In this paper, the historical changes in the Dow Jones Industrial Average index are examined. The distributions of index changes over short to moderate length trading intervals are found to have tails that are heavier than can be accounted for by a normal process. This distribution is better represented by a mixture of normal distributions where the mixing is with respect to the index volatility. It is shown that differences in distributional assumptions are sufficient to explain poor performance of the Black-Scholes model and the existence of the volatility smile. The option pricing model presented here is simpler than autoregressive models and is better suited to practical applications.
The Dow Jones Industrial Average (DJIA) has, for the past 100 years, been the single most important indicator of the health and direction of the U.S. capital markets. Composed of thirty of the leading publicly traded U.S. equity issues, the DJIA is reported in nearly every newspaper and newscast throughout the U.S. and the industrialized world. While the DJIA is not an equity issue itself, it has recently assumed this role through the advent of index mutual funds, depository receipts, and the DJX index option. Investors may "purchase" the DJIA through funds such as the TD Waterhouse Dow 30 fund (WDOWX) or through publicly traded issues such as the American Stock Exchange's "Diamonds," (DIA) a trust that maintains a portfolio of stocks mimicking the DJIA.
It is appropriate at the beginning of this new millennium to look back at the historic record of the DJIA to ascertain what information there might be in the record to assist analysts and investors.
This article advances the literature in three ways. The first contribution is to model the distribution of the DJIA over the past 100 years. The focus is on the relative frequency of index changes of various magnitudes - it is a tale about long tails. An analysis from theoretical, empirical, and practical perspectives leads to the conclusion that the distribution of changes over short to moderate length trading intervals (approximately one day to one month) can be represented by a mixture of normal distributions where the mixing occurs because the volatility of the index is not stationary (constant). Normally a mixture distribution is represented as the sum of several distributions weighted so the resulting sum is also a distribution. In our analysis the mixture is accomplished through a continuous mixing distribution on the index volatility and therefore the mixing is over an infinite array of normal distributions. If the mixing distribution for volatilities is a particular type of gamma distribution, the resulting distribution will be a member of the Student-t family of distributions as shown by Blattberg and Gonedes (1974). This result has important practical implications when one compares its ease of use to the stable Paretian family of distributions discussed by Fama (1965) and Mandelbroit (1963). The second contribution of this article is to develop and test a model of option prices based on the Student distribution. The model is simpler and thereby more suitable to practical applications than autoregressive models. Empirical tests demonstrate that this model is superior to the Black-Scholes model for pricing put options on the DJIA. The third contribution of this article is the development of a new method for estimating the parameters for the Student distribution. This new technique is based on the Q-Q plot and involves estimating the slope parameter as the value that maximizes the correlation between the observed log price relatives and the theoretical quantiles. While evaluating the statistical properties of this new method is beyond the scope of this paper, the new method is simpler and easier to use than maximum likelihood estimates. It also provides estimates in certain situations when maximum likely estimates can not be found. …