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    convolution


    1. 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


    2. the live wire mysteries destruction – explosions of convolution, discern an interest in


    3. 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


    4. 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


    5. 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


    6. ” The advantage of convolution is the simplicity of realization and the possibility to adjust the extent of influence of each criterion on optimization results


    7. The main drawback of convolution is an unavoidable loss of information that occurs when many criteria are transformed into a single one


    8. When calculating the convolution value, we must remember that criteria may be measured in different units and have different scales


    9. Let us consider an example of applying the convolution concept to the basic delta-neutral strategy


    10. 1 shows the optimization space of minimax convolution

    11. 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


    12. Hence, the application of convolution did not solve the optimization problem completely


    13. Since all of them have approximately the same altitudes (convolution values), the selection must be based on a different principle


    14. 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


    15. 2 demonstrates two transformations of the convolution shown in Figure 2


    16. The reason is that optimal areas of the original convolution represent narrow ridges and high peaks


    17. 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)


    18. 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


    19. 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


    20. 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

    21. 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


    22. When calculating the convolution value, it should be taken into account that different indicators can be measured in different units and have different scales


    23. The former of these methods is more appropriate for the additive convolution; the latter, for the multiplicative convolution


    24. 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


    25. After values of all indicators are normalized and convolution values are calculated, we can calculate the weight of each combination in the portfolio


    26. Capital allocation among 20 short straddles using the convolution of two indicators: expected profit and VaR


    27. 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)


    28. 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


    29. 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


    30. 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

    31. The optimized function is represented by the convolution of expected profit and index delta calculated for the whole portfolio:


    32. 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


    33. 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


    34. Two common methods will be used: additive and multiplicative convolution


    35. 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


    37. (It will be called the minimax convolution


    38. ) As shown here, this convolution produces the most unambiguous result


    39. This can be done only through application of different convolution techniques


    40. On the contrary, the minimax convolution represents a unimodal function with the unique optimal threshold value of 4% (Figure 5

    41. Three types of convolution of five utility functions shown in Figure 5


    42. 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


    43. 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


    44. In contrast, minimax convolution is again unimodal and gives the result that is easy to interpret (Figure 5


    45. When combining only two functions, minimax convolution is equivalent to a multiplicative one


    46. To obtain the optimal n value, these functions are converted to a single one using multiplicative convolution


    47. 4 shows the example of two utility functions and their convolution for the short strangle/straddle strategy and the EPLN criterion


    48. The convolution is rather smooth and unimodal


    49. We apply two techniques of the multicriteria analysis: the Pareto method and the convolution


    50. The sum of criteria represents an additive convolution














































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