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Options: What Is Known and What Is Not

In the 1980s, when the first option exchange and the first pricing model emerged, options markets began to develop so fast that the existing theoretical background could not satisfy increasing practical needs. The facilities required to store and to process information incoming from trading floors were not yet established. As a result, statistical data processing and theoretical developments could not satisfy growing demands of market professionals.

However, as time goes by the stream of information grows and the scope of theoretical research widens. Every year brings more and more professional publications on the subject. Options are thoroughly studied at universities, becoming one of the most popular topics of economical, mathematical, and interdisciplinary research. Option exchanges arrange seminars popularizing basic knowledge among beginners and organize advanced-level courses intended for market professionals.

A significant bulk of knowledge on options has been accumulated. These attainments are systematized and published in popular and professional sources. The literature on options can be divided into two main categories.

The first category includes theoretical research on the basis of financial mathematics. A substantial number of scientific articles and books are devoted to the development of advanced option pricing models. They apply probability theory and discuss various complicated issues, including volatility abnormalities, nonlinearities, and interrelationships between parameters.

A strict and extensive mathematical background of option theory is given by Peter James (James, 2003). This book represents the basis for researchers entering the world of options, though its complexity makes it comprehensible only to specialists with deep knowledge of mathematics. The basics of derivatives theory are perfectly described by John Hull (Hull, 2008). This is a textbook that covers all essential issues from basic terminology to complicated problems of financial mathematics. Option pricing is widely discussed in Espen Haug's book (Haug, 2006). It can be used as a universal handbook covering up-to-date progress in price modeling (see also Achdou and Pironneau, 2005, Rouah and Vainberg, 2007). Mathematical fundamentals of derivatives theory (not only options) are widely covered by Salih Neftci (Neftci, 2000). Although the majority of theoretical works have not yet been implemented, some of the mathematical models are widely used by option exchanges, brokers, market-makers, and traders. The ability to apply theoretical attainments becomes increasingly essential and publications dedicated to this issue gain considerable practical value (Reehl, 2007).

Various aspects of volatility modeling and their implications on derivatives pricing were reviewed by Jim Gatheral and Nassim Taleb (Gatheral and Taleb, 2006). The authors examine all main properties of stochastic, local, and implied volatilities and describe many classical and advanced mathematical models. A special emphasis is placed on the dynamic properties of the volatility surface and its relationship to options valuation. The discussion of volatility derivatives, barrier end exotic options is of particular interest. Besides this work, the theoretical problems of volatility modeling and forecasting were comprehensively treated by Ser-Huang Poon and Riccardo Rebonato (Poon, 2005, and Rebonato, 2004).

The second category of publications is based on practical option trading and summarizes the experience accumulated by market practitioners. It discusses strategies based on combining different options and describes methods of building desirable profit profiles on the basis of option positions structuring (Banks and Siegel, 2007, Cohen, 2005, Cohen, 2009, Courtney, 2008, Vine, 2005). Strong emphasis is placed on methods of deriving arbitrage profit.

Lawrence McMillan is a widely known author of popular books on options. His publications (McMillan, 2002; Lehman and McMillan, 2003) include a detailed description of different option strategies and are extremely useful. The author highlights a multitude of versatile techniques indispensable for any option trader. Plentiful examples based on real market data, simple language, and broad coverage—those are the distinguishing features of his books. You can find not only an encyclopedic review of option strategies in McMillan's books, but also a comprehensive description of delta-neutral hedging, arbitrage, and other specific techniques.

An excellent example of a handbook covering most aspects of option trading is the work by Michael Thomsett (Thomsett, 2009). An option trader can find there a detailed description of many useful strategies. The problems of return calculation and risk evaluation are also discussed in detail. The author gives much attention to technical aspects of option trading—information sources, taxation, accounting for dividends, and so on.

Books in which authors do not limit themselves to mere review of option strategies but discuss serious theoretical and practical issues without involving complicated mathematics are also helpful.

Sheldon Natenberg (Natenberg, 1994, 2007) describes the key elements of option theory in a popular and yet precise language. He discusses the peculiarities of implied volatility behavior and investigates the characteristics of the Greeks and specifics of their application as the instruments of risk analysis. Without superfluous mathematics, the author investigates such important phenomena as volatility smiles and skews. Comprehension of complicated theoretical issues is facilitated by intelligible charts and tables.

The book by Allen Baird (Baird, 1992) targets the same audience. Being a fairly comprehensive introduction to option theory and practice, it spares the reader the wilds of complicated mathematics. Accurate description of risk management basics is among the main vantages of the book. The section devoted to the most typical mistakes made by trading beginners also deserves a special mention.

Certain books are dedicated to specific option strategies. For example, the idea of volatility trading is popularly described in the work of Kevin Connolly (Connolly, 1997). Without resorting to complicated mathematics, the author applies dynamic hedging to combinations consisting of options and their underlying assets.

The book by Nassim Taleb (Taleb, 1997) also discusses various aspects of dynamic hedging and peculiarities of delta-neutral volatility trading strategies. This is the work written by a professional with years of experience in risk management. Although containing some inevitable portion of mathematics, it is still comprehensible to the majority of readers. In most cases the author uses diagrams and tables instead of formulae.

The book by Leonard Yates (Yates, 2003) belongs to the same category. The author discusses interesting ideas and gives ground for original trading strategies based, in particular, on negative correlation between VIX and S&P indices. The strategy is tested using historical data and the results indicate its potential applicability.

Many particular features of options trading were recently covered in impressive depth. These include pricing and risks associated with exotic options (De Weert, 2008), application of foreign exchange (Wystup, 2007), and commodity options (Garner & Brittain, 2007), trading at expiration (Augen, 2009), intraday trading (Augen, 2009), protective strategies based on Put options, and so on.

Our knowledge on options goes beyond the literature dedicated to this narrow topic. Theory and practice of option trading apply various elaborations originating from different areas of finance, statistics, probability theory, and applied mathematics. For example, creating their classical option pricing model, Black and Scholes used the well-known lognormal distribution that was widely discussed and cited in statistical and mathematical literature. Later other authors created their own pricing models using other known distributions.

The potential benefit of adopting ideas from adjacent scientific fields is far from being exhausted. For example, in classical option pricing theory the assumption of randomness of underlying asset price changes is the most questionable issue. Basically, it follows from applying lognormal distribution and means that the underlying asset price moves according to geometrical Brownian motion laws. The work of Edgar Peters (Peters, 1996) represents an interesting example of a more sophisticated approach to the description of price behavior. It applies chaos mathematics, fractal theory, and nonlinear dynamics to account for asset price fluctuations. Peters claims that these models describe price behavior more accurately than standard probability distribution functions. Therefore, their application opens the gates for more accurate option price modeling. There is a lot of work to be done here, and new research of physicists and mathematicians will surely contribute to elaborating option theory.

The up-to-date achievements in the sphere of options theory can be summarized as follows. There is an adequate, albeit with certain drawbacks, option pricing model. Numerous versions of the basic model, eliminating some of its drawbacks and making the estimations more accurate, are also available. The basic principles of creating option pricing models, based on assumptions about the main underlying asset characteristics, are reliably established. Basic option risk indicators ("the Greeks") are grasped. We know their features, interrelationships, and applicability in different situations. Various aspects of implied volatility behavior, including its dynamics, specific relationships with different parameters, and numerous anomalies, are profoundly investigated. We also possess an extensive set of advanced option strategies allowing construction of almost any desired payoff profile.

Despite this impressive progress, some important aspects still remain beyond theoretical and practical studies. Next we summarize issues still requiring additional investigation.

The main topic of theoretical research (though directly related to investment practice) is the determination of the fair option value. The term fair value stands for the price that implies zero profit for both option sellers and buyers. This requires creation of realistic option pricing models (Katz and McCormick, 2005). It is common knowledge that apart from parameters that are objectively defined (current underlying asset price, strike, risk-free interest rate, and so on), the option price is determined by the forecast of underlying asset price dynamics. In the classical model this forecast is expressed by a probability density function of lognormal distribution that is specified by two parameters: variance derived from historical volatility and mean value that is usually considered to be equal to the current price. This form of forecast has a number of drawbacks, though attempts to use other probability distributions gave only local improvements and added new drawbacks. Hence the main gap in option theory can be defined as the absence of alternative methods for creating probability forecasts of the future underlying asset price.

If the price is assumed to be a continuous value, then the forecast can take the form of probability density function. The construction of such functions should be the principal topic of future research. We consider attempts to create one universal function for all cases to be unproductive. It should rather be a set of rules and algorithms for generating a whole class of density functions, each of which will be appropriate in certain conditions. The development of effective algorithms generating appropriate probability density functions will minimize the difference between modeled prices and fair values of options.

Apart from developing high-quality probabilistic forecasts, further research should target the development of optimization algorithms for parameters used in option pricing models. Even in the Black-Scholes model—which is relatively simple and contains only a few parameters—the outcome strongly depends on the variance value. Historical volatility, which is usually used to derive variance, depends on the length of the historical period used for its calculation. The value of this parameter can change the modeled option price considerably. As models become more complicated, the number of parameters increases and their combined influence becomes more pronounced.

Another essential drawback of option theory consists in the insufficient development of specific risk indicators. (Some alternative indicators are described in Izraylevich and Tsudikman, 2009d, 2010.) The majority of works on this issue are based on calculating the Greeks that are derivatives of the option price with respect to the underlying asset price (delta), volatility (vega), time (theta), and the interest rate (rho). (Derivatives of higher orders are also used.) Derivatives are calculated analytically using formulae of option pricing models. This implies that risk indicators obtained in this way inherit all the drawbacks of initial models. Such an approach to expressing option risks seems to be rather lopsided. Just as options market prices rarely match with the modeled ones, the Greeks calculated analytically almost never coincide with real changes in option values. We believe that future research of option risk management should focus on three main issues.

The first one relates directly to the problem of improvements in pricing models. The modeling formulae should be modified to include not only high-quality probabilistic forecasts (previously mentioned) but also to enable calculation of useful indicators (derivatives or any other coefficients) that accurately reflect corresponding risks.

The second issue represents the empirical study of option price increments in response to changes of underlying asset price, volatility, time to expiration, and risk-free interest rate. The patterns established in the course of these investigations can then be used (i) as independent risk indicators, (ii) for adjustment of the Greeks derived analytically, and (iii) to calibrate option pricing models.

The third issue corresponds to the estimation of risks of an option portfolio as a whole entity. Some risk indicators, such as theta and rho, are additive. Hence the dependence of the portfolio on time decay or interest rate change can be easily expressed as the sum of thetas or rhos of all options included in the portfolio. On the contrary, delta and vega are nonadditive. Therefore, if the portfolio consists of options on several underlying assets, summing separate deltas and vegas is meaningless. One of the possible ways to solve this problem is to present the delta of each option as a derivative with respect to some index (such as S&P500 or NASDAQ) rather than with respect to the price of a corresponding underlying asset. (This issue is discussed in Izraylevich and Tsudikman, 2009b.) In the same way vegas of separate options can be expressed as derivatives with respect to volatility index (such as VIX or VXN) rather than with respect to volatility of a separate underlying asset. These procedures produce additive deltas and vegas that enable calculation of risk indicators (by summation of additive deltas and vegas) characterizing the whole portfolio. Other ways to estimate risks of a complex portfolio should also be examined. Research in this field will certainly bring useful practical results.

In this review we outlined what is already known about options and how much still lies ahead of us. We defined the main lines of future research that are, in our point of view, of special interest. Some of the gaps in option knowledge are partially filled in this book.

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