of Working and of Schwert. In addition, an analysis of the properties of higher‐order lags indicates that the bias persists, a result not previously suggested in the literature. We also find that these statistical biases are time varying‐which has significant implications for empirical financial research.     

On the Distribution of CBOE Option Trade Prices Occurring Between Consecutive Stock Trades

This paper examines the volume distribution of option trade prices that occurs when the underlying stock price remains constant. The width of these option trade price bands provides direct evidence on the law of one price and the redundancy of options assumed in many option models. We find that index option bands are narrower than equity option bands. Furthermore, for both equity and index options, puts have narrower bandwidths than calls. In general, option price bandwidth is narrow and can be explained by the minimum price movement allowed by the Chicago Board Options Exchanges (CBOE). This supports the single price law and the redundancy assumption. The existence of bid/ask quotes on the option does not materially affect the above results although it does alter the frequency of multiple option trade prices for a given underlying stock price. We note that over 53% of option trading volume occurs without bid/ask quotes on the CBOE compared to less than 15% a decade ago. Our results suggest that the effective bid/ask spread on options is probably no larger than the minimum price movements allowed by the CBOE. Furthermore, the need for the liquidity services of market makers may be declining if the decline in quoting activity stems from cross trading (i.e. trades not involving market makers).     

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.     

Jackknifing Bond Option Prices

Prices of interest rate derivative securities depend cruciallyon the mean reversion parameters of the underlying diffusions.These parameters are subject to estimation bias when standardmethods are used. The estimation bias can be substantial evenin very large samples and much more serious than the discretizationbias, and it translates into a bias in pricing bond optionsand other derivative securities that is important in practicalwork. This article proposes a very general and computationallyinexpensive method of bias reduction that is based on Quenouille's(1956; Biometrika, 43, 353–360) jackknife. We show howthe method can be applied directly to the options price itselfas well as the coefficients in the models. We investigate itsperformance in a Monte Carlo study. Empirical applications toU.S. dollar swap rates highlight the differences between bondand option prices implied by the jackknife procedure and thoseimplied by the standard approach. These differences are largeand suggest that bias reduction in pricing options is importantin practical applications.     

Skewness and Kurtosis Implied by Option Prices: A Correction

Corrado and Su (1996) provide skewness and kurtosis adjustment terms for the Black‐Scholes model, using a Gram‐Charlier expansion of the normal density function. In this note we provide a correction to the expression for the skewness coefficient and illustrate the effect on call option prices of the error found.     

Investor Sentiment and Option Prices

This paper examines whether investor sentiment about the stockmarket affects prices of the S&P 500 options. The findingsreveal that the index option volatility smile is steeper (flatter)and the risk-neutral skewness of monthly index return is more(less) negative when market sentiment becomes more bearish (bullish).These significant relations are robust and become stronger whenthere are more impediments to arbitrage in index options. Theycannot be explained by rational perfect-market-based optionpricing models. Changes in investor sentiment help explain timevariation in the slope of index option smile and risk-neutralskewness beyond factors suggested by the current models.     

General Properties of Option Prices

When the underlying price process is a one-dimensional diffusion, as well as in certain restricted stochastic volatility settings, a contingent claim's delta is bounded by the infimum and supremum of its delta at maturity. Further, if the claim's payoff is convex (concave), the claim's price is a convex (concave) function of the underlying asset's value. However, when volatility is less specialized, or when the underlying process is discontinuous or non-Markovian, a call's price can be a decreasing, concave function of the underlying price over some range, increasing with the passage of time, and decreasing in the level of interest rates.     

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.     

The Cross-Sectional Relationship Between Trading Costs and Lead/Lag Effects in Stock & Option Markets

Prior empirical research has failed to settle the question of lead/lag effects between stock and option markets. This study investigates the relation between cross-sectional differences in trading costs and intraday lead/lag effects in stock and option markets. The data for the study comprise 19 firms sampled at five-minute intervals over a two-month period. Consistent with a trading cost hypothesis, results indicate overall stock market leading behavior. However, the lead appears to be related to option market trading costs. This study uses an error correction model framework to investigate the lead/lag effects. This approach provides information on both the long run equilibrating process as well as the short term interactions between stock and option markets. Information regarding the long run equilibrating process is important to the overall understanding of lead/lag effects and cannot be determined from time series models of differenced data. Specific criteria for assessing lead/lag effects in cointegrated series are also proposed. One advantage of these new criteria is their ability to identify leading behavior in the presence of feedback. All models are estimated with quote data and are constructed to eliminate overnight effects. Hence, the results are robust to previously identified distortions due to closing, overnight, and potential non-trading effects. However, caution should be employed in generalizing the results as the study covers a two-month trading period for a limited number of firms.     

Managing Employee Stock Option Expense: A Fair Value Approach

Owing to special characteristics, classic option pricing models are not well suited to the valuation of employee stock options (ESOs). This paper attempts to conduct a more general fair value estimation based on attaching performance targets to option vesting. Considering a setting that includes factors such as options that may be exercised early at employee discretion, employee exit rates and firm default risk, this paper presents a sensitivity analysis and empirical tests of option value. The results highlight the importance of considering the characteristics of ESOs in the design of performance‐vested option plans so as to provide the most attractive incentives for employees.     

The Determinants of Implied Volatility: A Test Using LIFFE Option Prices

This paper presents and tests a model of the volatility of individual companies' stocks, using implied volatilities derived from option prices. The data comes from traded options quoted on the London International Financial Futures Exchange. The model relates equity volatilities to corporate earnings announcements, interest-rate volatility and to four determining variables representing leverage, the degree of fixed-rate debt, asset duration and cash flow inflation indexation. The model predicts that equity volatility is positively related to duration and leverage and negatively related to the degree of inflation indexation and the proportion of fixed-rate debt in the capital structure. Empirical results suggest that duration, the proportion of fixed-rate debt, and leverage are significantly related to implied volatility. Regressions using all four determining variables explain approximately 30% of the cross-sectional variation in volatility. Time series tests confirm an expected drop in volatility shortly after the earnings announcement and in most cases a positive relationship between the volatility of the stock and the volatility of interest rates.