- 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

## Overview of Trading Opportunities

This book covers various aspects of dealing with trading opportunities provided by options: from their detection and investigation to selection and deriving profits. This statement *a priori* assumes the existence of trading opportunities. Although this is quite obvious to the authors, it would be more appropriate to demonstrate the permanent presence of various trading opportunities in the options market. Besides, it is also useful to investigate their dynamics and structure.

The aim of this section is to perform the statistical investigation of trading opportunities existing at different moments in time for various underlying assets. An overview of trading opportunities is also expedient because it provides you with analytical tools for evaluating the potential profitability of different options markets and underlying asset types.

### What Is a Trading Opportunity?

A *trading opportunity* is the deviation of the market price of an option (or any option combination) from its fair value. The *fair value* is the price that implies zero profit for both the seller and the buyer of the option. This interpretation is related to the efficient market hypothesis stating that any new information is immediately priced in and hence all traded assets are fairly valued, and extracting profit is impossible neither for sellers nor for buyers. It is common knowledge that the efficient market concept is an idealization unachievable at present days. Financial markets are ineffective; asset prices constantly fluctuate and deviate from their fair value thereby creating various trading opportunities.

Because we define the trading opportunity as the difference between the market price and the fair value, we need to establish the algorithms to estimate these variables.

At each moment *the market price* is characterized by three indicators: last, bid, and ask prices. The first indicator is of little importance because it deals with the past whereas we are interested in current trading opportunities. To discover them it is preferable to use bid and ask prices. If the investor prefers to be on the conservative side, the worst of these two prices should be used (that is, ask price—when buying options, and bid price—when selling them). This approach decreases the probability of a mistake but reduces the number of trading opportunities considerably. The less conservative investor can use the combination of bid and ask prices—their simple average or weighted average with different weights for the best and the worst price. In this case the probability of erroneous inclusion of the option into the category of "trading opportunities" is higher, but the sample is more representative.

Estimating *the fair value* of any asset is an extremely difficult task, and options are not an exception. Their distinctive feature is that, in contrast to other financial instruments, options have expiration dates. This enables us to evaluate the accuracy of the fair value estimate in a reasonably short time period. The common way to estimate the fair value of options is the Black-Scholes formula and other similar models. However, they have a number of significant drawbacks and cannot be used to obtain the fair values suitable for the estimation of trading opportunity. That is why we use another method to get more accurate fair value estimates.

### Method for the Evaluation of Trading Opportunities

The quantitative expression of trading opportunities can be obtained by subtracting the fair value of an option from its current market price. For comparability of results the difference should be normalized by the strike price (allowing us to express the differences between fair and market prices in percentage).

We assume the market price of an option to be equal to the average between the bid and the ask prices. The option fair value can be calculated using the method proposed by Ralph Vince (Vince, 1992):

where *p _{i}
* is the probability of outcome

*i*,

*a _{i}
* is the profit or loss of outcome

*i*, and

*N* is the number of possible outcomes.

To obtain the most accurate estimate of the fair value, Vince permits looking into the future. Knowing the underlying asset price at the expiration date, we can find out the exact fair value of any option. In this case we have the only outcome (*N*=1) with probability *p _{i}
* = 1. Accordingly, the fair value of the option is

*a*.

_{i}For the Call option *a _{i}
* is equal to the difference between the underlying asset price (UAP) at the expiration date and the strike price (SP) if UAP > SP, otherwise

*a*= 0. For the Put option

_{i}*a*is equal to the difference between the SP and the UAP at the expiration date if UAP < SP, otherwise

_{i}*a*= 0. To be fully accurate, we need to discount this fair value by the riskfree interest rate normalized by the time to expiration. However, within the framework of current research, this correction can be neglected.

_{i}Zero (or close to zero) difference between the market price and the fair value indicates absence of trading opportunities. The positive difference indicates that the option is overvalued and there is a trading opportunity to sell it. Similarly, the negative difference indicates that the option is undervalued and there is a trading strike price opportunity to buy it.

To avoid zero fair values we evaluated trading opportunities of simple option combinations (straddles) rather than of separate options. The market prices and the fair value of combinations have been calculated for each of the 2,500 stocks and each date of the period from January 2, 2001, to August 16, 2007. Straddles were created using contracts with the nearest expiration date and strike prices closest to the current underlying price. Thus we obtained a table of 2,500 lines (according to the number of stocks) and 1,564 columns (the number of dates). Each cell of this table contains the value of the difference between the market price and the fair value of the combination corresponding to a certain date and to a given stock. In total, we calculated 3,910,000 values characterizing the presence or absence of trading opportunities.

### Structure and Dynamics of Trading Opportunities

To demonstrate the intraday structure of strike price trading opportunities existing in the options market, we arbitrarily chose several dates (May 1, June 1, July 2, 2007) and analyzed the deviations between the market and the fair values on these days. The quantity and quality of trading opportunities are vividly depicted in Figure I.1 with the market price of a combination on one axis and its fair value on another. Each point on the figure relates to the combination corresponding to a certain underlying asset.

Figure I.1 Relation between market price and fair value of straddles observed as of May 1, June 1, and July 2, 2007, for 2,500 stocks. The line separates overvalued (points above it) and undervalued (points below it) combinations. All values are expressed as the percentage of the strike price.

On the whole we can observe a firm relation between the two indicators (*R*
^{2} = 0.32). Points at the line relate to combinations with no trading opportunities. Points to the north of the line represent overvalued combinations. Undervalued combinations are represented by points under the line. The scattering pattern of points in Figure I.1 indicates the presence of a considerable number of trading opportunities. Both overvalued and undervalued combinations are observed in large quantities. However, the extent to which they are over- or undervalued varies widely. Although many points are not situated exactly on the separating line, they are still very close to it, meaning that the trading opportunities in these cases are negligible.

We propose the following heuristic rule to separate combinations with trading opportunities from others that do not possess any trading potential or only have an insignificant one. Combinations with a difference between the market price and the fair value of no more than one percent (that is, within the range of –1% to 1%) are considered as lacking trading opportunities. Combinations with a difference that is outside this range are considered as having trading opportunities. Combinations with a difference of >1% are overvalued; combinations with a difference of <–1% are undervalued. Based on this classification we can analyze data presented in Figure I.1. (Only one date is featured in the following discussion because there is no significant difference between different days.)

The best way to represent the structure of trading opportunities is to build a frequency distribution of the differences between market and fair prices (see Figure I.2). The distribution of trading opportunities existing as of July 2, 2007, is characterized by the skew toward positive values. This indicates that overvalued combinations prevailed over undervalued ones. Only 19% of combinations fall into the –1% to 1% interval that we consider to be the range with negligible trading opportunities. Fifty-two percent of combinations are overvalued and 29% are undervalued. This means that more than 80% of stocks had the potential of realizing either short- or long-straddle strategies. Short positions could be opened for more than half of the combinations, whereas long straddles turned out to be profitable in slightly less than one-third of all cases.

Figure I.2 Two distributions of differences between market prices and fair values of straddles observed on July 2, 2007, for 2,500 stocks. Prices and differences are expressed as the percentage of the strike price. Positive differences correspond to overvalued straddles, negative differences—to undervalued straddles.

The frequency distribution of differences between market and fair values deviates from normal distribution considerably (see Figure I.2). Small differences corresponding to the absence of trading opportunities are more frequent than it is expected under the normal distribution. Medium differences are observed less frequently than under normal distribution. Comparison of two distributions reveals asymmetry in distribution of trading opportunities. Moderately overvalued combinations are more frequent than under normal distribution whereas moderately undervalued combinations are less frequent. However, the situation with the distribution of big differences is the contrary—highly undervalued combinations (left tail of the distribution) are more frequent than highly overvalued ones (right tail of the distribution) (see Figure I.2).

So far we have been analyzing the distribution of trading opportunities between different underlying assets (to be more exact, between combinations corresponding to these assets) within one day. At the next stage the time dynamics of trading opportunities will be considered for separate underlyings. We begin with one stock (AAPL will be used as an example) and calculate the difference between market prices and fair values of straddles for all dates within the period from January 2001 until August 2007. Figure I.3 shows these differences plotted against the corresponding dates. Visual analysis of these data reveals that undervalued periods alternate with overvalued periods. In general, we can say that the dynamics of this process is characterized by quasiperiodical cycles. Although at first sight these cycles have similar periodicity, their detailed investigation suggests that trading opportunities can hardly be forecasted on their basis.

Figure I.3 Dynamics of the differences between the market price and fair value of straddles for AAPL stock. Prices and differences are expressed as the percentage of the strike price.

Using AAPL as an example we illustrated the dynamics of trading opportunities for just one underlying asset (Figure I.3). However, the analysis of other stocks (not presented here) gives similar results. The overwhelming majority of underlying assets shows similar behavior—more or less regular fluctuations between overvalued and undervalued areas. As in the AAPL case, there are periods when trading opportunities are negligibly small.

Previously we mentioned that the intensity of trading opportunities can vary widely. Although we agreed to consider the difference between the market and the fair values of more than 1% as indicating the presence of a trading opportunity, the profit potential may be quite low if the differences exceed 1% by just a slight margin. On the other hand, the difference of 5% and over has a strong profit potential. As it follows from Figure I.3 (and other research not presented here), medium trading opportunities (with a profit potential of approximately 2% to 4%) are prevailing in the market. However, it should not have a negative impact on our evaluation of trading opportunities because these medium deviations of market prices from their fair values occur quite frequently.

The dynamics of trading opportunities analyzed by the example of AAPL indicates the approximate equality of periods when the options of a certain underlying asset are overvalued and when they are undervalued. Does such uniform distribution of trading opportunities (between over- and undervalued periods) reside in all stocks? Are there underlying assets with options that are undervalued or overvalued most of the time? To answer these questions we calculated the number of days when options were overvalued and undervalued for each of the 2,500 stocks. We divided the obtained values by 1,564 (the total number of days) to express them as the percentage fraction of time. This data was used to build two distributions of time fractions: when options were overvalued and when they were undervalued (see Figure I.4).

Figure I.4 Distribution of time fractions when options were undervalued and overvalued.

Two distributions are shifted relative to each other: overvalued—toward the longer fractions of time, undervalued—toward the shorter fractions of time (see Figure I.4). This means that options are more often overvalued than undervalued. In general, we can conclude that most of the options are undervalued for no more than 25% to 40% of the time and are overvalued for 45% to 60% of the time (see Figure I.4). Moreover, options relating to approximately 5% of stocks are overvalued for more than 70% of the time. On the other hand, options relating to only 3% of stocks are undervalued for just 50 % to 60% of the time. This means that options of some stocks are permanently overvalued, whereas constantly undervalued options are relatively rare.

As shown in Figure I.1, a multitude of trading opportunities consisting of both undervalued and overvalued options exists in the market simultaneously. To get a detailed notion of their dynamic structure, we can analyze the proportions of combinations possessing trading potential and those lacking it. This investigation demonstrates how the ratio of overvalued, undervalued, and fairly valued combinations changes in time. To clarify these issues we use 2,500 straddles (one for every underlying asset) for each of the 1,564 dates. For every combination we calculate the difference between the market price and its fair value. Based on this indicator, the straddles are classified into three categories in accordance with the heuristic rule previously proposed. Combinations with a difference of >1% are considered to be overvalued; those with a difference of <–1% are undervalued. Combinations with a difference between –1% and 1% are considered to be fairly valued (that is, lacking any trading opportunities). For each date we calculate the proportion of combinations belonging to each of the three categories and observe the dynamics of their changes in time.

The proportion of fairly valued combinations is relatively stable in time—it fluctuates slightly within the range from 10% to 20% (see Figure I.5). Overvalued combinations prevail over undervalued ones. The former constitute approximately 50% to 60% throughout most of the time, whereas there are only 30% to 40% of undervalued combinations. (This corresponds with the conclusions drawn from the analysis in Figure I.5.) At the same time there are periods when the proportion of undervalued combinations rises sharply (see Figure I.5).

Figure I.5 Dynamics of proportions of undervalued, overvalued, and fairly valued combinations.

The following conclusions can be drawn from this statistical investigation. Considerable trading opportunities consisting of overvalued and undervalued options are constantly available in the market. However, their relative ratios have complicated time dynamics that can hardly be forecasted by discovering persistent cycles with regular periodicity. Consequently, an accurate prediction of future trading opportunities—whether most of the options will be overvalued or undervalued—seems to be unfeasible. Therefore, at every moment in time the investor should determine the trading potential of each separate combination and create corresponding strategies based on selling overvalued, buying undervalued, and excluding fairly valued options and their combinations.