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ALFRED
P.
WORKING PAPER
SLOAN SCHOOL OF MANAGEMENT
Forecasting Foreign Exchange Rates
Subject to De-volatilization
by
Bin Zhou
MIT
Sloan School Working Paper 3510
December 1992
MASSACHUSETTS
INSTITUTE OF TECHNOLOGY
50 MEMORIAL DRIVE
CAMBRIDGE, MASSACHUSETTS 02139
Forecasting Foreign Exchange Rates
Subject to De-volatilization
by
Bin Zhou
MIT
Sloan School Working Paper 3510
December 1992
M
jwn
1983
Forecasting Foreign Exchange Rates
Subject to De-volatilization
Bin Zhou
Sloan School of
Management
Massachusetts Institute of Technology
Cambridge,
MA
02139
December, 1992
Abstract: There
change
rates.
a considerable literature analyzing behavior of ex-
However, modeling and forecasting exchange rates have not
been very successful.
cial
is
time series
is
One
of the obstacles to effective
heteroscedasticity.
modeling of finan-
Recent availability of high frequency
data, such as tick-by-tick data, provides us extra information about the
market.
Considering vast amount of data, this paper proposes to analyze
a homoscedastic subsequence of such data.
such homoscedastic subsequence
is
The procedure
called de- volatilization.
volatilization can help us to detect trends of the
of obtaining
Apparently, de-
market much
forecasting results indicate that the exchange market
is
fast.
not efficient and can
be forecasted to certain extend.
Key Words:
Our
heteroscedasticity; high frequency data; volatility.
Introduction
1
Empirical studies have shown
rates using structural
One
little
and time
success at forecasting foreign exchange
models (Meese and RogofF 1983a,b).
series
of the obstacles to effective modeling and forecasting of exchange rates
The
ARCH
model addresses heteroscedasticity by estimating the conditional variance from historical data and has been used in modeling many financial time series. However,
forecasting exchange rates remains difficult. Recent availability of high frequency data creates new possibilities for forecasting exchange rates.
is
conditional heteroscedasticity (changing variance).
High frequency data, such as minute-by-minute or tick-by-tick data, has
been brought to attention recently by Goodhart and Figliuoli (1991) and
Zhou (1992). As reported in Zhou's paper, high frequency data behaves
differently
Zhou
from low frequency data.
has a significant noise component.
It
suggested the following process for high frequency exchange rates:
S{t)
where S{t)
is
=
d{t)
+
logarithm of price at time
t,
B{-)
a
noise,
independent of Brownian motion B{-).
volatility
and increment of r(6) — T{a)
return X{s,t)
=
S{t)
—
is
e,
is
(1)
standard Brownian motion,
increment function,
d{-) is
drift, r(-) is a positive
+
B{T(t))
tt is
Here
a
r(<)
mean
is
zero
random
called cumulate
called volatility in period
[a, b].
The
S(s) then has the following structure:
X{s,t)
=
^lis.t)
+
a(s,t)Zt
+
ct
-
e,
(2)
where Zt is a standard normal random variable and (T(s,t) = r(<) — t(s).
Returns of high frequency data are often negatively autocorrelated due to
the noises. This autocorrelation decreases as frequency decreases.
This paper presents a new approach to heteroscedasticity of financial time
series.
In Section 2,
we introduce a
de-volatilization procedure, one which
takes a homoscedastic subsequence from high frequency data. In Section
we
test the de- volatilization
volatilized
exchange
rates.
3,
procedure by examining various properties of deFinally, in Section 4,
procedure from the de-volatilized time
series.
we construct a
forecasting
De-volatilization
2
One of most
ticity.
significant characteristics of a financial
Heteroscedasticity tends to
become more severe
increases. This poses a great difficulty in
A
as
heteroscedas-
is
sampling frequency
modeling financial time
obvious shortcoming of equally spaced time series
time intervals and
sufficient in highly volatile
time series
is
series.
that information
is
One
is in-
redundant at other times.
time series with more data in highly volatile time and
less
data in other
is desirable.
Unfortunately no financial time series are recorded in
manner. However, availability of high frequency data allows us to sam-
times
this
ple a subsequence that has equal volatility apart.
The subsequence produced by
de-volatilization.
volatilized
time series or dv-strits
We
call
such a procedure
the procedure
is
called a de-
and differences of successive measurement
of dv-series are called dv-returns.
To carry out the
de-volatilization procedure,
volatility process r(/) first.
we need
to estimate the
Given high frequency data, {5(<,)}, Zhou (1992)
— T{a) for any
has proposed an estimator of the volatility increment t(6)
given period
[a, 6]
by:
T{b)-T{a)^\ J2
[XHt.-,,t,)
+
2X{t,_,,t,)X{t,_,,,t,_,)i
(3)
(.6[a.6]
where X{s,t)
is
—
S{t)
nearly unbiased.
—
and k
S{s),
is
a constant. This volatility estimator
For a given volatility estimator, we have following de-
volatilization procedure:
Algorithm
1
(De-volatilization):
Suppose that {5'(^)}
a series of observations
is
from process
(1).
This
algorithm takes a subsequence from the series and forms a dv-series, denoted
as Vt-
i)
ii)
The
Let
return of the dv-series has approximately the
initial
value Tq
=
same
volatility.
5(/o);
Suppose that we have obtained a dv-series data
at
time t^,
i.e.,
r^
=
S(tm);
iii)
Estimate the
for
i
=
1,...,
volatility
increment V{tm+i,tm)
until the
=
''"('m+i)
—
T{im) by (3)
increment V(im+i)^m) exceeds the
level v,
a
predetermined constant. Let
=
k
Tt+i
iv)
mm{i\T{t,n+,)-T{t^)
=
S{tm+k)
Repeat step
iii)
is
>v
and \Sitm+i)-S{tm+i-i)\
<
v], (4)
the next data in dv-series.
until
end of
series {5(<,)}.
Since the high frequency exchange rates are characterized by excessive
we add an extra condition in (4) to make the dv-series less sensitive to
the noise. Often, we see that price jumps back and forth due to noise. When
the first jump comes, it may significantly bias the volatility estimate. Waiting
noise,
for next
The
data point can minimize the impact of noise on our dv-series.
de-volatiHzation procedure
namic structure
easy to carry out because of the dy-
is
The parameter
of the volatility estimator.
trarily chosen to meet different needs of analysis.
large
enough so that the
volatility estimate
is
acceptable.
Var(et_)/u should also be small enough so that the noise
can be neglected
v can
However,
tt
The
in
be
arbi-
should be
it
noise ratio
the dv-series
.
US
(DM/$) exchange rates
are used to test our de-volatilization procedure. The same data set has also
been used in Zhou (1992). It has more than 2.1 million observations. Yearly
1990 tick-by-tick Deutsche mark and
volatility
is
dollar
estimated as .010.349, and average noise level Var(€)
Based on these
figures,
we choose
?;=3e-6.
hundred data points to estimate the
^
2.6e-8.
This gives us an average of
volatility
six
between two dv-series data
more than one hundred times the noise level. The k in (3)
The basic statistics of the returns of
the dv-series are listed in table 1. The statistics of bi-hourly series are also
give in the table for comparison. The variance of dv-return is always a little
bit larger than v. Both dv-series and bi-hourly series and their returns are
points,
is
and
it
is
chosen to be 6 as in Zhou (1992).
plotted in Figure
3
1
and
2.
Honioscedasticity of Dv-series
Under assumption that the process
rates, the dv-series should
mally distributed.
To
(1)
is
a good approximation of exchange
be homoscedastic and dv-returns should be nor-
visually inspect these properties,
we
plot
month by
Figure
1:
De- volatilized
1X0
DM/$
and
Its
Returns (1990)
Figure
voUohty uu(t)
tt\i(t)
2:
Bi-hourly
2000
1000
2000
voUohr^:
DM/S
and
Its
Returns (1990)
xn
1000
1
MO
hour
2000
ZXX)
3000
Table
1:
Summary
Statistics of the Returns: Dv-series
and Bi-hourly
Seires
Figure
3:
Monthly Sample Variance and Sample Kurtosis
Table
Month
2:
Testing Normality of Dv-returns
normal distribution
A
returns.
market
in
is
not a bad approximation of the distribution of the dv-
may indicate that
may be a forecastable component
small deviation from the Gaussian assumption
not totally efficient and that there
exchange
To
is
rates.
test homoscedasticity,
back as early as 1937
we use the Chi-square
([3], [27]).
It is
test
,
which can be traced
designed to test the null hypothesis of k
independent normal populations having the same variance. The test
statistic
is
k
fc
X'2
=
2.3026
logio-^^E("'
1
where
s]
and
n, are
-
=
1)
1
:
l)logio5?
(5)
=1
the sample variance and size of i-th sample and s^
pooled variance defined
is
the
as:
2
T^=^(n,-l)sJ
E,=i(".
Under the assumption that data
all
- L("- -
is
-
1)
normal and the variance
samples, \^ has approximately a chi-square distribution with
is
A-
—
equal for
1
degrees
of freedom.
Dividing the dv-returns into twelve monthly groups, we have
Xl(ll) = 22.35
Compare
to the hourly series,
(p
=
0.022)
which we also divide into nine monthly sub-
groups,
xLw,(ll) = 273.43
(p
=
0.000)
Clearly, heteroscedasticity exists in hourly series.
However,
it
is signifi-
cantly reduced in dv-series.
As do many other
financial time series recorded in calendar time, the bi-
hourly exchange rate shows autocorrelation in
From our assumption
of
exchange rates
autocorrelation of volatilities.
turns or
its
Box-Pierce
its
squared or absolute returns.
(1), this correlation
Therefore, autocorrelation in
comes from the
its
squared
absolute returns should also be removed in dv-returns.
Q
statistic in (7)
is
re-
The
chosen to test autocorrelation:
m
Qrn
=
n
Y:
1=1
r}
(7)
Table
3:
Testing Autocorrelation of Dv-series Returns
Qw
where
X.
dv-series
19.44
bi-hourly series
13.68
r, is
Xt
(p)
(.04)
16.75
(.19)
85.14
sample autocorrelation of time
|A^I
(P)
(p)
(.08)
20.70
(.02)
(.00)
169.42
(.00)
series
with lag
i,
n
is
size of data.
approximately X'^(m). By choosing m = 10, we calculated the BoxPierce Q statistics for both the dv-returns and the bi-hourly returns. The
Qm
is
statistics
with their p-values are listed
in
Table
3.
These
results
show that
the autocorrelations of returns, squared and absolute returns for the dv-series
are small.
In conclusion, the de- volatilization procedure
tic dv-series
closer to a
of the exchange rate.
The
ditional time series forecast
4
distribution of dv-returns
is
much
Gaussian than that of bi-hourly returns. These results indicate
that forecasting foreign exchange rates
we develop
produced a near homoscedas-
a
new
is difficult.
We
do not expect any
tra-
models to be successful here. In the next section,
forecasting procedure that utilized advantages of dv-series.
Forecasting Foreign Exchange Rates After
De- volatilization
Since the noise Ctau in dv-series
Xr
=
Sr
is
-
negligible, dv-returns
Sr-l
=
Hr
+
can be written as
(^Zr,
where a^ is the variance of the return, /<^ is the trend (which
and a^ is a constant. For 1990 DM/$ dv-returns obtained
When
is
often small),
in last section,
no trend, the return of dv-series ranges from
— 1.96(7 to 1.96cr. However, when the market receives external information,
a significant change occurs in drift and the return is likely out of —1. 96a to
o'^=3.578e-6.
1.96(7
band.
— 1.96(7
We
call
there
is
an '"event" occurred whenever a dv-return outside this
to 1.96(7 band.
If
the market
is
10
not efficient, a trend
may be formed
Table
4:
Correlation of Price Changes During and After the "Events"
k
To evaluate the
we use
forecast,
CI =
following criteria:
X^6.sign(x,)
(8)
= ^SrXr
CI I
(9)
CI is the difference between the number of right and wrong predictions and
CI I is total return assuming no transaction costs. The larger, they are the
better.
dv-returns are independent
If
dependents only on returns
signal Sj
mean
x,,
i
random
zero
<
noises, the forecast
and both
r
CI and CII
have
approximately normal distribution with
=
E[CI]
=
Var(C/)
and
np,
and
E[CII]
where n
is
the
number
forecast signals
=
and
Var(C//)
of non-zero returns
among
and p
= npa 2
is
the percentage of non-zero
these returns.
The only parameter in the procedure is the integer k, the length of the
The a is predetemined by the de- volatilization procedure and is not to
trend.
be estimated
14.
in
To
illustrate,
Table
5.
zero at the
procedure. Table 4 suggests that k
in this
we
When k =
5% level.
Although
this
is
rates.
5,
both
CI and CII
It is
more convincing
We obtained
1991
DM/$
DM/$
in the interval 3 to
for all k
=
I,
..,
15
are significantly greater than
a nonparametric procedure,
data retrospectively.
exchange
hst forecasting results of 1990
lie
it is
analyzed using the 1990
to forecast a succeeding year of
tick-by-tick data
from
J. P.
Morgan.
Using exactly the same de- volatilization procedure and forecasting procedure,
we show
forecast results of 1991
are not only significant at k
We
=
5,
DM/$
in
but at
many
Table
forecasting result
is
very
For 1991,
CI and CII
other choices of k as well.
The exchange
after the "event" and this trend is forecastable. The
encouraging. However profits are very slim if we take
conclude that the exchange market
rate often forms a trend
6.
is
not
efficient.
account of the bid-offer spread. Using the following simple trading program
with k
=
5
and bid
offer
spread .05% of the price, we have profit=2.1% for
1990 and profit=2.6% for 1991. Further improvement
the forecast more profitable.
12
is
necessary to
make
Table
5:
Forecasting 1990
k
DM/$
by Forecasting Procedure
I
Table
6:
Forecasting 1991
k
DM/$
by Forecasting Procedure
I
Algorithm 3 (Trading Program) This program assumes
of
US
•
amount
that fixed
dollars are traded at each position.
At time
1.
If
6t+\
2.
If
^T+i
At time
•
• If
suppose that no position
r,
r,
= 1,
held.
is
take a long position;
= — li
take a short position;
suppose that one position
same
is
held,
1.
If
Sr+i6r >0, keep the
2.
If
6j+i6r=0, terminate the position;
3.
If
St+iSt <0, reverse the position;
one position bought
p^^^^
position;
and
at time Tq
sold at time Tj,
^ 6.Jexp(.,)-expK)) _
^^Q^j ^ ^^^^^
^jQ^
exp(.sj
Recent stock market studies suggest that price has
less
autocorrelation
during period of large volume or large volatility (LeBaron 1990, Campbell,
If this is also true for currency exchange markets, an event
etc., 1992).
occuring during a period of extremely high volatility
trend.
We
Algorithm
4 (Forecasting procedure
11)
Given a dv-series
downward,
price St
•
is
flat
and upward trends.
1.
—
if
.St-i|
>
(7y[t(T)
Sj+i
2.
{st, r
=
0,
....
n},
Sj
recorded and VkT be average hourly volatility,
Initialize all 6^ to 0, r
• If \st
not form a future
G { — 1,0,1} corresponding to
Let <(r) be the time (in second) that
procedure generates forecasting signals
this
may
therefore modify our forecasting procedure as follows:
\f
1.96cr,
=
1, ...,
n;
and
-t{T -\)]< auhr/3600,
=
s'ign{sT
a'^/{t(T)-t{T
-
1)]
—
Sr-i),
>
Qt'/,r/3600, set all
be zero;
15
i—l,...k, and t
nonzero
+
?
<
«5^+,,
n;
i
>
0, to
Table
k
7:
Forecasting 1990
DM/$
by Forecasting Procedure
II
Table
k
8:
Forecasting 1991
DM/$
by Forecasting Procedure
II
Figure
5:
Forecast 1990
DM/$
by Procedure
Sv-t<ii<i: 1990
C.
K^
Foi(cistSi([uls
I
Piofit/Loss
u«(l)
18
II
{k
=
10)
Figure
6:
Forecast 1991
DM/S
Dv-suio:
by Procedure
1991
toll)
Foi«utSi(n>h
•^
1
*
II
tin
II)
PiofiVLsii
iiiO)
19
II
{k
=
10)
Table
{k=
9:
10)
Year
Distribution of Events and Profits in Different Sections of the Market
exchange rates
is
extremely
De-volatilization
is
an
difficult.
efficient
way
to use high frequency data.
It
not
only reduces the noise effect in the data, but reduces heteroscedasticity as
well.
The
de-volatilization procedure takes
market and helps to detect trends
early.
It
more observations
an active
corresponde closely to the way
sophiscaticated traders "look'' at exchange markets.
used in any market with high frequency observations.
21
in
The procedure can be
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24
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Date Due
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