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.     

Why Option Prices Lag Stock Prices: A Trading-based Explanation

While many studies find that option prices lead stock prices, Stephan and Whaley (1990) find that stocks lead options. We find no evidence that options, even deep out-of-the-money options, lead stocks. After confirming Stephan and Whaley's results, we show their results can be explained as spurious leads induced by infrequent trading of options. We show that the stock lead disappears when the average of the bid and ask prices is used instead of transaction prices. Hence, we find no evidence of arbitrage opportunities associated with the stock lead.     

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.     

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.     

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.     

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.     

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.     

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.     

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.     

Generalised Geske--Johnson Interpolation of Option Prices

Abstract:  This paper describes four separate option types as special cases of Bermudans with general inter–exercise and time to final maturity. This produces a surface with European, finite American, infinite Bermudan and infinite American options as special cases. This allows Geske–Johnson (1984) two–point pricing to be extended to consider time–to–maturity as well as time–between–exercise opportunities. Due to their position on this 'map', infinite Bermudans are christened Arctic options and their pricing solution is presented. Numerical comparisons to benchmark methods are made for call prices under GBM although the results here hold for other processes and for both puts and calls when symmetry arguments are invoked.     

对股票期权的思考--股票期权在我国企业的适用性研究

起源于美国的股票期权制度在其本土已经得到了较为广泛的应用，我国在这方面也进行了积极的探索，一些企业也进行了实践。但股票期权究竟是否适合我国的情况呢?本将以主板市场为基础，通过对股票期权的实施条件的分析来对其在我国企业的适用性进行研究。     

Option Prices Under Generalized Pricing Kernels

In this paper analytical solutions for European option prices are derived for a class of rather general asset specific pricing kernels (ASPKs) and distributions of the underlying asset. Special cases include underlying assets that are lognormally or log-gamma distributed at expiration date T. These special cases are generalizations of the Black and Scholes (1973) option pricing formula and the Heston (1993) option pricing formula for non-constant elasticity of the ASPK. Analytical solutions for a normally distributed and a uniformly distributed underlying are also derived for the class of general ASPKs. The shape of the implied volatility is analyzed to provide further understanding of the relationship between the shape of the ASPK, the underlying subjective distribution and option prices. The properties of this class of ASPKs are also compared to approaches used in previous empirical studies. JEL Classification: G12, G13, C65 Erik Lüders is an assistant professor at Laval University and a visiting scholar at the Stern School of Business, New York University.     

Recovering Probability Distributions from Option Prices

This article derives underlying asset risk-neutral probability distributions of European options on the S&P 500 index. Nonparametric methods are used to choose probabilities that minimize an objective function subject to requiring that the probabilities are consistent with observed option and underlying asset prices. Alternative optimization specifications produce approximately the same implied distributions. A new and fast optimization technique for estimating probability distributions based on maximizing the smoothness of the resulting distribution is proposed. Since the crash, the risk-neutral probability of a three (four) standard deviation decline in the index (about ?36 percent (?46 percent) over a year) is about 10 (100) times more likely than under the assumption of lognormality.     

True Spreads and Equilibrium Prices

Stocks and other financial assets are traded at prices that lie on a fixed grid determined by the minimum tick size. Observed prices and quoted spreads do not correspond to the equilibrium prices and true spreads that would exist in a market with no minimum tick size. Using Monte Carlo Markov Chain methods, this paper estimates the equilibrium prices and true spreads. For large stocks, most of the quoted spread is attributable to the rounding of prices and the adverse selection component is small. The true spread and the adverse selection component are greater for mid-sized stocks.     

Comparison of Option Prices in Semimartingale Models

In this paper we generalize the recent comparison results of El Karoui et al. (Math Finance 8:93–126, 1998), Bellamy and Jeanblanc (Finance Stoch 4:209–222, 2000) and Gushchin and Mordecki (Proc Steklov Inst Math 237:73–113, 2002) to d-dimensional exponential semimartingales. Our main result gives sufficient conditions for the comparison of European options with respect to martingale pricing measures. The comparison is with respect to convex and also with respect to directionally convex functions. Sufficient conditions for these orderings are formulated in terms of the predictable characteristics of the stochastic logarithm of the stock price processes. As examples we discuss the comparison of exponential semimartingales to multivariate diffusion processes, to stochastic volatility models, to Lévy processes, and to diffusions with jumps. We obtain extensions of several recent results on nontrivial price intervals. A crucial property in this approach is the propagation of convexity property. We develop a new approach to establish this property for several further examples of univariate and multivariate processes.