From the June 01, 2010 issue of Futures Magazine • Subscribe!

Options and the Pareto set

The individual equity options market provides a wide choice of potential trading opportunities. To make investment decisions in this varied arena, traders generally apply complex rules to different criteria that estimate the value and appropriateness of assets.

Although the selection process may be based on a single criterion, in practice more valuation procedures are executed in the course of portfolio structuring. The most common example of multi-criteria analysis (MCA) selects assets based on two measures: expected return and risk.

These fundamental characteristics can be analyzed in turn by simultaneous application of numerous criteria (with each one expressing the qualities of return and risk in a slightly different way). This represents another example of MCA.

Types of MCA

Consecutive filtration is the simplest multi-criteria selection technique used by many investors. At the first stage of this procedure, the most important criterion is applied to select elements satisfying its requirements. Then, variants meeting the next most important criterion are selected, and so on.

The drawback of consecutive filtration is the necessity to classify criteria by importance. In most circumstances, objective classification is hardly possible, which leads to artificial assignment importance. Besides, elements underestimated by the first criterion are filtered out at the initial stage and will not be considered during the consecutive steps, even if they have high values for all other criteria.

Convolution is a more sophisticated method of MCA. Here, the set of criteria is reduced to a single indicator. The sum or the product of values corresponding to all criteria included in the analysis represents the convolution (referred to as additive or multiplicative). More complex algorithms of combining multiple criteria into a single one are also available.

Among the drawbacks of convolution are problems associated with integration of positive and negative values (multiplicative convolution), non-conformity of different measurement units and scales for different criteria, and the inevitable loss of information, while switching from the multidimensional criteria vector to the single indicator.

Pareto-optimality is the most efficient approach to multi-criteria selection of investment alternatives. In contrast to successive filtration and convolution methods, application of Pareto principles provides optimal solution to analyzing investments that are superior to other criteria but inferior according to others. Such situations are common, especially when evaluation is based on criteria expressing risk and return simultaneously. (Usually, alternatives with high-return potential are less attractive due to their excessive risk level, and vice versa.)

Finding the Pareto set

We assume that higher criterion values indicate better alternatives. Suppose that for each alternative a belonging to the set of alternatives A there is an n-dimensional vector of criteria x(a) = (x1(a),..., xn(a)). Using values of n criteria, it is possible to find elements with superior vector coordinates. Comparing two alternatives a and b, we decide that alternative a dominates over alternative b, if xi(a)xi(b) for all i = 1,..., n, and there is at least one criterion j, for which xj(a)>xj(b). If domination is established, this unambiguously defines which of the two elements is better. However, if domination can not be determined (preceding inequalities do not hold), the problem of determining the best element is not resolved. In this case, we decide that none of the alternatives dominates over another.

Using this reasoning, we can formulate the multi-criteria selection problem as follows: find the set among all available alternatives that includes only non-dominated elements and doesn’t have any alternatives dominating over them. This is a Pareto set. Each element of this set can be regarded as “the best” in the sense defined above. The number of alternatives in the Pareto set may vary. In an extreme case, a single element can dominate all others. At the other extreme, this set can include the whole initial set. In practice, however, the Pareto set consists of a reasonably limited number of non-dominated elements.

A pairwise comparison of all available alternatives represents the most straightforward method to perform the Pareto MCA. To establish the optimal set, we run consecutively through all elements of the initial set, discarding dominated alternatives and adding non-dominated to the target list.

Consider an initial set, A={a1,...,am}, consisting of m alternatives, evaluated by n criteria with values x(a) = (x1(a1),...,xn(am)). Each element ak has to be compared with all other alternatives. If we find element al, which ak dominates over, al is discarded. If ak is dominated by am, then ak is discarded. If none of the remaining elements dominates over ak, the latter is included in the Pareto set. Afterward, the next element is compared with all alternatives that remain.

The maximum number of pairwise comparisons is about 0,5m*(m-1), which is quite reasonable in most practical applications. However, faster algorithms would be required if a large number of alternatives is simultaneously compared by numerous criteria.

The total number of elements falling into the optimal Pareto set is out of our control and intent. This is a substantial restraint. If the initial set contains an element for which values of all criteria are higher than for other alternatives, the optimal set will consist of only this element. Such a strict limitation on the number of investment objects may be unacceptable (for diversification reasons). However, this difficulty can be overcome by applying the “layer” concept, which enables the enlargement of the Pareto set up to the desired limits.

Within the framework of this concept, all elements constituting the Pareto set belong to layer 1. Then these elements are excluded from the initial set and the Pareto set is reestablished for remaining alternatives. The elements selected during the second iteration form layer 2. This procedure can be continued until we reach a predetermined number of layers or until the cumulative number of elements in all layers reaches a certain desirable level.

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