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Reading (and writing) about statistics is a challenge. Providing the necessary background and describing the information in different ways helps, but ultimately a picture is worth a thousand words. Using an example, we will drive home two key assumptions of the probability calculator and show how these assumptions impact the end result. Since the tool allows for user-defined inputs, it’s critical to understand certain aspects of the tool.

When incorporating trade and price probabilities into our analysis, we add a tool for trade selection. The tool still requires some input from the user and does not represent the success rate for a trade—it represents what will likely happen, assuming x, y and z. The tool will provide a valid statistical analysis if those assumptions are correct. Of course if all of our market assumptions were correct, trading would be easy.

Statistics and the Probability Calculator can be used by the trader as additional criteria for trade selection. Yes, a downside to these tools is that they look back in time and the markets don’t exactly repeat themselves. However, given two trades with similar risk profiles, which would you prefer, one with a higher probability of a profitable outcome or one with a lower probability? There is certainly a chance a stock won’t behave similar to the way it has in the past, but there is also a chance it will. If nothing else, it’s worth investigating—but only if you understand how the tool works.

Statistics & Price

When analyzing price history, there are a variety of off-shoot data sets we can consider:

1. Price or index levels,
2. Changes in price from a range or difference perspective, and
3. Change in price from a percentage or return perspective.

Each data set will look somewhat different when plotted on a graph. Because of this, the data set selected will determine which assumptions are reasonable and what calculations can be made from a statistical standpoint.*

Significance

What’s the significance of this information? It means we can generate expected returns, price levels and price changes from a statistical standpoint. The standard deviation bands provided on Optionetics Platinum provide the user with an endpoint range of values for a security based upon its historical movement over a certain period of time. Here is an image of expected price levels 127 days in the future using the 100 day statistical volatility [SV]:

Figure 1: Standard Deviation Price Projection Optionetics Platinum

In this example we have a stock with a closing price of 31.59, a 100-day SV of 28.13 and a Probability of Profit of 46.76%. The gray horizontal bands represent the trade’s upper and lower breakeven points and within these bands is the trade profitability zone. So in the example provided, the trader has a neutral or sideways outlook for the stock. The labels A and B represent the end of period price point for the stock [A] and the end of period price projection range for the stock [B]. A can also be viewed as the black horizontal line on the graph. Note that the distribution bands appear to be centered by this black line.

This forward looking distribution image is created using two critical assumptions:

1. The volatility for the 127 day period will remain at the 100-dy SV throughout the entire time (28.13), and
2. The closing price the day we completed this analysis is the expected mean at the end of the 127 day period (31.59).

In order to get from point A to point B, the process makes use of the lognormal distribution of the data and creates the red and blue standard deviation bands by using constant values for volatility (28.13) and price (31.59), and progressing forward D days.

Red Bands, 1 standard deviation:
1 StdD Up D days in the future = Price * e^{[V * sqrt (D/365.25)]} 
1 StdD Down D days in the future = Price * e^{[-V * sqrt (D/365.25)]} 

Blue Bands, 2 standard deviations:
2 StdD Up D days in the future = Price * e^{[2 * V * sqrt (D/365.25)]} 
2 StdD Down D days in the future = Price * e^{[-2 * V * sqrt (D/365.25)]} 

Given a trade that is profitable within the gray breakeven bands, the Probability of Profit is determined by calculating the extent to which these values fall within the standard deviation bands shown on the right axis, or point B, the end of period price projection range.

Keep in mind that the standard deviation bands are not constructed in the same way as Bollinger Bands—they do not make use of a “moving” data set. The annualized volatility value is constant. Also, we do not generate a new average value as we progress D days, the mean expected value in the future is constant. Each day, from day 1 through day 127 in the future, this calculation is completed with the same V and Price.

The bands are centered by a future mean value, with a slight upward bias. This bias will be discussed next week, but for now let’s look at what happens when changes are made in the assumptions. Recall that the Probability of Profits for this first set of assumptions is 46.76%.

Changing Volatility, Expected Price or Both

What if we changed the constant volatility value for volatility? That is, the volatility remains constant throughout the projected period, but this volatility level was actually lower than our original projection. Will the bands be wider or more narrow than those in the original image? The volatility has been changed from 28.13 to 15 and hopefully you correctly anticipated a narrowing of the end of period price distribution. Note the changing values on the right vertical scale.

Figure 2: Standard Deviation Price Projection with Decreased Volatility Optionetics Platinum

Now the gray breakeven bands are no longer completely within the red standard deviation bands; however, a greater portion of the prices within the 1 standard deviation level make for a profitable trade. The probability of profit rises to 70.5%. Graphically, the price range between the two bands has increased as a total percent of the range between the upper and lower red bands. In Figure 1 the area between the two breakeven points “filled” a lesser portion of the red bands. This makes sense from a strategy standpoint; if profits are captured when the stock moves sideways, decreased volatility in the underlying improves the prospect of pure sideways movement.

What happens if we expect an upward bias in price? Again, from a strategy point of view this bias should be modest, but we’ll exaggerate it for example purposes to a level that is above the upside breakeven. Figure 3 displays the same trade with the 100-day SV of 28.13, but a mean price of 35 for the stock in the future.

As an aside, how exactly might one determine a projected price level? This can be accomplished using fundamental or technical techniques, such as valuations based on growth, basic support and resistance levels or Fibonacci projections, among others. Those readers familiar with Elliott Wave Theory [EWT] have a tool available that provides such projections. ProfitSource charting software will calculate a Time & Price Projection [TAPP] using EWT that can be a user defined input value for the stock mean at a future date. Additionally, the time projection component also provides the user with a system oriented approach to select appropriate expiration months for the option strategy. As with statistical analysis, EWT does not guarantee a future price, but it’s a reasonable starting point for those who sufficiently understand the technique.

Figure 3: Standard Deviation Price Projection with a Change in Predicted Price Optionetics Platinum

Now the distribution is centered by the new expected price value and the distribution’s upward bias is even more skewed. The price distribution with one standard deviation has increased. At the same time, the lower portion of the profitability zone has move below – 1 standard deviation. Now the conditions reflect a decrease in the Probability of Profits to 36.2%.

Figure 4: Standard Deviation Price Projection Changing Volatility and Price Optionetics Platinum

Figure 4 incorporates both changes. The decreased volatility favors sideways movement in the stock while the projected price skews that movement upward. The graph reflects the fact that this situation decreases the chance of a downward movement that will capture more of the downside profit zone. As a result, the Probability of Profit further decreases to 32.0%.

Summary

Although the topic of understanding statistics may be “something nice” on its own, the primary goal for this article series is to provide the background necessary to gain a deeper sense of what underlies option pricing so we can distinguish relatively superior trades. It is an important step in evaluating single option positions, as well as combination positions. In addition, this allows us to make the most of analytical tools at our disposal.

In order to effectively use any tool the trader must understand the assumptions made by the tool and any shortcomings—i.e. a very high statistical probability of profits level does not guarantee profits. The trader also needs to understand how any changes to the model’s assumptions affect the outcome. The Optionetics Platinum Probability Calculator has two key assumptions: 1) the closing price the day the model is generated is the default for the expected mean future price which remains constant, and 2) the volatility level selected by the user (initial default to the more volatile 6-day SV) remains constant throughout the entire projection period. Next week we’ll run through specific trade examples.

* The Jerry Marlow text referenced in a previous article and on the Optionetics web site provides a great view of distinct distributions that can result when different calculations are applied to price. The book is worth the read if you wish to better visualize how price data looks statistically. The name of the text is “Black-Scholes Made Easy,” published by Wiley Trading.

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

To see other articles written by Clare White, 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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