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Home » Blog » 2014 » 02 » Enhancing Option Portfolio Returns Using Probability and Statistics - Part 5
By Craig Hilsenrath

For information on Fat Tail Distributions and The Empirical PDF refer to Enhancing Option Portfolio Returns Using Probability and Statistics -  Part 4.

The Empirical CDF

Using the empirical PDF a CDF can be constructed. The following chart shows the CDF derived from the PDF above. The red line depicts the empirical CDF and the blue line shows the log normal CDF for comparison. One difference between the two that is immediately apparent is that the log normal CDF is smooth and continuous whereas the empirical curve neither smooth nor continuous. The $SPX PDF used to construct the CDF is missing some bars in some of the upper end buckets. Even if all the buckets contained data the empirical CDF would still not be continuous. Recall that to calculate the expected profit a very small increment of standard deviations was used. Since the log normal CDF is a continuous function a value for any number of standard deviations can be computed. This is not the case for the empirical CDF. For values that are not points on the empirical CDF curve an interpolation method will need to be used to determine the probabilities.

One of the tradeoffs between using an empirical CDF versus the log normal CDF is that the former offers an accurate approximation of “tail risk” while the latter offers a better approximation of the probabilities. Another point to consider is what data are used to construct the PDF. If the $SPX PDF were constructed using only 3 years of daily data the 2008 financial crisis would not have been included. There are valid reasons for selecting both time frames depending circumstances. One could also choose to use weekly or monthly returns to construct the PDF. All of these choices will affect the shape of the CDF.

In summary, a fat tail distribution can offer a better estimate of expected profit but there is not one, correct fat tail distribution. An option trader who would like to use a fat tail distribution to calculate expected profit should be aware of what assumptions were made in constructing the PDF.

Cumulative Density

Expected Return

While having a positive expected profit is preferable it is not always sufficient. A trade with a positive expectation can still result in a loss so attention must be paid to risk. Expected return, the expected profit divided by the risk, tells an option strategist whether the expected profit is worth the risk.

In the long call example above the risk is $2.50, the cost of the call. So the expected return is approximately 30.54%. Suppose now that the cost of the call was actually $3.75. That would reduce the expected profit to approximately $0.14 for an expected return of 3.63%. Even though the trade still has a positive expectation, a return of 3.63% does not seem worth the risk. This conclusion is subjective and has to be weighed against alternative uses of the trading capital. Perhaps 3.63% is not a bad return given general market conditions. This type of comparison highlights another use of expected return.

The expected return of positions with different strikes, expiration dates and even different underlying assets can be compared using expected return. The final step is to normalize the return. Suppose a trader wants to compare the long call analyzed above with a 60 day call for a price of $1.85. The expected profit of the new call position is approximately $0.39 for an expected return of 21.26%. The question is would the trader prefer a return of 21.26% over 60 days or 30.54% over 90 days. The reason this needs to be considered is that the proceeds from the 60 day trade can be reinvested for the thirty days between the two expiration dates. Equation 4 shows how to calculate the compound annualized return.

Equation 4

equation 4

where:

re = The expected return.
t = The number of days over which the expected return is realized.

Using equation 4, the original 90 day trade yields 194.69% on an annualized basis and the 60 day trade yields 223.01%. This shows that the 60 day call has an edge over the 90 day call and the trader should prefer the new position.

Conclusion

Using expected value can help people to predict possible outcomes in many different areas. Casinos use probability and the expected value of the games that gamblers play to give the house an edge. In the same way option strategists can use expected return to establish an edge in their portfolios. By carefully establishing and maintaining positions with positive expected returns, option traders can use probability and statistics to enhance portfolio returns, and in a sense, be the house.

For powerful option trading software that calculates expected return, subscribe to Option Workbench. To download the fullEnhancing Option Portfolio Returns Using Probability and Statistics white paper in its entirety, visit the Option Workbench page and fill out the form at the bottom-right section of the page.