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.
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.
