Fractal Time Seres Biased Random Walks
In Chapter 2, we discussed the Efficient Market Hypothesis EMU , which basically states that, because current prices reflect all available or public information, future price changes can be determined only by new information. With all prior information already reflected in prices, the markets follow a random walk. Each day's price movement is unrelated to the previous day's activity. EMH implicitly assumes that all investors immediately react to new information, so that the future is unrelated...
Hursts Simulation Technique
Perhaps the best way to understand how Hurst's statistics can arise, and what they mean, is to examine Hurst's own method for simulating a random walk. Hurst was working in the 1940s, when computers were only a theoretical possibility and were certainly not available in Egypt. Hurst tried to simulate random walks by flipping coins, but found the process slow and tedious. Instead, he constructed a probability pack of cards. The cards in this pack were numbered -1, 1, -3, 3, -5, 5, -7, and 7. The...
Estimating The Hurst Exponent
By taking the log of equation 7.3 , we obtain log IVS - H log N log a 7.5 Finding the slope of the log log graph of R S versus N will therefore give us an estimate of H. This estimate of H makes no assumptions about the shape of the underlying distribution. FIGURE 7.2c Fractal noise Cumulative observations. H 0.90. FIGURE 7.2c Fractal noise Cumulative observations. H 0.90. For very long N, we would expect the series to converge to the value H - 0.50, because the memory effect diminishes to a...
Tests Of Normality
The first complete study on daily returns was dpqe by Fama 1965a , who found that returns were negatively skewed more observations were in the left-hand negative tail than in the right-hand tail. In addition, the tails were fatter, and the peak around the mean was higher than predicted by the normal distribution, a condition called leptokurtosis. Sharpe also noted this in his 1970 textbook, Portfolio Theory and Capital Markets. When Sharpe compared annual returns to the normal distribution, he...
The Hurst Exponent
Hurst was a hydrologist who began working on the Niie River Dam project in about 1907 and remained in the Nile region for the next 40 or so years. While there, he struggled with the problem of reservoir control. An ideal reservoir would never overflow a policy would be put in place to discharge a certain amount of water each year. However, if the influx from the river were too low, then the rese if vel would become dangerously low. The problem was What policy, of discharges could be set, such...
Modern Portfolio Theory
Meanwhile, Modern Portfolio Theory MPT was also being developed. Markowitz 1932 made the distribution of possible returns, as measured by its variance, the measure of riskiness fiC tfcyportfolio. Formally, the population variance is defined by the following formula rM - mean return r return observation At the limit, the variance would measure the dispersion of possible returns around the average return. The square root of the variance, or standard deviation, measures the probability that the...
Rs Analysis Of The Sunspot Cycle
Before we analyze the capital markets in the next chapter, it would be useful to apply R S analysis to a time series of real data from a natural system. Perhaps the most widely known natural system with a nonperiodic cycle is the sunspot cycle. Sunspot numbers have been recorded since the mid-18th century, when Wolf began a daily routine of examining the sun's face through his telescope and counting the number of black spots on its surface. When he died, the Zurich Observatory continued this...