An Empirical Investigation of the Black-Scholes Model: Evidence from the Australian Stock Exchange

By McKenzie, Scott; Gerace, Dionigi et al. | Australasian Accounting Business & Finance Journal, December 2007 | Go to article overview

An Empirical Investigation of the Black-Scholes Model: Evidence from the Australian Stock Exchange


McKenzie, Scott, Gerace, Dionigi, Subedar, Zaffar, Australasian Accounting Business & Finance Journal


ABSTRACT

This paper evaluates the probability of an exchange traded European call option being exercised on the ASX200 Options Index. Using single-parameter estimates of factors within the Black-Scholes model, this paper utilises qualitative regression and a maximum likelihood approach. Results indicate that the Black-Scholes model is statistically significant at the 1% level. The results also provide evidence that the use of implied volatility and a jump-diffusion approach, which increases the tail properties of the underlying lognormal distribution, improves the statistical significance of the Black-Scholes model.

Keywords: Black-Scholes model, probability

(ProQuest: ... denotes formulae omitted.)

1. INTRODUCTION

Published in the Journal of Political Economy 1972 Fisher Black and Myron Scholes develop a model to price a European call option written on non-dividend paying stock. Rubinstein, (1994) states the Black-Scholes option pricing model is the most widely used formula, with embedded probabilities, in human history.

Since development of the model authors have consistently searched to test its empiricism1. Empirical investigations concede that the Black-Scholes model produces bias in its estimation. The assumptions of a historical instantaneous volatility measure and an underlying lognormal distribution do not hold. Yang (2006) suggests an implied volatility approach. Duan (1999) suggests the tail properties of the underlying lognormal distribution are too small.

One facet of the Black-Scholes model which has received scant attention is the underlying probabilities attributed to the model, in particular the probability that an option will be exercised. The Black-Scholes model estimates the probability of a European call option being exercised through the calculation of N(d^sub 2^) ; which is the probability relating to the strike price. To our knowledge no test seeks to explicitly test the underlying probabilities within the Black Scholes model with evidence from the ASX, providing a future reference to potential model misspecification.

This paper looks to empirically examine the accuracy and statistical significance of the factors within the Black-Scholes model, with evidence from the ASX. The investigation uses qualitative regression; logit and probit models; and a maximum likelihood approach. If as hypothesised the value of N(d^sub 2^) is the probability of an option being exercised, factors within the Black Scholes model should exert levels of statistical and economical significance, when regressed on a data sample of ASX option contracts.

The paper is organised as follows. Section 2 reviews the relevant literature of option pricing. Section 3 describes the data used in this investigation. The model is presented and compared to the Black-Scholes model in Section 4. Section 5 presents the empirical results. A conclusion is presented in Section 6.

2. LITERATURE REVIEW

Despite the extant literature on the Black-Scholes model the following is a brief review of empirical developments related to the central theme of this paper. Starting with Black and Scholes (1973) empirical investigations conclude bias within the Black-Scholes model in terms of moneyness and maturity. Successive papers document similar bias regardless of boundary conditions2.

Studies have also noted volatility bias in the Black-Scholes model. Black and Scholes (1973) using S&P 500 option index data 1966-1969 suggest the variance that applies over the option produces a price between the model price and market price. Black and Scholes (1973) propose evidence, volatility is not stationary. Galai (1977) confirm Black and Scholes (1973) that the assumption of historical instantaneous volatility need be relaxed.

MacBeth and Merville (1980) compare the Black-Scholes model against the constant elasticity of variance (CEV) model, which assumes volatility changes when the stock prices changes. …

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