**Multi-layer set: an example**

Consider a step-by-step algorithm for creation of a three-layer Pareto set. To demonstrate this procedure, 500 short straddles were constructed using stocks from the S&P 500 index as underlying assets. All combinations were created 10 trading days before the expiration date (Oct. 16, 2009) using strikes closest to the current prices of respective underlying stocks. The volume of the position for each straddle was determined as $10,000 divided by the price of the underlying asset.

All straddles were evaluated by two criteria: “Expected profit on the basis of lognormal distribution” (EPLN) and “Expected profit on the basis of empirical distribution” (EPEM). These criteria are integrals of the payoff function of the combination over the probability density of lognormal (EPLN) and empirical (EPEM) distributions. The features of the empirical distribution and their implication for valuation of options are discussed in “An empirical solution to option pricing” (May 2009). For the sake of simplicity, criteria values may be estimated roughly by summing the products of probabilities of each underlying price outcome and the value of the payoff function respective to this outcome.

“Case study” (above) shows criteria values for 20 best combinations (out of 500 that were analyzed). Each combination is denoted by the ticker of its underlying stock. Applying the procedure described above, we reveal that values of both criteria for CIT are higher than for all other tickers. This means that CIT dominates over all elements of the initial set. Therefore, it is included in the Pareto set and all other alternatives are discarded. Thus, in this case, the first layer consists of one single element.

To determine the elements belonging to the second Pareto layer, we exclude CIT from the initial set and repeat the pairwise comparisons of remaining alternatives. The first ticker ACAS is dominated by FDO (EPLN_{FDO}=344 > EPLN_{ACAS}=210, EPEM_{FDO}=242 > EPEM_{ACAS}=224) and should, therefore, be discarded.

The next element, AES, is discarded because it is dominated by IPG (EPLN_{IPG}=202 > EPLN_{AES}=-139, EPEM_{IPG}=365 > EPEM_{AES}=307). AMD is not dominated by any of the remaining elements and is included in the second layer. APH is discarded because it is dominated by FDO. Ticker DYN is dominated by FHN. FDO is not dominated by any of the remaining alternatives. Therefore, we include it in the second layer and at the same time discard all elements, which it dominates over: GOOG, HOG, HSY, MAT, UNH, WLP, XLNX.

Next, we find that FHN is dominated by IPG. Ticker INTC is discarded, because it is dominated by MCO. The next ticker, IPG, is non-dominated (included in the layer 2). SWY is discarded as being dominated by IPG. The remaining non-dominated tickers MCO and TLAB are included in the second layer of the Pareto set, which finally consists of five elements (AMD, FDO, IPG, MCO, TLAB).

After excluding the first two layers from the initial set, we get only 14 tickers for further analysis. Following exactly the same algorithm, we determine a new Pareto set, which represents the third layer (AES, FHN, INTC). Thus, consecutive execution of three selection procedures produced partial three-layer sorting of the initial set (again, see “Case study,” above).

“Structure of the Pareto set” (above) shows all 500 combinations in the two-dimensional coordinate system. The position of each point on this chart is determined by values of two criteria (these values are used as coordinates for relevant combinations). Positioning of Pareto layers relative to each other demonstrates that elements belonging to top layers (that is, elements with higher criteria values) are grouped in the upper right part of the chart. Subsequent layers are situated lower and to the left (where criteria values are lower). Such distribution of layers implies that principles of Pareto optimality enable us to find a compromise area corresponding to relatively high values of both criteria.

However, successful identification of the optimal area with high criteria values does not prove by itself that MCA has a significant advantage over a single-criterion selection of investment alternatives. In the next part of this series, we will present the results of practical MCA application and explore factors influencing the effectiveness of multi-criteria selection.

**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. Their website is www.sys-options.com.**