In general, a PDF of the log of price ratios p /p is nearly normally distributed, and so the popular Black-Scholes pricing formula assumes price movement is log-normally distributed. However, this assumption is not totally accurate, as it implies yesterdays price changes are wholly unrelated to todays price changes. This assumption is appropriate for a time series that is 100 percent random (e.g., Brownian motion). If, however, the price time series for a traded asset is not random 100 percent of the time-and it is not-then the normal distribution hypothesis is inappropriate.

The difference between linear and nonlinear assumptions can be visualized as the difference between two probability distributions. The former, in the case of Black-Scholes, is the assumed normal distribution and the latter, in the case of the actual underlying equity, is the empirical and more stable distribution. Table 15.2 summarizes these features.

A comparison of these two curves is illustrated in Figure 15.1. With regard to the relative likelihood of how an underlying securitys price may change, we see a normal (Gaussian) distribution curve, a stable (real-world) distribution curve, and an overlay of both curves. Note the stable distribution is narrower, but has larger values as it tails. The overlay shows shaded regions where the two curves differ.

To examine how this affects pricing, look at the curves tails at the extreme right of the overlay (region A) in Figure 15.4. Because the Gaussian tail is lower than the stable tail, classical pricing formulas expect a smaller frequency of occurrence of very large positive price changes than you would find in the real world. It also expects a larger frequency of occurrence of moderately positive price changes than there really are in regions and C. To correct this problem, we would need to move a portion of classical models expected occurrences from region to region A, as shown by the arrow. However, because Gaussian models produce a bias toward smaller positive price changes than reality calls for in the A-B regions of the chart, the underlying security will tend to be underpriced in this region. Consequently, call options with a strike price within this region will tend to be underpriced and put options overpriced.

A similar argument can be made for regions and D but with the opposite effect. Because the Gaussian curve is higher than the stable curve in region C, classical pricing formulas expect a larger frequency of occurrence of moderately positive price changes than you would find in the real world. It also expects a smaller frequency of occurrence of tiny positive price changes than there really are in region D. To correct this problem, we would need to move a portion of models expected occurrences from region to region D, as shown by the arrow. However, because Gaussian models produce a bias toward larger positive price changes than reality calls for in the C-D regions

TABLE 15.2 RELATION OF TIME SERIES AND CORRESPONDING ASSUMPTIONS AND DISTRIBUTION TYPE

Figure 15.4 A normal distribution curve of price changes, a stable

DISTRIBUTION CURVE, AND AN OVERLAY OF BOTH. THE HORIZONTAL AXIS REPRESENTS THE SPREAD OF POSSIBLE FUTURE PRICES OF AN UNDERLYING

SECURITY, WITH THE CENTER OF THE CURVE REPRESENTING TODAYS PRICE. THE DIFFERENCE BETWEEN THESE TWO DISTRIBUTIONS MAY ALLOW

FOR ARBITRAGE OPPORTUNITIES AS IT CHANGES, A FUNDAMENTAL ASSUMPTION IN OPTION AND RISK PRICING AS WELL AS FINANCIAL THEORY.

Areas for Potential Mispricing

0.14

of the chart, the underlying security will tend to be overpriced in this region. Consequently, call options with a strike price within this region will tend to be overpriced and put options underpriced.

Similarly, Gaussian models will tend to underprice underlying securities in the small-to-medium negative price change regions (D-E) and overprice underlying securities in the medium-to-large negative price change regions (F-G).

Remember that nonlinear pricing applies to any traded asset, not just to options, and almost any time frame: minute bars, daily bars, monthly, and so on. However, it is folly to use just one valuation method (nonlinear pricing included) mechanistically, since reality is far more complex than any one model can depict.

Because current mispricing is based on the overly restrictive assumption of Gaussian returns used by Black-Scholes to model PDFs, it stands to reason that information

regarding the unique PDF of each equity would enable a superior way to value an option. However, there are shortcomings. For example, no clear link exists between probability and a stable time series with Hurst exponent H < 0.5. By definition, these series are antipersistent. To a mathematician then, positing certain nonlinear arguments may lack rigor because theory is incomplete.

Modeling Tools

The previous section discussed various measurements of a time series suitable for nonlinear pricing. The following section presents software technology that can process these variables and model their relationships in nonlinear ways. They include fuzzy logic, genetic algorithms, and neural networks.

Fuzzy Logic

Because the markets are extremely nonlinear, multidimensional, and time varying, attempts to simplify analysis by using fewer dimensions and standard econometric and portfolio analysis offers little help. What is needed is a practical way for users to devise models in low-dimensional space and transfer their knowledge about the subject matter to a mathematical function that yields nonfuzzy results. Fuzzy logic provides that capability.10

Fuzzy logic (FL) is a universal approximator. FL offers a way to both capture human knowledge mathematically and perform logical operations on it that makes intuitive sense. It is ideal for processing statements of reality that are neither 100 percent true or 100 percent false.

A simple way to picture the power of FL is to visualize the thermostat and air conditioner (AC) in your office. Although the thermometer gradually moves through its normal range, the AC is either off or on full force. When the temperature rises above 78 degrees F, the AC comes on full force until the temperature drops. It would be more effective to design a FL-based system with the two rules "If it is a little warm, then the AC comes on a little bit" and "if it is a lot warm, then the AC comes on a lot." The concept of a little bit warm and a little bit of AC is similar to the concept of a financial instrument being a little bit cheap or expensive, or to buy or sell a little bit more.

FL has some great advantages: it is fast and it is auditable. Unlike neural nets where the relative weightings between variables are hidden, with FL you can actually see the input-output relationship. This should give fund managers and investors an additional degree of comfort.

Figure 15.5 is an example of a fuzzy logic models response to two variables: Risk as a function of Volatility (V) and the Hurst coefficient (W). Input values W and V trigger a graded risk response called "Grade." Note that if this were a plain "yes/no" or "on/off" response map, then the peaks would not have sloped sides.

This chart was created with Mathematicas Fuzzy Logic™ toolbox. The Math-works also provides a Fuzzy Logic Toolbox™ for MATLAB users.