Besides demonstrating the long-run advantage of MCA, these results show significant differences between two criteria pairs in regard to the growth rate of their cumulative profits. Criteria based on profit probability (PPLN and PPEM) generated considerably higher excess profit. Moreover, growth of profit for these criteria was much smoother, which means that the advantage of MCA was more consistent. Hence, the advantage of multi-criteria selection can show up to a different extent, depending on the specific combination of criteria.

**Time, volatility & correlation
**To examine the influence of time left to options expiration on the effectiveness of MCA, we grouped differences between profits of combinations selected by MCA and SCA into weekly intervals. “Effect of time” (below) shows that near the expiration, the effectiveness of two selection methods was similar (the difference in profit of combinations selected by MCA and SCA is close to zero). However, at longer time intervals, the superiority of MCA became more evident, reaching its maximum at the longest time horizons. This relationship is statistically significant for both criteria pairs.

How can we explain this phenomenon? Previous research suggests that time left to options expiration influences the degree of criteria interdependence. “Time and correlation” (below), demonstrates a strong inverse relationship between the coefficient of determination (correlation coefficient squared) and the number of days to expiration. (Only one pair of criteria, EPLN-EPEM, is shown; the form and the strength of the relationship for the second pair was similar.) When there is only one day left until expiration, the coefficient of determination ranges from 0.6 to 0.9. As the time interval grows, correlation decreases non-linearly. The coefficient of determination stabilizes at the 0 to 0.2 level.

Taking into account the preceding data, we can argue that the advantage of MCA is more evident when the correlation of criteria is low. This assumption is based on simple logic: values of highly correlated criteria for any given combination are close to each other. Therefore, the information contained in such criteria is overlapping and they would select almost the same set of combinations. Consequently, the advantage of multi-criteria selection vanishes and MCA reduces to SCA. At the extreme, when two criteria are perfectly correlated, each Pareto layer will consist of one element (or several elements if values of both criteria for them coincide), and the ordering of combinations by the Pareto method will match the ordering by a single criterion.

“Effect of correlation” (right) supports this argumentation. The inverse relationship between the profit difference and the determination coefficient shows that MCA outperforms SCA only when correlation of criteria is low. Moreover, when criteria are highly correlated, the difference of profits becomes negative, which means that in such conditions SCA is preferable.

Interestingly, the regression lines in “Effect of time” and “Effect of correlation” are almost parallel and the line corresponding to the pair of criteria based on profit probability (PPLN-PPEM) is lower than the line of the criteria based on expected profit (EPLN-EPEM). Parallelism means that the form of the relationship between the profit difference and the time left to expiration (and correlation) is similar for both criteria pairs. At the same time, advantages of multi-criteria selection for EPLN-EPEM criteria exceed advantages for PPLN-PPEM criteria at the whole range of time to expiration (and correlation) values. Trends presented in “Advantages of multi-criteria selection” support these conclusions.

“Effect of volatility” (below) demonstrates the influence of market volatility on the effectiveness of multi-criteria selection. We grouped all data into six intervals of implied and historical volatility. For each interval, the average difference between profit of combinations selected by MCA and SCA was calculated.

For the pair of criteria based on expected profit (EPLN-EPEM), we detected a straight non-linear relationship between the profit difference and volatility prevailing at the moment of entering positions. This means that at high volatility levels, the superiority of MCA is more evident. The form of this relationship is similar for both implied and historical volatility.

However, we did not find any statistically significant relationship for the second pair of criteria (PPLN-PPEM). Hence, the effect of volatility on the effectiveness of MCA may depend on specific criteria used in the analysis and, perhaps, on other factors that we did not consider here.

Based on all the relationships examined in this research, we can conclude that multi-criteria selection enables traders to achieve better results vs. selection on the basis of a single criterion. However, the effectiveness of MCA depends on many factors, including: the specific criteria used in the analysis; the timing of entering the positions (the more time left to expiration, the more advantageous MCA); criteria interrelationships (the lower the correlation, the more advantageous MCA is); and, in some cases, MCA effectiveness depends on market volatility (MCA can perform better when volatility is high).

**Sergey Izraylevich, Ph.D., and Vadim Tsudikman are authors of “Systematic Options Trading” (to be published by Financial Times Press in 2010). They are principals of High Technology Invest Inc. and Integral Option Strategy Fund Ltd. Contact the authors at izraylevich.hti@numericable.fr.**