ABSTRACT

Nikkinen, Jussi (2001). Skewness and kurtosis adjusted Black-Scholes model: Small sample properties of MLE and test statistics and empirical evidence from

Finland.

Acta Wasaensia No. 94, 135 p.

The purpose of this dissertation is two-fold. First, the study uses the Monte Carlo method to evaluate the small sample properties of the maximum likelihood estimator (MLE) in the estimation of the Black-Scholes (1973) / Merton (1973) model with skewness and kurtosis adjustment terms. The results of the study show that the MLE of volatility, skewness and kurtosis and the MLE of the option price are not significantly biased, whereas the estimated asymptotic standard errors are substantially downward biased. Consequently, the exact sizes of the t-statistics substantially exceed the nominal levels. For the null hypothesis of the Black-Scholes model, the Lagrange multiplier, Wald and likelihood ratio statistics are badly oversized. Second, the study provides an empirical analysis of option implied risk-neutral stock return densities. The normality of the risk-neutral density is statistically assessed by testing restrictions regarding the skewness and kurtosis of the implied distribution and by applying the traditional test approach that involves a comparison of the pricing errors calculated using alternative models. It is found that both the approaches yield similar results, whereas the former method has the advantage that the significance of the estimated parameters can be statistically tested. Using data from the small Finnish market, the normality of the distributions is soundly rejected as expected based on the theoretical framework by Damodaran (1985). The implied distributions appear to be leptokurtic and in the most cases negatively skewed. Moreover, the study investigates the term structure behavior of the option implied risk-neutral probability density function. It applies the study by Das and Sundaram (1999) by providing statistical tests of their hypotheses about the term structure behavior of the conditional skewness and kurtosis of the return distribution. The results of the study reveal that the excess kurtosis decreases with maturity. Neither of the class of jump-diffusion nor stochastic volatility models is able to fully capture all features of the data generating process. Finally, the study investigates the effects of the abolition of foreign ownership restrictions on the stock price generating process. For that purpose, option implied probability density functions are examined using Finnish stock index derivatives data that cover continuously four and half years before and six years after the abolition of the ownership restrictions. The empirical results of the study reveal that the option implied distributions are significantly less leptokurtic after the abolition of the restrictions and support the hypothesis that the properties of the stock price generating process change when the foreign ownership restrictions are removed.

Jussi Nikkinen, Department of Accounting and Finance, University of Vaasa, P.0. Box 700, FIN-65101 Vaasa, Finland.

Key words: Black-Scholes model, skewness, kurtosis, implied distribution, maximum likelihood estimator