The Dow Jones Industrial Average in the twentieth century--implications for option pricing.

Hora, Stephen C. ; Jalbert, Terrance J.

ABSTRACT

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.

INTRODUCTION

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.

The remainder of the article is organized as follows. In the following section, the data and methodology are discussed. Next, the mixture distribution model for index changes is presented. The analysis continues by examining the empirical distribution of the DJIA as compared to the normal and Student theoretical distribution functions. When the predictions from the mixture probability model for index changes are compared to the historic record of changes the quality of the fit is much better than one could obtain with a normal distribution without the mixing. This is in contrast to the findings of Blattburg and Gonedes (1974) who find that monthly returns are nearly normal. Next, an application of these findings is provided. The Black-Scholes model is examined in light of the theoretical arguments and empirical findings. An alternative model is introduced that is based on the Student family of distributions is. The model is tested using data on DJIA put options.

DATA AND METHODOLOGY

To examine the historical record of changes, data on the daily level of the DJIA were obtained. Data were obtained from the Carnegie Mellon University SatLib Library, and from Sharelynx Gold. Carnegie Mellon University provides historical data on the DJIA from 1900 through 1993, including Saturday data when trading occurred on those days. This data is supplemented with recent data from Sharelynx Gold. The final data set extends from January 1, 1900 through December 31, 1999.

The historical record of changes is examined through the use of Q-Q plots. Q-Q plots are used to analyze distributions by comparing theoretical distribution functions to empirical distribution functions. The Q-Q plot, described by Wilk and Gnanadesikan (1968), provides a visualization of the fit between an assumed distribution and data. By convention, the theoretical quantiles of the assumed distribution are plotted on the horizontal axis against the ordered values of the data plotted on the vertical axis. When the data are a random sample originating from the theoretical distribution, except for a possible linear transformation of the data, the plot will be approximately linear. Departures from linearity indicate that the data have a parent distribution other than that of the theoretical quantiles. When empirical values are related to the theoretical distribution such that the data are realizations of the random variable X = [mu] + [sigma] Z. and Z has the theoretical distribution, the plotted line will have a slope of approximately s and will cross the vertical axis at approximately [mu]. To estimate the parameters for the Student distribution, we use maximum likelihood estimates. In addition, the parameters are estimated using a technique new to the literature. This new technique is based on the Q-Q plot and involves estimating the slope parameter by the value that maximizes the correlation between the observed log price relatives and the theoretical quantiles. One weakness of Q-Q plots is that they can hide extreme values near the origin which are the case in our analysis.

To examine these observations in additional detail, P-P plots are prepared. The P-P plot treats both ends of the spectrum equally showing the theoretical cumulative probabilities of the observations (vertical axis) plotted against the cumulative relative frequencies of the observations. To test the pricing precision of the option pricing model developed in this paper, data on put options on the DJIA were collected for a five year period commencing in November 1997 and ending in October 2002. Put option price data were collected from the Wall Street Journal. Prices were collected for each month, for options expiring in twenty-three trading days. Only put options with trading activity on the 23rd day prior to expiration have been included in this analysis. This procedure yielded 832 usable put option prices covering a time period of 60 months. Both the normal and Student models were optimized for the options prices of that month. The normal model was optimized with respect to the volatility while the student model was optimized with respect to both the volatility and the degrees of freedom parameter, v. The optimization criterion was to minimize the relative error of the model's evaluations where the relative error is given by (model value - market value)/market value.

The raw relative errors, by themselves, do not provide a test of the inconsistency of the normal model relative to the Student model. To construct such a test, the inverse of the degrees of freedom parameter, say [upsilon] = 1/v, is used to write the null hypothesis H0: [upsilon] = 0. When this hypothesis is true, the normal model is correct. The alternative considered here is that [upsilon] > 0 indicating that the normal model is inconsistent with the data relative to the Student model. Gallant (1975) shows that an approximate test of the hypothesis that a parameter's value is equal to zero can be obtained by examining the sum of square residuals of the constrained and unconstrained models. Moreover, this test is quite analogous to the reduced model test commonly used in regression analysis. Let SS0 and SS be the sum of squared residuals for the constrained model ([upsilon] = 0) and the unconstrained model. Then F = (n-p)[SS.sub.0]/SS, where n is the number of observations and p is the number of parameters determined by the data in the unconstrained model, will be approximately distributed as an F random variable with 1 and n-p degrees of freedom. For our purpose, p will always be 2 but n will vary from month to month depending on the number of different put options being traded.

THE MIXTURE DISTRIBUTION MODEL FOR INDEX CHANGES

A distribution function is the best guess of how future events will actually occur. It is a mapping of the possible outcomes from an event. The many different possible maps of the future that can be hypothesized have given rise to many different distribution functions in the literature, each with its own properties. A distribution function can be described based on its mean, variance, skewness and other higher order moments. The most basic of these distributions is the normal distribution, which appears as the well known bell curve. The normal distribution is specified by the mean and variance. Here, the focus is on the variance of the distribution function.

During the past two decades, a number of articles have appeared in the finance literature related to behavior of the variance (or its square root, the standard deviation or volatility) over time. Some investigators have attempted to model the behavior of the variance as a time series in order to predict its expected value at a future point in time. Most notable is the generalized autoregressive conditionalized heteroscedacity model (GARCH) presented by Bollerslev (1986). Integrating the GARCH framework into the valuation of options has been accomplished by Heston and Nandi (1997) up to the point of an integral equation requiring numerical evaluation. The valuation equation is derived by inverting the characteristic function of the distribution of the future value of the underlying asset.

Hull and White (1987) propose that variance be modeled as a stochastic process and they conclude that the value of an option is given by the expectation of the conditional value of the option given the volatility where the expectation is taken with respect to the probability distribution of the average volatility over the duration of the option. An essential difference in their approach vis-a-vis that given here is that we account for the changing variability in the distribution of the future value of the underlying asset by marginalizing the conditional distribution of log price relatives with respect to the distribution of the variance. The marginal distribution is then used to recast the option evaluation model.

A frequently used model in Bayesian statistics and decision analysis that accounts for uncertainty in the variance of the process is the normal-gamma natural conjugate relation. Briefly, this relation allows that a joint posterior distribution for the mean and variance of a normal process be in the same family as the joint prior distribution when the information is updated by a sample of values from a normal process (Raiffa and Schlaifer 1961). The marginal density of the uncertain variance V, up to a constant, is given by:

f(V|[alpha], [beta])[[varies]e.sup.-[beta]/V] [V.sup.-[alpha]-1]. (1)

This density is termed an inverted gamma density as h = 1/V will have the usual gamma density, which up to a constant, is given by:

f(h|[alpha], [beta])[[varies]e.sup-[beta]h][h.sup.[alpha]-1]. (2)

The parameter h is called the precision of the process.

Next consider a sequence of independent random variables each drawn from a normal distribution with mean [mu], but each having a variance independently drawn from the inverted gamma distribution. This sequence of random variables will be indistinguishable from a similar sequence of student random variables having a centrality parameter of [mu], a precision parameter of h = [beta]/[alpha], and a shape parameter (degrees of freedom) of v = 2[alpha]. The density of each of these random variables is:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

What is important here is that modeling the uncertainty about the variance applicable to any price relative through the inverted gamma distribution leads to a distribution of price relatives different from that usually assumed. Moreover, the distribution of price relatives will have thicker tails as the Student density has greater kurtosis than the normal density.

The conditions necessary for the distribution of log relative prices to be a member of the Student family will be given for both ex post and ex ante perspectives. Ex post, consider a sequence of log relative prices [Y.sub.1], [Y.sub.2] ... such that the sequence consists of subsequences of independent normal values with a constant variance in each subsequence and a mean common to all subsequences. Denote the length of the ith such subsequence by [n.sub.i]. Assume that the variance of the normal distribution generating values in the ith subsequence is drawn randomly and independently (with respect to the variances of other subsequences) from the distribution given in equation (1). Let the total number log prices in the sequence be m = [n.sub.1] + [n.sub.2] + ..... Then, if for each i, ni/m approaches zero as m grows without bound, the sequence [Y.sub.1], [Y.sub.2], ... will have an empirical distribution function that converges to a member of the student family whose parameters depend on the values of [alpha] and [beta] in equation (1).

The essence of the condition stated above is that the volatility changes over time but remains fixed within time periods that are asymptotically negligible with respect to the length of the sequence. The lengths of the subsequences are arbitrary and restricted only by the negligibility assumption. This assumption is much weaker than those imposed by Garch models and the resulting model is simple enough to have practical application. From the ex ante perspective, the following assumptions lead to the Student model for the future value of an asset: a.) The distribution of the log of the future price relative to the current price has a normal distribution with a known mean but uncertain variance and b.) The uncertainty about the variance is expressed by the density in equation (1). In the following sections, both the ex post and ex ante perspectives will be examined empirically. First, the historical record of the DJIA is examined and compared to the student model to provide an evaluation from the ex post perspective. This is followed by an examination of the pricing of puts from an ex ante perspective where the valuations provided by the market are compared to valuations made using the Student model.

THE HISTORIC RECORD

In this section, the historical record of changes in the level of the DJIA is examined. The section begins with an examination of the daily price relatives given by [Y.sub.i] = ln([X.sub.i]/[X.sub.i-1]) where [X.sub.i] is the closing value of the DJIA on the ith day. Note that the price relatives calculated here ignore any returns from dividends. Over the past century there have been 27,425 of these price relatives. One of these price relatives has been dropped from this analysis. This was done because the New York Stock Exchange was closed for a period of several months during World War I. The price relative from this close to the subsequent reopening has been eliminated because of the excessive period between prices. For other closings, such as weekends or holidays, the price relatives have been computed on the closing values of the consecutive trading days without adjustment for any intervening non-trading days. Lawrence Fisher suggested that the interposition of nontrading days could explain the thickness of the tails for stock price relatives (as noted in Fama, 1965). Such a model would employ a mixture of distributions differentiated by the presence and number of nontrading days between trading days. Fama (1965) however, found no empirical support for this argument. Examining a random sample of eleven stocks from the Dow Jones Industrial average, Fama (1965) found that the weekend and holiday variance is not three times the daily variance as is suggested by the mixture of distributions model. Rather, the weekend variance is found to be about 22 percent greater than the daily variance.

Figures 1a and 1b are the normal Q-Q plot and the Student Q-Q plot, respectively, for the 27,474 daily price relatives. The shape or degrees of freedom parameter for the Student plot was found using the method of maximum likelihood and is 2.985.

[FIGURE 1a OMITTED]

[FIGURE 1b OMITTED]

Nonlinearity is apparent in both Figures 1a and 1b but the amount of nonlinearity is much greater in Figure 1a than 1b indicating a poorer fit of the data to the theoretical distribution. The lack of fit is particularly pronounced in the tails in Figure 1a. A straight line appears in both figures. This line is the linear regression of the order observations (log price relatives) on the theoretical quantiles. The intercept provides an estimate of the location of the distribution while the slope provides a measure of the scale (standard deviation when it exists) of the data. The generalized log likelihood ratio test of the hypothesis of normality as compared to the alternative of a Student density produces a chi-squared statistic with one degree of freedom of [[chi].sub.1.sup.2] = 121,447 clearly favoring the alternative.

Obtaining maximum likelihood estimates for the Student density is somewhat tricky. The Solver optimizer in Excel 2000 often failed to converge to the correct estimates. This failure was detected by examining the derivatives of the likelihood function at the estimates. If these derivatives were not zero, the maximum likelihood estimates had not been found. A change to Premium Solver (Frontline Systems, 2001) consistently produced usable results.

Another, simpler, method for estimating the shape parameter, ?, of the Student distribution was developed. This method is based upon the Q-Q plot. The shape parameter is estimated by the value that maximizes the correlation between the observed log price relatives and the theoretical quantiles. This method is new to the literature and at this time, the statistical properties (sampling distribution and confidence intervals) associated with this method have not been developed. The method is very easy to apply relative to maximum likelihood estimation. It can be implemented on a spreadsheet using native Excel functions and the solver distributed with Excel.

Table 1 contains both the maximum likelihood estimates and correlation-based estimates for ? for three holding periods; 1 day, 23 days (approximately one month), and 274 days (approximately 1 year.) When estimating v for 274 day holding periods, it became apparent that one observation was particularly influential in determining the estimate of v. The corresponding period was mid 1931 to mid 1932. Eliminating this value and repeating the estimation process lead to a substantial increase in the estimate of v as seen in Table 1. Table 1 contains both the maximum likelihood estimates and correlation-based estimates for v, the shape parameter, for three holding periods; 1 day, 23 days (approximately one month), and 274 days (approximately 1 year).

Moment estimators, when available, often provide a simpler route to obtaining estimates. Although a moment estimator for v can be constructed from the fourth and second central moments (roughly the kurtosis and variance) such estimators fail for values of v [less than or equal to] 4 as the kurtosis fails to exists for v [less than or equal to] 4 just as the variance fails to exist for v [less than or equal to] 2. But it is this range of values that is of interest in describing the price changes for DJIA and thus we have not employed moment estimators.

Another path to obtaining an estimate of v is to examine the empirical volatility and to estimate the parameters of the gamma density from the empirical distribution of volatilities. While the historical record of daily closing values does not permit one to estimate one-day volatilities, as only one observation is available for each period, it does permit estimation for longer holding periods. Consider a 23 trading-day holding period, approximately one month. (Note: There are 1191 complete 23 day periods in the one-hundred year record versus 1200 months. During the early part of the 20th Century, the NYSE was open on Saturdays and thus there were more trading days per month during that period. Twenty-three days was chosen as the most representative integer number of days for a month for the entire period and consistently adhered to throughout the study.) We assume that in each 23 day holding period there is a constant volatility but the underlying volatilities differ from period to period according to the inverted-gamma process described earlier. Precisely, during each 23 day holding period there is a precision, say h, so that the daily price relatives during the period are normal with mean [mu] and standard deviation [h.sup.1/2]. Moreover, if the relative price changes in each holding period are independently and identically distributed normal random variables, the empirical volatilities, [S.sub.23], are related to the chi-square random variable [[chi].sub.k.sup.2] = by [[chi].sub.k.sup.2] = k h [S.sub.23.sup.2] where k = n -1 and n is the number of trading days in the holding period, in this case 23. The value k is the number of degrees of freedom for [[chi].sub.k.sup.2].

Now, [chi]sub.k.sup.2/ [(n-1) h] = [S.sub.23.sup.2] so that [S.sub.23.sup.2] depends on both Y and h. The joint distribution of [[chi].sub.k.sup.2] and h is given by:

g(x,h)[varies] x.sup.k/2-1] [h.sup.k/2 + [alpha]-1] [e.sup.-h]([[beta] + xk/2]). (4)

From this joint density, the unconditional density of S232 is easily found and is given by:

f(s) [varies] s.sup.k/2-1]/[(2[beta]/k + s).sup.k/2+[alpha]]. (5)

The unconditional density of the holding period variances, [S.sub.23.sup.2], is known as an inverted beta-density with parameters k/2, [alpha], and 2[beta]/k. (Raiffa and Schlaifer, 1961). The quantiles of this density maybe found by direct transformation from the standard beta density with parameters k/2 and [alpha]. The required transformation is s = 2[beta]x/[k(1-x)] where x is a quantile of the beta distribution and s is the resulting quantile of the distribution of [S.sub.23.sup.2].

Figure 2a displays a Q-Q plot of the 1191 values of [S.sub.23.sup.2] against the theoretical quantiles of the inverted beta-2 distribution with k = 22 and [alpha] = 2.18. The plot shows good linearity with exception of the two most extreme values which are both somewhat smaller than one might expect. The value of [alpha] was found by maximizing the correlation between the ordered data values and the theoretical quantiles.

[FIGURE 2a OMITTED]

The companion figure, 2b, shows the inverses of the empirical variances, the empirical precisions, plotted against their theoretical quantiles which are just the inverses of the quantiles of the inverted beta-2 distribution for the 1191 values with 23-day holding periods. Here, the linearity is even stronger. This Q-Q plot "hides" the two extreme values identified in Figure 2a near the origin, however. It is clear that each of the two plots compresses a different end of the spectrum of values, accentuating one end at the cost of sensitivity in the other end of the spectrum. A plot that treats both ends of the spectrum equally is the P-P plot which shows the theoretical cumulative probabilities of the observations (vertical axis) plotted against the cumulative relative frequencies of the observations.

Figure 2c is the corresponding P-P plot for the empirical variances. The plot for the precisions would be identical except the order would be reversed. For the P-P plot, it is necessary to estimate the parameter [beta], for the plot to be meaningful. This was not the case for the Q-Q plot in which [beta] determined the slope, but not the linearity, of the regression. The parameter [beta] was estimated by maximizing the correlation between the theoretical cumulative probabilities and the cumulative relative frequencies. The resulting value is [beta] = .00011. Alternative estimates of both [alpha] and [beta] can be obtained using the methods of moments. Designating the ith central moment as mi we have [m.sub.1] = [beta]/([alpha]-1) and [m.sub.2] = [m.sub.1.sup.2](n-1)/2 + [alpha] 1](2/k)/([alpha]-2). Solving for [alpha] and [beta] in terms of the moments gives [alpha] = [k(2r-1)-2]/(rk-2) and [beta] = ([alpha]-1)[m.sub.1]. Examining the expression for [m.sub.2] we see that the moment will not exist if [alpha] [less than or equal to] 2. This limits the usefulness of the moment estimators as, recalling that the degrees of freedom for the student distribution is twice [alpha], it is this range of values that are of interest for the 23 day holding period.

[FIGURE 2b OMITTED]

Figures 3a and 3b are the normal and Student Q-Q plots for the 23 day holding periods. Figures 3a and 3b are the normal and student Q-Q plots for the 23 day holding periods of the Dow Jones Industrial Average Index from 1900-2000 respectively. Again the behavior of the price relatives is better modeled by the Student density than the normal density. This is particularly true of extreme changes, both positive and negative. The generalized log likelihood statistic is again highly significant (chi-squared with one degree of freedom with a value of 2884) leading to the conclusion that the distribution of price changes is better represented by the Student density than the normal density.

[FIGURE 2c OMITTED]

[FIGURE 3a OMITTED]

[FIGURE 3b OMITTED]

Finally, the historical record for 274 day holding periods is examined. Figures 4a and b display the Q-Q plots for the normal and Student densities, respectively.

[FIGURE 4a OMITTED]

[FIGURE 4b OMITTED]

The Student density has 3.58 degrees of freedom which maximizes the correlation between the theoretical and empirical quantiles. The case for the mixture densities is not as strong here as it was for the 23-day holding periods. Examination of the companion normal Q-Q plot shows reasonably good fit in the upper end of the distribution but poorer fit in the lower tail with one price relative being much larger than is consistent with the normal distribution. The Student Q-Q plot partially corrects for the most extreme observation and has better fit in the entire lower tail compared to the normal. Still, this extreme observation, which represents the period from mid 1931 to 1932, appears to be extraordinary. It is interesting to note that this extreme value is nearly five sample standard deviations below the sample mean. Using the maximum likelihood estimates of the parameters of the normal and Student distributions, gives cumulative probabilities for this observation of .0000005582 for the normal model and .0012 for the Student model. Once again the likelihood ratio test soundly rejects the hypothesis of normality with a chi-squared statistic of 79.

THE BLACK-SCHOLES MODEL

The Black Scholes Option Pricing Model (Black and Scholes, 1973) can be used to compute the value of an option. Consider an option with a strike price x and time to maturity of t, on a stock with a current asset price of p, t days before expiration, and the volatility of the log price relative over the entire t day period is s. With a risk free rate of interest of r, the Black Scholes model prices call and put options respectively as follows where n(d) is the value of the cumulative normal distribution evaluated at d1 or d2:

Vc = n(dl)p-x([e.sup.-rT])n(d2)

Vp = x([e.sup.-rT])n(-d2)-pn(-dl)

where: dl = ln(P/x) + [r + [s.sup.2]/2]t/s[square root of t] and d2=d1-s[square root of t]

In its raw form, the Black Scholes model is only applicable to non dividend paying European options. However, many revisions of the model have been developed to handle other situations and special applications. Merton (1973) modified the Black Scholes model to accommodate continuous dividends. Black (1975), Roll (1977), Geske (1979, Whaley (1981) and Broadie and Glasserman (1997) all developed models for valuing American options. Models for valuing options on futures have been developed by Black (1976) and Ramaswamy and Sundaresan (1985). Other models have been developed for pricing options on stock indexes (Chance, 1986), options on currencies, (Amin and Jarrow, 1991, Bodurtha and Courtadon 1987, and others), and options on warrants (Lauterbach and Schultz, 1990)

Development of the Black and Scholes model was based on a number of assumptions. One of the assumption inherent in the usual formulation of the Black-Scholes model (Black and Scholes, 1973), is that the log of the ratio of successive prices of an underlying asset follow a Weiner process (Feller, 1971). This, in turn, requires that successive changes over equal time intervals are independently and identically distributed normal random variables. In this paper, the primary concern is the assumption of identical distributions. Such a condition, often called stability, requires the mean and variance of returns to be constant over the period of concern. Suppose, in contrast, that the variance of the log of successive price-relatives varies so that the distribution of changes is not constant. One potential result is that the distribution will have thicker tails (greater kurtosis) than one would otherwise expect.

THE EVALUATION OF DEEP OUT OF THE MONEY OPTIONS

Deep out of the money options are those having a small value due to the strike price being much larger or smaller than the underlying asset's current value relative to the volatility of the asset's price over the remaining term of the option. For a call option, the strike price that is much greater than the current price relative to the volatility means that the option is deep out of the money. Conversely, a put option is deep out of the money if the strike price is much lower than the current price relative to the volatility. The pricing of such options is sensitive to the tail behavior of the underlying asset's price--the upper tail for deep out of the money call options and the lower tail for deep out of the money put options. While the well known Black-Scholes option pricing model has been shown to provide good estimations of option prices overall (See Black and Scholes, 1972, Galai 1977 and 1978), Macbeth and Merville (1979) and Rubenstein (1985) show that the Black and Scholes model miss prices deep out of the money options. That said, Rubenstien compares the Black and Scholes model to the jump model from Cox and Ross (1975), the mixed diffusion jump model from Merton (1976), the constant elasticity of variance model from Cox and Ross (1976), the compound option diffusion model of Geske (1979b) and the displaced diffusion model from Rubenstein (1983). He finds that none of the alternative pricing models consistently performed better than the Black and Scholes model. The evidence regarding the distributional properties of the DJIA presented above implies that pricing errors might be reduced by utilizing models that incorporate different distributional assumptions. The paper continues by developing such a model. Consider a theoretical European put option that has a strike price of x, a current asset price of p at t days before expiration, and drift of m for the t-day period. Further, assume that the volatility of the log price relative over the entire t day period is s. To be clear, s is the standard deviation of the log of the ratio of the price of the underlying asset t-days hence to the current price of the underlying asset. If we assume that the log price relative follows a normal distribution with mean m and standard deviation s, the present value of the expected return of the put option is given by the integral expression:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

where r is the risk free interest rate, t is the time until expiration of the option, and [PHI] is the standard normal distribution function. This expression is equivalent to Black-Scholes option pricing model if one makes the substitutions m = rt - [s.sup.2]/2 and s = [[sigma]t.sup.1/2]. Similarly, if the log price relative follows a Student distribution with parameters m, h, and v, the value of the option is:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

The price of the option is affected by changes in the underlying parameters in the same direction as the Black-Scholes model. Like the Black-Scholes expression, this expression involves integration and cannot be stated in simple terms. However, numerical evaluation of the integral is fairly straightforward. Here, Simpson's extended rule is used for evaluation (Press et al., 1992). The intention is to show that 1) the use of the student distribution vis-a-vis the normal distribution makes a significant difference in evaluating out of the money put options and 2) the well known volatility smile can be accounted for by the tail behavior of the student distribution.

For the example, consider a put on an underlying asset with an annual volatility of [sigma] = .2, a risk free interest rate of 0.1, and a current value of $100. To highlight the differences attributable to the differences in distributions, we will select parameters for the Student distribution that yield the same expected log price relative and the same variance of the log price relative as the normal distribution. Thus, we choose m = (rt - [[sigma].sup.2]/2)(T), h = v/[(v - 2) [[sigma].sup.2]. For the demonstration we will use n = 4 and T =1/12, corresponding approximately to a one month put on the DJIA. Exercising the normal and Student models for the value of the put option at various strike prices from $85 to $110 produces the values shown in Figure 5. Options that are out of the money appear on the left hand side of the graph. Options that are at the money occur at a strike price of $100, and options that are in the money appear on the right hand side of the graph. It is clear that the Student model provides higher values for deep out of the money put options and lower values for options with strike prices near the current price. The longer tails of the Student density then provide an explanation for the phenomena of the under pricing of deep out the money put options by the BlackScholes model. This further suggests that the problem can be corrected by altering the distributional assumptions utilized in the Black and Scholes model.

[FIGURE 5 OMITTED]

Figure 6 shows the implied volatilities needed to bring the Black-Scholes model (normal distribution) into equality with evaluations provided by the Student model. We note that the curve is similar to what analysts call a volatility smile curve (Hull, 1989), reinforcing the idea that the market prices options in a manner more similar to the Student model than the normal model.

[FIGURE 6 OMITTED]

Surprisingly, the Student model cannot be used in a risk neutral setting to price call options. The required integrals do not converge implying an infinite value to any call option. More precisely, E([e.sup.x]) does not exist if X is a Student random variable. This holds for any finite degrees of freedom. Conversely, E([e.sup.x]) does exist if X is a normal random variable. There are several possible explanations to reconcile the Student model and the obvious fact that these options have finite values in the market. First, an examination of the Q-Q graphs for 1-day, 23-day, and 274-day holding periods show some lack of symmetry in the tails of the distributions with the upper tail being somewhat less fat than the lower tail. If the upper tail were to have a distribution that approaches zero sufficiently fast, faster than a Student tail, the value of the option would be finite. Alternatively, the market may not evaluate options in a risk neutral manner. If the market were sufficiently risk averse, an argument could be constructed that would allow finite evaluations. Whether either of these explanations or some other explanation will bear fruit is an open question.

While the Student model can not be used in a risk neutral setting to price call options directly, all is not lost. Because put options can be valued in the risk neutral setting, put-call parity conditions can be utilized to price call options. Put-call option parity was first introduced by Stol (1969). Others have confirmed and refined the approach (Gould and Galai, 1974, Merton, 1973b). In order to price call options using put call parity, information on the current market value of a put option on the same asset with the same strike price and time to maturity, the strike price, the risk free rate of interest and the current market value of the underlying asset are needed. The put-call relationship is specified as C = P + S - PV(X). Where X is the exercise price, P is the current price of the put option as estimated using Equation 7 and S is the current market value of the underlying security. The put-call parity relationship can be utilized to compute the implicit price of any call option given the implicit price of the put option.

As a final demonstration, the normal (Black-Scholes) and Student models were applied to put options on the DJIA during a five year period commencing in November 1997 and ending in October 2002. Put option price data were collected from the Wall Street Journal. Prices were collected for each month, for options expiring in twenty-three trading days. Only put options with trading activity on the 23rd day prior to expiration have been included in this analysis. This procedure yielded 832 usable put option prices covering a time period of 60 months. The Treasury Bill rate for each month was used as the risk free rate and as the drift rate. Both the normal and Student models were optimized for the options prices of that month. The normal model was optimized with respect to the volatility while the student model was optimized with respect to both the volatility and the degrees of freedom parameter, v. The optimization criterion was to minimize the relative error of the model's evaluations where the relative error is given by (model value market value)/market value.

The results are presented in Table 2. The table contains pricing errors for the Black Scholes and Student models. MO is the option expiration month, N is the number of put options expiring in that month with trading on the 23rd trading prior to expiration, NV is the volatility that optimizes the normal model, NE is the average pricing error as computed by the Normal Model, SV is the volatility that optimizes the Student Model, NU is the degrees of freedom, SE is the average pricing error as computed by the Student Model, and RE is the error of the Student Model in relation to the Normal model. The spread parameter in the normal model is [sigma], the volatility rate. For the student model, we have reported [{v/[(v - 2)h]}.sup.1/2] which is the annualized standard deviation of the log price relatives when that standard deviation exists (i.e. v > 2.). This value is equivalent to s for infinite v

The average of the absolute values of these errors for the normal model is .2649 (26.49% error) while the Student model had an average error of 0.1458 (14.58% error). On average, the student mode error is 56.00% of the normal model error. Much of the error associated with both models is accounted for by options that are deep out of the money. Prices for options are quoted in discrete units ($1/16 increments prior to September of 2000 and $.01 increments after that date) and options that are worth very little will tend to exhibit a large relative error because of the relative lumpiness of prices at these low price levels.

Of course, the Student model must perform as least as well as the normal model because the normal model is a special case of the Student model with one less parameter--that is, the normal model is nested within the Student model. Thus, the raw relative errors, by themselves, do not provide a test of the inconsistency of the normal model relative to the Student model. To construct such a test, the inverse of the degrees of freedom parameter, say [upsilon] = 1/v, is used to write the null hypothesis H0: [upsilon] = 0. When this hypothesis is true, the normal model is correct. The alternative considered here is that [upsilon] > 0 indicating that the normal model is inconsistent with the data relative to the Student model. Gallant (1975) shows that an approximate test of the hypothesis that a parameter's value is equal to zero can be obtained by examining the sum of square residuals of the constrained and unconstrained models. Moreover, this test is quite analogous to the reduced model test commonly used in regression analysis. Let [SS.sub.0] and SS be the sum of squared residuals for the constrained model ([upsilon] = 0) and the unconstrained model. Then F = (np)[SS.sub.0]/SS, where n is the number of observations and p is the number of parameters determined by the data in the unconstrained model, will be approximately distributed as an F random variable with 1 and n-p degrees of freedom. For our purpose, p will always be 2 but n will vary from month to month depending on the number of different put options being traded.

The test described above has been run for each of the sixty months. The sample sizes (number of unique put contracts available) range from seven to twenty-four with a median of fourteen. In Table 3, we provide an analysis of the frequency distribution of 60 p-values for the test of H0: [upsilon] = 0, where [upsilon] = 1/v. The figure in each cell is the number of months having a p-value within the indicated range. Our conclusion is that the evidence is quite strong against the normal model relative to the Student model. In only three of the sixty months, using a significance level of .05, would one not be able to detect the inappropriateness of the normal model.

CONCLUSIONS

In this paper, the historical changes in the DJIA for the last 100 years are examined. There appears to be strong evidence that the log price relatives of the DJIA average do not follow a normal distribution - at least for one day to one month holding periods. A logical explanation of this non-normality is provided by the mixing model which accounts for changing volatility. The empirical record supports the use of a gamma type density for modeling the changing volatility. This has been show three ways: a.) Through Q-Q plots and likelihood tests of daily and monthly prices, b.) By examining the distribution of the variance of prices within 23 day periods and c.) Analyzing puts with varying strike prices by comparing normal (Black-Scholes) valuations and valuations using Student densities.

A practical conclusion that one can draw from the analysis is that the poor performance of the Black-Scholes model is due to the tail behavior of price changes. This behavior can be included in options pricing models to better reflect the behavior that markets price into options. The option pricing model developed here is much simpler than autoregressive formulations and is therefore better suited to practical applications. There is strong evidence to the support the Student model in favor of the normal model, from both ex post and ex ante perspectives. There are still open questions. While the Student model fits better for short and moderate periods, it has not been shown that this is the best model. Further, while the model indirectly provides finite prices for call options, it does not directly provide finite prices for call options. This issue suggests the opportunity for further research. To complete the analysis it was necessary to develop 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 by the value that maximizes the correlation between the observed log price relatives and the theoretical quantiles. 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. Fully investigating the statistical properties of this new method is another opportunity for future research.

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Table 1: Estimates of the Shape Parameter v

 Maximum Likelihood Correlation Estimate
Holding Period Estimate from Q-Q Plot

One day 2.82 2.98
23 Day (monthly) 3.89 3.95
274 Day (annual) 4.15 3.58
274 Day with One
Observation Removed 10.24 8.62

Table 2

Analysis of Pricing Errors for Black-Scholes and Student Models

 MO N NV NE SV NU SE RE

Nov-97 12 .2755 .1034 .3372 2.862 .0745 .7206
Dec-97 15 .3289 .2292 .4371 2.905 .147 .6414
Jan-98 9 .2738 .162 .3503 2.928 .1389 .8573
Feb-98 13 .2379 .2984 .441 2.266 .178 .5967
Mar-98 14 .2312 .353 .3741 2.251 .1348 .3817
Apr-98 10 .2006 .2879 .4104 2.179 .1102 .3828
May-98 14 .243 .346 .3458 2.429 .1619 .4679
Jun-98 14 .2127 .2372 .3327 2.376 .1636 .6897
Jul-98 14 .2219 .1402 .2967 2.836 .1218 .8683
Aug-98 10 .2023 .2171 .3367 2.348 .1042 .4799
Sep-98 12 .2831 .18 .3725 2.839 .1532 .8512
Oct-98 21 .3888 .2471 .5209 2.726 .1991 .8056
Nov-98 24 .3811 .3226 .5735 2.411 .1979 .6135
Dec-98 19 .3026 .3285 .5101 2.239 .1697 .5166
Jan-99 19 .2963 .2472 .4459 2.480 .1858 .7518
Feb-99 15 .317 .1337 .4478 2.586 .0944 .7063
Mar-99 18 .281 .1937 .4194 2.474 .1567 .809
Apr-99 17 .2791 .2307 .4 2.419 .1716 .7441
May-99 14 .3078 .2303 .4766 2.354 .0836 .363
Jun-99 14 .2947 .4459 .4365 2.402 .3051 .6843

 MO N NV NE SV NU SE RE

Jul-99 14 .2949 .4454 .4365 2.403 .3045 .6838
Aug-99 15 .1954 .5625 .352 2.31 .4359 .775
Sep-99 8 .2603 .2317 .3819 2.289 .1126 .4857
Oct-99 12 .2401 .2144 .3289 2.553 .1608 .7499
Nov-99 20 .2707 .2548 .4111 2.388 .1647 .6464
Dec-99 15 .2665 .4849 .3706 2.31 .3367 .6944
Jan-00 14 .2833 .4556 .5045 2.192 .1797 .3945
Feb-00 8 .2315 .1777 .3337 2.489 .0881 .4959
Mar-00 10 .2636 .1959 .3952 2.358 .1234 .6299
Apr-00 18 .2452 .3479 .4553 2.206 .1184 .3404
May-00 15 .3011 .1911 .452 2.398 .0875 .4578
Jun-00 13 .2711 .1806 .4494 2.268 .0593 .3282
Jul-00 7 .2591 .2551 .4572 2.225 .0907 .3554
Aug-00 7 .1826 .2047 .3183 2.289 .0712 .3477
Sep-00 8 .1837 .1962 .3285 2.211 .0574 .2927
Oct-00 14 .2098 .1257 .2528 2.845 .0809 .6442
Nov-00 11 .3079 .1066 .3965 2.671 .081 .7594
Dec-00 17 .2953 .337 .335 2.694 .1393 .4133
Jan-02 15 .2565 .1876 .4048 2.383 .1088 .5803
Feb-02 8 .2599 .2221 .3180 2.579 .0814 .3664

 MO N NV NE SV

Mar-02 10 .2284 .2431 .3295
Apr-02 21 .2813 .5261 .3219
May-02 15 .2731 .3277 .4178
Jun-02 16 .2481 .3759 .362
Jul-02 15 .2093 .3184 .4137
Aug-02 8 .247 .1542 .4381
Sep-02 10 .2350 .3358 .3579
Oct-02 15 .3659 .1353 .4649
Nov-02 14 .3536 .2195 .4806
Dec-02 20 .3346 .2399 .4956
Jan-02 10 .2501 .3074 .4766
Feb-02 12 .2727 .2154 .4100
Mar-02 12 .2369 .496 .4459
Apr-02 15 .1958 .2798 .3523
May-02 14 .1892 .2503 .3476
Jun-02 13 .2345 .3833 .41
Jul-02 7 .4283 .2003 .5929
Aug-02 17 .3608 .0843 .4319
Sep-02 22 .3485 .305 .5383
Oct-02 19 .3916 .1869 .5101
 Mean 14 .272 .2649 .4091

Table 3

Frequency Distribution of p-values for the Test of H0: u = 0

p [less than or .001 < p [less .01 < p [less .05 < p [less p >.1
equal to] .001 than or equal than or equal than or equal
 to] .01 to] .05 to] .1

 33 17 7 1 2