1. Price change over time interval n.-

V,=P.-P,

2. The maximum fluctuation during the Inierva! «-

V, = max(P„ )-aar(P„P,.„,,)

3. The sum of the absolute pnce changes over the interval n.-

v, = I \P.-P.

In (1), the volatility is entirely dependent on the value of the two endpoints P, and P,-„.i. regardless of the price activity that occurred duiirig the days between them. Over longer lime periods, a predictable change tan be expected, but in the short lertn this approach is not as good as either (2) or (3).

The maximum ratige (2) corrects for the dependence on only two points and will produce a more consistent measure of volatility, which may be used as an estimate of risk. This method may be effective as the basis for a stop-loss, because it defines the extent of the price fluctuation at each price level Because prices end near where they began in Figure 20-2&. the close-to<lose method (1) would show iinle or no volatility

The sum of the absolute price changes (3) is the mosl descriptive measurement of volatility, although It is nondirectional. It clearly shows that prices are more or less active in cases that are not apparent to cither (1) or (2). A price sequence that moves from highs to lows alternately each day is much more volatile than one in which prices move slowly to the highs and then back to the lows only once during the same time period. This measure is useful for indicating a change in market character, an increase or decrease in activi.

Bookstaber presents a volatility measurement V which is a ratio of successive closing prices, the high and low. or a combination of the two as follows,

where C, is the dosing price on day r U, is the high price on day ( I, is the low on day* V, is the wJatJity on day /

(a) <yuseo-ctose votatilitfi

okmber.lbeCoTnplelelmvstment Book (Scan, Foresnun. Clenneni IL, 198$, p. 349).

FIGURE 20-3 Three wajs of measuring volatility.

then

V? = i(ln«, ,-A)

(b) HIgb-low volatility

(c) Higb-lotv-close volatility

Note that the current t may be a time litterval rather than a single clay. Then, Q is the last price of the period, and H, and L, are the highest and lowest prices of the interval.

In the closc-tD-close estimation, the volatility V, is the standard deviation of the closing price ratios. Bookstaber states that this measurement will follow a distribution and that

the actual volatility during the current period / can be set within the error bounds defined by the distribution.

Volatility System

The daily volatility can be used in a trading strategy, presented by Bookstaber," as follows:

where D, is the maximum of: a. \ , Q-A b,H,-L. \L,-C..A and H, is the high on day t L, is the low on day / C, is the close on day (

Note that D, is the extended range ("true range") concept thai appears in many other worte. V, is the average extended range over the -day interval. The trading strategy presented with this is

Buy if the close Q . i rises by more than k * V,(w) from the previous close C,. SeU if the ciose C,., falls by more than k * V,(«) from the previous close Q.

The voiatiiity constant is given as approximately 3. but it can be varied higher or iower to make the trading signals iess or more frequent, respectively.

Voiatiiity Filters

High voiatiiity is cleariy related to greater risk, but low voiatiiity may also mean that there is no chance of profits This is especially true for frend-following sjstems. The following are reasonable expectations for selecting trades based on volatility-Entering on very high voiatiiity wiil increase rid;. It may be best to simply avoid those trades.

Entering on low voiatiiity seems safe, but prices often have no direction and produce losses. Waiting for an increase in activity before entering mieht improve retums.

Exiting a position when prices become very volatiie should reduce both profits and risk, but may come after-the-fact. Ibis is an issue best resolved by testing.

Filter or Delay

Whenever a high- or low-volatility situation occurs at the time of an entrj signal, there are two choices. The trade can be completely eliminated by filtering, or it can be delayed until the high voiatiiity drops or the iow volatilitj increases to an acceptable level, if the trade ends during the waiting period, the trade is eliminated. Both of these cases have been shidied and the results follow.

Consfructmg a Voiatiiity Fitter

Calculating the voiatiiity is simple to program using any eadaieet or strategj-testing software. The followmg stms were used here:

Ibid.I, 2 --236