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137

This wait-and-see approach reduces volatilitj

Figure 20-la shows that the base price can be found by detrending the data and formulating the volatility based on the detrended values. Although Figure 20-lb does not indicate a time period, volatility only makes sense when measured over some interval. Once detrended, a scatter diagram of price versus volatility should show a relationship similar to Figure 20-lb. Figure 20-lc is a reminder that the magnitude of the volatility and the pricevolatility relationship, is directly proportional to the time interval over which the volatility is measured.

Considering the price-volatility relationship of gold since 1976 gives a tjpical example of how to find the base price. Figure 20-2a is a scatter diagram of monthly gold prices versus the monthly change in price, taken as positive numbers. A more sophisticated study would look at the price range over the month, rather than the net price change. The range will approach zero a little more slowly than the price change. Note that there is a cluster of dots in the price range from $100 to $175 per ounce and then at $300 to $450 per ounce. The lower values can be related to the pre-1980 price levels, while the higher grouping shows low volatility during periods of higher prices since 1980 Again, the use of monthly price changes can yield a value of zero even if there was significant volatility during the month, simply because prices ended at the same place at they began. Using a price range will show a smoother pattem of price versus volatility

Detrending the price of gold using the Consumer Price Index, a technique applied to gold in Chapter 3, improves the uniformity of the results, seen in Figure 20-2b. Instead of two separate clusters of dots at lowvolatilitf levels, there is only one cluster in the range from $110 to $250 per ounce. A curved line has been drawn to represent the pattem of declining volatility in relationship to declining price. This line could be straight if prices were fiirther adjusted using a log or exponential function. According to this curve, volatility approaches zero at a slower rate as prices drop below $200 per ounce.

The time period over which volatility is measured is also a significant factor in the price-volatility relationship. Longer measurement periods give higher volatility values. Regardless of the number of dajs used to determine volatility volatility will increase as price increases. This direct relationship will be very uniform for exchangetraded markets except when prices are very high. Exchange rules may limit trading when prices move too quickly; therefore, charts of real market prices will show that volatility stops expanding when it collides with these artificial constraints.

The magnitude of price movement will also increase as the period of measurement gets longer. During a volatile interval, prices will move farther in one direction during 3 or 4 dajs than they will in I or 2 dajs. As this interval gets very long, the volatility does not keep increasing at the same rate; it tends to slow, as shown in Figure 20-Ic. Thefiattening

FIGURE 20-1 Measuring volatility from a relative base price, (a) Prices become less volatile relative to a long-term deflator (detrending line), (b) Volatility as a function of the detrended price, (c) Change in volatility relative to the interval over which it is measured.



FIGURE 20-2 Finding the base price where volatihty is zero, gold 1976-1993. (a)Volatility versus price, (b) Volatility versus deflated prices.



Gold (US$ per ounc

of the curve occurs at the point equal to the duration of the maximum sustmned price move.

Altemate Measures

The three most used measures of volatility all satisfy the previous lognormal relationship; that is, ihey all expand at an increasing rate as prices rise. The one discussed so br has been net price change over time. This is equivalent to a momentum calculauon @abs(P, - -«*1>, where volatility is calculated over n days. This measure is most useful for establishing the probability of a price change at different levds.

Markets that do not have a significant price change may also be volatile. Many sjstems use the daily or weekly high-low range to define rid; and volatility. This technique can woit, although it does not always capture a full measure of activity The sum of the absolute price changes is a more descriptive value. Using Figure 20-3, the three measures can be diown as:



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