英语论文网

mid-1982. Since that
time it has traded within a range of approximately $250–$500, with the price being
around $300 during the .rst half of 2002.
2. Detrended uctuation analysis of the gold price
The time series of the gold price showninFig. 1 is obviously non-stationary, with
clear trends being visible for both short and long periods, although these local trends
change rather erratically. Detrended Fuctuation analysis (DFA) attempts to overcome
the statistical diGculties introduced by such irregular local trends by linear detrending
using non-overlapping windows. Thus, if we denote the level of the index at time t
(t running from 1 to N = 8237) as xt , and choose a window length L, thenthere willT.C. Mills / Physica A 338 (2004) 559 – 566 563
0
1
2
3
4
5
6
7
8
-10 -8 -6 -4 -2 0 2 4 6 8 10
Logs of P(l)
l- l+
Fig. 4. Histogram for step lengths of monotonous index changes, l− and l+ denote the length of increasing
and decreasing series. The solid lines are exponential .ts of Eq. (4).
The distributionis therefore asymmetric about zero and the probability of a run
of positive changes of length l is less thanthe probability of a negative runof the
same length. (Note that this is consistent with the predominance of signi.cant negative
autocorrelations showninFig. 3.) These models imply that, for example, the probability
of getting an all-positive sub-series of length l after observing an all-positive sub-series
of length l − 1 is exp(−0:82) ≈ 0:44, while the corresponding probability for an
all-negative sub-series is exp(−0:66) ≈ 0:52.
3. Distribution of gold returns
We now consider the behaviour of gold returns, de.ned as
rt = log xt+1 − log xt (5)
(Since the mean of rt is almost zero, de-meaning is unnecessary.) Panel (a) of Fig. 5
shows a quantile-quantile plot of rt against a Gaussiandistributionwith the same mean
and standard deviation. If returns were Gaussian, all points should fall on a straight
line. Panel (b) shows the empirical density function of rt , with the Gaussianden sity
superimposed. It is seen that the tails are much ‘fatter’, and the central part of the
empirical density more ‘peaked’, than a Gaussian distribution and hence gold returns
display the leptokurtic behaviour that is typical of asset returns. An interesting questionT.C. Mills / Physica A 338 (2004) 559 – 566 565
0
4
8
12
16
20
50 100 150 200 250 300 350 400 450 500
K (L)
L
Fig. 6. Kurtosis, K(L), of cumulative returns, as a function of L, computed according to Eq. (7). The value
K(L) = 3 (horizontal line) of a Gaussian distribution is reached for L ≈ 235 days.
where L is de.ned as above, so that we have m=N=L non-overlapping variables of this
type. For each L we cancompute the kurtosis of the cumulative returndistribution[ 5]
K(L) =
r4
t (L)
r2
t (L)2 : (7)
For a Gaussiandistribution K(L)=3, and Fig. 6 plots K(L) against L, showing that
the distributionof rt(L) recovers the Gaussianshape after a time lag of around 235
days, i.e., after approximately elevenmon ths.
4. Volatility analysis of gold returns
留学生论文网Given a window length L, we canestimate the volatility for the m = N=L time
windows as [6]
O 2i
(L) =
1
L
L
j=1
(ri; j − O ri)2; O ri =
1
L
L
j=1
ri; j; i= 1; : : : ; m :
With this vola