- What the Book Is About and Who Should Read It
- Introduction
- Options: What Is Known and What Is Not
- The Concept of the Systematic Approach
- Overview of Trading Opportunities

## The Concept of the Systematic Approach

This section introduces the basic concept of the systematic approach including its philosophy, objectives, and methodology. Here we overview the essence of operations required for consecutive execution of valuation, comparative analyses, and selection procedures. We strongly recommend you get acquainted with this material as it represents an all-embracing description of the general framework for systematic options trading.

### The Goals and Objectives

One of the main issues in the option trading is the problem of selecting the best variants among many available alternatives. The choice is wide and the objects to examine and assess are compound structures. Although continuous functioning in such complicated environment hampers the investment process significantly, it provides at the same time a broad spectrum of promising trading opportunities.

In the literature and in multiple services offered by brokerage firms and Internet sources, the problem of choice is generally solved through application of different market scanners and rankers. A typical scanner screens the market for underlying assets that currently have extreme characteristics, such as divergence between historical and implied volatilities, daily volatility fluctuations, changes in trading volume, and so on. Afterward a ranker orders underlying assets according to the suitability for a particular option strategy. Then suitable combinations should be designed for all chosen underlying assets. Because a great number of combinations can be constructed for a given underlying asset within a given strategy, it is usually advised to use combinations' payoff charts (the functional relationship between the price of the underlying asset and a combination's profit estimated for a certain future date) as a basis for their comparison and decision making. However, in most cases visual analysis is quite unfeasible if a large quantity of option strategies and underlying assets have to be compared simultaneously.

We regard the choice of suitable underlying assets for the *a priori* defined strategy as a differential approach. It is a forced measure resulting from the imperfection of analytical tools limited to simple scanning and visual analysis of payoff functions. Differential selection deprives the investor of the potential to utilize the whole spectrum of various trading opportunities provided by the market completely and effectively.

What we oppose to a differential approach is an integral systematic approach based on the strictly formalized assessment criteria, universal procedures of multicriteria analysis, and well-structured selection algorithms. The systematic approach enables simultaneous processing of a considerable number of option strategies and underlying assets. Without such an integral system, the investor has little or even no chance to make prompt selection decisions and to adapt successfully to changing market conditions.

The main goal of the systematic approach is to create a complex portfolio containing a potentially unlimited number of option combinations corresponding to a variety of strategies and underlying assets. Its application ensures that all trading opportunities appearing at any particular moment will be thoroughly estimated and none of the variants worth considering will be omitted. The systematic approach is absolutely indispensable for turning option trading into a long-term continuous process of income generation with controllable parameters of risk and profitability.

### Valuation, Comparative Analysis, and Selection

The systematic approach is realized through consecutive execution of the following procedures: valuation, comparative analyses, and selection. These procedures are applied to the multitude of option combinations. The combination represents a complex structure consisting of any number of long and/or short individual options corresponding to certain underlying assets. Each option combination can be characterized by the shape of its payoff function. When referring to the *option trading strategy*, we will imply a certain definitive shape of the payoff function that is inherent to all combinations belonging to the same strategy and that is qualitatively distinguishable from payoff functions of combinations not belonging to this strategy. The set of option combinations available at any given moment in time for valuation, analyses, and selection of promising trading opportunities will be referred to as the *initial set*.

### Valuation

Option combinations are valuated through the application of strictly formalized criteria developed specially for this purpose and expressing potential profitability and risk of the assessed variants in different ways. Criteria represent mathematical constructions with different degrees of complexity and one or many parameters. Optimization of parameters is performed either by means of statistical analyses of historical time series or by expert forecasts. Because parameters optimized on historical data are inclined to suffer from the disadvantage of curve fitting, close attention should be paid to the validity of statistical patterns used to determine their optimal values. Expert forecasts also have significant drawbacks because they reflect opinions of particular specialists and thus represent rather subjective estimates. A systematic approach, applying both statistical analyses and expert forecasts, allows diminishing their drawbacks while amplifying advantages of these two parameterization methods.

Development of sophisticated criteria capable of valuating option combinations adequately, and optimization of their parameters, are the crucial issues that determine the practical success of systematic approach. The first part of this book discusses the basic principles of criteria construction and parameterization; the main criteria are described and analyzed in detail.

After being valuated by criteria, every combination receives a numerical characteristic reflecting its investment attractiveness. Option combinations can be valuated by one or several criteria. In the latter case the number of characteristics attributed to each combination is equal to the number of criteria.

### Comparative Analysis

Following the completion of the valuation stage, the characteristics attributed to combinations need to be analyzed. During the analysis every combination is compared with all the others according to their characteristics. As a result, all variants constituting the initial set are ordered according to their quality indicators.

If the valuation was based on several criteria, then the analysis generates several orderings. In this case the same combination can have different positions in different orderings. For example, a combination can be the best one according to its expected profit, but at the same time it can be at the end of the list in an ordering obtained by the application of some risk-related criterion. Subsequently all orderings can be either used separately or combined into a unified one.

The unified ordering can be either complete or partial. Usually partial ordering appears when a complete one is unachievable. This may happen if some items turn out to be incomparable by certain criteria or if they are valued differently according to different criteria. In such cases the entire set of alternatives is divided into groups, and these groups are consequently ordered as joint entities. Different methods appropriate for execution of such procedures are discussed in Chapter 7, "Basic Concepts of Multicriteria Selection as Applied to Option Combinations."

### Selection

At the next stage the results of the comparative analysis are used to select a limited number of combinations possessing superior quality characteristics. This procedure needs to be arranged thoroughly because it leads to the irreversible decision as to which combinations will enter the portfolio and which ones will be rejected.

You need to consider three main principles when choosing combinations suitable for inclusion into the portfolio.

- The number of combinations selected should be large enough to maintain diversification of the portfolio above some minimum level. Like in the classic portfolio theory, it minimizes specific risks related to individual underlying assets.
- Criteria values of selected combinations should exceed the values of the rejected ones. The minimal threshold for this excess should be established for each particular situation.
- The relative superiority of some combinations over others (resulting from the comparison of their criteria values) should not be considered as the sufficient reason for selection. The absolute criteria values must also be taken into account. For example, between two combinations with an expected return of –$2 and –$10, the first variant is preferable and, in principle, can be selected as the one with relatively better characteristics. However, the absolute value of the expected return corresponding to the first combination is negative and hence this combination, just as the second one, cannot be selected to enter the portfolio.

In practice, however, these principles contradict one another. Thus, following the second and the third principles an investor endeavors to decrease the number of combinations selected. At the same time the principle of portfolio diversification induces the opposite tendency—to increase this number. Thereby the structure of the resulting portfolio represents a compromise (trade-off) between all three principles.

The selection procedure represents a set of rules determining how to draw the line separating potentially profitable combinations from those that lack such potential. Consider a simple situation: The initial set consists of *N* combinations ordered according to the values of a certain criterion; the investor must select *N*' best variants out of *N* alternatives. This problem may be solved by creating one or several utility functions. The argument of such function is the number of selected combinations (numerical value of the place occupied by the last selected combination in the ordering). The value of the utility function is an indicator reflecting the measure of utility arising from the selection of this particular number of combinations. In other words, the utility function may be defined as the relationship between an average return (the maximum drawdown, the Sharpe ratio, or any other characteristic reflecting the investor's satisfaction) and the number of combinations selected.

Analytical methods are not applicable to the majority of utility functions because no formulae establish the relations between the value and the argument of these functions. Hence the values of utility functions are usually derived empirically from historical time series using different statistical techniques.

If several utility functions are used simultaneously, they need to be combined into one unified function. Such unification is possible because all utility functions have the same argument (the number of selected combinations). The main requirement for the methods used to combine different utility functions is the unambiguity of the outcome that must be consistently interpretable. It means that the resultant function should be unimodal with a single evident maximum corresponding to the optimal number of combinations to be selected. Statistics offers several methods to combine empirical functions; the most popular are multiplicative and additive convolutions. 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.

### Sequence of Operations, Notion of a "Matrix" and Its Reduction

The initial set consists of a huge number of option combinations that must be processed during the execution of valuation, analyses, and selection procedures. Suppose that at any time moment there are *m _{i}* options traded for every underlying asset

*i*. Assuming that any option can either be absent or present in the combination (in the latter case it can be either long or short) and that the proportion of different options is the same in all combinations, the number of possible combinations for one underlying asset is determined as Accordingly, the total number of combinations for

*n*underlying assets can be estimated as follows:

Even if only 1,000 underlying assets are available for trading, and on average only 20 different options are traded for every underlying asset, then the procedures of valuation, analyses, and selection will cover more than 300 billion combinations! Moreover, the possibility to use unequal proportions of different options within one combination—which is quite realistic—generates a truly enormous number of variants to process. This number is so gigantic that computational procedures become unrealizable even for the most advanced computer hardware. Therefore, the initial set needs to be decreased to some reasonable quantities that are possible to work with on personal computers. You can achieve this decrease through the creation of combination-generating algorithms that produce only potentially appropriate combinations instead of generating all possible variants. (Their appropriateness is determined by specific requirements and limitations of the particular investor.)

First, the investor must decide what strategies to use and then generate combinations corresponding exclusively to these strategies. This can significantly decrease the number of variants in the initial set. Then additional reasonable limitations should be applied within every strategy. For example, the following limits can be used for the *short strangle strategy*: The strike of the Call option must be greater than the strike of the Put option; the difference between Call and Put strikes must not exceed 25% of the underlying asset's price; the ratio of Put options to Call options must be between 0.8 and 1.2. Such limits, on the one hand, are well founded and, on the other hand, they do not prevent an investor from taking full advantage of the majority of opportunities appearing in the options market. At the same time, these limitations reduce the initial set to such an extent that makes it processible for personal computers.

Further facilitation of computational procedures can be achieved if selection is realized as a series of consecutive subselections. We propose to adhere strictly to the following sequence of operations. Before initiating any selection procedure, a range of potentially suitable underlying assets and trading strategies should be determined. At the same time the algorithms used to generate option combinations must be established. After that the initial set of variants available for trading can be represented as a three-dimensional space ({underlying assets x strategies x combinations}) on which the consecutive subselection procedures are executed.

If the algorithms used to generate option combinations allow creating only one combination for every underlying asset within every strategy (that is, one single combination corresponds to each {underlying assets x strategies}), then the three-dimensional space of the initial set turns into a two-dimensional space. The two-dimensional initial set can be visualized as a table with lines corresponding to underlying assets and columns—to strategies. Each cell of this table contains one option combination relating to a given underlying asset and to a certain strategy. Such a table can further be referred to as a *two-dimensional matrix*.

If several option combinations are created for every underlying asset within every strategy, then each cell in the table will contain more than one combination for any {underlying assets x strategies}. In this case the two-dimensional matrix becomes a three-dimensional one. (Its elements form the initial set.)

The procedure of selection can be viewed as a *reduction of the three-dimensional matrix*. We propose to realize it as a sequence of three consecutive operations (each of which can be regarded as a subselection).

*The first operation* represents selection of one or several best option combinations corresponding to a specific underlying asset and a given strategy. Suppose that the initial set consists of 30,000 elements (1,000 underlying assets, 5 strategies, and 6 combinations for every {underlying assets x strategies}). If during the first subselection procedure only one combination is chosen out of the 6 possibilities, then the initial set of 30,000 items decreases to 5,000 and the three-dimensional matrix becomes two-dimensional. Chapter 4, "Selection of Option Combinations," discusses the methodology of this operation.

*The second operation* consists of choosing one or several superior option trading strategies for every underlying asset. In our example this leads to a further decrease of the initial set. If only one strategy is chosen, the initial set declines to 1,000 combinations. The result of this operation is the ultimate reduction of the matrix because after the execution of this subselection a unique list of underlying assets corresponds to every strategy. Therefore, the remaining part of the initial set cannot anymore be presented as an entire table without gaps. This operation is discussed in Chapter 5.

*The third operation* is intended to select the best variants from the lists of underlying assets corresponding to every strategy. If this procedure selects approximately 10% of combinations out of those that were chosen during two previous operations (we assume this percentage as an average estimate though in practice it can vary substantially), then the initial set is finally reduced to just 100 variants. Chapter 6, "Selection of Underlying Assets," describes this operation in detail.

This sequence of operations represents only one possible way to perform the procedures of valuation, analyses, and selection. Some other, more complicated approaches to the reduction of the initial set matrix can be developed. However, the scope of this book is limited to the preceding scheme because even such a relatively simple algorithm has more than enough particular features and specific peculiarities.