Extreme-value Volatility in Emerging Markets

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Extreme-value Volatility in Emerging Markets

T. A. Besselink (1333534) University of Groningen Faculty of Economics June 19, 2008 Abstract

This research tries to identify if extreme-value volatility models are preferred over standard volatility when forecasting future volatility. Because of the illiquidity of immature option markets the results are especially interesting for estimating and forecasting volatility in emerging markets. The dataset contains daily open, high, low and closing prices from 4 emerging and 2 developed markets which are used to calculate monthly volatility. The data is divided in an in-sample model estimation period (1996-2000) and an out-of sample forecasting period (2001-2005). Results derived from standard forecasting accuracy statistics show that extreme-value models are preferred over a standard model when forecasting future volatility. These results are consistent in both emerging and developed markets.

JEL Classification: C22, C52, C53

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I. Introduction

If you are in the business of pricing and trading options you would like to estimate a price as accurate as possible. Volatility is an essential part in pricing options and Poon and Granger (2003) argue in their review on volatility papers that better and more accurate estimations would save financial institutions money and would improve the stability of financial markets. Nowadays, the forecasting of volatility is almost entirely done using options implied volatility or the historical volatility (Vipul & Jacob, 2007). If volatility models show improved results using extreme-value models, further research should not focus on historical data purely based on closing prices. The extreme-value method of forecasting volatility is known as a reliable method when there are no settlement prices available (Natenberg, 1994). Several extreme-value models have evolved true time and will be further examined in this thesis.

Because this thesis is so heavily focused on volatility, it is helpful to spell out what volatility actually is. If uncertain variables are involved, volatility concerns the spread of all likely outcomes (Poon, 2005). When focused on financial markets, investors are often concerned with the spread of asset returns. Hull (2006) defines volatility as the standard deviation within a sample, calculated by:

(1) 2 1 ) ( 1 1

∑

= − − = n t t r r n

σ

where _      = −1 ln t t t C C

r , as the natural logarithm of the closing prices of the current and previous day t and

1 − t .

∑

= = n t t r n r 1 1

is the average yield over an n-day period.

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Volatility is that important because it is a key variable for many important financial applications such as option pricing and risk management. In this thesis extreme-value volatility estimators are introduced to determine their use on volatility data from emerging markets.

This thesis on volatility estimation and forecasting is different from most papers which examine volatility because it introduces extreme-value models. These models incorporate open, high, low and closing prices to calculate the volatility, where standard volatility calculations are based merely on closing prices. There are some papers available that research the effectiveness of extreme-value models but this area remains less touched than forecasting based on standard volatility. This thesis stands out from the available papers on extreme-value volatility as it explicitly looks at the link with emerging markets where its implementations could be most useful and effective.

The objective of this thesis is threefold. First, identify if extreme-value volatility models are more accurate than a standard volatility model for forecasting volatility. Second, determine which of the existing extreme-value models is best in predicting future volatility. Third, identify if the findings of this research are consistent throughout different evolved and emerging markets. Note that the actual interest of this thesis is to identify which volatility model is best when forecasting volatility, not which forecasting model is best.

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II. Theoretical Framework

Black and Scholes (1973) introduced the modern option pricing theory to the financial literature. In the final version of the model they introduced there are six parameters to determine the price of an European option; the stock price, strike price, the time left to expiration, interest rate, dividends and volatility. Volatility gained a central role in valuing the price of the option and is also the most difficult to grasp1. Volatility is the key input when it comes to option pricing and risk management. The problem with volatility is that, in contrast to the other five parameters, it is not directly observable. Hence, there need to be a way to estimate volatility. A trader should use his knowledge of historical, forecast, implied and, in case of commodities, seasonal volatility to make an intelligent decision about future volatility (Natenberg, 1994). In this thesis the focus will be on estimating volatility. The estimated historical volatility will than be used for forecasting future volatility.

Poon and Granger (2003) show that during the last decade, the field of modeling and forecasting volatility has evolved progressively. In their survey, most research forecasts are made using the options implied volatility or the historical volatility from time series data. However, there are some problems when using implied volatility. If you want to calculate the implied volatility of an underlying, there need to be options available to calculate the implied volatility from. This is also mentioned by Vipul and Jacob (2007). They continue that for calculating an options implied volatility the option market needs to be liquid and developed sufficiently. However, this is not the case for all markets. Especially in emerging markets, without a developed option market, the use of extreme-value models can be of great help.

Except for the conclusion that in most cases implied volatility is the best predictor of future volatility, Poon and Granger (2003) note that it is difficult to assign a single best instrument for forecasting volatility. This is because researchers used different data sets and evaluation techniques. Another shortcoming when using implied volatility is that the volatilities themselves are stochastic. This is caused by the volatility of volatility. Poon and Granger (2003) continue that option implied volatility is biased. Studies show that it under forecasts low volatility and over forecasts high volatility. At the end, the average, implied-volatility estimates are higher than the actual realized volatility.

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Opposed to the implied volatility models, researchers have developed alternative models for estimating volatility. Extreme-value volatility models are first introduced by Parkinson (1980). Extreme-value estimators are assuming that stock prices follow a geometric Brownian motion. Volatility (one of the two included parameters, the other is drift) can then be estimated from the range of the observed prices. The principal idea is that these price ranges include more information about volatility as the historical volatility which includes only the closing prices. Garman and Klass (1980) developed a more sophisticated estimator based on the model of Parkinson (1980). A third estimator, developed by Rogers and Satchell (1991) is claimed to be the most efficient when there is non-zero drift. Extreme-value volatility estimators are said to be between 5 and 14 times more efficient than standard close-to-close volatility models (Beckers 1983, Vipul & Jacob 2007). Intuitively extreme-value models provide better estimations as standard close-to-close models because it takes the price range into account and not just a mere price shift at the end of a trading day (Wiggins 1991). Extreme-value estimators are relatively easy to implement and only require open, high, low and closing prices.

Besides the three models introduced above there are more attempts made to model volatility. Kunitomo (1992) used the model developed by Parkinson (1980) as a basis but assumes that the underlying asset follows geometric Brownian motion with non-zero drift. Non-zero drift indicates an expected return in the long run which is not zero. Kunitomo’s estimator needs the tick-level data as input to construct a Brownian-bridge process. Because of the unavailability of tick-data Kunitomo’s model is omitted in this study. The same accounts for the model developed by Yang and Zhang (2000). Their model does not give a volatility estimate for a day and is therefore also omitted in this research.

When the theoretical basis for extreme-value volatility models was laid by Parkinson (1980) and Garman and Klass (1980) empirical implications of the models would soon follow2. Beckers (1983) was the first to test the models on an empirical basis. He finds evidence that the extreme-value estimators appear to be more accurate than traditional close-to-close measures. His research also indicates that predications can be improved by including the standard deviations implied in option prices.

Wiggins (1991) shows that the Parkinson model performances 2.5 times better in a data set of NYSE and NYMEX stocks. Later research by Wiggins (1992) on S&P futures confirms the out performance of extreme-value estimators and shows that extreme-value volatility provides a better estimate of realized volatility than the current used implied or historical volatility.

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Research by Bali and Weinbaum (2005) confirmed the predictive power of extreme-value estimators. They found that when testing 8 different extreme-value models all outperformed the traditional close-to-close method, except the model by Rogers and Satchell (1991). The Garman-Klass model was the best overall performer. The extreme-value models performed better using daily data compared to weekly or monthly data. The study by Vipul and Jacob (2007) also shows that forecasting models increase their power if they use volatility based on extreme-value models instead of historical volatility merely based on closing prices. However, they see the greatest improvement in weekly and monthly data.

To identify which of the volatility models is best for forecasting future volatility, forecasting models are introduced. Common forecasting models are the random walk, moving average, exponentially weighted moving average (EWMA) and generalized autoregressive conditional heteroskedasticity (GARCH 1,1) which are used in many volatility studies (Poon & Granger 2003, Balaban & Bayer 2005). The first two are so-called “naive models” but those models are not capable of handling the natural tendency of volatility to auto correlate. The EWMA and GARCH (1,1) model can cope with this characteristic as part of the model is estimated based on the behavior of previous observations.

The GARCH (1,1) model, as proposed by Bollerslev (1986), is calculated from a long run variance rate. It takes the mean reverting and serial correlation characteristics of volatility into account when forecasting volatility (Natenberg, 1994). The EWMA model is essentially a special case of the GARCH (1,1) model without a long run variance rate. Without this long run variance rate the EWMA model does not mean-revert. The problem of using GARCH on monthly data is that it is difficult for the parameters to converge to a satisfying equation. According to Figlewski (1997) there rests a premium on forecasting models and volatility estimators which are more robust to small fluctuations. He continuous that the most sophisticated models, which capture every small move in the in-sample set, are the ones which are more likely to perform worse when taken out-of sample. This is especially true for longer horizons and therefore GARCH (1,1) and other ARCH-type models are not preferred without the input of high frequency data.

Now the volatility forecasting models are introduced a hypothesis can be formulated.

H0: Forecasts made on extreme-value volatility are superior over traditional analysis using standard historical volatility.

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III. Data

One of the great advantages of extreme-value models is that they only require readily available open, high, low, and closing prices. In contrast to implied volatility models, where a mature and liquid option market is needed, the extreme-value models are easy to implement. Therefore, extreme-value models could be of great use in underdeveloped or emerging markets. MSCI Barra facilitates a free float adjusted market capitalization index on emerging markets. As of June 2007, the MSCI Emerging Markets Index consists of 25 emerging market country indices. Because of the unavailability of the required data this led to the selection of four markets. The developed American S&P 500 and the British FTSE 100 serve as a robustness check.

The dataset runs from 1:1996 till 12:2005 and contains daily, open, high, low and closing prices. Separate time-periods of five years (1995-2000; 2000-2005) are constructed to check the performance of the models. The period from 1:1996 till 12:2000 will be used for estimation purposes. The timeframe 1:2001 - 12:2005 is used for testing the forecasts out-of-sample. Eventually, four emerging market indices had sufficient data observations to build the models. From Datastream, the Price Indices from these four emerging markets are collected. These are the national indices from China (Shanghai A-shares), India, Malaysia and Korea.

Table 1 presents the descriptive statistics from the selected indices. These statistics are based on 21 trading day monthly observations. This practice is also used in comparable studies concerning volatility forecasting (e.g. Dunis et al. 2003).

Table 1: Summary statistics of monthly log-returns (January 1996 – December 2005)

1996-2005 CHINA INDIA KOREA MALAYSIA FTSE S&P500

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The descriptive statistics show that the mean monthly log-returns of the indices are all positive, except for Korea. The highest mean returns are found for China (+0.78%) and India (+0.63%). The low mean log-return for Korea (-0.47%) could be related to the severe impact of the Asia Crises for this country. This is also reflected by a median of -1.32% which indicates that most of the monthly returns in this sample period where negative. Table 2 and 3 show the descriptive statistics for the different time-periods used in this thesis. The period from 1996-2000 reflects the data used to construct forecasting models and the period 2001-2005 is the out-of-sample component used for evaluation.

Mean 2.67% 0.21% -2.14% -0.12% 0.48% 0.89% Median 2.05% 0.29% -2.42% 1.34% 0.70% 0.77% Maximum 26.96% 15.58% 34.34% 35.20% 13.53% 9.15% Minimum -21.04% -24.83% -30.18% -28.46% -10.61% -15.02% Std. Dev. 0.0862 0.0880 0.1163 0.1138 0.0421 0.0448 Skewness 0.3528 -0.3336 0.5076 0.3490 -0.3200 -0.7527 Kurtosis 3.8324 2.7058 3.8537 4.1999 4.3648 4.0291 Jarque-Bera 2.9773 1.3294 4.3980 4.8176 5.6806 8.3136 Probability 0.2257 0.5144 0.1109 0.0899 0.0584 0.0157 Observations 60 60 60 60 60 60

Mean -1.10% 1.06% 1.20% 0.38% -0.37% -0.32% Median -1.78% 1.61% 2.26% 0.62% 0.19% -0.23% Maximum 13.47% 16.33% 21.05% 10.92% 8.30% 7.84% Minimum -13.81% -23.45% -13.03% -11.31% -11.63% -12.96% Std. Dev. 0.0594 0.0759 0.0762 0.0490 0.0395 0.0443 Skewness 0.1480 -0.6351 0.0217 -0.3263 -0.7646 -0.6109 Kurtosis 2.2940 3.9145 2.4812 3.1553 3.9178 3.5143 Jarque-Bera 1.4652 6.1243 0.6775 1.1247 7.9527 4.3933 Probability 0.4807 0.0468 0.7127 0.5699 0.0188 0.1112 Observations 60 60 60 60 60 60

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For most indices the difference between the highest and lowest observation is much wider dispersed in the first period compared to the second. This is reflected in the standard deviation which is lower for all six indices in the second period. The standard deviations of the monthly log returns show that returns from emerging markets fluctuate more as those from developed indices. This observation is also reported in other studies about the behavior of returns in emerging markets (Bekaert & Harvey, 1997). Although an interesting observation, the distribution of the monthly returns has not the full interest of this thesis. The next part of the data description will describe the statistics from the different volatility models. In this discussion, no knowledge about the exact formulation of the models is required. After the discussion of the descriptive statistics, these will be discussed thoroughly in the methodology section.

Table 4: Summary statistics of standard historic monthly volatility (January 1996 – December 2005)

Historic volatility Mean 23.57% 23.88% 31.30% 19.82% 16.09% 16.95% Median 19.40% 21.31% 29.02% 13.65% 14.31% 15.64% Maximum 75.97% 63.26% 85.56% 150.28% 51.27% 42.80% Minimum 8.56% 8.91% 9.58% 4.53% 6.91% 7.18% Std. Dev. 0.1197 0.1079 0.1488 0.1817 0.0818 0.0712 Skewness 1.4708 1.2169 0.9991 3.9587 1.5701 1.0888 Kurtosis 5.5874 4.3615 3.9638 25.0428 5.9865 3.9687 Jarque-Bera 76.7421 38.8849 24.6092 2742.8420 93.8988 28.4002 Probability 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 Observations 120 120 120 120 120 120

Table 4 shows the results derived from standard historic volatility using the close-to-close model. Returns from emerging markets are more dispersed from the mean resulting in a higher standard deviation as the volatility in developed countries. The maximum volatility is also higher in all the emerging markets and is not normally distributed for all countries.

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Appendix 3 gives a clear view of the outliers present in one of the datasets.3 Eventually, only the Malaysian far outlier is reduced to the second highest observation to not disturb the prediction capabilities of the models.Table 5 continues with the descriptive statistics from the extreme-value models.

Table 5: Summary statistics of Extreme-Value monthly volatility (January 1996 – December 2005)

Panel A: Parkinson Mean 18.90% 19.63% 23.23% 16.18% 14.29% 14.75% Median 16.51% 18.02% 21.01% 11.84% 12.94% 13.74% Maximum 48.25% 51.32% 49.24% 103.28% 41.80% 36.71% Minimum 7.83% 8.18% 10.14% 5.12% 5.60% 7.29% Std. Dev. 0.0781 0.0827 0.0936 0.1236 0.0714 0.0572 Skewness 1.1140 1.4844 0.6462 3.5684 1.3833 1.1864 Kurtosis 4.0684 5.5845 2.5195 22.5918 5.2121 4.4245 Jarque-Bera 30.5273 77.4695 9.5058 2173.8580 62.7355 38.2954 Probability 0.0000 0.0000 0.0086 0.0000 0.0000 0.0000 Observations 120 120 120 120 120 120

Panel B: Garman-Klass Mean 17.42% 18.62% 22.26% 15.19% 13.68% 14.20% Median 15.74% 17.30% 20.64% 11.38% 12.53% 12.93% Maximum 42.03% 48.65% 44.59% 85.14% 38.00% 34.86% Minimum 7.11% 7.26% 9.27% 5.22% 4.75% 7.02% Std. Dev. 0.0683 0.0792 0.0890 0.1085 0.0682 0.0547 Skewness 1.0299 1.5464 0.6238 2.9712 1.3448 1.2736 Kurtosis 3.9256 5.8546 2.3597 16.6725 5.1016 4.6784 Jarque-Bera 25.4992 88.5676 9.8321 1111.2510 58.2517 46.5255 Probability 0.0000 0.0000 0.0073 0.0000 0.0000 0.0000 Observations 120 120 120 120 120 120

Panel C: Rogers-Satchell Mean 25.09% 25.21% 30.90% 21.52% 19.23% 20.03% Median 22.42% 23.64% 27.87% 16.74% 17.40% 18.40% Maximum 62.86% 60.26% 63.51% 129.54% 53.01% 51.91% Minimum 10.84% 9.41% 13.19% 7.73% 7.21% 9.68% Std. Dev. 0.1072 0.1011 0.1217 0.1597 0.0933 0.0773 Skewness 1.0809 1.4392 0.6781 3.3205 1.2713 1.3736 Kurtosis 3.8279 5.4881 2.5555 19.7849 4.6391 5.3285 Jarque-Bera 26.7935 72.3827 10.1848 1629.1760 45.7601 64.8469 Probability 0.0000 0.0000 0.0061 0.0000 0.0000 0.0000 Observations 120 120 120 120 120 120

3_{The data from the four different estimation techniques all show approximately the same outlier characteristics with}

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Again, the calculated mean volatility is higher in emerging markets than in developed markets. In this dataset, the volatility in emerging markets reaches higher maxima and standard deviations. As mentioned before, this could be caused by the relative wider distribution of returns in emerging markets, creating volatility. One interesting observation is that the mean and median values from the Rogers-Satchell model are all severely larger as the numbers presented for the Parkinson and Garman-Klass model. The summary statistics over the total sample period (1996-2005) show that all four models used for volatility calculation are not normally distributed and often fat tailed.

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IV. Methodology

This study can be separated in two parts. The first step is to estimate the volatility of the dataset using the proposed competing volatility estimators. The second step is to use these volatility estimates to forecast future volatility. Results of the first step are presented as descriptive statistics in the previous section.

This section starts with introducing the models which are used for estimating volatility. These are the standard historical volatility (HV) mentioned by Hull (2006) and the extreme-value models by Parkinson (PK), Garman and Klass (GK) and Rogers and Satchell (RS). Later in this section, three different models for forecasting future volatility are introduced. Finally, the forecasting methods are evaluated with accuracy statistics to distinguish if forecasting techniques are improved by using extreme-value models. Estimating volatility

Because the focus of this research is on monthly realized volatility, a measure of realized historical volatility is based on 21 trading days. A fixed rolling window approach is used which uses daily data for constructing the monthly volatility. Therefore, no intraday data is needed for constructing daily volatility and the research uses only the monthly realized volatility. This measurement is also used by Dunis et al. (2003) and Dunis and Chen (2005). For calculating the historic volatility Hull (2006) defines:

(2) 2 1 ) ( 1 1

∑

= − − = n t t hv r r n

σ

as mentioned in the introduction, _

     = −1 ln t t t C C

r , as the natural logarithm of the closing prices of the

current and previous day t and t−1.

∑

= = n t t r n r 1 1

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When the return in a certain time period is zero, the standard deviation provides a good estimate of the rate of movement experienced by the underlying variable (Vipul & Jacob, 2008). The extreme-value estimators are based on the belief that price movement of the underlying captures volatility more effectively than the standard deviation. An improvement in the efficiency of the volatility could be realized by using a proper scaled squared log range. The first extreme-value model was proposed by Parkinson (1980) which assumes that prices experience no drift. Equation 3, shows the PK model,

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(

)

_      − =

∑

= 4ln2 1 * 1 2 1 n t t t pk H L n

σ

where:

σ

: The estimated standard deviation of the underlying t

H : The log transformed high of the day t

L : The log transformed low of the day

n: The number of observations

Soon after the development of the first extreme-value model Garman and Klass (1980) tried to improve efficiency by introducing open and closing prices to the model. Their GK model claims to have a lower standard deviation as the PK model. This claim is supported by the descriptive statistics from the dataset used in this research. Equation 4 shows the GK model,

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[

2 2

]

1 ) ( * ) 1 2 ln 2 ( ) ( * ) 5 . 0 ( 1 t t t t n t gk H L C O n − − − − =

∑

=

σ

where as an addition to the symbols mentioned above, t

C : The log transformed close of the day t

O: The log transformed open of the day

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(5) 1

[

( )*( ) ( )*( )

]

1 t t t t t t n t t t rs H C H O L C L O n − − + − − =

∑

=

σ

Now the volatility estimators have been established, the next step is to identify the forecasting models.

Forecasting volatility

The forecasting methods include the random walk (RW), simple moving average (MA) and exponentially weighted moving average (EWMA). The RW model assumes that the last actual observation is the best predictor of future volatility. Equation 6 shows the RW model,

(6) RW :

σ

ˆt =

σ

t−1

where

σ

ˆt is the forecasted volatility by the model and

σ

t−1 is the previous period actual observation. Another model commonly used when forecasting is concerned is a moving average model. This method averages the last n observations and is according to Makridakis et. al (1983) particular useful when time series with a slow changing mean are concerned. Equation 7 shows the moving average model,

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∑

= + − = n i i t t n q MA 1 1 1 ˆ : ) (

σ

where q is the length of the period being averaged. Optimizing techniques could be used to come up with a single best length of q or a wide range of q’s can be tested to find a preferred length. To stay objective, q is arbitrarily set to the last 10 observations. There have now been two naive forecasting methods considered but the nature of volatility is that it creates non-random patterns and tends to auto correlate. If there are non-random patterns, there is a great change that items within the series are related to one another4. If there is a consistent relationship the series have some capability to forecast themselves. This characteristic is captured by the EWMA model as weight is distributed to a portion of previous data acting as a moving average and the last observation.

(8) : ˆ (1 )1 :0 1 1 1 1+ − ≤ ≤ =

∑

= −+ −

α

σ

α

ασ

σ

n i i t t t n EWMA

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For EWMA-

α

estimation the principles used by Balaban et al. (2006) are adopted. To find the optimal value of

α

, the one step ahead prediction error is optimized for the four volatility models5. The prediction error is defined as the absolute difference between the forecast and the actual observation, also known as the mean absolute error (MAE). The moving average part of the equation will be the average of the last 12 observations, making up one year. The in-sample period 1996-2000 is used to optimize

α

. This figure is than used for forecasting in the 2001-2005 out-of-sample period.

Figlewski (1997) discovered during his extensive study on volatility that GARCH models do not fit monthly data. He mentioned that statistically GARCH needs much more data points opposed to historical volatility. For properly estimating GARCH, a large dataset or high frequency data is needed. According to Figlewski (1997) a monthly dataset of five years is not enough and the GARCH parameters will only accidentally converge to acceptable values. Based on these arguments forecasts with GARCH (1,1) or other ARCH-type models are omitted.

Evaluation of volatility forecasts

The evaluation of volatility predictors will be done using standard accuracy statistic also used in other academic studies on volatility forecasting (e.g. Dunis et al, 2003). A full range of economic accuracy statistics is used to interpret the data which is the result of the data generated by the forecasting models.

Root mean squared error (RMSE)

The RMSE measures the average size of the error (Pindyck & Rubinfeld, 1998). It is a quadratic scoring rule which squares the difference between forecasts and corresponding observed values and takes the average over the total sample. The final step involves taking the square root of the average. Large errors are given high weight to the RMSE as these errors are squared before they are averaged. Because of this characteristic, the RMSE is most practical when large errors are unwanted.

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(

)

5 . 0 1 2 , , 1       − =

∑

+ + = n t t m a m f n RMSE τ

σ

Where: m f ,

σ

: The forecasted monthly volatility m

a,

σ

: The actual observed monthly volatility

5

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Mean Absolute Error (MAE)

For measuring the average size of the errors, without taken their direction into account, the MAE is used (Brooks, 2002). In essence, the MAE and RMSE are relative similar measurement statistics, but the MAE is less sensitive to large forecast errors. One problem of mean errors is that they may be close to zero as large positive and negative errors cancel each other out. To avoid this problem, the absolute value of the error is taken. Mean errors are often used to look for a systematic bias in the forecasts (Pindyck & Rubinfeld, 1998). (10)

∑

+ + = − = t n t m a m f n MAE 1 , , 1 τ

σ

The results from the MAE and RMSE may vary from zero to infinite. They are negatively-oriented which implies that lower values are indicating good forecasts. The score is unbounded and needs to be compared to the score of other models for the same data and forecast period (Brooks, 2002). In practice the MAE and RMSE are often used together to diagnose the deviation in the errors. The evaluation of the forecasts will therefore be made on a range of statistics and not merely one. When evaluating the results, the RMSE will always be larger or equal to the MAE. The difference between them gives a glance on the variance of the individual errors in the forecast sample. If the RMSE equals the MAE, than all errors in the dataset are of the same size.

Mean absolute percentage error (MAPE)

In contrast to the RMSE and MAE statistics, which are scale-dependent measures, the MAPE and Theil-U statistics are independent of the scale of the variables and are presented in percentages. The MAPE, introduced by Makridakis (1993), measures the difference between the actual value and the forecasted value and is then divided by the actual value.

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∑

+ + = − = t n t am m a m f n MAPE 1 , , , 1 τ

σ

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Theil-U statistic

A low value of the Theil-U statistic indicates a more accurate forecast. If the Theil-U statistic has a level of zero, it indicates a perfect fit of the forecast with the out-of-sample data. (Pindyck & Rubinfeld, 1998).

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(

)

(

)

_       +         − = −

∑

+ + = + + = + + = n t n t m a n t n t m f n t t m a m f n n n U Theil τ τ τ

σ

2 , 2 , 1 2 , , 1 1 ) ( 1

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V. Empirical analysis

In this section of the thesis the forecasts from the models are tested on the out-of-sample dataset. Eventually, an evaluation is made which volatility estimator and which volatility forecast model have the best characteristics for forecasting volatility.

Below the results from optimizing the in-sample EWMA-α parameter by minimizing the Mean Absolute Error are presented. A low α will result in less weight on the last observation and more weight to the moving average part of the EWMA equation.

Table 6: In-sample static EWMA-α estimations

1996-2000 HV PK GK RS China 0.3910 0.4193 0.2461 0.7947 0.0941 0.0643 0.0571 0.0803 India 0.5143 0.4573 0.6304 0.5634 0.0348 0.0286 0.0276 0.0368 Korea 0.5873 0.6996 0.8271 0.6436 0.0944 0.0528 0.0507 0.0763 Malaysia 0.7778 0.7734 0.7870 0.7253 0.0938 0.0588 0.0563 0.0844 FTSE 0.6760 0.5580 0.3940 0.6953 0.0348 0.0286 0.0276 0.0368 S&P 500 0.4322 0.5847 0.5708 0.7679 0.0430 0.0315 0.0300 0.0432

HV: Standard historic close-to-close volatility, PK: Parkinson (1980), GK: Garman-Klass (1980), RS: Rogers-Satchell (1991).

Numbers in Italic are the MAE minimizations.

The α parameter is not consistent within one index or volatility estimator. The lowest (highest) value is observed in China, 0.2461 (Korea, 0.8271) for the GK model. The most consistent and on average highest α-parameters are found in Malaysia. After optimizing, most weight is transferred to the last observation. This indicates that the moving average part performs relatively poor in the in-sample data of this emerging market.

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Table 7, 8 and 9, 10 give the out-of-sample accuracy statistics of the forecasting models on the different volatility estimators. Derived from Balaban et al. (2006), a ranking system is used to evaluate the performance of the models. The ranking system assigns 1 point to the best performer, 2 to the second and 3 to the worst. During the evaluation the results from the combined accuracy statistics and the individual results per accuracy statistic are concerned.

Table 7: RMSE statistics of the forecasting models

2001-2005 HV PK GK RS Panel A: China RW 10.23% 5.71% 5.05% 8.58% MA (10) 9.43% 6.08% 5.33% 8.70% EWMA 8.77% 5.23% 4.84% 7.97% Panel B: India RW 12.18% 9.27% 8.61% 10.09% MA (10) 10.23% 8.09% 7.55% 9.30% EWMA 10.30% 7.86% 7.55% 8.83% Panel C: Korea RW 9.83% 5.28% 4.92% 5.83% MA (10) 7.84% 5.01% 4.83% 6.17% EWMA 8.57% 4.89% 4.66% 5.51% Panel D: Malaysia RW 5.95% 4.12% 3.90% 5.26% MA (10) 4.76% 3.26% 3.13% 4.11% EWMA 5.41% 3.75% 3.57% 4.70% Panel E: FTSE RW 7.89% 6.03% 5.55% 7.35% MA (10) 8.55% 7.01% 6.48% 8.93% EWMA 7.29% 5.69% 5.43% 6.94% Panel F: S&P500 RW 6.10% 4.15% 3.91% 5.87% MA (10) 6.23% 4.84% 4.73% 6.80% EWMA 5.54% 3.92% 3.75% 5.54%

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Table 8: MAE statistics of the forecasting models 2001-2005 HV PK GK RS Panel A: China RW 7.37% 4.21% 3.73% 6.25% MA (10) 6.14% 4.16% 3.65% 5.81% EWMA 6.40% 3.99% 3.72% 5.90% Panel B: India RW 9.26% 7.15% 6.63% 8.11% MA (10) 6.97% 5.31% 4.91% 6.23% EWMA 6.89% 5.32% 4.99% 5.92% Panel C: Korea RW 7.06% 3.89% 3.59% 4.38% MA (10) 5.83% 3.58% 3.40% 4.41% EWMA 6.50% 3.70% 3.43% 4.23% Panel D: Malaysia RW 4.10% 2.88% 2.69% 3.79% MA (10) 3.29% 2.25% 2.22% 3.04% EWMA 3.68% 2.69% 2.51% 3.47% Panel E: FTSE RW 5.09% 3.75% 3.41% 4.57% MA (10) 5.40% 4.30% 3.91% 5.47% EWMA 5.00% 3.77% 3.56% 4.50% Panel F: S&P500 RW 4.38% 2.84% 2.68% 4.03% MA (10) 4.21% 3.13% 3.04% 4.29% EWMA 4.18% 2.70% 2.61% 3.88%

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Table 9: MAPE statistics of the forecasting models 2001-2005 HV PK GK RS Panel A: China RW 36.66% 25.65% 24.47% 27.35% MA (10) 36.82% 28.96% 27.35% 30.21% EWMA 33.63% 24.66% 24.84% 25.97% Panel B: India RW 36.06% 29.79% 28.46% 23.86% MA (10) 41.24% 34.50% 32.83% 31.01% EWMA 35.03% 29.74% 27.62% 24.87% Panel C: Korea RW 28.07% 20.15% 19.12% 16.54% MA (10) 27.39% 21.44% 21.02% 19.65% EWMA 27.40% 20.01% 18.65% 16.70% Panel D: Malaysia RW 33.19% 26.08% 25.49% 26.96% MA (10) 32.52% 24.89% 25.00% 25.31% EWMA 30.52% 24.58% 23.87% 25.02% Panel E: FTSE RW 27.98% 21.81% 20.71% 19.95% MA (10) 37.17% 32.12% 30.17% 29.87% EWMA 29.26% 24.73% 24.96% 21.11% Panel F: S&P500 RW 25.52% 18.81% 18.44% 18.69% MA (10) 28.96% 24.12% 23.86% 24.21% EWMA 25.92% 18.84% 18.76% 18.51%

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Table 10: Theil’s-U statistics of the forecasting models 2001-2005 HV PK GK RS Panel A: China RW 23.23% 16.10% 15.29% 18.01% MA (10) 22.33% 17.65% 16.60% 18.90% EWMA 20.64% 15.10% 15.05% 16.93% Panel B: India RW 26.45% 23.21% 22.55% 19.79% MA (10) 22.41% 20.41% 19.92% 18.24% EWMA 22.60% 19.84% 19.91% 17.39% Panel C: Korea RW 17.93% 13.30% 12.95% 11.08% MA (10) 14.09% 12.39% 12.43% 11.48% EWMA 15.50% 12.24% 12.21% 10.36% Panel D: Malaysia RW 22.49% 18.23% 18.03% 17.31% MA (10) 17.74% 14.13% 14.19% 13.26% EWMA 20.42% 16.51% 16.44% 15.39% Panel E: FTSE RW 20.70% 18.00% 17.46% 16.52% MA (10) 22.99% 21.31% 20.70% 20.37% EWMA 19.45% 17.25% 17.38% 15.78% Panel F: S&P500 RW 16.64% 13.38% 12.95% 13.88% MA (10) 17.08% 15.61% 15.64% 16.10% EWMA 15.21% 12.68% 12.47% 13.15%

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Table 11: Score based results of accuracy per volatility estimator Volatility estimators HV PK GK RS All Indices 283 159 102 176 3.93 2.21 1.42 2.44 Emerging Markets 192 106 68 114 4.00 2.21 1.42 2.38 Developed Markets 91 53 34 62 3.79 2.21 1.42 2.58

HV: Standard historic close-to-close volatility, PK: Parkinson (1980), GK: Garman-Klass (1980), RS: Rogers-Satchell (1991). Numbers in Italic are average scores.

The score based system shows that the GK volatility estimator is the most efficient estimator to use when all accuracy statistics are given the same weight. In both emerging and developed markets, this estimator ends on top of the list. This is not consistent with the research of Vipul and Jacob (2007) who found the RS model to come in first but confirm the findings by Bali and Weinbaum (2005). The PK estimator comes in second, just before the RS model. The standard historical close-to-close model closes the list and gives a first indication that extreme-value estimators should be preferred over the standard model when forecasting volatility. These results are both consistent in emerging en developed markets. A per country analyses can be found in appendix 7a. Table 12 concentrates on the performance of the different forecast models.

Table 12: Score based results of accuracy per forecast model

Forecast models RW MA-10 EWMA

All Indices 225 217 135 2.34 2.26 1.41 Emerging Markets 169 122 94 2.64 1.91 1.47 Developed Markets 56 95 41 1.75 2.97 1.28

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could be found in the descriptive statistics of the volatility estimates. Volatility fluctuates more heavily in emerging markets compared to volatility in developed markets. If fluctuations are high and volatility can jump or decline suddenly between months, the RW model has a hard time forecasting as it uses the last months’ observation to forecast the next. In a more stable market, as in developed countries, the RW model has a better change of making a clever prediction as volatility fluctuates less between connecting months as in emerging markets. Appendix 7b shows the per country accuracy statistic scores of the volatility forecasting models.

The score based approach puts the GK volatility estimator and the EWMA forecasting model in front of the rest. A more thorough analysis will now be made to interpret the accuracy statistics and see if the conclusion made by the score based approach hold.

As already mentioned in the methodology section, the RMSE and MAE are mean error statistics. These are used to look for a systematic bias in the forecasts. The results from these statistics show a clear preference for the GK model, followed by the PK model. The RS and HV model are respectively third and fourth. A low reading indicates a low systematic bias. On the other hand, if we change our perspective from mean errors to statistics that are independent of the scale of the variables, the RS model is performing much better. This is shown in Table 13, which sums the different rankings of the estimation models compared to the accuracy statistics.

Table 13: Score based results of individual accuracy statistic per volatility estimator

HV PK GK RS RMSE 69 36 18 57 3.83 2.00 1.00 3.17 MAE 70 36 18 56 3.89 2.00 1.00 3.11 MAPE 72 44 31 33 4.00 2.44 1.72 1.83 Theil-U 72 43 35 30 4.00 2.39 1.94 1.67

Numbers in Italic are average scores.

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that according to the relative MAPE and Theil-U statistic emerging and developed markets are indifferent in choosing the GK or RS model.

Table 14: Market type results of individual accuracy statistic per volatility estimator

HV PK GK RS Panel A: RMSE Emerging Markets 48 24 12 36 4.00 2.00 1.00 3.00 Developed Markets 21 12 6 21 3.50 2.00 1.00 3.50 Panel B: MAE Emerging Markets 48 24 12 36 4.00 2.00 1.00 3.00 Developed Markets 22 12 6 20 3.67 2.00 1.00 3.33 Panel C: MAPE Emerging Markets 48 28 20 24 4.00 2.33 1.67 2.00 Developed Markets 24 16 11 9 4.00 2.67 1.83 1.50 Panel D: Theil-U Emerging Markets 48 30 24 18 4.00 2.50 2.00 1.50 Developed Markets 24 13 11 12 4.00 2.17 1.83 2.00

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VI. Conclusion and recommendations for further research

The first objective of this thesis was to identify if extreme-value volatility models are more accurate when forecasting volatility opposed to the standard historical close-to-close model. Several tests with three different forecasting models and a wide range of accuracy statistics show that extreme-value models outperform the standard volatility model. These findings are consistent through developed and emerging markets. The results in the empirical analyses do not reject the null-hypothesis and these finding open the door for more extensive research at the practical use of extreme-value models.

The second objective, determine which extreme-value model is best to use when forecasting future volatility, is a lot harder to answer. The accuracy statistics show that it matters if you look at absolute or relative forecast errors. The results from the empirical analyses assign the Garman-Klass (1980) model as overall winner. However, the second place in the competition is a photo finish between the Parkinson (1980) and Rogers-Satchell (1991) model. The Parkinson model performs better if absolute forecast errors are concerned, but gets beaten by the RS model if percentage errors are preferred.

It seems that there are no large discrepancies between the results from emerging and developed markets when volatility models are tested. Both market types favor extreme-value models over the standard volatility model and assign the Garman-Klass model as the best performer. When it comes to forecasting models, developed and emerging markets choose the EWMA model as the best forecasting model.

The findings of this research show that standard volatility is not the preferred volatility measurement when forecasting volatility. Forecasting errors can be reduced by using extreme-value volatility models. When placed in a real-life situation, this could mean that your monthly volatility forecasts for risk management purposes can be made more accurate. Another practical application can be found in the pricing of options. In illiquid option markets, where implied volatility is not the preferred volatility estimate, extreme-value models should get the preference above standard volatility estimators when volatility forecasts are made. This thesis finds similar results for developed and emerging markets indicating that the findings are robust through different market types.

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Another recommendation can be made about the data frequency. This research looks at monthly volatility calculated by daily observations. If high frequency data is available, the same research can be done on daily volatility, calculated with for instance 5 minute interval observations. The high level of observations makes it more interesting to implement Arch-type models, which could be added to the selection of forecasting models.

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V

.

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Internet

http://www.mscibarra.com/products/indices/equity/definitions.jsp visited 10-03-2008

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Appendix 1

To visualize the importance of the volatility parameter, an example is given. Table 15 provides information over a certain non-dividend paying stock with a current price of € 100. A client likes to buy options with a strike price of € 90. The annual risk free rate is 5% and the time to maturity half a year. The only parameter not known to calculate the value of the option is the volatility.

Table 15: Summary of parameters in example

Parameter Symbol Values

Stock Price S € 100

Strike Price K € 90

Risk free rate (annualized) r 5%

Time to maturity (annualized) T 0.5

Volatility (annualized) σ 5% -- 60%

Figure 1 shows that a mistake in volatility estimation can have a huge impact on the price of the option. Figure 1: Influence of volatility on option price

Option price at different volatility levels

€ 0.00 € 5.00 € 10.00 € 15.00 € 20.00 € 25.00 5.00 % 10.0 0% 15.0 0% 20.0 0% 25.0 0% 30.0 0% 35.0 0% 40.0 0% 45.0 0% 50.0 0% 55.0 0% 60.0 0% Volatility P ri c e ( e u ro ) Call Put

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In absolute terms, the change in value of calls and puts is identical as the volatility increases. However, relatively seen the difference is huge. Imagine that the client buys call or put options and the volatility is a modest 20%. Time goes by and after a few months the stock has moved around, increasing volatility to 30%, but eventually returned at € 100. The loss of time-value has made the call option decrease in value but the put option has increased because of the higher volatility. Speculating on a volatility increase with out-of-the-money options is called a volatility play (Natenberg, 1994). The results from the scenario mentioned above are shown in Table 16.

Table 16: Influence of volatility on option price

Parameter Symbol Values (t) Values (t+0.25)

Stock Price S € 100 € 100

Strike Price K € 90 € 90

Risk free rate (annualized) r 5.00% 5.00%

Time to maturity (annualized) T 0.5 0.25

Volatility (annualized) σ 20% 30%

Price Call option C € 13.50 € 12.86

Price Put option P € 1.28 € 1.74

Change in Value Call ∆f -4.74%

Change in Value Put ∆f 36.34%

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Appendix 3

Figure 2: Outliers in close-to-close model

0.0 0.4 0.8 1.2 1.6 C H IN A IN D IA K_O R E_A M A LA_Y S IA FT_S E S_& P

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Appendix 4a

Table 18: Summary statistics of monthly volatility (January 1996 – December 2000)

Panel A: Historic volatility

Mean 26.90% 27.26% 37.21% 27.69% 15.95% 17.45% Median 21.98% 25.42% 36.05% 22.06% 15.79% 16.01% Maximum 75.97% 57.98% 85.56% 150.28% 33.32% 34.82% Minimum 9.15% 10.47% 11.43% 6.22% 7.38% 7.18% Std. Dev. 0.1378 0.1025 0.1670 0.2258 0.0600 0.0644 Skewness 1.1348 0.7382 0.5764 3.1604 0.6250 0.9622 Kurtosis 4.3541 3.1969 3.0234 16.1364 3.0150 3.6407 Jarque-Bera 17.4622 5.5460 3.3236 531.2938 3.9071 10.2854 Probability 0.0002 0.0625 0.1898 0.0000 0.1418 0.0058 Observations 60 60 60 60 60 60

Panel B: Parkinson Mean 20.94% 21.11% 27.58% 21.91% 14.05% 15.38% Median 18.30% 19.24% 29.42% 19.16% 14.21% 14.09% Maximum 48.25% 43.02% 49.24% 103.28% 28.85% 29.91% Minimum 8.38% 8.18% 10.14% 6.78% 5.60% 8.01% Std. Dev. 0.0912 0.0809 0.1031 0.1500 0.0580 0.0519 Skewness 0.7600 0.9000 -0.0295 2.9426 0.4816 1.1034 Kurtosis 2.8796 3.6824 1.9909 15.7080 2.7472 3.8174 Jarque-Bera 5.8128 9.2648 2.5544 490.3255 2.4795 13.8441 Probability 0.0547 0.0097 0.2788 0.0000 0.2895 0.0010 Observations 60 60 60 60 60 60

Panel C: Garman-Klass Mean 19.09% 19.80% 26.34% 20.38% 13.50% 14.61% Median 16.96% 18.68% 28.06% 18.47% 13.70% 13.06% Maximum 42.03% 41.87% 44.59% 85.14% 31.96% 29.65% Minimum 8.13% 7.26% 9.27% 6.23% 4.75% 7.28% Std. Dev. 0.0795 0.0799 0.0993 0.1297 0.0582 0.0491 Skewness 0.7399 0.9989 -0.0944 2.3767 0.6248 1.1843 Kurtosis 2.8786 3.9865 1.8305 11.7206 3.4245 4.1637 Jarque-Bera 5.5108 12.4099 3.5085 246.6087 4.3538 17.4099 Probability 0.0636 0.0020 0.1730 0.0000 0.1134 0.0002 Observations 60 60 60 60 60 60

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Appendix 4b

Table 19: Summary statistics of monthly volatility (January 2001 – December 2005)

Panel A: Historic volatility

Mean 20.23% 20.49% 25.40% 11.94% 16.24% 16.46% Median 18.72% 17.15% 23.74% 10.82% 12.58% 15.28% Maximum 47.89% 63.26% 56.13% 32.95% 51.27% 42.80% Minimum 8.56% 8.91% 9.58% 4.53% 6.91% 7.38% Std. Dev. 0.0876 0.1032 0.0984 0.0551 0.0995 0.0777 Skewness 1.4932 2.1074 0.7656 2.0415 1.6033 1.2119 Kurtosis 5.4087 7.9939 3.3164 8.2931 5.0816 4.1133 Jarque-Bera 36.8003 106.7585 6.1112 111.7185 36.5378 17.7859 Probability 0.0000 0.0000 0.0471 0.0000 0.0000 0.0001 Observations 60 60 60 60 60 60

Panel B: Parkinson Mean 16.86% 18.14% 18.87% 10.45% 14.53% 14.12% Median 16.16% 16.07% 17.44% 9.66% 11.49% 12.19% Maximum 35.22% 51.32% 33.71% 25.64% 41.80% 36.71% Minimum 7.83% 9.37% 10.96% 5.12% 6.35% 7.29% Std. Dev. 0.0561 0.0825 0.0564 0.0407 0.0831 0.0619 Skewness 1.0207 2.2015 0.6936 1.8062 1.5684 1.3408 Kurtosis 4.7458 8.6590 2.8314 6.9160 4.9537 4.8668 Jarque-Bera 18.0379 128.5271 4.8820 70.9621 34.1413 26.6899 Probability 0.0001 0.0000 0.0871 0.0000 0.0000 0.0000 Observations 60 60 60 60 60 60

Panel C: Garman-Klass Mean 15.76% 17.43% 18.17% 10.00% 13.87% 13.79% Median 15.62% 15.83% 16.93% 8.91% 11.27% 12.26% Maximum 30.97% 48.65% 32.56% 25.37% 38.00% 34.86% Minimum 7.11% 9.28% 10.51% 5.22% 5.95% 7.02% Std. Dev. 0.0502 0.0773 0.0523 0.0384 0.0774 0.0600 Skewness 0.7983 2.2311 0.7307 1.8631 1.5666 1.3801 Kurtosis 3.9836 8.9567 2.9693 6.9930 5.0313 4.8875 Jarque-Bera 8.7910 138.4839 5.3417 74.5717 34.8580 27.9536 Probability 0.0123 0.0000 0.0692 0.0000 0.0000 0.0000 Observations 60 60 60 60 60 60

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Appendix 5a

Table 20: Autocorrelation of the in-sample (1996-2000) China Close-to-Close model

Autocorrelation Partial Correlation AC PAC Q-Stat Prob

. |*** | . |*** | 1 0.404 0.404 10.316 0.001 . |** | . |*. | 2 0.318 0.185 16.816 0.000 . |** | . |*. | 3 0.297 0.146 22.587 0.000 . |*. | . | . | 4 0.169 -0.030 24.476 0.000 . |** | . |*. | 5 0.280 0.189 29.771 0.000 . |*. | . | . | 6 0.166 -0.035 31.669 0.000 . |** | . |*. | 7 0.212 0.107 34.816 0.000 . |** | . |*. | 8 0.252 0.092 39.375 0.000 . |*. | . | . | 9 0.141 -0.030 40.821 0.000 . |*. | .*| . | 10 0.099 -0.082 41.557 0.000 . |*. | . |*. | 11 0.139 0.078 43.025 0.000 .*| . | **| . | 12 -0.090 -0.273 43.658 0.000 . | . | . |*. | 13 0.038 0.086 43.770 0.000 . | . | .*| . | 14 -0.037 -0.108 43.880 0.000 .*| . | . | . | 15 -0.082 -0.032 44.439 0.000 .*| . | **| . | 16 -0.124 -0.218 45.745 0.000 .*| . | . |*. | 17 -0.151 0.078 47.707 0.000 . | . | . | . | 18 -0.050 -0.015 47.927 0.000 .*| . | . | . | 19 -0.139 -0.025 49.672 0.000 . | . | . |** | 20 0.010 0.205 49.682 0.000 . | . | . |*. | 21 0.012 0.086 49.696 0.000 .*| . | .*| . | 22 -0.086 -0.126 50.419 0.001 .*| . | . | . | 23 -0.113 0.016 51.699 0.001 . | . | . |*. | 24 -0.019 0.136 51.736 0.001

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Appendix 5b

Table 21: Autocorrelation of the in-sample (1996-2000) FTSE Close-to-Close model

Autocorrelation Partial Correlation AC PAC Q-Stat Prob

. |****** | . |****** | 1 0.716 0.716 32.336 0.000 . |**** | . |*. | 2 0.550 0.077 51.749 0.000 . |**** | . |*. | 3 0.462 0.089 65.669 0.000 . |** | .*| . | 4 0.323 -0.110 72.604 0.000 . |** | . | . | 5 0.234 -0.000 76.296 0.000 . |** | . |*. | 6 0.229 0.112 79.902 0.000 . |** | . | . | 7 0.203 0.022 82.807 0.000 . |** | . |*. | 8 0.248 0.158 87.192 0.000 . |** | . |*. | 9 0.291 0.072 93.377 0.000 . |** | .*| . | 10 0.226 -0.125 97.180 0.000 . |** | . | . | 11 0.201 0.009 100.24 0.000 . |** | . |*. | 12 0.254 0.161 105.25 0.000 . |*. | .*| . | 13 0.184 -0.092 107.94 0.000 . |*. | .*| . | 14 0.106 -0.092 108.85 0.000 . |*. | . | . | 15 0.108 0.033 109.82 0.000 . |*. | . | . | 16 0.080 -0.000 110.35 0.000 . | . | . | . | 17 0.040 -0.053 110.49 0.000 . | . | **| . | 18 -0.042 -0.199 110.65 0.000 .*| . | . | . | 19 -0.074 0.041 111.14 0.000 .*| . | .*| . | 20 -0.125 -0.115 112.59 0.000 .*| . | .*| . | 21 -0.155 -0.084 114.89 0.000 **| . | .*| . | 22 -0.236 -0.143 120.35 0.000 **| . | . | . | 23 -0.266 -0.012 127.49 0.000 **| . | . | . | 24 -0.244 -0.024 133.64 0.000

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Appendix 6

Figure 3: FTSE Monthly volatility

FTSE Monthly volatility (Jan. 1996 - Dec. 2005)

0.00% 10.00% 20.00% 30.00% 40.00% 50.00% 60.00% 31-1 -96 31-7 -96 31-1 -97 31-7 -97 31-1 -98 31-7 -98 31-1 -99 31-7 -99 31-1 -00 31-7 -00 31-1 -01 31-7 -01 31-1 -02 31-7 -02 31-1 -03 31-7 -03 31-1 -04 31-7 -04 31-1 -05 31-7 -05 Date A n n u a li z e d V o la ti li ty CC Vol. % PK Vol % GK Vol % RS Vol %

Volatility calculations are made with the standard close-to-close historic model and the three extreme-value models. Figure 3 shows that all volatility models create some sort of the same pattern with spikes and troughs at similar places.

Figure 4: FTSE Monthly volatility estimates

Simple Volatility Forecast

0.00% 10.00% 20.00% 30.00% 40.00% 50.00% 60.00%

Jan-96 J ul-96 Jan-97 J ul-97 Jan-98 Jul-98 Jan-99 Jul-99 J an-00 Jul-00 J an-01 Jul-01 J an-02 Jul-02 J an-03 J ul-03 J an-04 J ul-04 Jan-05 J ul-05

Date A n n u a liz e d V o la ti li ty C-C Vol. RW MA (10) EWMA

(40)

Appendix 7a

Table 22: Per country analyses accuracy statistic scores of the volatility estimators

HV PK GK RS China 48 23 13 36 4.00 1.92 1.08 3.00 India 48 29 19 24 4.00 2.42 1.58 2.00 Korea 48 29 19 24 4.00 2.42 1.58 2.00 Malaysia 48 25 17 30 4.00 2.08 1.42 2.50 FTSE 46 28 20 26 3.83 2.33 1.67 2.17 S&P 500 45 25 14 36 3.75 2.08 1.17 3.00

The GK model comes in first in all countries. As a second, the PK model is preferred in the dataset from China, Malaysia and the S&P 500. The RS model is the second pick in India, Korea and FTSE.

Appendix 7b

Table 23: Per country accuracy statistic scores of the volatility forecasting models

RW MA-10 EWMA China 37 39 20 2.31 2.44 1.25 India 43 32 22 2.69 2.00 1.38 Korea 41 31 24 2.56 1.94 1.50 Malaysia 48 20 28 3.00 1.25 1.75 FTSE 26 48 22 1.63 3.00 1.38 S&P 500 30 47 19 1.88 2.94 1.19