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The August article series was initiated by a discussion board question on the Kelly Bet Fraction. This one value in Optionetics Platinum only scratches the surface on statistical applications in the platform. Given the underlying data for which the software used—stock volatility and option pricing factors—this makes sense. Financial market analysis in general is a statistician’s dream; the data sets available are endless.

The goals of the next few article series are to provide the options trader with more background on statistical applications in trading and the Platinum user with a better understanding of what drives the platform. The graphs and displays used are from Optionetics Platinum and StatSoft Statistica, among others as noted.

Statistics Review

In statistics, a random sampling of data from a population is often used to infer something about that population. In order to reliably do this, the sample set must be large enough to truly be representative of the larger group and as stated, be random. Even when these two requirements are met, conclusions about the data include statements like, “95% confidence level” and “not rejecting the hypothesis.” The statements are not absolute.

At Optionetics we discuss putting the odds in your favor. The context may describe selecting a strategy where the market can move in 2 of 3 directions, rather than 1 of 3 directions or one that avoids options that are near expiration and far out of the money. In either case though, it’s possible for anything to happen; the market can move in the one direction that is unprofitable to the position or an out of the money, near term option can have intrinsic value at expiration. The same is true when statistics is applied to projections, expected values or reversion to the mean concepts. We seek to analyze past results to gain a sense of what may happen in the future always recognizing that ultimately anything can happen.

Expected and Future Values

Last week a coin toss example was provided and it was noted that each flip has one of two possible outcomes. When repeating the coin toss a large number of times (i.e. 1000) a random sampling is gathered with each separate toss serving as an independent result—a heads on one toss does not impact the result for the next toss. In the Platinum Probability of Profit calculations, the data set created includes a collection of projected stock prices using the stock’s statistical volatility each of which are obtained independently of the other.

Recall the statistical volatility [SV] is a value that provides an annualized standard deviation for a specific security. This means that stock prices are used over a set interval in the past to identify (measure) how dispersed the different prices are around an average or mean value. An additional calculation is then completed to normalize this value over a year’s time. Using current prices, past mean values and dispersion, a program then randomly projects future values for the security. This process is repeated over and over to obtain data set that is sufficiently large to represent the whole population of future values for the stock. Presumably, this means a lower limit of zero and an upper limit of infinity.

Assuming a stock price of 44 that changes an average of $0.11 each day in the last 50 days, the program may obtain a value of 0 or 95 for the stock 20 days in the future. However, given the past movement, we will likely find that more of the projected values are closer to one that results from $0.11 moves on a daily basis. So an expected value is not a target prices identified through the wisdom of experienced, professional traders, it’s a value that is obtained by collecting a sample set using past performance, then calculating a variety of measurements on the set.

Distributions

Mean and standard deviations are terms used to describe the shape of the normal distribution curve which is also referred to as the normal frequency distribution and the Gaussian distribution. The most universally understood statistical measures make use of this distribution, but it’s not the only one out there. Using S&P 500 monthly results over a 40-year period, we can use closing prices and price changes from the previous month (Returns) to provide two frequency distribution charts. The data is shown on a histogram with closing value on the x-axis and the number of times a particular value appears (frequency) displayed on the y-axis.


Figure 1: Monthly Closing Levels for S&P 500 (10/1965-7/2005) Statistica

Figure 2: Monthly Changes for S&P 500 (10/1965-7/2005) Statistica

As you can see the index level monthly close data is not normally distributed, but price changes are. The shape of the curve in Figure 2 also provides additional information—the S&P 500 is relatively non-volatile. Look at all of the observations that collect around the mean value. The bell shaped curve displayed is high and narrow (smaller deviation) rather than wide and flat (larger deviation).

The next couple of images display the Normal Probability Plots for Closes & Returns. Data that is normally distributed will exhibit a nice fit to the line added to the chart. Both charts have a blue line of best fit along with red standard deviation bands. Again, it becomes clear that price change (also thought of as returns) is normally distributed while price alone generally is not.


Figure 3: Normal Probability Plot of Closes for S&P 500 (10/1965-7/2005) Statistica



Figure 4: Normal Probability Plot of Returns for S&P 500 (10/1965-7/2005) Statistica

Ideally this provides you with a compelling argument for the validity of projecting future prices in the manner described, or does it? We’ve been describing price changes in percent terms rather than the system application which uses the difference between today’s price and yesterday’s price. So how does the difference data stack up? First, a histogram is displayed in Figure 5—note this represents the difference in index level changes on a monthly basis, not a daily basis. The exercise still holds value.


Figure 5: Monthly Differences for S&P 500 (10/1965-7/2005) Statistica

There definitely seems to be some reasonable potential here, so a check on the Normal Probability Plot is completed. Unfortunately, that does not appear to be as conclusive. How can what appears to be bell shaped data be off so far? 
 
Figure 6: Normal Probability Plot of Differences for S&P 500 (10/1965-7/2005) Statistica

While these tools may not readily be at your ready disposal, further testing of the data provides insight. It’s not uncommon to have to transform the data in order to obtain a workable distribution—one that can make us of valid relationships. In this case, testing a lognormal distribution immediately improves the picture. Although additional work is needed, it’s probably a good time to distinguish between logarithmic data, normal distributions and lognormal distributions.


Figure 7: Lognormal Probability Plot of S&P 500 Monthly Differences Statistica

Logarithms, Normal Distributions & Lognormal Distributions

Wikipedia.org offers a nice description in this area—it allows the reader to branch into other topics for additional information or clarification. As a summary, multiplication is to division what exponential form is to logarithms. So in the more commonly viewed exponential form, 2³ = 8 makes sense to us. The logarithmic inverse of this is log ₂ (4) = 3 with a log base of 2 as denoted by the subscript. One challenge comes in when standard nomenclature makes base assumptions (i.e. the use of the constant “e” for natural logarithms and the use of base 10 in the Arabic number system as an assumed value if nothing appears in the subscript).

The math review is all well and good, but what does it mean for data to have a lognormal distribution? It means that if you take the log of each data point you have a transformed set of data that is normally distributed. Certain calculations used for such data sets will be slightly different than the ones used with normal frequency distributions.

Standard Deviation Bands

Platinum 4.1 has the ability to overlay standard deviation bands on the stock price chart in Stock Charts and in Risk Graph II. Standard deviation bands assume the future stock price is log normally distributed and the stock price difference is normally distributed. These two assumptions are also used by the Black-Scholes option model to find a stock's implied volatility. The S&P 500 index level differences provided in Figure 7 need to be reconciled with this result.

The stock price in the unknown future is assumed to have a statistical distribution based on a volatility value, V. Given the volatility, V, the stock price in a future, D days from today and 1 standard deviation (StdD) from the stock price today can be computed using these equation (See Natenburg, Option Volatility and Pricing, Appendix B).

1 StdD Up D days in the future = Price today * e^{[V*sqrt(D/365.25)]} 

1 StdD Down D days in the future = Price today * e^{[-V*sqrt(D/365.25)]} 

The stock price 2 V standard deviations D days from today are:

2 StdD Up D days in the future = Price today * e^{[2*V*sqrt(D/365.25)]}

2 StdD Down D days in the future = Price today * e^{[-2*V*sqrt(D/365.25)]} 

Look at the left side image in this picture taken from the Help Guide: "Explain Odds and Probability of Profit." 

Figure 8: Price Prediction Function within Standard Deviation Bands Optionetics Platinum

The red parabolic curve intersects the chart boundary on the right side at two Future Stock prices +/– 1 standard deviation and 60 days from the current stock price. These two point intersections come directly from the above two 1 standard deviation equations.

The blue parabolic curve intersects the chart boundary on the right side at the two Future Stock prices +/– 2 standard deviations and 60 days from the current stock price. Those intersections would form the upper and lower limit of a standard deviation chart band predicted 60 days into the future.

These upper and lower stock standard deviation band values can be used to draw a continuous standard deviation chart band around the historical stock prices. Today’s stock close price is used as the Price today in the above equations and is fixed. The value chosen for V in the historical band has a number of options.

There are 5 ATM Implied Volatility values that could be used and 5 Statistical Volatility values that also could be used. And whichever V value we select, we could use the V values that existed back in time as we draw the stock band. The standard deviation stock band would show us how well the past stock values have stayed within today's stock price band.

An example of the standard deviation bands in Risk Graph II chart is shown next:

Figure 9: Standard Deviation Bands and Risk Graph for Condor Strategy Optionetics Platinum

The Historical Standard Deviation bands shown above are unique to Optionetics Platinum and not available elsewhere. 

The following provides details on how to turn the standard deviation band on and select its future days and volatility options.

1)     Go to Settings and select General Settings

2)     At far lower left is "Price Band." You have four options for the standard deviation band. Pick one.

§ 1 & 2 Stand Dev IV Bands - draws 1 and 2 standard deviation bands using IV as V.

§ 1 & 2 Stand Dev SV Bands - draws 1 and 2 standard deviation bands using SV as V.

§ 1 Stand Dev IV & SV Bands - draws two 1 standard deviation bands using IV as V and SV as V. 

§ 1 Stand Dev IV & Bollinger Band - draws two 1 standard deviation bands using IV as V and also draws the 20 day Bollinger Band around the stock price

3)     Look on the left side of General Settings for "Statistical Volatility". The selection shown there is also the selection used for SV in the standard deviation band.

4)     Look on the left side of General Settings for "IV Numerator". The selection shown there is also the selection used for IV in the standard deviation band.

5)     Now go to Charts and select Stock Charts.

a.   Stock Charts has its own saved settings for band selection.

b.   Select one of the above bands under the band selector option shown.

c.    Stock Charts uses the General Settings values for IV or SV bands

d.   The standard deviation band future days (D) is controlled by the day’s value under Bollinger Bands which has a 20 days default value. 

6)     Now go to Trade Tools and select Risk Graph II.

a.  Your General Settings values are used in this stock chart.

b.  The standard deviation band future days (D) is controlled by the days to expiration in the trade.

c.   If you have a lot of days left in the trade, the bands will be huge.

A lower limit of 7 days is used. So when your trade nears expiration, the standard deviation band D day’s value will not go below 7 days.

Summary

Option pricing models make use of different price data assumptions and distribution attributes to generate theoretical values. In a similar manner, these underlying relationships can also be used for additional price analysis to identify projected values. Once this is accomplished, Probability and Odds calculations can also be run to evaluate a trade. Ideally the combination of statistical review and application via the Optionetics Platinum software provides clarification for trade analysis you regularly complete.

Copyright© 2006 Optionsanalysis, Inc. and Optionetics, Inc. All Rights Reserved.

To see the other articles by these authors, please click here.


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

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