Global Economy and Finance Journal

Vol. 8. No. 1. March 2015 Issue. Pp. 46 – 66

Performance of Range and Return Based Volatility Estimators: Evidence from Indian Crude Oil Futures Market

Vivek Rajvanshi1

This study compares the performance of various daily and intraday range based estimators for crude oil futures, which is one of the most liquid futures traded at Multi Commodity Exchange India Limited (MCX) for the period from July 2005 to July 2011. Findings show that the intraday realized range based estimator performed best as compared to the daily range based and classical estimator in terms of both efficiency and bias. Results of volatility estimation and forecasting performance show that realized range based estimators seems to be more economical and efficient to estimate the ‘true’ volatility.

Key words: High frequency, Commodity futures, two scales realized volatility, discreteness JEL Classification: C13, C14

1. Introduction Return volatility is not only a parameter of return distribution but also a key input in asset pricing, portfolio management and risk management. Volatility is not observable; therefore, unbiased and consistent estimation and accurate forecast of volatility is critical. This study compares the performance of various range based measures of volatility by using tick-by-tick data of crude oil futures contracts traded at multi commodity exchange India Ltd. (MCX). Various volatility measures based on returns (e.g, absolute returns), using information content in past returns (Generalized auto regressive conditional heteroscedasticity (GARCH) family models, e.g, Bolerslev 1986; exponential GARCH by Nelsen, 1991 among others), range based models (e.g, Parkinson, 1980; Garman and Klass, 1980; among others) have been proposed in finance literature. Price range is defined as the difference in the high and low price observed during a time interval. Studies show that range based estimators are more efficient than the return based estimators (e.g., Rogers and Satchell, 1991: Alizadeh, Brandt and Diebold, 2002 and Bali and Weinbaum 2005). In the last three decades a number of range based estimators have been proposed, details of these estimators are given in the later part of the paper. However, range based estimators are proposed under the strong assumption that the price generating process is continuous and follow lognormal distribution. Theoretically these estimators may be more efficient and unbiased than the return based estimators but in reality prices are not observed continuously and therefore observed high and low during the trading interval may be different from the true high and low and may induce bias. In real market conditions, in the presence of market microstructure

1 Vivek Rajvanshi is an Assistant Professor at Indian Institute of Management Lucknow, Prabandh Nagar, Off

Sitapur Road Lucknow (India), PIN – 226013, email: vivekrajvanshi@gmail.com, vivekr@iiml.ac.in ; Ph. (+91) 9956 333 688 Author would like to thank Mr. Rakesh Jangili, PGP 27

th Batch Student, at IIM Lucknow for helping in writing

code for the estimation of various range based volatility estimators and for his valuable suggestions.

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noise such as discretization and bid-ask bounce effect, range based estimators may be biased and inconsistent. Therefore, it is important to test the performance of these estimators in different real market conditions. Studies using high frequency data in recent years has provided enormous insights about the reaction of prices to information and volatility behavior. Martens and Dijk (2007) and Chriestensen and Podolskij (2007) proposed intraday range based estimators with bias correction procedures for the potential distortions caused by the market microstructure noise. Estimators based on high frequency data are computationally expensive and requires large data set and for some assets, high frequency data is not easily available (e.g, real estate market). Further, in emerging markets, where liquidity in trading and infrequent trading is a common phenomenon, it is interesting to compare the performance of intraday range based estimators with the daily range based estimator. In this paper, we compare the performance of intraday and daily range based estimators for the crude oil futures contracts traded at MCX for a period from 2005 to 2011. We contribute the extant literature in several ways. First, Commodity markets as an alternative investment class has gained enormous popularity among the investors as it provides diversification benefits and helps in reducing risk in portfolio management. Most of the research in this area are concentrated on the index performance or forex markets, but, given the importance of the commodity markets it has not been explored effectively. For the emerging markets like India, very limited studies has been carried out though such studies are required to understand the characteristics of these markets. Second, This study examines the performance of various intraday and interday range based volatility estimators with the traditional absolute return. This is important to see in the light of newly developed intraday range based estimators whether daily estimators are still relevant, given the fact that infrequent trading is a major concern for the emerging market like Indian crude oil futures market. Also, daily range data are easily available as compared to high frequency data so it is important to see what gains, if any, are there if practitioners use estimators based on daily range . Third, this study have used tick-by-tick data which is most informational. Very few studies has used such data set for Indian crude oil futures market. Lastly, For the benchmark volatility this study uses two scale realized volatility (TSRV) proposed by Zhang et al. (2005), which make use of both low and high frequency data and it is corrected for the microstructure noise over the realized volatility. Our findings show that all the range based estimators are better than the traditional absolute return measure of volatility in terms of both unbiasedness and efficiency. Intraday range based estimators are superior than the daily range based estimators. We use MBIAS, MRB to test the bias and MAE, RMSE for the efficiency of the estimators by taking TSRV as benchmark. AR, MA, ARMA, three volatility forecasting filters are used for evaluating forecasting performance of the volatility estimators. ARMA (1,1) filter provides the minimum bias and highest efficiency for both in-sample and out-of-sample forecast. This work has implication in risk management.

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This study is organized as follows. Section 2 provides relevant literature review. In section 3 we discuss the methodology for the estimation and forecasting of return volatility. Section 4 provides the data description. Section 5 discussed the results and finally section 6 concludes.

2. Literature Review Parkinson (1980), Garman and Klass (1980) and Rogers and Satchels(1991) proposed volatility estimators based on daily price range. These estimators are 5 to 14 times more efficient than the historical volatility estimator. Akay et al. (2010) finds that in a market where inter predictable patterns in volatility are expected Parkinson measure is more efficient and reduce the impact of microstructure noise than the absolute return. Despite the fact that these estimators are build with the assumption of continuous trading, studies show the efficiency of these estimators over the volatility estimators based on daily returns (Wiggins, 1991). However, for the illiquid assets and in the presence of microstructure noise, such as, discrete trading these estimators may be biased (Marsh and Rosenfeld, 1986). Another issue with these studies is with the choice of benchmark volatility, these papers uses absolute returns as the benchmark for the true volatility which itself a noisy estimator of volatility. High frequency data provides more information about the price discovery process and has been used extensively to develop more efficient volatility estimators. Andersen et al. (1998) proposed the concept of realized volatility and defined it as a sum of squared returns of intraday returns as a proxy for the true volatility. But, studies show that sum of squared returns are not efficient estimator in the presence of market microstructure noise. Ait-Sahalia, Mykland, and Zhang (2005) proposed two scale realized volatility (TSRV) by using both high and low frequency and find that even in the presence of market microstructure noise TSRV perform better in terms of both bias and efficiency. Both realized volatility estimator and TSRV have been extensively used in finance literature as a benchmark (e.g, Bali and Weinbaum, Vipul and Jacob 2007). Christensen and Podolskij (2007); Martens and van Dijk(2007) proposed volatility estimators by using intraday range and corrected for the microstructure noise. In a study comparing the performance of intraday and interday range based estimators, Jacob and Vipul find that the estimation performance of range based estimators is not affected by the levels of liquidity and volatility (Jacob and Vipul, 2008). However, in a study of the stocks traded at CAC, Todarova and Husman find that bias and efficiency of the range based estimators depend upon the stocks’ liquidity (Todarova and Husman, 2012). Chou et al. (2010) provides an excellent review of range based volatility estimators and their application in the field of finance. Intraday range based estimator requires more intensive data set and difficult to implement as compared to easily computable daily range based volatility estimators. Correction procedures for microstructure noise in intraday range based volatility estimators depends upon the sampling frequency and vary from asset to asset with liquidity level and other market characteristics. Most of the literature on volatility estimation is focused on forex markets, indices and stocks. Commodities are not supposed to be as liquid as indices or forex market. Therefore, it is required to investigate further in different market settings to test the robustness and to get more insight of the results. Given the importance of the futures market,

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studies on emerging commodity futures markets using intraday data are rare. Rajvanshi, V. (2013) used tick-by-tick data for testing the role of the trading numbers and order imbalance for explaining volume-volatility relationship for energy and metal futures. In Indian context, studies on crude oil futures contracts volatility are very important as India is the fourth world’s largest importer of crude oil and it is used as a hedging tool by several industries. In 2011-12 India imported oil of around $140 billion which accounts a significant percentage of India’s total import.

3. Methodology

3.1 Range Based Estimators Let Ot, Ct, Ht and Lt denoting the log of opening, closing, highest and lowest price respectively during the day ‘t’. Range based estimators are proposed with the assumption that asset price at time ‘t’, St follows the geometric Brownian motion

(1) Where the Brownian motion assume constant drift µ and constant volatility σ>0. The solution

of stochastic equation given in equation (1) is given by (

) .

The classical estimator (CL) is computed as the absolute returns from open-to-close price of the trading day ‘t’. (2) Parkinson (1980) defined a range based volatility estimator (PK) by using log of high and low price obtained during the trading day and assuming zero drift in prices:

(3)

Where, factor 1/4 ln 2 is equal to the reciprocal of the second moment of the range of a standard Brownian motion of a continuous price series. It can be shown that in the absence

of drift (i.e.

) Parkinson’s measure is five times more efficient than the classical

estimator of volatility,. Garman and Klass (1980) proposed a minimum variance unbiased estimator (MVUE) of volatility under the same assumption of the Parkinson (1980). In addition to Parkinson’s measure, Garman and Klass (GK) incorporated the opening and closing prices. GK measure is defined as

( )

(4)

Jacob and Vipul (2008) pointed out that middle terms in the above expression is much smaller than first and last terms. Therefore, the Garman Klass estimator (GK) is in fact a

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weighted average of the Parkinson estimator (PK) and the classical estimator (CL). Theoretically, GK estimator is seven times efficient than the classical estimator. Unlike the PK and GK measures, Rogers and Satchell (1991) relaxed the zero drift assumption and developed a new volatility estimator and show that this estimator perform better than PK and

GK measures in presence of drift (i.e.

)

(5)

However, Rogers and Satchell estimator fails to capture the true variation in prices when opening or closing price are also the high or low of the day. Rogers and Satchell (1991) also proposed a correction in RS and GK measures for the continuous price assumption, as in actual trading, prices are observed discontinuously. For a trading day ‘t’ with N documented prices, the Adjusted Rogers and Satchell’s (ARS) estimator of volatility is the positive root of the following quadratic equation

√

(6)

and the adjusted GK is the positive root of the following quadratic equation

√

√

(7) Christensen and Podolskij (2007) developed realized range based volatility estimator (RR), by replacing each squared intraday returns in Andersen et al. (1998) realized volatility model with the intraday range

∑

(8)

Where ‘I’ is the number of intraday intervals and and denote log price of highest and

lowest prices observed in the i-th intraday interval of day ‘t’, respectively. In the presence of microstructure noise this estimators may be upwardly biased as compared to realized variance proposed by Andersen et al. (1998). In particular, because of the discretization and bid-ask bounce effect, observed highest price during the trading interval is likely to be at the ask and observed lowest price is likely to be at bid price which increases the range by the bid-ask spread as compared to the true range and induces upwardly bias in RR estimators. Also, infrequent trading may induce downward bias in realized range based estimator therefore to avoid the impact of infrequent trading we use 5, 10, 15 and 30 minutes intervals for the computation of RR estimator. In order to improve the bias and efficiency in RR estimators, Martens and van Dijk (2007) proposed a bias correction procedure by using daily range, as, daily range is much less contaminated by the microstructure noise than the intraday range. They scaled the realized range estimator with the average level of daily range in order to correct for the downward

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bias. The scaled realized range estimator proposed by Martens and van Dijk(2007) is defined as

(

∑

∑

) (9)

Where ‘q’ is the number of days with which the realized range estimator is scaled. The choice of ‘q’ depends upon the trading intensity and spread. If spread and trading intensity is changing then ‘q’ should not set too large. We have calculated the scaled realized range variance estimators using 10 to 120 days period in an increment of 10 days. RR0510 denotes the realized range volatility corrected for bias by using the last 10 days daily range. 3.2 Measuring Benchmark Volatility

While judging the performance of volatility estimators in estimation and forecasting, choice of benchmark volatility is important. Andersen et al. (1998) proposed realized volatility as the sum of squared returns but in the presence of microstructure noise and Jumps realized volatility does not provide efficient estimator of true quadratic variation. Ait-Sahalia, Mykland, and Zhang (2005) proposed two scale realized volatility (TSRV) which uses a linear combination of the quadratic variances at the two frequencies. TSRV is consistent and unbiased estimator of true volatility even in the presence of microstructure noise. The intuition behind this estimator is that the realized variance at highest frequency consistently estimates the noise variance that may be used to reduce the bias from the low frequency estimator. TSRV is estimated as:

(

) (10)

Where

is the average realized variance, obtained by using a certain low frequency (e.g.,

5-min, 10-min etc.) which is corrected, by realized variance obtained with the higher available

sampling frequency

(e.g., 1-seconds, 10-seconds etc, depending upon the liquidity of the

trading instruments). M is the number of daily returns and n is average number of returns over the low frequency. TSRV is used as benchmark for realized volatility in number of studies (e.g.; Martens & van Dijk, 2007; Vipul & Jacob, 2007; Jacob & Vipul, 2008; Christensen, Podolskij & Vetter (2009); Todorova and Husmann (2011)). We have considered 5-minutes returns for low frequency and 10-seconds return for high frequency. 3.3 Forecasting Techniques

In this study, to test the forecasting performance of the volatility estimators, we have used autoregressive AR(p), Moving average MA(q), Autoregressive moving average ARMA(p, q) methods.

AR(p) process for a time series can be defined as

∑ (11)

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Where are the parameters, is a constant and is a white noise process.

Autoregressive moving average ARMA(p, q) is presented as,

∑ ∑

(12)

While simple moving average forecasting method is defined as

∑

(13)

3.4 Performance Evaluation

Unbiasedness and efficiency both are desirable properties of a good estimator. To evaluate the meaningful comparison for the estimation and forecasting performance of various range-based estimators we have used both efficiency measures (Root Mean Square Errors (RMSE) and Mean absolute error (MAE)) and bias measures (Mean Relative Bias (MRB) and Mean BIAS). Choice of MAE is important for the robustness of the results as the outliers in time series may have significant impact on RMSE whereas, MRB and MBIAS provide the magnitude of bias.

√

∑

(14)

∑ | |

(15)

∑

(16)

∑

(17)

Where TSRV is the two scale realized volatility proposed by Zhang, Mykland and Ait Sahalia, (2005) and it is used as benchmark volatility and is the volatility estimated by using

different estimators.

4. Data and Formation of Time Series

We obtained tick-by-tick data from Systrade, RCS Poitiers, France, for crude oil contracts from July 2005 to July 2011 traded at MCX. This data provides the price, quantity traded and time stamp in milliseconds. Crude oil contracts are one of the most liquid contracts among the commodities traded at MCX. Table 1 provides the information about the traded volume, contract expiration months of the top commodities traded at MCX.

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Table 1: The current state of trading of most liquid commodity futures in India

Category/ commodity

Trading venue

Unit Year started

Contract expiration month

Daily average volume

% Share of total of the exchange (by value*)

Precious Metals

Gold MCX KG 2003 F,A,J,A,O,D 44751 34.88

Silver MCX MT 2003 J,M,M,M,S,N 1415 17.31

Copper MCX MT 2004 F,A,J,A,O,D 97057 12.55

Zinc MCX MT 2006 F,A,J,A,O,D 66120 2.87

Nickel MCX MT 2004 F,A,J,A,O,D 8027 3.19

Energy

Crude Oil MCX BBL 2005 J,F,M,..D 47339 20.32

Agricultural

Cumin (Jeera) NCDEX MT 2005 J,F,M,..D 7193 3.45

Guar seed NCDEX MT 2004 J,F,M,..D 29732 24.77

Pepper NCDEX MT 2004 J,F,M,..D 1484 2.73

Soy oil NCDEX MT 2004 J,F,M,..D 83708 14.92

Soy oil MCX MT 2004 J,F,M,..D 12469 0.29

Chick peas (Chana)

NCDEX MT 2004 J,F,M,..D 11818 10.92

R/M seed NCDEX MT 2003 J,F,M,..D 77860 9.49

* These figures represents percentage share of the specific commodity in value in the specific exchange during the year 2009. The total value of traded contracts in the year 2009 for MCX is INR 59.57 trillion and in NCDEX is INR 8.04 trillion. For the year 2009, the top four commodities in terms of value of traded contracts in MCX are gold (34.88%), crude oil (20.88%), silver (17.31%) and copper (12.55%).

We form the time series of prices by using nearest month expiration contracts. We formed continuous time series by rolling over contract to the next contract when the daily volume of the last month exceeds the daily volume of the current month. In other words, we formed the time series by using the active contracts only. During the analysis we found that the volume of the last month contracts exceeds the front month contracts on an average four days before the contract expiration. It might be due to the fact that participants in the futures markets rollover their position to the next contracts few days before the expiration of the contracts. This is particularly important in the Indian context as only nearest month contracts are liquid enough to provide the insight about the price discovery and other characteristics of the returns. Time series produces 13,779 average number of tick per day during the study period. Table 2 provides, year-wise daily average number of ticks, median and standard deviations. It is interesting to see that average number of ticks has increased significantly

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from July 2005 to July 2011. This shows that the trading interest in this market has increased significantly in the last few years.

MCX provides about 13 and a half-hour trading window (from 10:00 AM to 11:30 PM). We have not considered those trading days in which the volume in any 30 minute was not available or was equal to zero. In addition, during some holidays trading in energy and metal futures starts at 5:00PM and continues till 11:30 PM. Liquidity during these days are usually significantly low as compared to normal days, therefore we have not considered those trading days in this study. We have considered data from Monday to Friday only as on Saturdays, trading timing is different from other working days.

Finally, we arrived at 1481 trading days for the analysis. Further for the estimation of the TSRV we converted data into nearest to second time. If price information was not available in the preceding second nearest previous tick information is used. This produces 48 600 price information per day and overall 71 976 600 data points.

Table 2: Year wise basic characteristics of the ticks and daily returns

Open to Close Abs. Return Daily Number of Ticks

First Order Autocorrelation Year

Daily Drift Mean St. dev Mean Median St. dev

2005 -0.0016 0.0109 0.0088

6,319 5,882 2,667 -0.3723

2006 -0.0012 0.0102 0.0076

4,544 4,363 1,867 -0.2987

2007 -0.0004 0.0112 0.0087

8,704 8,566 2,821 -0.3352

2008 -0.0026 0.0185 0.0160

14,523 14,923 5,852 -0.2764

2009 -0.0005 0.0174 0.0155

20,711 21,860 6,467 -0.3482

2010 -0.0004 0.0101 0.0077

19,263 20,143 5,960 -0.4046

2011 -0.0011 0.0126 0.0108 23,162 23,238 6,884 -0.3757

All -0.0010 0.0132 0.0119 13,779 11,825 8,295 -0.3371

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Figure 2: Histogram of Daily Return Series for period July 2005-July 2011

-10

-50

510

Re

turn

s(%

)

01jan200501jan2006

01jan200701jan2008

01jan200901jan2010

01jan2011

Time

Daily raw return series

Figure 1: Daily raw returns series from July 2005 to July 2011

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Figure 1 shows the daily raw return series from July 2005 to July 2011. It is clear from the graph that there is a presence of volatility clusters. From July 2008 to July 2009 volatility was very high as compared to other periods. This might be because of the turmoil in financial markets during that time. To get more insights about the return distribution we plot the histogram and compared with standard normal distribution. Figure 2 provides the histogram of raw returns. Histogram show that returns are concentrated around the mean, but at the same time returns exhibits fatter tails and higher peaks. It is also clear from the histogram that the number of extreme losses is higher than the number of extreme gains (as captured by negative skewness). Histogram provides the support of our findings that returns are not normally distribution.

5. Results and Discussion 5.1 Estimation Performance of Different Volatility Measures Table 3 provides the mean, standard deviation, kurtosis, skewness, minimum and maximum values for all volatility estimators used in this study. The mean volatility is highest for the TSRV estimator followed by the Garman-Klass estimator corrected for the discreteness in the trades. The results are in line with the theoretical expectations due to the fact that TSRV uses all the information available in high frequency data and estimate the true variations in the returns. Classical measure open-to-close return volatility is the lowest among the estimators estimated. Adjusted Garman Klass measure (AGK1) shows an improvement in the mean volatility estimate over simple Garman Klass measure of volatility (GK) as it become more near to our benchmark volatility measure (i.e. TSRV). An improvement in volatility estimation is also noticed after the correction for bias (ARS1) in the Rogers and Satchels estimate of volatility (RS) estimated by using daily range. These findings shows that there is a presence of microstructure noise in the data and adjustment procedures are useful in correcting for that bias. One interesting finding is that the mean volatility estimated by realized range of intraday intervals (RR05 to RR30) for different intervals (05, 10, 15, 30 minutes) increases as we move towards the lower frequency and it is very close to the benchmark volatility estimator (TSRV) when estimated by using 30 minute interval range. This may be due the infrequent trading. Martens and Dijk (2007) mentioned that in the presence of infrequent trading volatility estimators estimated by intraday realized range are likely to be downwardly biased and they proposed the bias correction through scaling. If this is the case then by scaling with daily realized range would improve the volatility estimators. We applied bias correction procedure proposed by Martens and Dijk (2007). Volatility estimators obtained after bias correction (RRst0510 to RRst05120) through scaling of last ‘q’ days (‘q’ vary from 10 to 120 days). Realized range volatility corrected for bias by using past 10 days realized volatility show a close resemblance to TSRV (average RRst0510 is 1.628 as compared to the average TSRV 1.624). However, volatility increases consistently as we scaled the realized range volatility (RR05) with realized volatilities of longer length. This may be because of the fact that trading intensity and bid-ask spread change rapidly during the period of study as it can be seen from the Table 2 that the trading intensity has increased significantly in the past years and first

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order correlation shows the presence of bid-ask bounce noise. Descriptive statistics shown in Table 3 indicates that volatility distribution is not normal for all the volatility estimators but kurtosis reduces as we move from daily to intraday realized range volatility estimators and improved further when corrected for bias.

Table 3: Descriptive statistics for the volatility estimators

Mean Median St. Dev Kurtosis Skewness Min Max

TSRV 1.624 1.409 0.746 4.478 1.930 0.311 5.680

CL 1.319 1.008 1.191 4.334 1.794 0.019 7.976

PK 1.576 1.347 0.874 4.939 1.909 0.260 6.393

GK 1.591 1.365 0.862 6.239 2.070 0.237 7.385

AGK1 1.603 1.375 0.866 6.225 2.068 0.239 7.413

RS 1.569 1.337 0.893 6.062 2.067 0.001 7.353

ARS1 1.580 1.347 0.896 6.057 2.067 0.043 7.381

RR05 1.563 1.349 0.729 4.032 1.879 0.338 5.155

RR10 1.591 1.375 0.742 4.172 1.891 0.346 5.441

RR15 1.601 1.380 0.752 4.125 1.886 0.332 5.376

RR30 1.616 1.388 0.767 4.176 1.879 0.328 5.954

RRst0510 1.628 1.413 0.771 4.394 1.873 0.327 5.641

RRst0520 1.636 1.413 0.761 4.155 1.833 0.308 5.333

RRst0530 1.640 1.425 0.760 3.944 1.810 0.323 5.521

RRst0540 1.644 1.421 0.760 3.809 1.799 0.313 5.379

RRst0550 1.645 1.421 0.761 3.762 1.799 0.324 5.270

RRst0560 1.648 1.422 0.762 3.656 1.788 0.330 5.259

RRst0570 1.648 1.412 0.764 3.548 1.776 0.335 5.272

RRst0580 1.651 1.415 0.766 3.550 1.777 0.337 5.309

RRst0590 1.655 1.420 0.769 3.528 1.772 0.339 5.303

RRst05100 1.658 1.423 0.770 3.523 1.771 0.339 5.295

RRst05110 1.664 1.428 0.771 3.529 1.778 0.342 5.317

RRst05120 1.669 1.430 0.772 3.500 1.772 0.347 5.317 Estimation performance of the volatility estimators are estimated by using four loss functions viz. MBIAS, MAE, MRB and RMSE. Table-4 reports sample estimates of the loss functions for all volatility estimators. Mean bias is negative for all volatility estimators except for the realized range estimators scaled for bias-correction. This indicates daily range based estimators (PK, GK and RS) underestimates the true volatility. Realized range estimators estimated by using 10, 15 or 30 minute interval range (RR10, RR20 and RR30) are very close to benchmark TSRV and this difference is narrowed down further when these measures are corrected for bias through daily range (RRst0510 to RRst05120). Results are further confirmed by mean absolute error (MAE) and mean relative bias (MRB). Earlier studies (see e.g., Marsh & Rosenfeld, 1986:

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Wiggins, 1991) finds that the daily range based estimators are generally negatively biased as compared to the classical volatility estimator CL. Our findings support the results obtained in earlier studies which uses the realized variance (e.g., Bali and Weinbaum, 2005). and TSRV as a proxy for the true volatility (e.g., Vipul and Jacob, 2007; Todorova and Husmann, 2011). Classical estimator CL is inferior to all daily and intraday range based estimators in terms of both bias and efficiency. As far as efficiency is concerned, RMSE shows that intraday realized range volatility estimators estimated from 10, 15, 30 minute intervals (RR10 to RR30) perform best among the volatility estimators under comparison. Table 4: Estimation performance of all volatility estimators using TSRV as benchmark

MBIAS MAE MRB RMSE

CL -0.305 0.864 -0.185 1.100

PK -0.048 0.343 -0.037 0.453

GK -0.032 0.276 -0.030 0.379

AGK1 -0.021 0.277 -0.023 0.380

RS -0.055 0.308 -0.046 0.442

ARS1 -0.044 0.308 -0.039 0.442

RR05 -0.061 0.099 -0.038 0.134

RR10 -0.032 0.074 -0.021 0.104

RR15 -0.022 0.073 -0.016 0.104

RR30 -0.008 0.085 -0.008 0.121

RRst0510 0.002 0.127 -0.001 0.173

RRst0520 0.010 0.108 0.006 0.149

RRst0530 0.012 0.099 0.007 0.136

RRst0540 0.014 0.095 0.008 0.131

RRst0550 0.015 0.093 0.009 0.128

RRst0560 0.015 0.091 0.009 0.126

RRst0570 0.015 0.089 0.010 0.124

RRst0580 0.015 0.088 0.010 0.122

RRst0590 0.016 0.088 0.010 0.122

RRst05100 0.016 0.088 0.010 0.122

RRst05110 0.017 0.087 0.011 0.123

RRst05120 0.017 0.087 0.011 0.123 Among the daily range-based estimators, bias corrected, Adjusted Garman-Klass estimator (AGK1) is less biased to the benchmark followed by the Garman-Klass estimator. Parkinson estimator performs worst among the range-based estimators when both efficiency and biasness are considered. Scaled realized range based estimators (RRst) of Martens and van Dijk (2007) estimated by using the five minute returns show significant improvement over the realized range based estimator (RR) in terms of bias measured by the MBIAS and MRB and efficiency measured by the MAE and RMSE. Results for RRst reported in Table 4 shows one interesting finding that the mean bias is positive compared to the benchmark volatility (TSRV). The correction

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procedure for microstructure noise as proposed by the Martens and van Dijk (2007) is successful in improving the efficiency of the realized range estimator. The optimal value of the RRst is obtained when 5 minute realized range volatility estimator is scaled by using 10 days daily range and beyond that there is no improvement till 120 days. In short, daily range based estimators are less biased and more efficient than the classical estimator CL. Intraday range based estimators corrected for microstructure noise performs better than the daily range based estimator and classical estimator. 5.2 Forecasting Performance Out of sample volatility forecasting of the classical estimator CL, range based estimators PK, GK, AGK, RS, ARS, RR, scaled realized range based estimator RRst and TSRV has been examined by forecasting 1-day ahead volatility for the 481 days. 1000 days rolling window has been used for the estimation of the AR, MA and ARMA process. Results of volatility forecasts are reported in Tables 5 to 7. Selection of order for AR, MA and ARMA process is chosen on the basis of the Akaike Information Criteria (AIC ) and Baysian Information Criteria (AIC ). We also looked at the correlation structure for the best fitting of the forecasts made by different volatility estimators. AR(3) process provides the minimum MBIAS and RMSE for all the volatility estimators among the AR process applied from order 1 to 10. Forecasts deteriorate for the higher order of the auto regressive process, for the brevity of space; those results are not reported. Findings show that TSRV and realized range based estimators of Martens and Dijk performs better than the other volatility estimators. Classical estimator CL performs worse among all the estimators compared. Given the expensive tick-by-tick price information required to compute TSRV estimator, intraday realized range based estimators seems more economical and efficient to estimate the ‘true’ volatility. Five minute realized range based estimator provides most accurate forecast that the lower frequency intervals. It seems that five minute frequency is the optimal interval for the estimation of the realized volatility. These findings support the findings of the Martens and van Dijk (2007) and Christensen and Podolskij (2007).

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Table 5: Forecasting Performance of volatility estimators by using AR process

MBIAS MAE MRB RMSE

TSRV AR(1) 0.298 0.946 0.454 1.567

AR(2) 0.183 0.884 0.357 1.525

AR(3) 0.146 0.854 0.332 1.497

CL AR(1) 1.188 1.574 1.239 1.395

AR(2) 0.949 1.380 1.003 1.449

AR(3) 0.765 1.277 0.840 1.549

PK AR(1) 0.804 1.242 0.856 1.384

AR(2) 0.444 1.047 0.526 1.592

AR(3) 0.359 0.993 0.462 1.614

GK AR(1) 0.809 1.300 0.873 1.615

AR(2) 0.414 1.063 0.516 1.730

AR(3) 0.336 1.011 0.455 1.709

AGK1 AR(1) 0.847 1.325 0.897 1.616

AR(2) 0.445 1.077 0.534 1.734

AR(3) 0.365 1.024 0.473 1.713

RS AR(1) 0.841 1.373 0.907 1.832

AR(2) 0.462 1.138 0.563 1.897

AR(3) 0.373 1.072 0.491 1.832

ARS1 AR(1) 0.877 1.397 0.931 1.835

AR(2) 0.492 1.152 0.581 1.902

AR(3) 0.402 1.083 0.508 1.836

RRS(5) AR(1) 0.105 0.860 0.318 1.549

AR(2) 0.046 0.830 0.272 1.502

AR(3) 0.012 0.807 0.251 1.484

RRS(10) AR(1) 0.195 0.899 0.378 1.582

AR(2) 0.119 0.854 0.316 1.521

AR(3) 0.082 0.830 0.292 1.497

RRS(15) AR(1) 0.230 0.915 0.398 1.566

AR(2) 0.140 0.867 0.324 1.524

AR(3) 0.104 0.840 0.301 1.494

RRS(30) AR(1) 0.291 0.952 0.448 1.563

AR(2) 0.176 0.883 0.351 1.513

AR(3) 0.129 0.850 0.321 1.489

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Table 5: Forecasting Performance of volatility estimators by using AR process…cont’d

MBIAS MAE MRB RMSE

RRST(10) AR(1) 0.284 0.955 0.415 1.610

AR(2) 0.212 0.919 0.359 1.584

AR(3) 0.175 0.904 0.336 1.572

RRST(20) AR(1) 0.276 0.951 0.400 1.646

AR(2) 0.215 0.915 0.353 1.594

AR(3) 0.183 0.896 0.333 1.571

RRST(30) AR(1) 0.277 0.935 0.398 1.622

AR(2) 0.208 0.899 0.345 1.572

AR(3) 0.167 0.878 0.318 1.552

RRST(40) AR(1) 0.287 0.935 0.406 1.614

AR(2) 0.222 0.900 0.355 1.560

AR(3) 0.183 0.880 0.331 1.543

RRST(50) AR(1) 0.293 0.935 0.410 1.617

AR(2) 0.230 0.901 0.361 1.559

AR(3) 0.188 0.879 0.335 1.538

RRST(60) AR(1) 0.289 0.930 0.409 1.603

AR(2) 0.227 0.898 0.361 1.549

AR(3) 0.191 0.877 0.338 1.528

RRST(70) AR(1) 0.284 0.926 0.406 1.611

AR(2) 0.218 0.895 0.355 1.555

AR(3) 0.181 0.874 0.333 1.532

RRST(80) AR(1) 0.280 0.926 0.408 1.605

AR(2) 0.211 0.893 0.355 1.552

AR(3) 0.167 0.870 0.327 1.527

RRST(90) AR(1) 0.281 0.928 0.410 1.611

AR(2) 0.209 0.893 0.355 1.557

AR(3) 0.165 0.871 0.328 1.532

RRST(100) AR(1) 0.269 0.923 0.405 1.605

AR(2) 0.189 0.885 0.344 1.551

AR(3) 0.138 0.861 0.312 1.527

RRST(110) AR(1) 0.277 0.926 0.413 1.601

AR(2) 0.198 0.889 0.353 1.547

AR(3) 0.147 0.864 0.320 1.524

RRST(120) AR(1) 0.280 0.927 0.417 1.596

AR(2) 0.198 0.889 0.355 1.543

AR(3) 0.147 0.863 0.322 1.520

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Table 6: Forecasting Performance of volatility estimators by using MA(1) process

MBIAS MAE MRB RMSE

TSRV 0.063 0.266 0.104 0.365

CL -0.131 0.306 -0.026 0.412

PK 0.054 0.279 0.100 0.371

GK 0.061 0.286 0.107 0.388

AGK1 0.070 0.289 0.114 0.388

RS 0.048 0.298 0.101 0.413

ARS1 0.057 0.300 0.108 0.413

RRS(5) 0.029 0.257 0.078 0.365

RRS(10) 0.048 0.261 0.092 0.366

RRS(15) 0.054 0.263 0.097 0.364

RRS(30) 0.060 0.266 0.102 0.364

RRST(10) 0.068 0.284 0.105 0.387

RRST(20) 0.076 0.281 0.110 0.380

RRST(30) 0.076 0.276 0.110 0.376

RRST(40) 0.081 0.276 0.113 0.374

RRST(50) 0.083 0.274 0.116 0.371

RRST(60) 0.083 0.274 0.115 0.370

RRST(70) 0.082 0.273 0.115 0.370

RRST(80) 0.080 0.273 0.114 0.370

RRST(90) 0.080 0.274 0.114 0.370

RRST(100) 0.075 0.271 0.111 0.369

RRST(110) 0.079 0.273 0.114 0.369

RRST(120) 0.080 0.274 0.115 0.369 Results are not conclusive about the choice of volatility forecasting method but in most of the cases ARMA (1,1) process seems to be superior. Results for the forecasting suggest that intraday realized range based estimators performs best followed by the TSRV. Among the daily range based estimators GK estimator perform superior followed by the PK estimator. The realized range sampled at five and ten minutes appear to be more efficient which supports the results drawn for the estimation performance in the previous section.

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Table 7: Forecasting Performance of volatility estimators by using ARMA(1,1) process

MBIAS MAE MRB RMSE

TSRV 0.0024 0.2427 0.0558 0.3611

CL -0.2385 0.3099 -0.1110 0.3911

PK -0.0350 0.2476 0.0329 0.3645

GK -0.0260 0.2518 0.0418 0.3750

AGK1 -0.0180 0.2523 0.0478 0.3750

RS -0.0429 0.2619 0.0319 0.3889

ARS1 -0.0350 0.2618 0.0379 0.3888

RRS(5) -0.0151 0.2404 0.0435 0.3612

RRS(10) -0.0012 0.2413 0.0535 0.3611

RRS(15) 0.0021 0.2433 0.0558 0.3611

RRS(30) 0.0022 0.2427 0.0566 0.3613

RRST(10) 0.0245 0.2661 0.0686 0.3835

RRST(20) 0.0329 0.2605 0.0748 0.3750

RRST(30) 0.0212 0.2559 0.0649 0.3732

RRST(40) 0.0323 0.2559 0.0739 0.3704

RRST(50) 0.0346 0.2544 0.0760 0.3677

RRST(60) 0.0386 0.2552 0.0792 0.3660

RRST(70) 0.0316 0.2530 0.0744 0.3661

RRST(80) 0.0260 0.2521 0.0705 0.3657

RRST(90) 0.0233 0.2519 0.0688 0.3664

RRST(100) 0.0090 0.2532 0.0581 0.3709

RRST(110) 0.0127 0.2511 0.0614 0.3673

RRST(120) 0.0124 0.2508 0.0616 0.3668

6. Conclusions Studies comparing the performance of the volatility estimators suggest that the range-based estimators are more efficient than the return based estimators though these estimators have some limitations. These estimators require adjustment for the market microstructure noise, particularly, bid-ask bounce, presence of drift and discreteness which tend to distort the volatility estimators. This study compares the performance of various daily and intraday range based estimators considering TSRV as the benchmark volatility for the crude oil futures traded at Multi Commodity Exchange India Limited (MCX) for the period from July 2005 to July 2011. All the daily and realized range based estimators are negatively biased except the realized range estimator proposed by Martens and van Dijk (2007). These findings support the earlier findings of Todorova and Husmann (2011) for CAC index. Our findings show that the intraday realized range based estimators perform best as compared to the daily range-based and classical estimator in terms of both efficiency and bias. Garman-Klass and Rodgers-Satchell estimators corrected for discreteness

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reduce the bias in volatility estimation but are not successful in removing the bias completely. Parkinson measure performs the worst among all daily range based estimators. We use AR, MA, ARMA, three volatility forecasting filters for evaluating the estimation and forecasting performance of the volatility estimators. ARMA (1,1) filter provides the minimum bias and highest efficiency for both in-sample and out-of-sample forecast. Estimation and forecasting performance shows that intraday realized range based estimator proposed by Martens and van Dijk (2007) appears to capture the volatility process better than the other volatility estimators. Given the expensive tick-by-tick price information required to compute TSRV estimator, realized range based estimators seems to be more economical and efficient to estimate the ‘true’ volatility.

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