Trade selection is usually based on applying several criteria and evaluating potential profitability, risk and other characteristics. Last month, we described the main approaches to multi-criteria analysis (MCA) and presented the method of finding the optimal Pareto set. This represents the most appropriate approach to handling many potential trading alternatives. Here, we examine practical advantages of MCA vs. single-criterion selection, and analyze how these advantages are affected by different factors inherent in options trading.

To test the effectiveness of MCA, we used the price history of options and their underlying assets from January 2003 to October 2009. For each expiration date, we created 50 sets of option combinations corresponding to different time intervals left until expiration. The most distant set was constructed 50 trading days before the expiration, the next one was 49 days before, and so on, up to the last set with only one day left to expiration.

In total, we created 4,100 sets of option combinations (50 sets for each of 82 expiration days). Each set consisted of 500 short straddles related to the underlying stocks that make up the S&P 500 index. All combinations used the strike closest to the current stock price. The volume of the position for each straddle was determined as $10,000 divided by the price of the respective stock.

Four criteria were used in two pairs. The first pair consisted of “Expected profit on the basis of lognormal distribution” (EPLN) and “Expected profit on the basis of empirical distribution” (EPEM) criteria. The second pair involved “Profit probability on the basis of lognormal distribution” (PPLN) and “Profit probability on the basis of empirical distribution” (PPEM) criteria.

EPLN is the integral of the payoff function of the combination over the probability density of lognormal distribution. Its value may be approximately estimated by summing the products of probabilities of each underlying price outcome and the value of the payoff function corresponding to these outcomes.

The EPEM criterion is calculated in a similar manner, using empirical probabilities (valuation of options on the basis of the empirical distribution was discussed in “An empirical solution to option pricing,” May 2009).

Values of PPLN and PPEM criteria roughly can be estimated as the sum of probabilities (based on lognormal and empirical distributions, respectively) related to underlying prices for which the payoff function of the combination is positive.

The values of the four criteria were calculated for all 500 straddles within each combination’s set. Then, for both criteria pairs, we determined elements relevant to the first 10 Pareto-optimal layers (the method of establishing the Pareto layers was described in the previous article). The profit or loss of selected combinations was recorded at the expiration date.

To estimate the advantage of the multi-criteria selection, we calculated the difference between the average profit of combinations selected by the Pareto method and the average profit of the same number of combinations selected by these criteria independently. A positive difference indicates that the profit of combinations selected by MCA is higher than the profit of combinations selected by single-criterion analysis (SCA). Accordingly, a negative difference shows the opposite.

**In the long run
**For each date, we calculated the cumulative difference between profits of combinations selected by two alternative methods (MCA and SCA). “Advantage of multi-criteria selection” (below) shows excess profits that can be accumulated thanks to MCA. The time dynamics of this indicator reflect the uncontestable advantages of MCA over selection based on a single criterion.

Although absolute profit is not representative (because it depends on the investment volume and money management system), the steady growth of cumulative profit is obvious. This means that in most cases, combinations selected by an MCA method generated higher profits than combinations selected by SCA.