TESTING STATISTICAL HYPOTHESES In order to apply different stochastic models like Black-Scholes, it is necessary to check the two basic assumption: • the return rates are normally distributed • the return rates are uncorrelated. We mention that using the Black-Scholes model we get, as a conclusion, the log-normal distribution of the stock price.
AMERICAN AIRLINES CASE STUDY As an example, we use the historical data from American Airlines. The data are listed chronologically, on a weekly basis, for the time period 1/2/87 - 9/20/96. For each date we have the corresponding closing stock price. We mention that there are some missing data, most of them due to holidays.
STEP I - USE ALL AVAILABLE DATA (1/2/87 - 9/20/96) NORMALITY TEST For the normality test we use the D’Agostino tests. Mathematical details are presented in the appendix. Departures from normality may be caused by skewness, kurtosis, or both. •
When we test for departures from normality due to skewness, the output includes the skewness coefficient (computed using the usual formula and the EXCEL one), the Z statistic and the corresponding p-value. If we reject the normal distribution hypothesis, we have a probability equal to p to make an error. Particularly for our study case, if we reject the normal distribution hypothesis we make an error with probability 2.7 x 10-10. This error is very small, we are of course ready to take such a small risk, and therefore we conclude that the distribution is not normal due to skewness.
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When we test for departures from normality due to kurtosis, the output includes the kurtosis coefficient (computed using the usual formula and the EXCEL one), the Z statistic and the corresponding p-value. If we reject the normal distribution hypothesis, we have a probability equal to p to make an error. Particularly for our study case, if we reject the normal distribution hypothesis we make an error with probability 2.2 x 10-16. This error is very small, we are of course ready to take such a small risk, and therefore we conclude that the distribution is not normal due to kurtosis.
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When we test for departures from normality due to either skewness or kurtosis, the output includes the chi-square statistic and the corresponding p-value. If we reject the normal
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distribution hypothesis, we have a probability equal to p to make an error. Particularly for our study case, if we reject the normal distribution hypothesis we make an error with probability 1.8 x 10-23. This error is very small, we are of course ready to take such a small risk, and therefore we conclude that the distribution is not normal due to either skewness or kurtosis. •
Based upon the D’Agostino tests, because we are faced with both skewness and kurtosis, we conclude that Black-Scholes provides just a rough estimate.
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The above conclusion is provided directly, in plain English, by the software, whenever we know the risk we are ready to take rejecting the normality hypothesis. Particularly for our study case, for a risk of 5%, the message is “Black-Scholes provides just a rough estimate”. For other cases, other potential answers are: • Black-Scholes provides a good estimate • Black-Scholes overprices out-of-the-money calls and in-the-money puts. It underprices out-of-the-money puts and in-the-money calls. • Black-Scholes overprices out-of-the-money puts and in-the-money calls. It underprices in-the-money puts and out-of-the-money calls. • Black-Scholes underprices out-of-the-money and in-the-money calls and puts. • Black-Scholes overprices out-of-the-money and in-the-money calls and puts.
For graphical purposes, we provide a graph with the real histogram and the theoretical normal histogram. We pick up the desired number of classes (always an even number), and as an output we get for each class its mid-point, the real and the theoretical frequencies. Of course we may use the chi-square test to compare the real and theoretical histograms. However, we do not recommend this test, because it is not sensitive enough. The D’Agostino tests presented above are by far more powerful. Particularly for our study case, we can see the significant departure from normality due to both skewness and kurtosis, a fact already diagnosed by the software. Once we have diagnosed a significant departure from normality, we are interested to know which dates are responsible for this fact. If the distribution is really normal, plotting the return rates as a function of the corresponding scores should result in a diagram where all points lie on a straight line. Additional details are listed in the Appendix. We can decide what type of scores we want to use (i.e., Blom, Tuckey, or Van der Waerden). Visually inspecting the “return rates - scores” diagram we can identify the outliers and the high leverage points. In order to do this on a statistical basis, we may use the output provided by the software: it includes the leverage, the standardized residual, the Jacknife residual, the Cook distance, the Welsch & Kuh distance, and the Belsley, Welsch & Kuh distance. One choice is to get the numerical values of the above listed statistics: in this case the user has to identify for each date the correct diagnostic. Another choice is to get directly the diagnostic, instead of the numerical values of these statistics: whenever we are faced with a normal point, the output is zero, while the abnormal points are flagged by an output equal to one. Although the first choice is by far more informative, the second choice may be more useful for the user. In order to easily make a decision, the data may be sorted either chronologically or by scores. Particularly for our study case, the return rates computed on a weekly basis jump from +948% to 1622% (per annum). The dates with huge absolute value return rates are flagged by most tests. It © Montgomery Investment Technology, Inc. / Sorin Straja, PhD Page 2
seems that the middle period exhibits a volatility significantly higher than the beginning or ending period. Using only the statistical tools is not possible to explain why we are faced with this behavior. A direct analysis of the history of the company or the industry may provide the answer. However, for computations affecting future decisions, we should not use all the available data. We have to acknowledge that significant changes took place, and therefore the company we are dealing with in 1996 is significantly different with respect to the company we dealt with in 1987. Based upon this conclusion, we decide to discard all data prior to 5/22/92. We have to repeat our statistical tests using only the data from 5/22/92 until 9/20/96.
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CORRELATION TEST To decide wether or not the return rates are correlated, when the dates are evenly spaced, we may use the autocorrelation function. Details are listed in the Appendix. When the return rates are really uncorrelated, the autocorrelation function should be zero for all values of the lag-time excepting the zero lag-time case. Of course, under real circumstances we are faced with autocorrelation functions that match more or less this ideal case. In order to take a decision we may visually inspect the shape of the autocorrelation function. In addition to this, the software provides the maximum lag-time value to be considered, the Q-statistic, and the corresponding p-value. If we reject the hypothesis that the return rates are uncorrelated, we make an error with probability equal to p. Particularly for our study case, the maximum lag-time to be considered is 22 weeks, the Q-statistic is 20.8, and the corresponding p-value is 0.53. We are not willing to take a risk of 53%, therefore we cannot reject the hypothesis that the return rates are uncorrelated. We have to point out that the autocorrelation function is estimated assuming evenly spaced data. In our case there are some missing dates, therefore the dates are not always evenly spaced, and henceforth the conclusion should be treated with circumspection. In order to bypass the restriction of evenly spaced data we may use the Lomb periodogram. The software provides the length of the output arrays, the Lomb periodogram, and the corresponding pvalue. If we reject the hypothesis of an uncorrelated noise, we have a probability equal to the p-value to make an error. We have to pick up a significance level, i.e. the risk we are willing to assume when rejecting the non-correlation hypothesis. The Lomb periodogram resembles to a cardiogram: it presents many peaks, some of them may be significant peaks, and others may be just background noise. A horizontal straight line corresponds to our significance level: whenever a peak is above this line it is a significant peak, otherwise it is just background noise. If the Lomb periodogram exhibits at least one significant peak, than we should reject the non-correlation noise. Particularly for our study case, the array length is 1964, and the selected significance level is 5%. All peaks are well below the horizontal line corresponding to this significance level, therefore we conclude that the return rates are uncorrelated. The output p-value is 90%, i.e. if we want to reject the non-correlation hypothesis we make an error with probability 90%.
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STEP II - USE RECENT DATA ONLY (5/22/92 - 9/20/96) NORMALITY TEST We basically repeat the same tests using recent dates, only. For the normality test we use the D’Agostino tests. •
If we reject the normal distribution hypothesis due to skewness, we make an error with probability 19.30%. This error is quite high, we are of course not ready to take such a high risk, and therefore we conclude that the distribution is normal.
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If we reject the normal distribution hypothesis due to kurtosis, we make an error with probability 37.43%. This error is quite high, we are of course not ready to take such a high risk, and therefore we conclude that the distribution is normal.
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If we reject the normal distribution hypothesis due to either skewness or kurtosis, we make an error with probability 65.25%. This error is quite high, we are of course not ready to take such a high risk, and therefore we conclude that the distribution is normal
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Based upon the D’Agostino tests, because we are not faced with neither skewness nor kurtosis, we conclude that Black-Scholes provides a good estimate.
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The above conclusion is provided directly, in plain English, by the software, whenever we know the risk we are ready to take rejecting the normality hypothesis. Particularly for our study case, for a risk of 5%, the message is “Black-Scholes provides a good estimate”.
For graphical purposes, we provide a graph with the real histogram and the theoretical normal histogram. We pick up the desired number of classes (always an even number), and as an output we get for each class its mid-point, the real and the theoretical frequencies. Of course we may use the chi-square test to compare the real and theoretical histograms. Once more, we do not recommend this test, because it is not sensitive enough. The D’Agostino tests presented above are by far more powerful. Particularly for our study case, we can see that there is no significant departure from normality, a fact already diagnosed by the software. Of course, there are still outliers and high leverage points, but just by visual inspection of the “return rates - scores” diagram we can see that their effect is quite limited. In our case, the return rates computed on a weekly basis jump from +460% to -627% (per annum), which is quite narrow when compared with the initial case. Therefore, for computations affecting future decisions, we may use the data from 5/22/92 until 9/20/96.
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CORRELATION TEST Particularly for our study case, the maximum lag-time to be considered is 14 weeks, the Q-statistic is 11.9, and the corresponding p-value is 0.61. We are not willing to take a risk of 61%, therefore we cannot reject the hypothesis that the return rates are uncorrelated. Once more, we have to point out that the autocorrelation function is estimated assuming evenly spaced data. In our case there are some missing dates, therefore the dates are not always evenly spaced, and henceforth the conclusion should be treated with circumspection. In order to bypass the restriction of evenly spaced data we may use the Lomb periodogram. Particularly for our study case, the array length is 876, and the selected significance level is 5%. All peaks are well below the horizontal line corresponding to this significance level; therefore, we conclude that the return rates are uncorrelated. The output p-value is 40%, i.e. if we want to reject the non-correlation hypothesis we make an error with probability 40%.
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CONCLUSION During the period 1/2/87 - 9/20/96 the company seems to have undergone significant changes. Part of the data should be discarded as past history, and only recent data should be considered as relevant to the today performance of the company. Based upon statistical tests we conclude that the return rates for the period 5/22/92 - 9/20/96 are normally distributed and uncorrelated. These data can be used for computations affecting future decisions.
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APPENDIX Refer to MITI Working Papers: B-S Valuation Normality and Correlation Stochastic Stock Prices Excel Worksheet with Calculations: AmrCaseStudy.xls
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