Forecasting stock market volatility
Abstract
Volatility analysis of Stock markets is an important area of study. There have been various academic studies in past on the effectiveness of time series models in estimating and forecasting the volatility of the stock markets of various developed and developing countries. Some scholars chose to model volatility of stock index by using all ARCH and non ARCH models while some of them only used one model. The data used by these scholars also differed in composition. Depending on the objective of the study, data from only one country was analyzed or multiple stock exchanges' data was modeled and compared.
This paper examines the use of GARCH type models for modeling volatility and explaining financial market risk on the historical data of Nigerian Stock exchange. We have used data for 9 years from DataStream sourced from Nigerian Stock Exchange (LSE All share index) for our analysis. Four time series are employed viz. GARCH, Threshold GARCH, Exponential GARCH and ARCH8. The analysis on the data gives strong evidence that the four models can be used to characterize the daily returns (to be verified)..
Chapter 1
Introduction
In the past few years, considerable uncertainty and volatility has been observed in the emerging and mature financial markets worldwide. Financial analysts and investors are concerned about the fluctuating returns of their investments due to the market risk and variation in the market price speculation as well as the instable business performance (Alexander 1999).
Quantitative models are used in financial econometrics to decipher the investor's attitude towards the risks and returns as well towards the volatility as well. It is important for any investors or prospective investor to be cognizant of the risks associated with the market volatility and the techniques to manage those risks. The study of market volatility involves application of models which are capable of handling the volatility of the market and the time series. Financial Analysts model and explain the behavior of stock market and its returns as well as volatility by using the time series econometric models. The reasons for using econometric models are many viz. uncertainties in returns and the prices, variance in the financial markets which is nonconstant and unexpected events in the country or world which has the potential of influencing investors for example September 11.
One of the most prominent tools for capturing such changing variance was the Autoregressive Conditional Heteroskedasticity (ARCH) and Generalized ARCH (GARCH) models developed by Engle (1982), and extended by Bollerslev (1986) and Nelson (1991). Two important characteristics within financial time series, the fat tails and volatility clustering (or volatility pooling), can be captured by the GARCH family models. A series with some periods of low volatility and some periods of high volatility is said to exhibit volatility clustering. Volatility clustering can be thought of as clustering of the variance of the error term over time: if the regression error has a small variance in one period, its variance tends to be small in the next period, too. In other words, volatility clustering implies that the error exhibits timevarying heteroskedasticity (unconditional standard deviations are not constant).
In this paper, we capture financial time series characteristics by employing GARCH (p,q) model and its EGARCH, Threshold GARCH (TGARCH), and ARCH8 extensions. These models have the advantage of permitting investigation of the potentially asymmetric nature of the response to past shocks.
Several studies investigate the performance of GARCH models on explaining volatility of mature stock markets (e.g. Sentana and Wadhwani, 1992; Kim and Kon, 1994; Kearney and Daly, 1998; Floros, 2007; Floros et al., 2007), but few have tested GARCH models using daily data from African markets. Frimpong et al. (2006) examine the behavior of stock returns as well as the market efficiency and volatility effects in the Ghana stock exchange using GARCH models. They conclude that GARCH (1,1) model outperformed the other models under the assumption that the innovations follow a normal distribution.
Few studies have been conducted on the Middle East markets as well. Alberg et al. (2006) estimate stock market volatility of Tel Aviv Stock Exchange indices, for the period 19922005, using asymmetric GARCH models. They report that the EGARCH model is the most successful in forecasting the TASE indices.
Finally, Meric et al. (2007) study the comovements of the US, UK and Middle East stock markets (Egyptian, Israeli and Turkish) during the period 1996 to 2006, and report a very low correlation. They also present the average weekly returns and volatility of returns. According to Meric et al. (2007), Israeli's average weekly returns are 0.24%, while the Egyptian stock market has a high average weekly return (0.45%). Also, the Egyptian stock market has the highest return per unit of volatility risk (0.1%), while the Israeli stock market has the lowest return per unit of volatility risk (0.06%) over the examined period.
The purpose of this paper is to forecast the stock market volatility by using the four time series GARCHfamily models for the last 9 year's stock index data from Nigerian stock market.
The analysis focuses on Nigerian stock index i.e. the LSE All share index and data for 09 years is captured for conducting the analysis. Four GARCH family models i.e. GARCH, Threshold GARCH, Exponential GARCH and ARCH8, are used to conduct the analysis.
The rest of this paper is organized as follows. Chapter 2 contains the details on the Nigerian Stock market with focus on history, fundamental driver and the recent trading activity and the happening in the Nigerian Stick exchange. Chapter 3 is devoted to literature review while Chapter 4 deals with the data collection, analysis and results. Chapter 5 is the concluding chapter in this paper.
Chapter 2
Overview of Nigerian Stock Market
According to the latest study, Nigerian Stock Market ranked fourth in the world in 2007, on the basis of sheer growth. It is behind China's Shanghai (Hongyu, P. and Zhichao, Z, 2006) Ukraine's PFTS and Slovenia. Its growth was 73% which is measured by All Share Index as it acts as natural pointer to the growth of economy. Within Nigeria all its companies are publicly quoted and this index contains shares of them. Equity market was small before 2004 in Nigeria because of low value of equity trade as a proportion to both market capitalization and GDP. Index of NSE has grown with time from 134.6 in 1986 to 65005.48 in 2008.
NSE has automatic trading system which means that data for the companies that are listed is presented monthly, weekly, daily, quarterly and annually. Government has abolished laws that used to prevent flow of foreign exchange in the Nigeria to increase foreign investment in the country.
History of the Nigeria Stock market
Stock market of Nigeria is called Nigerian Stock Exchange (NSE) that was started in 1961 with 19 securities that increased to 264 securities in 1998 (Olowe , 2008). Today there are 262 securities listed on The Exchange, made up of 11 Government Stocks, 49 Industrial Loan (Debenture/Preference) Stocks and 194 Equity / Ordinary Shares of Companies.
The Nigeria Stock Exchange took on its present name in 1977 as by then the Stock Market already had branches in the most important business centers of the nation. The branches of The Nigeria Stock Exchange are as given below:

Lagos opened in 1961

Kaduna opened in 1978

Port Harcourt opened in 1980;

Kano opened in 1989;

Onitsha opened in February 1990;

Ibadan opened in August 1990;

Abuja opened in October 1999;

Yola opened in April 2002.
Lagos however remained the headquarters of the Nigeria Stock Exchange. The business of trading in Nigeria stock exchange Market is conducted during the weekdays from 11 o' clock in the morning to 1 o' clock in the afternoon local time. The goods, which are kept as security while trading are, corporate bonds, shares and government bonds.
The trading of shares and bonds are done by the stock brokers in large halls by shouting out loud or by making phone calls which each branch having its own trading hall. The Nigeria Stock Exchange now uses an Automated Trading System which makes trading easier, faster and safer. The Nigeria stock exchange was started with a small capital and in the year 1997 the capital went up to a total of about $ 3 billion. Nigeria Stock Exchange has been one of the most successful businesses in Nigeria. The AllShare Index (AMI) is the stock market index of the Nigeria Stock Exchange which for its calculation uses only common stocks (ordinary shares), was developed in 1984. Its market capitalization was 5.12 trillion naira at the end of 2006 compared to 2.9 trillion the previous year.
In the quest of attaining world standard in stock marketing, Nigeria stock exchange became member of the FIBV or “Federation of International Stock Exchange”. Also with the current fight for fraud and corruption, public trust in the Nigeria stock market has grown tremendously, with about three million individual investors and hundreds of institutional investors (including foreigners who own about 47% of the quoted companies) using the facilities of The Nigeria Stock Exchange. The Nigeria Stock Exchange's 39year history is devoid of any fraud, shocks, scandals or insider dealings.
The body that governs the Nigeria Stock Exchange is the Securities and Exchange Commission (SEC). Clearing, Settlement and Delivery of transactions on The Nigeria Stock Exchange are done electronically by the Central Securities Clearing System Limited (CSCS), a subsidiary of The Nigeria Stock Exchange. The CSCS Limited (“the Clearing House”) was incorporated in 1992 as part of the effort to make the Nigeria stock market more efficient and investorfriendly. Apart from clearing, settlement and delivery, the CSCS Limited offers custodian services. Charges included in all transactions are a 3% commission on the traded value of shares and a 1% Securities and Exchange Commission fee. Withholding tax on dividend and interest remains at 10%; corporate income tax, 35%, capital gains tax, 10%.
The Fundamental driver of the Nigeria Stock Market
In 1971 to 1987 government and industrial loan stocks dominated the market and there were hardly any trading transactions in equity market. But after 1988 value of equity transaction increased and was around 0.0624. It has always been fluctuating as it was 0.0516 in 1998, 0.1059 in 2004 and 0.878 in 2005.
When Central Bank increased the capitalization of Nigerian Banks to 25 billion, then banks raised around 406.4 billion from capital bank. This affected quoted securities on NSE. Market Capitalization was increased to 81.0104 in 2007 due to recapitalization of banking industry that also increased value of equity trade. If we take value of equity trade to be proportional to GDP, then it was 0.0023 in 1988, 0.00001 in 1989, 0.0033 in 1998, 0.0192 in 2004 and finally 0.0171 in 2005. The data indicates a fluctuating trend.
In 2004, Central bank of Nigeria announced new capital requirements of 25 billion for banks of Nigeria. The new capital requirements increased number of activities in NSE and brightened the confidence of those who invested in Nigerian economy and stock market. This affected the volatility and increased capital formation.
In 2005, recapitalization of insurance and reinsurance companies was announced by federal government of Nigeria. This increased number of securities in Nigerian stock market that increased public awareness and confidence for stock market. Since 2008, stock prices were declining of Nigerian stock market.
Trading activities and recent happening in the Nigeria stock market
Every major bank in Nigeria is raising funds through Global depository receipts; this can help other world states to invest indirectly. Many of the GDR holders settled abroad are playing in a manner that they can affect prices of Nigeria. If they exit the market by disposing their shares, this will mark the market down. Capital market or banking industry is not the only source for the growth of Nigerian economy. More local people are joining its stock market day by day. This increase in participation locally and internationally will create a situation where there will be more bids and less number of offers. A situation of market correction can also occur where there could be sudden market exit of the people leading to the crash. Due to increased participation of international investors and regulators, global market may behave in irrational ways and can lead to volatility, herding and contagion. This means it may happen that all the global markets may pull all the funds from market one day and major stakeholders sell off from market then the small shareholders who were unaware may lose big amounts and incur heavy losses.
It is believed that there will be not much volatility in coming years in Nigeria and its stock market will mature over time. A slowdown is necessary at some point so that bond market can become vibrant. To increase market capitalization and capital growth, there should be ensured from repositioned banks that more companies are getting into stock market. Since crude oil prices are surpassing at $100, this has developed confidence for global investors. Nigeria will be financially stable if it keeps racking up its foreign reserves. Government also needs to ensure that social roles are discharged effectively.
Noise traders had an impact on Nigeria, since rational traders were drowned, but it failed to establish derivative market like US's CMA. They bring in more companies for quoting and increase in exchanges but the size of the economy is still too small. There is a need of creating strong derivative market.
For the past few years, the Nigerian Stock Exchange (N.S.E) has been a hive of activity, receiving tremendous patronage from both corporate and individual investors. But on 24 July 2008 its market capitalization sank to N10.03 trillion compared with the high of N12.64 trillion on 5 March, slightly rallying by 5 August to N10.64 trillion. This made investors to wonder if the Global slowdown has affected the Nigerian Stock Index as well.
Chapter 3
Literature Review
The returns of the African and other emerging markets have been extensively written on and tested for anomalous stock market seasonal or crosssectional behavior of their stock returns using annual returns (Ayadi, 1998). Even though tests of stock market anomaly focus more on seasonal or crosssectional behavior of stock returns and these tests differ from time series tests which look at the predictability of rates of return over time (Claessens et. al, 1995), the presence of anomalies in stock markets generally indicates predictability of returns. The tests applied on emerging markets' returns to determine the presence of anomalies are similar to those applied on developed markets' stock returns. The presence of anomalies in returns of common stocks has intrigued researchers since the last century challenging the appropriateness of the Capital Asset Pricing model (CAPM) and the whole theory of market efficiency.
In an anomalous turnoftheyear study of stock return seasonality in lowincome African emerging markets using monthly market indices for the Ghanaian stock market (19911996), Nigerian stock market (19841995), and Zimbabwean stock market (19871995), Ayadi, (1998) found that the results of both the KruskalWallis and Friedman tests suggest the absence of seasonality in stock returns on the Nigerian and Zimbabwean stock markets while the Friedman test confirms the presence of seasonality in stock returns for Ghana. Furthermore, the WilcoxonMannWhitney test and the dummyvariable regression analysis show the presence of the "January effect" for Ghana but not for Nigeria and Zimbabwe.
In a more recent study, using weekly index returns adjusted for thin trading as a nonlinear autoregressive process with conditional heteroscedasticity, AppiahKusi and Menyah (2003) used the EGARCHM model to investigate the weakform pricing efficiency of eleven African stock markets. Their findings reject evidence in prior studies that the Nigerian stock market is weakform efficient. They confirm ex ante results that the markets in Egypt, Kenya, and Zimbabwe are efficient while that of South Africa is not weakform efficient. Their findings indicate that stock markets in Mauritius and Morocco may be efficient while the stock markets in Mauritius and Morocco, Botswana, Ghana, Ivory Coast, and Swaziland are not consistent with weakform efficiency. The application of the EGARCH model enabled them to capture how conditional volatility affects the pricing process without imposing undue restrictions on the parameters of the conditional variance equation. It is obvious that the question of efficiency of the African financial markets is still unresolved as conflicting research findings prevail.
According to Piesse and Hearn (2002), studying African markets integration have suggested that the univariate EGARCH model suggested by Nelson (1991) are appropriate for the analysis of African market since they can successfully model asymmetric impacts of good news (market advances) and bad news (market retreats) on volatility transmission with high levels of accuracy. Using weekly market data from January 1993 to 2000 for Ghana, they found no evidence of asymmetry (i.e. Leverage effect).
Importance of ARCH and GARCH methods
Methods of using mean and variance to calculate volatility is unconditional and does not recognize patterns of asset volatility i.e. time varying and clustering properties. We use ARCH and GARCH models where we consider volatility return to be a central issue. Many banks and other financial institutions have used the idea of value at risk as a way so that they can measure risks that their portfolios are presently facing. If there is one percent value at risk then any losses for the next day can exceed around ninety nine percent.
GARCH (1, 1) model is used where there is conditional variance.
Its equation is: S_{t} = b_{o} + έ_{t}, έ_{t}/Φ_{t1}~N (0,σ_{t}^{2})
GARCH model (refer to Engle, 1982 to know more about the model) can be generalized into a GARCH (p, q) model having additional lag terms. These kind of higher models are used when we are dealing with large data. Using these additional lags we can have both fast and slow decay of the information. There are certain disadvantages of GARCH model. We also have GARCH (2, 2) model that is also called the component model. We also have another version for GARCH model that takes in asymmetric view. It estimates negative and positive returns separately. Higher volatilities generally follow negative returns more than positive returns. Variance calculated is dependent on just previous error term and not on its sign. It does not tell anything about asymmetry. It cannot tell anything about negative shocks.
Because of these shortcoming family of GARCH model have been created that can deal with asymmetry. These are Exponential GARCH and GJR GARCH models. Besides this there are other forms of GARCH model. These are: IGARCH (Integrated generalized ARCH), GARCHM (GARCH in Mean), QGARCH, TGARCH (Threshold GARCH), APARCH, FIGARCH (Fractionally Integrated GARCH), FIEGARCH (Fractionally integrated EGARCH), FIAPARCH, FCKARCH, HYGARCH, NGARCH (Non Linear GARCH), NAGARCH (Non Linear In Mean Asymmetric GARCH) and FGARCH (Family GARCH) models. These models were proposed by Engle and Bollerslev (1986).
Equation for GJRGARCH model:
σt = ὠ + + α_{i} έ _{tI} ^{2} + + β_{j} σ _{tj}^{2} + + ϒ_{k} έ _{tk}^{2} I^{}_{tk}
To calculate Stock market Volatility we have used EGARCH (Generalized Autoregressive Conditional Heteroskedasticity) in mean model. It has been developed from generalization of ARCH (Autoregressive Conditional Heteroskedasticity) model. ARCH model relates conditional variance to the linear combination of squared frequencies, but after generalization conditional variance is related to both lagged values and squared values.
There should be a positive relation between volatility and stock return. According to Leon (2007) high premium risk should be paid for high volatility in stock market. GARCH examines relationship between volatility and stock return so that risk return tradeoff can be measured. We use non parametric techniques to find relationship between risk and return. We can incorporate exogenous variables in the equations that are provided for the GARCH models.
We will use mean and variance equations for full sample, pre stock market crash, stock market crash, pre global and global crisis.
Mean and variance equations for full sample are:
R_{t}=b_{0} + b_{1}R _{t1} + b_{2}σ_{t} + b_{3}BR + b_{4}ISR + b_{5}SMC + b_{6}GFC + έ_{t}
έ_{t}/Φ_{t1}~N (0,σ_{t}^{2},v_{t})
Here v_{t} is degree of freedom
Equation for mean for pre stock market:
R_{t}=b_{0} + b_{1}R _{t1} + b_{2}σ_{t} + b_{3}BR + b_{4}ISR + έ_{t}
έ_{t}/Φ_{t1}~t (0,σ_{t}^{2}, v_{t})
Equation for variance for pre stock market:
έ_{t}/Φ_{t1}~N (0,σ_{t}^{2},v_{t})
Mean equation for pre global financial crisis:
R_{t} = b_{0} + b_{1}R _{t1} + b_{2}σ_{t} + b_{3}BR + b_{4}ISR + b_{5}SMC + έ_{t}
έ _{t}/ Φ _{t1}~t(0,σ_{t}^{2}, v_{t})
Variance equation for pre global financial crisis:
έ_{t}/Φ _{t1}~N (0,σ_{t}^{2},v_{t})
Mean equation for stock market crash:
R_{t}=b_{0} + b_{1}R _{t1} + b_{2} σ_{t} + b_{6}GFC + έ_{t}
έ_{t} / Φ _{t1}~t (0,σ_{t}^{2},v_{t})
Variance equation for stock market crash:
Log (σ_{t}^{2}) = ὠ + α_{i} (έ _{ti}/σ _{ti})(2/π) ^{1/2} + β_{1} log (σ_{t1}^{2}) + ϒ  έ _{ti}/σ _{ti} 
It has the presence of ARCH in it.
Mean equation for global financial crisis:
R_{t} =b_{0} + b_{1}R _{t1} + b_{2}σ_{t}
έ_{t}/ Φ _{t1}~t (0,σ_{t}^{2},v_{t})
Variance equation for global financial crisis:
log (σ_{t}^{2}) = ὠ + α (έ _{t1}/σ _{t1})(2/π) ^{1/2} + β_{1} log (σ _{tj}) + ϒ  έ _{t1}/σ _{t1} 
It also has presence of ARCH.
Here α, β, ϒ, and ὠ are volatility parameters.
Chapter 4
Data and Sample description
The main steps for the process are:

Collection of Data and Samples.
We collect daily results of stock markets over 9 years from 2000 to 2008 using the DataStream.
To calculate monthly volatility we need to compute daily returns using the following formula:
R _{m, t}=ln (I _{m, t}/I _{m, t1})
Where I _{m, t} is value of stock market index and R _{m, t} is compound return on month.
Monthly realized volatility is computed as:
µ= (1/n) R _{m, t}
σ _{a, m=} [ (R_{m, t} µ_{m}) ^{2}]^{0.5}
Number of trading days is given by n. Now if we get n observations then half of them are used for estimation and other half for forecasting.
Stock return can be computed as:
R_{t} = log [(NSI_{t})/ (NSI_{t1})]
Where, R_{t} stands for stock return at time t
NSI_{t} and NSIt_{1} refers to Nigerian Stock Index at time t and t1 respectively

Use forecasting techniques.
Forecasting models are chosen by keeping in mind their complexity and range of measures. We employ four time series models to analyze the data. The models are summarized below:

GARCH (1, 1) Model. It stands for Generalized Autoregressive Conditional Heteroscedastic model. If we have ARMA (Auto Regressive Moving Average) model for error variance then the result is GARCH model. In this model conditional volatility depends on yesterday's conditional volatility and yesterdays squared forecast error. While estimating total number of lags we use LJung Box Test. LJung Statistics follow X^{2} distribution, having n number of degree of freedom. If we reject null then we have no conditional variance error. It is computed as follows:
h_{t} = α_{o} + φ έ_{t1}^{2} + β_{1}h_{t1}

EGARCH (1, 1) Model (Nelson, 1991). It stands for Exponential Autoregressive Conditional Heteroscedastic model. It is an asymmetric model. In some cases we can have value of result as negative also. Its equation is:
Ln (h_{t}) = α_{o} +ϒ (έ_{t1}/ ( h_{t1})^{1/2}) + φ[(έ_{t1}/ ( h_{t1})^{1/2})(2/π)^{0.5}] + βln (h_{t1})

TGARCH model. It stands for Threshold GARCH model. It is quite similar to GJRGARCH model. But here we use conditional standard deviation instead of using conditional variance. This model takes in asymmetric approach. Its equation is given as: σ_{t}= K + £σ _{t1} + α_{1} ^{+}σ _{t1}^{+} + α_{1} ^{}σ _{t1}^{}

ARCH8 Model. This is a class of semiparametric ARCH(∞) models that includes as a special case the partially nonparametric (PNP) model introduced by Engle and Ng (1993) and which allows for both flexible dynamics and flexible function form with regard to the 'news impact' function.

Comparing Results:

Symmetric Error Statistics (Brailsford and Faff 1996).
Mean Error (ME): Formula for calculating ME:
1/60 (σ _{f, m}  σ _{a, m)}
Mean Absolute Error (MAE): Formula for calculating MAE:
1/60 σ _{f, m}  σ _{a, m} 
Root Mean Squared Error (RMSE): Formula for calculating RMSE:
1/60 [(σ _{f, m}  σ _{a, m) 2}]^{0.5}
Mean Absolute Percentage Error (MAPE): Formula for calculating MAPE:
1/60 (σ _{f, m}  σ _{a, m)}/ σ _{a, m}
Here σ _{f, m} refers to forecast volatility and σ _{a, m} refers to realized volatility in month

Asymmetric Error Statistics. In pricing of options over prediction of volatility is undesirable for buyers and under prediction of volatility is undesirable for sellers therefore Mean Mixed Error Statistics (MME) is employed as follows:
Here O is the number of over predictions and U is number of under predictions.

Volatility forecasts:
Volatility forecasts can be computed from symmetric and asymmetric models by using four Models: ARCH (p), GARCH (1, 1), GJR GARCH (1, 1) and EGARCH (1, 1). Conditional mean function is computed as: R_{t} = c + θ_{i} R _{tI} + έ_{t}
Following symmetric models are used as:
ARCH (p): h_{t}^{2} = α_{o} + α_{i} έ ^{2}_{t1}^{2}
GARCH (1, 1): h_{t}^{2}= α_{o} + α_{1} έ_{t1}^{2} + β_{1}h ^{2}_{t1}
Following asymmetric models are used:
EGARCH (1, 1): ln (h_{t}) = α_{o} +ϒ (έ_{t1}/ ( h_{t1})^{1/2}) + λ [(έ_{t1}/ ( h_{t1})^{1/2})(2/π)^{0.5}] + βln (h_{t1})
This shows asymmetric effect is exponential and not quadratic.

To test relationship between return and volatility:
Forecast model produces a regression that gives relationship between return and volatility:
µ_{w} = α + β_{f} σ _{f, w} + e_{w}
If β_{f} = 0 then µ_{w} (market returns) And volatility are unrelated
If α =0 and β_{f} > 0 then, market returns and volatility are proportional
Relationship between market returns and unexpected volatility is given as:
µ_{w} = α + β_{f} σ _{u, w} + e_{w}
Using the above procedure we get generalized results as follows:

Exponential Smoothing approach provides superior forecasts for monthly volatility where there are different market conditions and contexts.

NonARCH models are superior to ARCH models.

If we have sub groups of ARCH type models, then the complex models are generally more superior.
Chapter 5
Conclusions
This section would be written after the data analysis.
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