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If the answers to all these questions are “yes”: then we can conclude, without any doubt, that the information contained in the Bible is not just some mythical convolution of ideas conjured up by weird and fanatical people from the past and put together in a collection over many centuries
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the live wire mysteries destruction – explosions of convolution, discern an interest in
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This biological-chemical-environmental intricate convolution of earthly creation we call evolution, gives added weight and further credit to the possibility of the existence of a Supernatural Power and Intellect as the designer of such amazing living architecture
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In the convolution approach, selecting a particular technique to combine criteria and setting weights (especially if they are introduced to account for the relative importance of different criteria) is highly subjective
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Giving up simultaneous application of several criteria and substituting them with a new and only criterion (which is a function with initial criteria serving as arguments) constitutes the approach called “convolution
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” The advantage of convolution is the simplicity of realization and the possibility to adjust the extent of influence of each criterion on optimization results
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The main drawback of convolution is an unavoidable loss of information that occurs when many criteria are transformed into a single one
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When calculating the convolution value, we must remember that criteria may be measured in different units and have different scales
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Let us consider an example of applying the convolution concept to the basic delta-neutral strategy
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1 shows the optimization space of minimax convolution
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In this case, multicriteria analysis based on the convolution of objective functions did not allow establishing a single optimal solution since each of the three areas has its own local maximum
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Hence, the application of convolution did not solve the optimization problem completely
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Since all of them have approximately the same altitudes (convolution values), the selection must be based on a different principle
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Distribution of optimal areas obtained by applying the convolution method is quite similar to the distribution observed when optimal solutions were determined using the Pareto method (compare Figure 2
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2 demonstrates two transformations of the convolution shown in Figure 2
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The reason is that optimal areas of the original convolution represent narrow ridges and high peaks
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2 contains three areas with altitude marks higher than 10 (remember that this optimization space represents transformation of the initial space obtained by convolution of three utility functions)
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To make optimization results comparable (and to enable the creation of the convolution of several optimizations), the values of objective functions in each case should be of approximately the same scale
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When steadiness is estimated by comparing the large number of optimization spaces (as it is in our example), the convolution method can be used to facilitate the comparison
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If spaces differ from one another considerably (that is, optimization is unsteady), their convolution will have an appearance of a surface with a large number of optimal areas scattered over it irregularly
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At the same time, if optimization spaces are similar, with optimal areas having approximately the same form and situated at the more or less same locations (indicating steadiness of optimization), the convolution will have a limited number of easily distinguishable optimal areas
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When calculating the convolution value, it should be taken into account that different indicators can be measured in different units and have different scales
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The former of these methods is more appropriate for the additive convolution; the latter, for the multiplicative convolution
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Since one indicator expresses expected profit and the other one expresses potential loss (VaR), the multiplicative convolution should be calculated as a ratio of expected profit to VaR
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After values of all indicators are normalized and convolution values are calculated, we can calculate the weight of each combination in the portfolio
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Capital allocation among 20 short straddles using the convolution of two indicators: expected profit and VaR
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In one case capital was allocated by the convolution of two indicators (expected profit and VaR); in another case, by the single indicator (expected profit)
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3, we analyzed the relationship between the profits/losses realized when the capital was allocated using the convolution of two indicators and the profits/losses realized when the portfolios were constructed on the basis of one indicator
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1 profit generated under the capital allocation scenario using convolution is on the vertical axis, and profit generated when the portfolio was created using the only indicator is on the horizontal axis
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When capital is allocated on the basis of a non-additive indicator or convolution of several indicators (either additive or non-additive), the maximization problem is usually not solvable by analytical methods
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The optimized function is represented by the convolution of expected profit and index delta calculated for the whole portfolio:
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When portfolios were created on the basis of convolution calculated for each separate combination (elemental system), the concentration index distribution was skewed toward the area of low index values (see Figure 4
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We have developed an additional method—a minimax convolution (see Chapter 5, “Selection of Option Strategies”) that in most cases brings more reliable and unambiguous results
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Two common methods will be used: additive and multiplicative convolution
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When each parameter value corresponds to a constant number of utility function values (in our case, it is constant and equals 5), then additive convolution is equivalent to the arithmetic mean and multiplicative convolution—to the geometric mean (Chapter 7, “Basic Concepts of Multicriteria Selection as Applied to Option Combinations,” discusses different types of convolutions in detail
36.
) We introduce an additional convolution type—developed specifically for the purpose of reducing many functions to a single one—the product of the maximum and the minimum values of the five utility functions corresponding to each threshold value
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(It will be called the minimax convolution
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) As shown here, this convolution produces the most unambiguous result
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This can be done only through application of different convolution techniques
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On the contrary, the minimax convolution represents a unimodal function with the unique optimal threshold value of 4% (Figure 5
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Three types of convolution of five utility functions shown in Figure 5
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In this case the additive convolution is absolutely useless because it is concave and has maximums at the two opposite ends of the range (Figure 5
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Multiplicative convolution also gives little information because its function does not possess any clear maximum and is rather flat across a wide range of threshold values
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In contrast, minimax convolution is again unimodal and gives the result that is easy to interpret (Figure 5
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When combining only two functions, minimax convolution is equivalent to a multiplicative one
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To obtain the optimal n value, these functions are converted to a single one using multiplicative convolution
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4 shows the example of two utility functions and their convolution for the short strangle/straddle strategy and the EPLN criterion
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The convolution is rather smooth and unimodal
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We apply two techniques of the multicriteria analysis: the Pareto method and the convolution
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The sum of criteria represents an additive convolution
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Multiplying criteria values by weight coefficients can modify the degree of their influence on the resultant convolution; the higher the criterion weight, the stronger its effect on the selection results
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Additive convolution is more appropriate for criteria that are close in their nature and are similarly scaled
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On the other hand, multiplicative convolution seems to represent a more appropriate solution for combining the criterion expressing expected profit with the criterion predicting profit probability (on the basis of any distribution)
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Calculating the convolution we should remember that criteria can be measured in different units and have different scales
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In this case the values of all criteria will lie within the range from zero to one, which facilitates their convolution considerably
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The former is more appropriate for the additive convolution; the latter is more suitable for the multiplicative one
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For each evaluated element j of the initial set, we calculate the convolution value R according to the following formula:
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For the research described next we used additive convolution
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In real trading, however, it is preferable to use more advanced convolution methods, similar to those previously described or others not mentioned here
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To make selection results obtained by the application of convolution comparable with Pareto results, for each of 55 criteria pairs we calculate average profit using the number of combinations equal to their number in each Pareto layer
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For example, if for some pair of criteria there are 5 combinations in the first layer, 12 in the second, 24 in the third (and so on up to the 20th layer), we calculate the average profit of combinations selected by convolution using 5, 12, 24 (and so on) best combinations as well
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As the result we obtain 1,100 average profits (20 layers × 55 criteria pairs) corresponding to the Pareto method and 1,100 average profits relating to the convolution
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Our next task is to verify whether the second multicriteria selection method (the convolution) also provides higher average profits than the monocriterion selection
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Average profits of combinations selected by convolution will be plotted on the horizontal axis, whereas monocriterion profits will be plotted on the vertical axis
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Totally, in 629 cases out of 1,100, selection by application of the convolution technique resulted in higher profits than monocriterion selection
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(a) Relationship between the average profit of combinations selected by one criterion and the average profit of combinations selected by two criteria using the convolution method
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The inclined dashed line divides the area where the profit of combinations selected by one criterion is lower (dots below the line) than the profit of combinations selected by the convolution method (dots above the line)
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The advantage of the Pareto method of multicriteria analysis over the monocriterion selection is more pronounced than the advantage of the multicriteria analysis realized via the convolution method
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Probably this phenomenon can be explained by the higher profitability of the Pareto method as compared with the selection by convolution
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Although such distribution of data is not random but statistically significant, 178 dots under the Pareto selection and 443 dots under the convolution selection (which is 16% and 40% of cases respectively) are still situated above this line
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For each pair the best combinations are selected by the Pareto method (20 layers) and by additive convolution
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Within each criteria pair the number of combinations selected by convolution corresponds to the number of combinations included in the Pareto layers
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To verify whether profits under two various methods of multicriteria selection are different, we build the regression where average profit under the Pareto selection is plotted on one axis and profit under the selection using convolution on the other axis
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The average profits of combinations selected by the Pareto method are plotted on the horizontal axis, and the profits corresponding to the selection via convolution are on the vertical axis
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Totally the Pareto selection was more profitable than selection via convolution in 774 cases out of 1,100, and in 96 cases the results were equal
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(a) Relationship between the average profit of combinations selected by convolution and the average profit of combinations selected by the Pareto method
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The inclined dashed line divides the area where the profit of combinations selected by convolution is lower (dots below the line) than the profit of combinations selected by Pareto method (dots above the line)
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Summarizing this study we conclude that the Pareto method of multicriteria analysis prevails over the convolution
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However, although in most cases application of the Pareto method gives better results than selection by convolved criteria; in 230 cases (21% of all cases) the situation was quite opposite—the performance of convolution was better than that of the Pareto
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It means that although in general the Pareto method is better than convolution, it is preferable to determine criteria and their combinations for which the selection of combinations by convolution may be more appropriate
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Quite often the results of the selection by convolution coincided with the Pareto selection results (96 cases that make almost 9%)
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Whatever the reason for this phenomenon, it would be useful to verify whether the extent of coincidence between optimal sets selected by Pareto and by convolution influences the difference between the profitability of positions formed by these two methods
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In other words, is it possible to explain the advantage of the Pareto method over the convolution (and, in fewer cases, inverse advantage of convolution over the Pareto method) by the extent of coincidence of two sets? To answer these questions we have arranged a series of investigations
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Thus, the greater the number of combinations required for positions opening, the lesser the difference between optimal sets formed via convolution and the Pareto method
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In all other cases the Pareto method can be used without any hesitation because, on the one hand, it selects the same combinations as the convolution by 90% and, on the other hand, in most cases it shows better performance than the convolution does (Figure 7
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The second stage of our investigation demonstrates that the total number of selected combinations is not the only factor affecting the extent of the coincidence between optimal sets obtained via the convolution and via the Pareto method
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Hence the area where we have to answer the question “When is the selection via convolution better than via the Pareto method?” gets even smaller
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Relationship between the percentage of combinations included in both optimal sets (Pareto and convolution) and the correlation coefficient of criteria
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The first variable should reflect the degree of convergence between profits realized via the Pareto method and via the convolution, whereas the second variable should combine the total number of selected combinations with correlations between criteria
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In most cases the Pareto method turned out to outperform the convolution
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However, inverse situations—when average profitability of combinations selected via the convolution is higher than of those selected by the Pareto method—also occur
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4 Criteria Correlation and Profitability of Selection Using the Convolution Method
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In contrast to the Pareto method, another technique of multicriteria selection, convolution, does not have the problem of uncertainty concerning the number of elements included in the optimal set
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Applying convolution we can control the number of selected combinations
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Although this property of convolution represents a definite advantage, at the same time it creates additional difficulties forcing the investor to decide on the number of selected combinations based solely on subjective reasons
96.
02) between the average profit of combinations selected by convolution and the correlation coefficient (Figure 8
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Convolution is just the ticket to meet that objective
98.
In mathematics, convolution is an operation on two functions that produces a third function
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Convolution is similar to cross-correlation between the two input functions—with a twist
