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Figure 15.5 response map of a set of fuzzy logic rules to input variables w and v.

Grade

Genetic Algorithms

Genetic algorithms (GAs) offer a way to optimize complex systems. Inspired by Darwins theories of evolution, GAs create a solution with the optimal mix of variables the same way nature "creates" a well-adapted animal for a given environment by allowing the fittest to pass along their good genes to a new generation. GAs create a virtual population of possible solutions and let the fittest survive to create offspring solutions. This process continues until the best solution evolves. Elegant in their simplicity, GAs are now being used by over half the Fortune 500 companies in scheduling, distribution, management, and any other problems that involve optimizing models with many interacting solutions.

GAs can optimize performance based on historical trading decisions and are applied to problems assuming that some a priori solution exists, such as finding optimal portfolios. For example, determining optimal weighting for a portfolio is analogous to the topographical problem of determining the highest mountain in a range where clouds cover many of the peaks. Rather than examine every mountain peak in search for the highest one, GAs jump through the cloud layer and take a repetitive sample of many peaks, yielding the correct answer relatively quickly. Money managers look to GAs to handle cases when stocks close at 54 and open at 48 the next morning limit down. Designing an algorithm to handle such jump-discontinuities is no trivial feat, yet GAs have an innate ability in this area as they naturally jump to possibly correct solutions.



Neural Nets

When modeling a financial instrument, we require a powerful tool that can process several independent (input) variables. Rule-based models were found insufficiently adaptive to deal with the markets nonstationary nature. Neural networks are more suitable because they are inherently nonlinear and adaptive.

The neural net is a modeling device that takes data regarding a previously solved example and learns to make new forecasts by modifying interconnections between processing elements, called "neurons" or "cells," analogous to the way the human brain connects neurons. There are numerous types of neural net architectures and learning paradigms. One popular architecture, the "Perceptron" net, has three or more layers of processing elements, and cells in one layer feed their output to cells in the next layer. The first layer receives input data and the last layer produces the models output.

Neural nets are "trained" to mimic the input-output performance of some target model. Training involves feeding a representative data set of previously solved examples and adjusting the weights or strengths of the cells interconnections to improve the nets performance. This process continues until the neural nets weights have either stabilized and will not change in any significant manner or the desired input-output performance is attained, whichever comes first. Once the net is trained, new predictions are feasible.

The problems that best lend themselves to this type of processing require "pattern recognition" and involve fuzzy variables. Modeled on the structure of the synapses of the human brain, neural nets attempt to ascertain patterns in the relationship between various kinds of data. Money managers do this intuitively when they attempt to distill meaning from the relationship between interest rates, bond yields, equity indices, gold and oil prices, and so on.

Neural networks can be trained to yield new types of indicators. They can combine not only historical data from the current financial instrument, but also cross-market indicators, which singly or together can predict buy-sell signals.

A neural net may be too brittle if it is either over trained or the training data is not sufficiently representative of what could actually occur (e.g., a crash).11,12

Conclusion

Some popular assumptions used in todays financial models are in error. Asset prices are linear and random for only a fraction of the time, and are mostly persistent and nonlinear. This says that, statistically, random walks occur only part of the time and there is a relationship between variables in time and that relationship changes over time. Because nonlinear pricing more accurately predicts empirical data, it has created an opportunity for arbitrage whereby the vigorish is the difference between the assumption of randomness (i.e., H = 0.5) and empirical results.

When adopting the nonlinear paradigm, which is more consistent with reality, new measurable aspects of time series become available. This chapter discussed



persistence, Hurst coefficient, wavelet coefficients, fractal dimension, and reflexivity. Modern nonlinear techniques are also available: genetic algorithms for optimization, fuzzy logic for representing human concepts on a "gray scale," and neural nets for function approximation. The growing number of investors that incorporate chaos, genetic algorithms, neural nets, and fuzzy logic is an indication of their success.

The reader is encouraged to consider using nonlinear pricing techniques. Two simple examples of patterns can be profitably exploited. One is the difference between the assumption of randomness and persistence (or antipersistence) in an equity time series (Figure 15.2) and the other is in the difference between the assumption of a normal distribution and stable distribution for an option (Figure 15.4).

We have entered an era where biology, which allows for the messiness of reality via chance and mutation, may be a more accurate paradigm for financial economics than one that assumes constituent variables do not change in statistical behavior. One wonders why investors have not exhibited, up to now, greater discomfort using linear mathematics in a clearly nonlinear environment.

Further advancement in financial modeling lies not only in looking further within the discipline of financial analysis, but also in seeking descriptions of phenomena from other fields. This should be a priority of all who renounce the arrogance of certainty. In that spirit, much remains to be described in light of a new interpretation. For a more complete treatment of nonlinear pricing, see May.13

REFERENCES

May, C.T., Nonlinear Pricing, New York: John Wiley & Sons, 1998.

May, C.T., "An Introduction to Nonlineat Pricing," Asia Risk Manager, pp. 45-50.

May, C.T., "Nonlinear Pricing: A New Investment Category," Private Asset Management, p. 8.

May, C.T., "An Introduction to Nonttaditional Pricing,"/. Hedge Fund Review, pp. 20-30.

Mirowski, P. "From Mandelbrot to Chaos in Economic Theory," Southern Economic Journal pp. 289-307.

Mirowski, P. "Tis a Pity Econometrics Isnt an Empirical Endeavot: Mandelbrot, Chaos, and the Noah

and Joseph Effects," Ricerche Economiche, pp. 76-99. Shiller, R. Market Volatility, Cambridge, MA: MIT Press, 1994. Vaga, T. Profiting from Chaos, New York: McGraw Hill, 1994.

Endnotes

1. Soros, G., Soros on Soros, John Wiley & Sons, 1995; Soros, G. The Alchemy of Finance, (2nd ed.).

2. Brockman, J., The Third Culture, New York: Simon & Schuster, 1995.

3. Cutland, N., Kopp, P., & Willinger, W, "Stock Price Returns and the Joseph EfFect: Fractional Version of the Black-Scholes Model," in Mathematics Research Reports, Decembet 1993, Hull: Univ. Hull.

4. Peters, E., Chaos and Order in the Capital Markets, (2nd ed.), New York: John Wiley & Sons, 1995. Peters, E„ Fractal Market Analysis, New Yotk: John Wiley & Sons, 1994.



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