Charles J. Corrado

The Specification and Power of the Sign Test in Event Study Hypothesis Tests Using Daily Stock Returns

This paper evaluates a nonparametric sign test for abnormal security price performance in event studies. The sign test statistic examined here does not require a symmetrical distribution of security excess returns for correct specification. Sign test performance is compared to a parametric t-test and a nonparametric rank test. Simulations with daily security return data show that the sign test is better specified under the null hypothesis and often more powerful under the alternative hypothesis than a t-test. The performance of the sign test is dominated by the performance of a rank test, however, indicating that the rank test is preferable to the sign test in obtaining nonparametric inferences concerning abnormal security price performance in event studies.

Event Studies: A Methodology Review

Originally developed as a statistical tool for empirical research in accounting and finance, event studies have since migrated to other disciplines as well, including economics, history, law, management, marketing, and political science. Despite the elegant simplicity of a standard event study, variations in methodology and their relative merits continue to attract attention in the literature. This paper reviews some of the fundamental topics in short-term event study methodology, with an attempt to add new perspectives to some pressing topics.

IMPLIED VOLATILITY SKEWS AND STOCK INDEX SKEWNESS AND KURTOSIS IMPLIED BY S&P 500 INDEX OPTION PRICES

The Black-Scholes (1973) option pricing model is used to value a wide range of option contracts. However, the model often inconsistently prices deep in-themoney and deep out-of-the-money options. Options’ professionals refer to this phenomenon as a volatility ‘skew’ or ‘smile.’ In this paper, we apply an extension of the Black-Scholes model developed by Jarrow and Rudd (1982) to an investigation of S&P 500 index option prices. We find that non-normal skewness and kurtosis in option-implied distributions of index returns contribute significantly to the phenomenon of volatility skews.

A note on a simple, accurate formula to compute implied standard deviations

Abstract We derive a simple, accurate formula to compute implied standard deviations for options priced in the classic framework developed by Black and Scholes (1973) and Merton (1973). When a stock price is equal to a discounted strike price, this formula reduces to a formula provided by Brenner and Subrahmanyam (1988). However, their formulas accuracy is sensitive to stock price deviations from a discounted strike price. The formula derived here extends the range of accuracy to a wide band of option moneyness.

Risk Aversion, Uncertain Information, and Market Efficiency

Abstract. We reexamine and extend tests of the uncertain information hypothesis (UIH) proposed by Brown, Harlow, and Tinic (1988, 1993). We find that their empirical results are sensitive to the sampling procedure employed and that their particular methodology does not sufficiently distinguish between event and nonevent periods. When the sampling procedure is modified to identify only relatively large, isolated events, the test results generally do not support the UIH. Instead, significant price shocks are consistently followed by short-lived price reversals. We observe this behavior following positive and negative events regardless of whether the event is classified as risk increasing or risk decreasing.

An Empirical Test of the Hull-White Option Pricing Model

INTRODUCTIONThe Black-Scholes (1973) option pricing model provides the foundationfor the modern theory of options valuation. In actual applications, how-ever, the model has certain well-known deficiencies. For example, whencalibrated to accurately price at-the-money options the Black-Scholes(1973) model often misprices deep in-the-money and deep out-of-the-money options. This model-anomalous behavior givesrisetowhatoptionsprofessionals call “volatility smiles.” A volatility smile is the skewed pat-tern that results from calculating implied volatilities across a range ofstrike prices for an option series. This phenomenon is not predicted bythe Black-Scholes (1973) model, since volatility is a property of the un-derlying instrument and the same implied volatility value should be ob-served across all options onthatinstrument.Volatilitysmilesaregenerallythought to result from the parsimonious assumptions used to derive theBlack-Scholes model. In particular, the Black-Scholes (1973) model as-sumes that security log prices follow a constant variance diffusion pro-cess. The constant variance assumption has been tested and rejected inearly studies by Beckers (1980), Black and Scholes (1972), Christie

The exact distribution of the maximum, minimum and the range of Multinomial/Dirichlet and Multivariate Hypergeometric frequencies

The exact distribution of the maximum and minimum frequencies of Multinomial/Dirichlet and Multivariate Hypergeometric distributions of n balls in m urns is compactly represented as a product of stochastic matrices. This representation does not require equal urn probabilities, is invariant to urn order, and permits rapid calculation of exact probabilities. The exact distribution of the range is also obtained. These algorithms satisfy a long-standing need for routines to compute exact Multinomial/Dirichlet and Multivariate Hypergeometric maximum, minimum, and range probabilities in statistical computation libraries and software packages.

A Nonparametric Distribution-Free Test for Serial Independence in Stock Returns: A Correction

A fundamental statistical test of serial independence developed by Ashley and Patterson (1986) to examine a possible form of serial dependence in daily stock returns is shown to be improperly constructed. As a consequence, the significance probabilities that they ob? tain are overstated. This paper presents a corrected version of their test. The test statistic obtained after correction is shown to possess the same limiting distribution as the Kolmogorov-Smirnov test statistic. Applying the corrected test procedure to data identical to that used by Ashley and Patterson, we find that their original null hypothesis can no longer be rejected at conventional significance levels.

Hurdle Rate: Executive Stock Options

Executive stock options with a rising strike price are a recent innovation in executive compensation in Australia and New Zealand. These options combine a dividend protection feature and a strike price that increases at a hurdle rate set with reference to a cost of capital estimate. With a constant dividend yield, the strike price becomes a path-dependent function of the stock price and exact analytic valuation becomes intractable. However, path-dependent American options can be valued using a Monte Carlo approach proposed in Longstaff and Schwartz (2001). We examine procedures for valuing these options and compare them with Black and Scholes (1973) and Merton (1973) formula valuations.

Tweaking Implied Volatility

Hallerbach (2004) derives an approximation formula to compute a Black-Scholes implied volatility. This formula is equivalent to equation (7) in Corrado and Miller (1996a), with the substitution of a geometric average of stock and strike prices in place of an arithmetic average. Ceteris paribus the same numerical values are obtained. Although useful in a pedagogic setting, even with tweaking neither formula has the robustness typically required for commercial or research applications.