FORECASTING VOLATILTY OF ISTANBUL STOCK
EXCHANGE
Ahmet ASARKAYA
Istanbul Bilgi University
Istanbul, Turkey
ahmet.asarkaya@student.bilgi.edu.tr
Abstract
This paper uses eight different models; random walk, historical mean, exponential smoothing,
ARCH(1), GARCH(1,1), EGARCH(1,1), APARCH(1,1) and GJR-GARCH(1,1) to forecast
weekly volatility of Istanbul Stock Exchange 100 Index between 2002 and 2008. Within-week
standard deviation of countinously compounded daily return is used as the volatility
measurement. Both symmetric and asymmetric loss functions are employed to measure the
forecasting accuracy of the models. According to symmetric statistics error, exponential
smoothing produces the best forecast. However, when under predictions are penalized more
heavily ARCH-type models outperform any other models. On the other hand, non-ARCH
models provide the best result when over predictions are penalized more heavily.
Key Words: Stock market volatility, GARCH, forecasting
JEL Classification: C22, C53, G15, G17
3. INTRODUCTION
Forecasting stock market return volatility has great importance for both investors,
traders as well as researchers, because predicting volatility might enable one to take
risk-free decisions including portfolio selection and option pricing. Recent financial
turbulance once again proved the importance of reasonable measurement of
uncertainity in financial markets. This uncertainity is usually known as volatility
which has crucial significance to financial decision makers as well as policy makers.
Forecasting volatility has attracted the interest of many academicians, hence various
models ranging from simplest models such as random walk to the more complex
conditional heteroskedastic models of the GARCH family have been used to forecast
volatility. GARCH was used to forecast volatility for the first time by Akgiray
(1989). Over the years different variations of the GARCH model has been used to
forecast volatility. These models include E-GARCH, GJR-GARCH, VSGARCH, QGARCH etc.
Purpose of this paper is to evaluate out of sample forecasting accuracy of various
models applied to Istanbul Stock Exchange, and to answer the question; Do
GARCH-type models have superiority in forecasting volatility?
Akgiray (1989) is the first one who uses GARCH model to forecast volatility and he
shows that GARCH produces better forecast than most of the other forecasting
methods such as Random Walk (RW), Historical Mean (HM), Moving Average
(MA) and Exponential Smoothing (ES) when applied to monthly US stock market
data. In addition, Cao and Tsay (1992) found that Threshold GARCH outperforms
ARCH, GARCH, and Exponential GARCH on monthly US stock market data. On
the other hand, Dimson and Marsh (1990) evaluate the forecasting accuracy of
simple models such as Random Walk, Moving Average, Exponential Smoothing and
Regression Models on UK stock market data and they conclude that Exponential
Smoothing and Regression Models provide the best forecast. It is important to note
that Dimson and Marsh (1990) did not include ARCH type models in their research.
In the literature, other papers reach different conclusions from one another. For
example, Tse (1991) shows the superiority of Exponentially Weighted Moving
Average (EWMA) over ARCH-type models by using weekly Japanese data. In
addition, Tse and Tung (1992) found the same result with the Singaporean data.
Brailsford and Faff (1996) prefers GJR-GARCH(1,1) as the best model to forecast
Australian stock market. Franses and Van Dijk (1996) use Random Walk,
GARCH(1,1) and other non-linear GARCH models to forecast the volatility of
thestock markets of Spain, Germany, Italy and Sweden and conlude that Quadratic
GARCH (QGARCH) produces the best forecast.
Balaban, Bayar and Faff (2002) investigate the forecasting performance of both
ARCH-type models and non-ARCH models applied to 14 different countries. They
include not only symmetric loss funtions but also asymmetric loss funtions to
measure the out-of-sample accuracy of these various models. By using symmetric
loss functions, they find that Exponential Smoothing is the best model to forecast
weekly volatility and MA models provide the second best forecast. In addition, they
observe that non-ARCH models usually produce better forecast than ARCH type
models. Finally, Exponential GARCH is the best among ARCH-type models. On the
other hand, when asymmetric loss functions are introduced which penalize over
prediction or under predictions more heavily, results differ. When under-predictions
are penalized more heavily, ARCH-type models produce the best forecast while
Random Walk is the worst. However, Exponential Smoothing provides the best
forecast when over predictions are penalized more heavily.
Pan and Zhang (2006) use Moving Average, Historical Mean, Random Walk,
GARCH, GJR-GARCH, EGARCH and APARCH to forecast volatility of two
Chinese Stock Market indices; Shanghai and Shenzhen. They reach a number of
several results. First, Moving Average model is preferred to forecast daily volatility
for Shenzhen stock market. However, they could not reach a conclusion for Shangai
stock market since different models are found to be the best under different
evaluation criteria. Among GARCH models, GJR-GARCH and EGARCH
outperforms other ARCH models for Shenzhen stock market.
Magnus and Fosu (2007) employed Random Walk, GARCH(1,1), TGARCH(1,1)
and EGARCH(1,1) to forecast Ghana Stock Exhange. GARCH(1,1) provides the best
forecast according to three different criterias out of four. On the other hand,
EGARCH and Random Walk produces the worst forecast.
4. DATA AND SAMPLE DESCRIPTION
The data sample used in this research is the daily closing prices of Istanbul Stok
Exchange 100 Index (ISE100) for the period between January 2002 and December
2008. The analysis presented on this exercise involves weekly volatility forecast.
Daily return is calculated as follows:
Ri  log( index i / index i 1 )
where indexi is the closing price on ith day and Ri is the continously compounded
return on ith day. Within-week standard deviation of weekly return is used as weekly
realized volatility:
n
 a ,t 
 (R
t 1
t
 t ) 2
n 1
Where µt is the mean return and n is the number of trading days excluding national
holidays. There are 365 weekly volatility observations in our data sample. First 183
observations are used for estimation whereas the second 182 are used for forecasting.
Descriptive statistics for within-week standard deviations of continuously
compounded weekly returns in the full period, estimation period and forecasting
period are shown in Table 1.
Table 1:
Summary Statistics: Within-Week Standard Devations
Mean
Median
Maximum
Minimum
Standard Deviation
Skewness
Kurtosis
Jarque-Bera Statistics
Probability
Observations
Full Period
Estimation Period
Forecasting Period
0.0185
0.0159
0.0823
0.0028
0.0107
1.98
9.48
877.8
0.0000
365
0.0196
0.0173
0.0823
0.0028
0.0111
2.09
10.25
533.8
0.0000
183
0.0173
0.0149
0.0712
0.0031
0.0100
1.82
7.92
284.3
0.0000
182
There is evidence of positive skewness which means that the distribution has a long
right tail in all periods, so deviations are non-symmetric. Since kurtosis is greater
than 3 in all sub-samples, standard deviations of weekly returns are leptokurtic
relative to the normal. According to Jarque-Bera statistic, we reject the null
hypothesis of a normal distribution; hence we can say that the series is non-normal.
Generally volatility has large differences between its maximum and minimum
values. Standard deviation is also high which indicates that high volatility exists in
Istanbul Stock Exchange.
1. FORECASTING TECHNIQUES
1.1 Random Walk Model
Supporters of efficient market hypothesis say that stock market indices are random
and any attempt to forecast these indices is fruitless. Under Random Walk Model, the
best forecast of this week’s volatility is the last week’s realized volatility.
 f ,t   a ,t 1
1.2 Historical Mean Model
According to this model, the best forecast of volatility at time t is the average of all
past realized volatilities.
 f ,t
1 t 1

  a ,i
t  1 i 1
1.3 Exponential Smoothing Model
An exponential smoothing model gives some weight to the last week’s observed
volatility and takes last week’s forecast into consideration when forecasting this
week’s volatility.
 f ,t  (1   ) a ,t 1   f ,t 1
The smoothing parameter, λ, should lie between 0 and 1. λ is calculated and it is
equal to 0.17.
1.4 ARCH(1) Model
Before the ARCH model introduced by Engle (1982), the most common way to
forecast volatility was to determine volatility using a number of past observations
under the assumption of homoscedasticity. However, variance is not constant. Hence,
it was inefficient to give same weight to every observation considering that the recent
observations are more important. ARCH model, on the other hand, assumes that
variance is not constant and it estimates the weight parameters and it becomes easier
to forecast variance by using the most suitable weights. Mean function of ARCH(1)
is a simple first order auto regression:
Rt  c  Rt 1   t
and the conditional variance equation is as follows:
 t2    t21
1.5 GARCH(1,1) Model
The GARCH model was developed by Bollerslev (1986) and Taylor (1986)
independently. In GARCH(1,1) model, conditional variance depends on previous
own lag. Mean equation of GARCH(1,1):
Rt  c  Rt 1   t
and the variance equation is:
 t2    t21   t21
Where ω is constant,
is the ARCH term and
is the GARCH term. As we
can see, today’s volatility is a function of yesterday’s volatility and yesterday’s
squared error.
1.6 The Exponential GARCH Model - EGARCH(1,1)
Nelson’s (1991) EGARCH(1,1) model’s variance equation is as follows:
log(  t2 )     (
 
 t 1
2
2
)   ( t 1 ) 
   log(  t 1 )
 t 1

  t 1
1.7 The Asymmetric Power ARCH Model – APARCH(1,1)
Taylor (1986) and Schwert (1989) introduced the standard deviation GARCH model,
where the standard deviation is modeled rather than the variance. It is a very
changable ARCH model and the model is specified as follows:
 t     (  t 1  t 1 )   t1
1.8 The GJR-GARCH Model – GJR-GARCH(1,1)
This model is proposed by Glosten, Jagannathan and Runkle (1993). Conditional
variance is given by;
 t2    t21   t21  t21I t 1
Where
I t 1  1
 t 1  0
if
and
I t 1  0
otherwise.
4. FORECASTING EVALUATION
4.1 Symmetric Loss Functions
Root mean squared error (RMSE), mean absolute error (MAE), mean absolute
percentage error (MAPE) and Theil inequality coefficient (TIC) are emloyed to
measure the accuracy of the forecasting models.
365
RMSE 
 (
t 184
  f ,t ) 2
182
365
MAE 
a ,t

t 184
a ,t
  f ,t
182
365

MAPE  100
184
 a ,t   f , t
 a ,t
182
365
 (
a ,t
  f ,t ) 2
184
TIC 
181
365

365
2
a ,t
184
182


2
f ,t
184
182
Where σa,t is the actual volatility and σf,t is the forecasted volatility.
Table 2 gives the actual forecast error statistics for each model. Ranking of the
models is also provided in the table (1 being the best forecast and 8 being the worst
forecast). In the case of RMSE, Exponential GARCH provides the best volatility
forecast where Historical Mean model produces the worst forecast by far. It is
interesting to note that all error statistics except Historical Mean are so close to each
other that no models has clear superiority over other models.
If we look at MAE and MAPE, Exponential Smoothing Model clearly produces the
best forecast, and it is followed by EGARCH. Historical Mean Model, once again, is
the worst forecasting model. According to these two criterias, ARCH-type models
provide better forecasting than non-ARCH models.
The Theil Inequality Coefficient (TIC) is a scale invariant measure that always lies
between zero and one, where zero indicates a perfect fit. Looking at this coefficient
we can say that Random Walk is the best forecasting model. It is interesting to note
that Exponential Smoothing is no longer the best model, on the contrary, it produces
the worst forecasting according to TIC. It is not easy to make a judgement if ARCH
models are better than non-ARCH models; however, if we calculate the mean of the
ranks of ARCH models, we can see that this mean is smaller than non-ARCH
models’ mean which indicates that ARCH models gives better forecasting results.
4.2 Asymmetric Loss Functions
Commonly used loss functions such as MAPE, ME and RMSE are symmetric which
means that they give the same weight to over and under predictions of volatility.
However, for traders, investors and decision makers, it is sometimes important to
know the direction of volatility. For example, under prediction of volatility is not
good for a seller, whereas over prediction is not good for a buyer. In order to
penalize under and over predictions the following Mean Mixed Error (MME)
statistics are introduced:
U
1 O

MME (U ) 









a
,
t
f
,
t
a
,
t
f
,
t

182  t 1
t 1

U
1 O

MME (O) 









a
,
t
f
,
t
a
,
t
f
,
t

182  t 1
t 1

Where U is the number of under predictions and O is the number of over predictions.
MME(U) penalize the under predictions more heavily where MME(O) penalize the
over predictions more heavily. In Table 3, forecast error statistics of each model and
their rank are presented. Since we know that ARCH-type models are likely to overpredict, these models’ forecasting performance are better than non-ARCH models
when their performance is assessed by MME(U). On the other hand, conventional
methods outperform ARCH models according to MME(O). Exponential GARCH
produces the best result among ARCH models regardless of MME(U) and MME(O).
According to MME(U), there is not much difference between ARCH models;
however, EGARCH is slightly better than the other models, and it is followed by
GJR-GARCH. On the other hand, in the case of MME(O), EGARCH is the best
model by far among ARCH family models. Asymmetric Power ARCH is the second
best.
5. CONCLUSION
Forecasting Volatility has attracted the interest of investors and researchers. Various
models has been employed for volatility forecasting ranging from simplest models
such as Random Walk and Historical Mean to more complex ARCH family models.
Some argue that simpler models produces the best forecast, because it is not possible
to construct a model to fully cover the dynamics of financial markets. On the other
hands, some others stress the importance of ARCH-type models in forecasting.
Researchs in the literature has several different findings. Some shows the superiority
of ARCH models whereas others show that simple models are the best.
In this study, we employed eight different models to forecast volatility; Random
Walk model, Historical Mean model, Exponential Smoothing, ARCH(1),
GARCH(1,1), EGARCH(1,1), APARCH(1,1) and GJR-GARCH(1,1). We used
within-week standart deviation to measure the volatility. In order to measure the outof-sample forecasting accuracy of the models, we used both symmetric and
asymmetric loss functions.
Findings of this research can be summerized as follows; first, when we employ
standard symmetric loss funtions to measure the accuracy of the models, we observe
that Exponential Smoothing model provides the best forecast and Historical Mean
model is the worst model in forecasting volatility. Among ARCH-type models,
Exponential GARCH produces the best forecasting results. Overall, ARCH family
models are better than non-ARCH models.
Second, when asymetric loss functions are introduced to penalize under predictions,
ARCH-type models’ performance is better than non-ARCH models’ performance.
On the other hand, Conventional models produce better forecast when over
predictions are penalized more heavily. EGARCH is the best forecasting model
among ARCH models, and GJR-GARCH is the second best.
6. REFERENCES
Akgiray, V. (1989) Conditional Heteroscedasticity in Time Series of Stock Returns:
Evidence and Forecasts, Journal of Business, 62, 55-80.
Balaban, E., Bayar, A. and R. Faff (2002) Forecasting Stock Market Volatility:
Evidence From Fourteen Countries, University of Edinburgh Center For Financial
Markets Research Working Paper, 2002.04.
Bollerslev, T. (1986) Generalized Autoregressive Conditional Heteroscedasticity,
Journal of Econometrics, 31, 307-327.
Brailsford, T. J. and R. W. Faff (1996) An Evaluation of Volatility Forecasting
Techniques, Journal of Banking and Finance, 20, 419-438.
Cao, C.Q. and R.S. Tsay (1992) Nonlinear Time-Series Analysis of Stock
Volatilities, Journal of Applied Econometrics, December, Supplement, 1S, 165-185.
Dimson, E. and P. Marsh (1990) Volatility Forecasting Without Data-Snooping,
Journal of Banking and Finance, 14, 399-421.
Engle, R. F (1982) Autoregressive Conditional Heteroscedasticity with Estimates of
the Variance of U.K. Inflation, Econometrica, 50, 987-1008.
Figlewski, S. (1994) Forecasting Volatility Using Historical Data, New York
University Working Paper Series, FD-94-32.
Franses, P. H. and D. V. Dijk (1996) Forecasting Stock Market Volatility Using
(Non-Linear) Garch Models, Journal of Forecasting, Vol. 15, 229-235.
Glosten, L., R. Jagannathan and D. E. Runkle (1993) On the Relation Between the
Expected Value and the Volatility of the Nominal Excess Return on Stocks, Journal
of Finance, 48, 1779-1801.
Magnus, F. J. and O. A. E. Fosu (2006) Modelling and Forecasting Volatility of
Returns on the Ghana Stock Exchange Using GARCH Models, MPRA Paper,
No.593.
Nelson, D. B. (1991) Conditional Heteroscedasticity in Asset Returns: A New
Approach, Econometrica, 59, 347-370.
Pagan, A. R. and G. W. Schwert (1990) Alternative Models for Conditional Stock
Volatility, Journal of Econometrics, 45, 267-290.
Pan, H. and Z. Zhang (2006) Forecasting Financial Volatility: Evidence From
Chinese Stock Market, Durham Business School Working Paper Series, 2006.02.
Poon, S.H. and C. Granger, (2003) Forecasting Volatility in Financial Markets: A
Review, Journal of Economic Literature, XLI, 478-539.
Taylor, S. J. (1987) Forecasting the Volatility of Currency Exchange Rates,
Internatioal Journal of Forecasting, 3, 159-170.
Tse, Y. K. (1991) Stock Returns Volatility in the Tokyo Stock Exchange, Japan and
The World Economy, 3, 285-298.
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Market, Asia Pacific Journal of Management, 9, 1-13.
APPENDIX
Table 2:
RMSE, MAE, MAPE and TIC of Forecasting Weekly Volatility
RW
HM
ES
ARCH (1)
GARCH (1,1)
EGARCH (1,1)
APARCH (1,1)
GJR-GARCH (1,1)
RMSE
Rank
MAE
Rank
MAPE
Rank
TIC
Rank
0.0102
0.0121
0.0101
0.0102
0.0101
0.0100
0.0101
0.0101
7
8
5
6
3
1
2
4
0.0081
0.0097
0.0068
0.0079
0.0077
0.0072
0.0077
0.0076
7
8
1
6
5
2
4
3
64.68
93.64
46.58
63.19
59.88
51.41
59.85
58.43
7
8
1
6
5
2
4
3
0.2588
0.2686
0.2867
0.2595
0.2614
0.2719
0.2614
0.2608
1
6
8
2
4
7
5
3
Table 3:
Mean Mixed Error (MME) Statistics of Forecasting Weekly Volatility
RW
HM
ES
ARCH (1)
GARCH (1,1)
EGARCH (1,1)
APARCH (1,1)
GJR-GARCH (1,1)
MME(U)
Rank
MME(O)
Rank
Underestimation (%)
Overestimation (%)
0.0195
0.0165
0.0201
0.0121
0.0119
0.0084
0.0105
0.0095
7
6
8
5
4
1
3
2
0.0249
0.0346
0.0186
0.1952
0.1454
0.0904
0.1193
0.1812
2
3
1
8
6
4
5
7
54.3
73.1
41.9
1.4
2.7
3.1
1.1
0.9
45.7
26.9
58.1
98.6
97.3
96.9
98.9
99.1
Figure 1:
Istanbul Stock Exchange National100 Index
70000
60000
50000
40000
30000
20000
10000
0
Jan-02 Nov-02 Sep-03
Jul-04 May-05 Mar-06 Jan-07 Nov-07 Sep-08
Figure 2:
Istanbul Stock Exchange National100 Index Daily Return (%)
15
10
5
0
-5
-10
-15
Jan-02
Nov-02
Sep-03
Jul-04
May-05
Mar-06
Jan-07
Nov-07
Oct-08