00.3610-
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
Revised January 16, 1993
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
Revised January 16, 1993
MAR 2 5
1993.
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
exchange
is
a considerable literature
analyzing the behavior of
However, the modeling and forecasting of exchange rates
rates.
has not been very successful.
financial time series
is
One
of the obstacles to effective
heteroscedasticity.
The
modeling of
recent availability of high fre-
quency data, such as tick-by-tick data, provides us with extra information
about the market.
However, even with today's computer power, analyz-
ing data with order of gigabits
is
extremely expensive and time consuming.
This article proposes to analyze a homoscedastic subsequence of such data.
The procedure
volatilization.
of obtaining such a homoscedastic subsequence
The
is
called de-
empirical evidence suggests that de- volatilization can help
us to detect trends of the market quickly.
Our
forecasting results indicate
that the exchange market can be forecast to certain extend.
Key Words:
heteroscedasticit}- high frequency data; volatility.
Introduction
1
Empirical studies have shown
and time
rates using structural
success at forecasting foreign exchange
little
models (Meese and Rogoff 19S3a,b).
series
and forecasting of exchange rates
is conditional heteroscedasticity (changing variance). The ARCH model addresses heteroscedasticity by estimating the conditional variance from historof the obstacles to effective modeling
One
ical
data and has been used
in
modeling many financial time
The
forecasting exchange rates remains difficult.
frequency data has created new possibilities
series.
However,
recent availability of high
for forecasting
exchange
rates.
High frequency data, such as minute-by-minute or tick-by-tick data, have
recent received attentionly from
(1992).
As reported
in
Goodhart and
suggested the following process
S{t)
d{-)
^
for
is
significant noise
d{t)
+
+
B{T{t))
t,
B{-)
is
and increment of
return Xis,t)
=
S{t)
-
r(6)
-
T{a)
is
(1)
tt
standard Brownian motion,
a positive increment function,
independent of Brownian motion B{-).
volatility
component. Zhou
high frequency exchange rates:
logarithm of price at time
a drift, r(-)
is
noise,
is
and Zhou
Zhou's paper, high frequency data behave differently
from low frequency data and they have
where S{t)
Figliuoli (1991)
tt
Here
is
a
r(<)
mean
is
zero
random
called cumulate
called volatility in period
[a, b].
The
S(s) then has the following structure:
XisJ) =
fi{sJ)
+
<7{sJ)Zt
+ et-es
(2)
where Zt is a standard normal random variable and a{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 article presents a new approach to heteroscedasticity of financial
time
series.
In Section 2,
I
introduce a de-volatilization procedure, which
takes a homoscedastic subsequence from high frequency data. In Section
various properties of
.3, I test the de-volatilization procedure by examining
de-volatilized
exchange
rates. Finally, in Section 4,
procedure from the de-volatilized time
series.
I
construct a forecasting
De-volatilization
2
One
of
most
significant characteristics of a financial time series
Heteroscedasticity tends to
ticity.
become more severe
as
is
heteroscedas-
sampling frequency
One
increases. This poses a great difficulty in modeling financial time series.
obvious shortcoming of equally spaced time series
time intervals and
sufficient at highly volatile
A
is
that information
is
redundant
time series with more data at highly volatile time and
times
data at other
less
Unfortunately no financial time series are recorded
desirable.
is
in-
is
at other times.
in
this manner. However, availability of high frequency data allows us to sam-
ple a subsequence that has equal volatility apart.
The subsequence produced by
de-volatilization.
volatilized time series or dv-senes
call
I
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,
I
need to estimate the
volatil-
Given high frequency data. {5(^)}, Zhou (1992) has
proposed an estimator of the \'olatility increment 7(6) — T{a] for any given
ity
process T(t)
period
first.
by:
[a, b]
-r(a) = i J2
7(6)
[-\''it,-,.t.]
+
-2X{t,_k,t,)X{t,_,kJ.-k)].
(3)
(.e[a,6]
where X{s,t)
=
S{t)
S{s), and k
is
a constant. This volatility estimator
For a given volatility estimator.
nearly unbiased.
is
—
I
have following de-
volatilization procedure:
Algorithm
1
(De-volatilization):
Suppose that {^(i,)}
is
a series of observations from process (1). This algo-
rithm takes a subsequence from the
r^.
The
i)
ii)
Let
iii)
initial
value Tq
Suppose that
rr
=
and forms a
i
I
=
dv-series,
same
denoted as
volatility.
5(^0);
have obtained an element of dv-series at time tm-
i-e.,
S{t^);
Estimate the
for
series
return of the dv-series has appro.ximately the
=
1....,
increment \'{tm+,Jjn)
=
—
T{tm) by (3)
until the increment V'(fm+,,^n,) exceeds the level v, a
volatility
3
r(^n,+,)
predetermined constant. Let
=
k
r^+i
iv)
min{i\r(t„^+,)-T(t^)
=
S{tm+k)
Repeat step
iii)
>
and |5(i^+,) -5(^m+,_i)| <
V
i-},
(4)
the next element in dv-series.
is
end of
until
series
{5(^)}.
Since the high frequency exchange rates are characterized by excessive
noise,
add an extra condition
I
the noise. Often,
the
first
in (4) to
see that price
jump comes,
it
may
make
the dv-series less sensitive to
jumps back and
forth
due to
noise.
When
Waiting
significantly bias the volatility estimate.
next data point can minimize the impact of noise on the dv-series.
for the
The
de-volatilization procedure
namic structure
is
easy to carry out because of the dy-
The parameter
of the volatility estimator.
v can be arbi-
chosen to meet the different needs of a variety of analyses. However,
trarily
it
I
should be large enough so that the volatility estimate
noise ratio Var(e(_)/i' should also be small
enough
is
acceptable.
so that the noise
Ct
The
in the
dv-series can be neglected.
(DM/$) exchange rates
procedure. The same data set has also
US
1990 tick-by-tick Deutsche mark and
are used to test our de-volatilization
been used
volatility
Zhou
in
is
(1992).
It
dollar
has more than 2.1 million observations. Yearly
estimated as .010349, and average noise level Var(e)
Based on these
figures,
hundred data points
I
choose L'=3e-6.
^
2.6e-8.
This gives us an average of six
to estimate the volatility
between two dv-series data
more than one hundred times the noise level. The k in (3)
The basic statistics of the returns of
is chosen to be 6 as in Zhou (1992).
the dv-series are listed in Table 1. The statistics of bi-hourly series are also
give in the table for comparison. The variance of dv-return is always a little
bit larger than v. Both dv-series and bi-hourly series and their returns are
plotted in Figure 1 and 2.
points,
3
and
it
is
Homoscedasticity of Dv-series
Under the assumption that the process
change
rates, the dv-series should
(1)
is
a good approximation of ex-
be homoscedastic and dv-returns should be
Figure
1:
De- volatilized
DM/$
and
Its
1000
2000
1000
Returns (1990)
20X
voltohcy: t4u(t)
Figure
2:
Bi-hourly
DM/$
and
Its
ll
Returns (1990)
3000
Table
1:
Summary
Statistics of the Returns: Dv-series
and
Bi-liourly Seires
Figure
3:
Monthly Sample Variance and Sample Kurtosis
Table
Month
2:
Testing Normality of Dv- returns
The normality should be
some
rejected by
test results indicate that
normal distribution
A
of the distribution of the d\'-returns.
assumption may indicate that market
may be
To
a forecastable
component
test homoscedasticity.
is
random walk and that there
exxhange
in
not a bad approximation
small de\'iation from the normal
not a
is
rates.
use the Chi-scjuare test
I
,
which can be traced
back as early as 1937 (Bartlett 1937; Ostle and Mensing 1975).
to test the null hypothesis of k independent
same
The
variance.
test statistic
=
2.3026
is
k
iogio-^'E("'
-i)-E(", -
1=1
where
s^
and
n,
niogio^^^
(5)
1=1
are the sample variance and size of i-th sample
pooled variance defined
designed
It is
normal populations having the
k
y
However the
tests for a large n.
and
s^
the
is
as:
=
-<'
^^^
E,=i(".
-
(6)
1)
Under the assumption that data are normal and the variances are equal
for all samples,
\'^
—
I
monthly subgroups,
I
has approximately a chi-square distribution with k
degrees of freedom.
Dividing the dv- returns into twelve monthly groups.
,X'i.(ll)
From
the hourly series, which
I
=
22.35
(p
=
I
have:
0.022)
also divide into twelve
have:
vLw,(H) =
273.43
(p
-
0.000)
Clearly, heteroscedasticity exists in hourly series.
cantly reduced
.As
do many other
From our assumption
financial
of
time
series
exchange rates
autocorrelation of volatilities.
its
Box- Pierce
it
is
signifi-
in dv-series.
hourly exchange rate shows autocorrelation
turns or
However,
recorded
in its
in
calendar time, the
(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
bi-
squared or absolute returns.
re-
The
chosen to test autocorrelation:
Til
Qr,.
-
n
Y:
1=1
rf
(7)
Table
Testing Autocorrelation of Dv-series Returns
3:
Qw
where
Qm
-v,
(p)
19.44
(.04)
16.75
(.OS)
20.70
(.02)
bi-hourly series
13. 6S
(.19)
85.14
(.00)
169.42
(.00)
sample autocorrelation of time
is
approximately \'^(w).
Q
Pierce
1X1
(p)
dv-series
/•,
is
xf
(p)
statistics
statistics for
series
By choosing
m
with lag
—
10,
I
i,
n
is
size of data.
calculated the Box-
both the dv-returns and the bi-hourly returns. The
with their p-values are
listed in
Table
These
3.
results
show that
the autocorrelations of returns, scjuared and absolute returns for the dv-series
are small.
In conclusion, the de- volatilization procedure
tic
dv-series of the exchange rate.
The
distribution of dv-returns
indicate that forecasting foreign exchange rates
any traditional time
I
is
much
normal distribution than that of bi-hourly returns. These results
closer to a
section,
produced a near homoscedas-
series forecast
propose a new
is
difficult.
I
do not expect
models to be successful here. In the next
forecasting procedure that utilizes the advantages
of dv-series.
4
Forecasting Foreign Exchange Rates After
De- volatilization
Since the noise
e^ in
dv-series
.IV
where
cr^ is
and a^
is
a constant.
When
to 1.96cr.
For 1990
there
band.
—
I
is
=
^r-1
/'t
fij is
DM/$
+
can be written as
CrZr,
the trend (which
is
often small),
dv-returns obtained in last section,
no trend, the return of dv-series ranges from
However, when the market receives external information,
a significant change occurs
1.96(7
Sr
negligible, dv-returns
the variance of the return,
<T^=3.578e-6.
— 1.96(7
=
is
in drift
and the return
is
likely
out of —1.96(7 to
say that an "event" has occurred whenever a dv-return
10
falls
Table
4:
Correlation of Price Changes During and After the "Events"
k
•
—
If \Sr
>
Sr-\
1.96(7
I
<5r+i
=
—
siga(^,-
Sr-i),
=
1
and
1, •••A\
t
+
i
<
n;
This forecast overwrites any previous forecast.
To evaluate the
forecast.
use following criteria:
I
CI
=
J2^rs\gnixr)
(8)
CI I = Y.^rXr
(9)
CI is the difference between the number of right and wrong predictions and
CII is total return assuming no transaction costs. The larger they are the
better.
mean zero random noises, the forecast
.r,,
< r and both CI and CII have
dv-returns are independent
If
depends only on returns
signal 6r
;
approximately normal distribution with
E[CI]
=
Var(C/)
and
=
np,
and
E[CII] =
where
?7
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
trend. The a is predetemined b\' the de-volatilization procedure and
to be estimated in this procedure. Table 4 suggests that k
to 14.
in
To
Table
illustrate,
5.
zero at the
I
When k =
5% level.
Although
this
is
rates.
I
forecasting results of 1990
.5,
CI and CII
both
It
is
more convincing
DM/§
obtained 1991
DM/$
lie
is
not
in the interval 3
for all k
=
1,
..,
15
are significantly greater than
a nonparametric procedure,
data retrospectively.
exchange
list
of the
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,
I
show
forecast results of 1991 DM/!*! in Table 6. For 1991,
not only significant at k
I
and
=
.5.
but
at
many
CI and CII
are
other choices of k as well.
conclude that the exchange rate often forms a trend after the "event"
this trend
is
forecastable.
The
forecasting result
12
is
very encouraging.
Table
5:
Forecasting 1990
k
DM/S
by Forecasting Procedure
I
Table
6:
Forecasting 1991
k
DM/S
by Forecasting Procedure
I
However
profits are very slim
if I
take account of the bid-offer spread. Using
=
the following simple trading program with k
of the price.
I
improvement
have profit=2.l7c.
is
necessary to
and bid
more
the forecast
.At
time
1.
If (*)r+i
2.
U
dr+\
At time
•
•
If
suppose that no position
r,
r,
= l.
=
take
a
spread .05%
profitable.
Algorithm 3 (Trading Program) This program assumes
ber of US dollars are traded at each position.
•
offer
1990 and profit=2.6% for 1991. Further
for
make
5
that a
fi.xed
num-
held.
is
long position;
—1. take a short position;
suppose that one position
same
is
held,
1.
If
Sr+\6r >0, keep the
2.
If
(5r+i^r=0, terminate the position;
3.
If
6r+\dr <0, re\'erse the position;
one position bought at time
position;
and sold
Tq
at
^ ^,(exp(s.J-exp(.J) _
time
Tj,
^QQC7,.
QQQ^j ^
(iq)
exp(.^ro)
Recent stock market studies suggest that price has less autocorrelation
during period of large volume or large volatility (LeBaron 1990. Campbell,
1992).
etc.,
If
this
is
also true for currency
exchange markets, an event
occuring during a period of extremely high volatility
trend.
I
Algorithm
4 (Forecasting procedure
II)
Given a dv-series
procedure generates forecasting signals 6r
this
downward,
price
.St
may
not form a future
therefore modify our forecasting procedure as follows:
is
flat
and upward trends.
=
n},
corresponding to
Let <(r) be the time (in second) that
recorded and Vhr be a\'erage hourly volatility,
•
Initialize all Sr to 0, r
•
If \Sr
1.
6 { — 1,0,1}
{s^, r
—
if
Sr-i\
>
a-7[^(^)
dr+i
1.960-.
=
;;;
1
and
-t{r - D] <
—
sign(.s,.
—
ar/.,/.3600,
.Sr_,
).
15
I
=
\. ...k.
and t
+
i
<
n\
\icr^/[tiT)-t{T
2.
-
>
1)]
nonzero
Qf,,, /3G00, set all
6,+,,
i
>
0, to
be zero;
This forecast overwrites any previous forecast.
Empirical results show that forecast procedure
CI and C I L
values of
=
with Q
5 are listed in Table 7
procedure
4
<
<
A:
but the profitability as
is
11.
arbitrary.
and
CI and CII
for
The
well.
choice of
results of forecasting
a =
5 in the forecasting
both 1990 and 1991 are significant at the
in
dv-.series; (b) forecast signal St
in
—
For k
10,
there
is
the market. In Figure 5 and
and
5%
9.03% profit in
the market and 18.62% profit in 1991
profits.
1990 with only 26.7% of total time
with only 25.5%i of total time
There
increases not only the
Results for a between 4 and 6 are very similar. For
Both years show sizeable
level.
The
8.
II
(c)
6,
I
show:
(a)
cumulate profit/loss curve.
result: most profits are from "events" that
European
and
the
happened in
US markets. The "events" in other markets are merely noises. Dividing the 24-hour market into four sections: Eu-
another interesting
is
rope only (3:00am
EST/EDT
-
EST/EDT)
day),
EST/EDT
-
8:00am EST/EDT), Europe and US (8:00am
12:00pm EST/EDT), US only (r2:00pm EST/EDT - 5:00pm
and .\sia/other (5:00pm EST/EDT - 3:00am EST/EDT next
calculate possible profits from the "events" in different section of the
I
market (Table
9).
The numbers may
not add up to the total in Table 7 and
8 because that a position taken in one section
may
hold into next section of
the market.
The
largest
number
of "events" occur during the period that both Eu-
ropean an .American markets are open, although
only lasts four hours.
ket
is
relevant to the
DM/$
exchange
correlation between the "events" in
events
in
market
Profit distribution in different sections of the
consistent with our expectation that only news from
is
markets
my
rates.
These
mar-
European or US
results indicate the
forecasting procedure
real
news
random
walk.
and
the market.
Discussion
5
I
this section of the
believe that foreign exchange markets do not follow a pure
It is
consisted with
many
small trends. These trends last only a day or two
16
Table
k
7:
Forecasting 1990
DM/$
by Forecasting Procedure
II
Table
k
S:
Forecasting 1991
DM/S
by Forecasting Procedure
II
Figure
5:
Forecast 1990 D.M/S
b}'
Dv-s<ii<s; 1990
Foicc>slSi([uls
Piofivlon
K«1
e
19
Procedure
11
{k
=
10)
Figure
6:
DM/S
Forecast 1991
by Procedure
Dv-s(2ics: 1991
FoKCistSifluls
__,
-
»v
'r-
ry~
I
I
Halt)
Piofit/Losi
K
i>«9)
20
II
[k
=
10)
and occur
in
random
exchange rates
is
extremely difBcult.
De-volatilization
is
an
efficient
only reduces the noise effect
well.
The
Consequently precise forecasting of daily
directions.
in
way
to use high frequency data.
de-volatilization procedure takes
market and helps
to detect trends early.
It
more observations
in
not
any market with high frequency observations.
21
in
an active
corresponde closely to the way
sophiscaticated traders "look" at exchange markets.
used
It
the data, but reduces heteroscedasticity as
The procedure can be
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McMahon
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The Foreign Ex-
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./.
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24
5939 G55
Date Due
JUL
08
199^
MIT
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