Statistics can be used to assess stock market moves and when certain assumptions are applied, to model option prices. This articles series include three related articles:

• Basic probabilities and the nature of stock market returns
• Probabilities and option prices
• Probability tools available in the Platinum platform

Understanding basic statistics as they apply to the stock market can provide both investors and traders with a reality check on returns, keeps risk management and important aspect of the plan and for option traders, provides a useful base for understanding pricing models. At all times, market participants need to recognize that probabilities never provide guarantees.

Interesting reading on the topic is provided at the end of the article.

Basic Probabilities

Perhaps the best place to start with probabilities is with two simple examples to lay the foundation. They both include tossing a standard coin which on a single toss provides two possible outcomes:

1. Heads (H) or
2. Tails (T)

Each time the coin is tossed these possible outcomes are the same – the new toss is in no way dependent on the previous one, the next one, etc. As a result, each toss is referred to as an “independent” trial.

• Chance of H: 1 of 2 possible outcomes (50%)
• Chance of T: 1 of 2 possible outcomes (50%)

If you begin to run some trials and record the results, you will find the actual percentages for H’s and T’s each move towards 50%. Assuming sufficient trials are completed you should expect your results to be very close to 50% H and 50% T. Figure 1a displays this result as a basic histogram.

Figure 1a Histogram for Single Coin Toss Trials

What if you simultaneously toss two coins, what are the potential heads and tails outcomes now?

1. H-H: 1 in 4 (25%)
2. H-T: 1 in 4 (25%)
3. T-H: 1 in 4 (25%)
4. H-H: 1 in 4 (25%)

For purposes of this discussion we are only concerned with the total number of heads and/or tails (combinations), not the order of them (not permutations). 

• H2: 1 in 4 (25%)
• H1T1: 2 in 4 (50%)
• T2: 1 in 4 (25%)

Once again when you perform a sufficient number of trials, you will find the percentage of actual results for each combination move towards the percentages provided. Figure 1b displays an updated view of the histogram for the two coin toss example.

Figure 1b Histogram for Two Coin Toss Trials

As you add coins the histogram progression in figures 1a and 1b will continue. The central combinations that split the numbers of heads and tails will represent the largest percentages and the combination that is all heads or all tails will be the smallest percentages. This is basically the foundation for a histogram view that fits a normal distribution, also known as the student’s bell curve.

Its value is allowing us to apply combination probabilities to much larger possible outcomes without actually running thousands of trials and to make other observations about expectations for the data.

Stock Market Returns

Instead of the number of heads of tails that come up during a coin toss, market participants can assess daily return data. Rather than counting the number of different combinations we can count the number of times the market moved different percentages, including for example:

• Gain of 5% or greater
• Gain between 1 and 5%
• Approximately unchanged (less than 1% move either way)
• Loss between 1 and 5%
• Loss of 5% or greater

If you collect data from the past and use a sufficient number of days, the histogram that results will also follow a distribution similar to the coin toss histograms. It won’t be the same, but returns do indeed follow a normal distribution when plotted this way.

Note that prices are not used in this exercise because price data poses a problem. Since price cannot go below $0 the data is bounded on the low end by this value. This will be reflected in the resulting histogram which actually follows a lognormal distribution. Return data does not have the zero level constraint and can be used as perhaps the quickest fix to the issues.

Still, stock market return data does not provide a textbook example of a normal distribution, all neat and symmetrical. However, even in its asymmetry, it provides investors and traders with useful information.

Historical Data View

I have provided market return data in this space already and want to use a slightly different approach this go around. Here I provide three data sets with each chart view to compare system oriented return data versus market return data for a similar holding periods. The data sets include holding period rate of change data for the following:

• Holding Period Returns for All Trading Days: 10-day Rolling Average (1975-2004)
• End of Month System (10-day Holding Period): All Months (1975-2004)
• End of Month System (10-day Holding Period): 4 Best Months (1975-2004)

So the system oriented data includes a 10-day period starting late in the month and holding through the beginning of the following month. The All Months data set provides results for each of the twelve months over a thirty year period and the 4 Best Months provides data for four of the twelve months. The Dow Jones Industrial Average (INDU) closing levels were used for back-testing purposes DIA, the widely traded exchange traded fund that closely tracks INDU was used in forward tests.

The remainder of this article displays and discusses return data from these three sets. StatSoft, Inc.’s Statistica^© was used to generate the histogram and normal probability curves, Microsoft^® Excel^® was used to create the scatter plots and data for both was downloaded from Worden Brothers, Inc.’s TeleCharts^® package.

Figures 2a-c provide scatter plots for the holding period returns. A regression line is highlighted in blue which makes the first scatter in particular a bit more descriptive. The primary focus is to display all of the return data for the system.

Figure 2a INDU 10-day Rolling Returns for All Days, All Months – Scatter Plot (1975-2004)

Figure 2b INDU System Returns for All Months – Scatter Plot (1975-2004)

While holding period returns are concentrated around the unchanged level, focusing on seasonally beneficial periods for the market (i.e. four best months) maintained some of the positive outliers while reducing the negative ones.

Figure 2c INDU System Returns for 4 Best Months – Scatter Plot (1975-2004)

A better plot to discuss probabilities is the histogram which is provided in figures 3a-c. In each chart the 0% holding period return level is highlighted by a vertical black line.

Figure 3a INDU 10-day Rolling Returns for All Days, All Months – Histogram (1975-2004)

This histogram is similar to daily return data for the period with some minor differences. In general, the majority of daily market returns are centered near the 0 level, but do favor positive returns (positive skew). Although difficult to see, the outliers are further out to the left side indicating that the negative outliers seem to have greater magnitude (fatter tail). This was better visible in the scatter.

Figure 3b INDU System Returns for All Months – Histogram (1975-2004)

While not as seemingly symmetric, figure 3b does display a similar positive skew for market returns. The most significant negative outliers occurred around the crash of 1987 which was not a holding period included in either system. While you do attempt to remove those instances when reasonably applying filters with a system, it seems to be more of a coincidence with these data sets.

In the case of figure 3c, the negative outliers are filtered out and the length of the left side tail reduced. In both cases the majority of returns are positive.

Figure 3c INDU System Returns for 4 Best Months – Histogram (1975-2004)

As a last view, Statistica allows a “line of best fit” type display for the data which allows the reader to see the extent to which a normal distribution applies. It’s not the exact way to describe the chart, but is reasonably close.

Figure 4a INDU 10-dy Rolling Returns for All Dys, All Mos – Normal Probability (1975-2004)

Although the negative returns in particular move of the line, you can see the majority of the return data fits the normal distribution well. Consider the data series includes over 7,000 values.

A better fit is displayed in figures 4b and 4c. The graph view removes the smoothness seen in distributions when there are a large number of data points.

Figure 4b INDU System Returns for All Months – Normal Probability (1975-2004)

Figure 4c INDU System Returns for 4 Best Months – Normal Probability (1975-2004)

Rather than advocating a system, the focus was to provide a view of the nature of US stock market returns:

• A normal distribution
• Returns centered near the 0 level
• A positive skew (or return bias) exists
• Fat tails (or outliers) are also present, particularly to the downside

Regardless of probabilities, remember that at the end of the day … anything is possible

Brightman, H.J. (1986). Statistics in Plain English. Cincinnati, OH: South-Western Publishing Company

Kase, C. (1996). Trading with the Odds:Using the Power of Probability to Profit in the Futures Market. New York, NY: McGraw Hill

Marlow, J. (2001). Option Pricing: Black-Scholes Made Easy. New York, NY: John Wiley& Sons, Inc.

Taleb, N.N. (2005). Fooled By Randomness. New York, NY: Random House, Inc. (Original work published in 2004)

Vince, R. (2007). The Handbook of Portfolio Mathematics: Formulas for Optimal Allocation & Leverage. Hoboken, NJ: John Wiley& Sons, Inc.

Clare White, CMT
Contributing Writer and Options Strategist
Optionetics.com ~ Your Options Education Site

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