cnc equity kaos

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Source: Bloomberg Financial Markets. Used by permission.

special case of the more common dis-equilibrium state in markets and of the more infrequent far-from-equilibrium state (e.g., crashes).

Reflexivity goes one step further, in describing the warp and woof of market activity. It states that practitioners expectations can actually influence the markets themselves. Because you and your competitors have the same information, you have to anticipate. If several firms do large trades, their collective views and resultant action will have a real but temporary effect on the market. For example, when views of a widely followed spread affect the market, and views of the market affect the spread, reflexivity is easily demonstrated. When reflexivity occurs, nonlinear pricing models have the advantage over linear pricing models.

Fractal Dimension

Fractal is a property, just like mass, or hardness. A fractal consists of self-similarity and a fractal dimension. A graphic example of scaling self-similarity is the fern leaf, where a small leaf is geometrically similar to a larger leaf. Some price time series exhibit statistical self-similarity. That is, when viewed without labels, the minute, daily, weekly, and monthly time series look alike.

Figure 15.2 hurst exponent versus time for CNC.

A fractal dimension is a non-integer dimension. A typical price time series drawn on a sheet of paper is somewhere between a straight line (ID) and a solid plane (2D). A random time series (Brownian motion or Bm) has a fractal dimension of 1.5D. A persistent time-series is closer in resemblance to a line and conversely, an antipersistent time series is closer to a solid plane. Persistent and antipersistent time series are fractional Brownian motion (fBm).7

Wavelets

Spectral analysis searches for periodicity and the systems characteristic scale. The cyclic behavior in time series is most commonly measured by using a fast calculating version of Fourier Analysis, called the FFT. It is based on the assumption that a time series captured within a window (a piece of a time series) repeats itself outside the window forever and that the segment within the window can be analyzed into constituent sine and cosine waves. Sine and cosine terms assume stationary (fixed) periodicities over time and do not do well for financial time series that are aperiodic or nonstationaiy.

Wavelet theory is a more suitable form of analysis and is now over a decade old.8 Wavelets may be likened to a formless lycra garment. The garment fits many similar body contours and derives its form from whatever it fits. Wavelets can be made to "fit" data in a window by means of translation and dilation, decomposing the original time series into component parts. These parts can be used to reconstruct the same time series. In part because of its flexible windowing capability, wavelets can be used to analyze multiple scales of fractal time series. Sometimes, feeding wavelet coefficients into a nonlinear pricing model is superior to feeding the prices themselves.

Unlike Fourier analysis, which is given to stationary data, time-frequency (e.g., Grossman-Morlet) wavelets are suited for quasi-stationary signals and time-scale wavelets are given to fractal structures. Figure 15.3 is an example of a wavelet generated using the least asymmetric Daubechies filters of order 11 in Mathematica 3.0™.9

Applications

The following are some applications based on nonlinear pricing. Asset Allocation

Asset allocation is potentially one of the most promising and least explored applications of nonlinear pricing. Rotating between asset classes or even evaluating fund managers are well within the purview of nonlinear pricing.

Conceptually, asset allocation is very similar to stock selection. Suppose we wish to rotate between equities, bonds, and cash. If each asset class is represented by an index, then the balancing between indices is much like looking at patterns in different equities.

Figure 15.3 A sample wavelet generated using Mathematical

FUNCTION "LEASTASYMMETRICFlLTER" TO GENERATE FILTER COEFFICIENTS, AND THE FUNCTION "WAVELET" TO RECONSTRUCT THE ORIGINAL "MOTHER" WAVELET.

Source: Wolfram Mathematic Research 3.0. Used by permission.

Options

Each equity poses a unique probability distribution that may be expressed as a probability density function (PDF). To assume that the returns of all equities have the same PDF-in this case, a Gaussian or normal bell shape curve-is tantamount to saying that all men wear the same size suit, or that all companies should be valued by the same methods, or that the loss distributions from one line of insurance, such as commercial automobile collision, may be used to price another line, such as flood insurance. And as with insurance underwriting, accurate risk pricing is the central issue of sustained profitability.

Improper modeling can lead to mispricing. To illustrate this effect, we will consider stock option pricing. A stock option is the right to buy or sell a particular stock at a certain price for a limited period of time. The price at which the stock may be bought or sold is called the strike price. The options intrinsic value is the difference between the agreed strike price and price of the underlying security.

An options market price is based on the publics perception of the options potential future intrinsic value, and therefore, to a large degree, on the perceived future price of the underlying security. One way for mispricing to occur is to misjudge likely future price changes of the underlying security.