Assessing Day-to-day Volatility : Does the Trading Time Matter ?

The aim of this study is to examine whether investors have dis tinct perceptions about the daily volatility of an asset. In order to capture th uncertainty faced by these investors, we define the volatility perceived by inves tors as the distribution of standard deviations of daily returns calculated from intra day prices collected randomly. We find that this distribution has a high degree of disp er ion. This means that different investors may not share the same opinion rega rding the variability of returns of the same asset. Moreover, the close-to-close vol atility is often less than the median of the volatility distribution perceived by inve stors while the open-toopen volatility is greater than that statistic. From a pract ical point of view, our results indicate that volatilities estimated using tradit ional samples of daily returns (i.e., close-to-close and open-to-open returns) may not do a g od job when used as inputs in financial models since they may not properly capt ure the risk investors are exposed.


Introduction
Since the seminal work of Markowitz (1952) the volatility of asset returns plays an important role in the modern nance theory, particularly in pricing models, portfolio selection, and risk management.The volatility is an unobservable variable that reects the degree of variation in prices of a given asset at a certain time period.Undoubtedly, it is the simplest way to quantify the uncertainty of an asset payo. 1 Several studies propose models to estimate volatility. 2 In general, when these volatility models are used in practical applications they are estimated with daily data, commonly using the closing price, as noted by Goodhart and O'Hara (1997).However, this procedure has some drawbacks.For example, Parkinson (1980) shows that the volatility estimated using the highest and lowest prices of the days is superior to the close-to-close volatility.Wood, McInish and Ord (1985) and Lockwood and Linn (1990) show that higher standard deviations are observed at the beginning and at the end of the trading session, i.e., the volatility has a U-shaped pattern throughout the day.Brown (1990) argues against the use of the closing prices since it can be inuenced by the lack of trading at the close or by \marked on the close" orders.Guillaume, Dacorogna and Pictet (1994) and Andersen and  Bollerslev (1998) observe that the series of intraday returns have dierent characteristics for dierent periods of the day.They warn that this intraday seasonality should be corrected to avoid distortions in the volatility estimation.
In this paper we revisit these criticisms by a dierent point of view.As in other studies, we calculate volatility from a series of daily prices.Therefore, we have the same amount of information used by models which estimate the volatility from closing or opening prices.However, instead of prices being collected at a xed time, we randomly select the time that it is observed in each day.With these prices, we calculate the realized volatility of the asset return in a given period (set as one month in the empirical exercise of Section 3). 3 Next we repeat this experiment, i.e., we draw another sequence of random daily prices and a new volatility is calculated for the same period.From a sequence of draws, we build a probability distribution of volatility.We denominated this distribution as the volatility distribution perceived by investors (VDPI).This distribution is the focus of our paper.
Note that the selection procedure described in the previous paragraph better approximates the daily volatility perceived by investors.An investor does not trade (or look at the market) only at the beginning or at the end of the trading session.In fact, there is no reason for the buying and selling decisions to occur at a specic time.In this sense we understand the volatility calculated from a random sampling of daily prices as the volatility perceived by investors.
A tale may shed some light on the reason why the volatility calculated with prices collected in random moments represents the volatility perceived by an investor.Suppose that an investor negotiates a single share over a month once a day.The time of the trade is determined, for example, when the price reaches thresholds (above an upper limit she sells and below a lower limit she buys).If at the end of this month we ask the investor what the asset volatility perceived by her is, the answer will be the standard deviation of daily returns calculated based on the prices actually negotiated.In other words, her perception of uncertainty is one draw from VDPI.If this volatility is much higher or much lower than the close-to-close volatility, the investor's perception of the uncertainty of this share is undoubtedly dierent from that calculated by a market analyst who uses closing prices.
Let's look at a more realistic example.Suppose that a hedge fund evaluates its risk by the Value-at-Risk (VaR) metric.Suppose further that VaR is estimated from a parametric model in which the volatility of the portfolio is obtained from the closing prices.However, the hedge fund does not necessarily negotiate assets at the closing of the market.The trades are conducted according to the manager's strategy and can happen at any time during the session.Therefore, the VaR model may not do a good job, i.e., it can not adequately capture the uncertainty faced by the hedge fund.
Our goal in this paper is to study the dispersion of the distribution of the daily volatility perceived by investors.Additionally, we compare the volatility calculated from daily prices randomly selected with the volatility calculated from opening and closing prices.More specically, we investigate the location of the open-to-open and the close-to-close volatilities on the VDPI.It allows us to evaluate whether this volatilities can be a good representation of the uncertainty perceived by an arbitrary investor.Harris (1986), Amihud and Mendelson (1987), Lockwood and Linn (1990) The database of this study consists of all intraday prices of 84 shares traded on the Brazilian stock exchange (BM&FBovespa) between July 2006 and April 2009.BM&FBovespa centers the negotiations of Brazilian stocks.The Brazilian stock exchange had a daily average turnover of $ 3.9 billion in June 2008, which places it as one of the largest stock markets in the world and the largest in Latin America according to World Federation Exchanges. 4n a nutshell, our results show that the volatility perceived by investors has a high degree of dispersion.The dierence between the lowest and highest volatility of the VDPI of a stock in the same month can be greater than 100%.Thus investors have dierent perceptions about the uncertainty of the same share.Regarding the open-to-open and close-to-close volatilities, both these volatilities can be far away from the median of VDPI.
Furthermore, the open-to-open is often located on the right side of the VDPI while the close-to-close is on the left side.For example, we nd that the frequency in which the close-to-close volatility is on the VDPI 5% percentile is 11% and the frequency in which the open-to-open is on the VDPI 95% percentile is 84.5%.
Our ndings imply that the usual practice of estimating volatility from opening and closing prices can distort the measures of risk since the day-to-day uncertainty perceived by an investor can be quite dierent from these volatilities.In this way, our results are important not only for investors and risk managers but also for nancial regulators.For instance, as capital requirement for market risk is based on the close-to-close volatility, it may not be enough to cover daily losses. 5In order to account for the dispersion of the VDPI, we propose to adjust capital requirements by the ratio between a high VDPI percentile (e.g.95%) and the close-to-close volatility.
The rest of the paper is organized as follows.Section 2 presents the data description.Section 3 describes the procedure proposed to assess the volatility perceived by investors.Section 4 discusses the empirical results.Section 5 provides some concluding remarks.

Database
The database was provided by BM&FBovespa.It consists of a series of all intraday trade prices of 84 stocks of Brazilian rms.These rms were chosen due to the liquidity of their shares.Together they represent more than 85% of BM&FBovespa traded volume.;7 The data cover the period from July 3, 2006 to April 30, 2009, corresponding to 739 trading days.Not all stocks are traded in every month and the average number of stocks is 77.03 per month. 8It is worth noting that we do not collect data according to xed time intervals.Instead of, we include all trades in the sample.Therefore, the size of the price series varies with the share liquidity.For example, the price series of Petrobras and Vale, the two most liquid stocks in the sample, contain 5,875,374 and 4,793,357 observations, respectively.Besides the price, the database includes the date, the time (recorded with precision of seconds) and the volume of each trade.To compute the volatilities, we adjust the price series of each stock to splits, inplits and dividends.
Another issue to consider about our database is that some records refer to the same order.Consider, for instance, that there are two limit orders for selling, one of 200 shares at $40.00 and another of 100 shares at $40.30.Assume also that both orders have the lowest sell prices in the limit order book.An order to buy 300 shares at $40.30 generates two trades in the BM&FBovespa database.We modify the database so that this order only generates one trade of 300 shares at $40.10 (average price per share of the trade).

Methodology
In this section we describe the selection mechanism of the daily prices as well as the volatility evaluation methodology.For each share, we generate several paths of daily prices.The price of each day is chosen according to the following procedure.First, we randomly generate a number from a discrete uniform distribution between one and the number of trades occurred in the day.Next, we select the price of the trade that corresponds to the number randomly drawn.The number of observations for each day is set as the number of trades held between 10:00am and 5:00pm (or 11:00am and 6:00pm during the daylight using of high frequency data.Currently, the realized volatility theory is the workhorse of this strand of the literature (see Andersen, 2000). 6For example, Petrobras has two types of shares, PETR3 (with voting rights) and PETR4 (without voting rights).Only PETR4 is in our sample because its liquidity is higher than that of PETR3. 7Liquidity is measured by the liquidity ratio dened by 100 p P p n N v V , where p is the number of days that there was at least a trade with the stock within the chosen period; P is the total number of days in the selected period, n is the number of trades of the stock within the chosen period, N is the number of trades of all stocks within the chosen period, v is the traded volume (in monetary units) of the stock within the chosen period, and V is the traded volume of all stocks within the chosen period.This denition of liquidity is used by BM&FBovespa to select shares that compose the Bovespa index (Ibovespa).
savings time).By repeating this procedure we obtain several daily prices paths in a way that each price represents a trade actually occurred.The next step is the construction of a series of daily logarithmic returns calculated using the prices drawn.From these series we calculate the realized daily volatility (the standard deviation of the daily returns) of each stock in each month of the sample.This is the volatility of one path and possibly the volatility observed by an investor who trades in each day during the trading hours.If P i t;1 ; : : : ; P i t;N is a series of stock i daily prices randomly selected in month t, the realized volatility of this path is

N :
For each stock in each month we generate 400,000 paths and calculate 400,000 daily realized volatilities according to the procedure described above.These volatilities represent the ones perceived by investors.The intuition is that investors trade at random times and not at xed times such as the opening and closing of the market.From the sequence of draws, we build a histogram of volatility.As a result, we have a probability distribution of the realized volatility of each stock for each month of the sample.We denominated this distribution as the volatility distribution perceived by investors (VDPI).
Our goal is to investigate the dispersion of the VDPI.It allows us to answer the following question: Can the uncertainty perceived by dierent investors (who trade daily but possibly at distinct times) be quite dierent?If so, it places an important issue: Do a regulator or a risk manager who require capital to cover market risk using volatilities computed from prices observed at specic times (such as at the closing or opening of the market) indeed capture the risk faced by investors?Moreover we want to compare the close-to-close and open-to-open volatilities with the median of the VDPI.For this reason, besides the VDPI, we also compute the daily returns volatilities from opening prices (open-to-open volatility) and closing prices (close-to-close volatility).
For example, Figure 1 shows the VDPI of Vale (VALE5) in January 2009. 9The numbers 1 and 2 on the histogram represent, respectively, the open-to-open (4.47%) and close-toclose (3.56%) volatilities.The fact that the open-to-open volatility is greater than the close-to-close is in agreement with many papers.For instance, Harris (1986), Amihud and  Mendelson (1987), Lockwood and Linn (1990) and Hong and Wang (2000) report that opening returns exhibit greater dispersion than closing returns.Note also that in this month both the close-to-close and open-to-open volatilities are far from the median of the VDPI (4.05%).
The volatility of daily returns of Vale in January 2009 ranges from 2.41% to 5.97%, depending on the prices path selected.The 5% and 95% percentiles of Vale VDPI in January 2009 are equal to 3.36% and 4.85%, respectively.This variation can yield a signicant dierence between the volatility perceived and the one used in models for quantifying risk such as Value-at-Risk (VaR).An investor could be trading their shares with the volatility of 5.97%, but considering, for purposes of risk measuring, the volatility of the closing price.The volatility of 5.97% is 68% higher than the close-to-close volatility.for example), i.e., occasionally this volatility is far from the median of the VDPI.Similarly, in other months the close-to-close volatility is very distant from the median of the VDPI.In January 2008 the close-to-close volatility is greater than the 95% percentile of the VDPI.On the other hand, in November 2006 it is less than the 5% percentile.In addition, note that the highest distance between the 5% percentile and 95% percentile of the VDPI occurs in October 2008 (the epicenter of the subprime crisis).If we had computed the volatility by the traditional way, i.e., using the closing price (Goodhart and O'Hara, 1997), we would have probably subestimated the risk of this share in this month.4

Results
In this section we present a detailed description of the results of the empirical exercise described in Section 3. Our results consist of a series of unbalanced panel data containing observations of VDPI percentiles for each stock in each month of the sample period.The panel has the form vol(i; t), where P f1%; 5%; 10%; 25%; 50%; 75%; 90%; 95%; 99%g is the percentile of the VDPI, i is the spatial dimension (stock) and t is the time dimension (month).For example, vol95(VALE5,August 2008) is the VDPI 95% percentile of the rm Vale in August 2008.From each path we calculate the realized volatility of the returns.

Spatial dimension analysis
We can observe that the median of percentiles has a high dispersion across the stocks.For example, the median of the 1% percentile ranges from 1.17% (CGAS5) to 3.25% (ECOD3), while the median of the 99% percentile ranges from 2.19% (CGAS5) to 5.34% (MRVE3).As we can see, the highest median of the 1% percentile is greater than the lowest median of the 99% percentile.We can also note a high dierence between the medians of the extreme percentiles (1% and 99%) for all stocks which suggests that the volatility perceived by two investors can be quite dierent.
In order to investigate further the dispersion of the VDPI, we compute the ratio between some VDPI percentiles for each stock in each month.Then, we calculate the median of this ratio for each stock, i.e.This indicates that the volatility perceived by an investor can be 73% higher than another investor in the same month for this share.Furthermore, the median of the dierence between the lowest and highest volatility of the VDPI for all stocks is greater than 100%.On the other hand, note that the percentiles ratios do not vary signicantly across the stocks.This indicates that the VDPI dispersion is almost the same for all stocks.10This table presents descripitive statistics of the median of the percentiles ratios.vol 1 vol 2 is a series of the median (calculated over time) of the ratios between the VDPI 1 and 2 percentiles.

Time dimension analysis
After we discuss the spatial dimension properties of the VDPI, we proceed to investigate the time behavior of the percentiles vol(i; t).We can note that the perceived volatility can uctuate through the months.For example, for the 50% percentile, it varies from 1.68% in December 2006 to 8.29% in October 2008 (the epicenter of the subprime crisis).
For each stock and for each month, we also compute the ratio between some percentiles.Then we calculate the median of these ratios over the stock dimension, i.e., vol 1 vol 2 (t) = Med vol 1 (i;t) vol 2 (i;t) : i Table 2 presents the results.The minimum of the median of the ratio vol99 vol1 (t) is 1.49, conrming by another point of view that the volatility perceived by investors can vary signicantly.Interestingly, this minimum ratio occurs in October 2008 (the epicenter of the subprime crisis).However, since in time of crisis the volatility is high, the maximum dierence between vol99(t) and vol1(t) is also reached in this month as we will show in Section 4.4.This table presents the median of the ratios between VDPI percentiles for all months of the sample period.vol 1 vol 2 is a time series of the median (calculated across the stocks) of the ratios between the 1 and 2 percentiles.To study the location of these volatilities on the VDPI, we computed, in each month, for all stocks, how many observations of them are below the percentiles of the corresponding VDPI.Then we sum the number of observations for all months and, nally, divided this sum by the sample size (2,619).
For example, let's analyze the close-to-close volatility.We compute in July how many close-to-close volatilities of all stocks are below the 10% percentile of the corresponding VDPI (each stock in July 2006 has its own VDPI).We do the same procedure for all months.After this we compute the number of observations of the close-to-close volatility that are below the 10% percentile of the corresponding VDPIs and divide this value by 2,619.The result is 18.56%.Thus, the close-to-close volatility often lies in the left tail of the VDPI.Table 3   Let us assume that the median is a good measure to summarize the information contained in a probability distribution.Then, observing the 50% percentile, we note that the close-to-close volatility seems to underestimate the VDPI while the open-to-open volatility seems to overestimate it.To conrm these conclusions we implement two t-tests.
The rst seeks to verify if the open-to-open volatility is greater than the median.The alternative hypothesis for each stock i is: Mean vol open (i; t)=vol median (i; t) > 1: t The second one tests if the close-to-close volatility is less than the median.The alternative hypothesis for the stock i is: H 1 : Mean vol close (i; t)=vol median (i; t) < 1: t The results are remarkable.With a 5% signicance level, in 38 of 84 stocks we can arm that the mean of the open-to-open volatility is higher than the median volatility.With a 10% signicance level, in 46 of these stocks, we conclude the same.For the close-to-close volatility, in 19 stocks we can infer that this volatility is lower than the median volatility on the basis of a 5% signicance level.With a 10% signicance level, this number increases to 25.Therefore, we conclude that these volatilities cannot be good proxies of the day-to-day volatility perceived by an investor who trades during the trading hours and not just in the opening or closing of the market. 11

Discussion
Our results presented in the three previous subsections bring some issues about the modern theory of portfolio selection, risk management and option pricing.The uncertainty of the economy drives all the nancial decision process.Although not immune to criticism, the asset volatility is the most famous risk measure. 12However, we show that the usual practice of calculating asset volatility using opening or closing prices may not capture the uncertainty that an investor is exposed. 13rom the portfolio selection point of view, an investor who buys or sells assets based, for example, on the mean-variance frontier built using closing prices can hamper her optimum return/risk allocation since her evaluation of risk (volatility) is misleading.On the risk management aspect, our results provide sound arguments to a strand of the literature which contests the traditional methodologies to gauge market risk (see, for instance, Danielsson,  2002, Taleb, 2007 and Vicente and Araujo, 2010).When a player uses a daily volatility measure which does not consider intraday prices, such as the close-to-close volatility, she may assess inaccurately the uncertainty.Consequently, she may bias the risk incurred.Let's look at an actual example.Consider a risk manager who estimates a 1-Day 95% VaR by the Delta-Normal model.The portfolio includes only VALE5 shares and its market value is $1 (long position).If she uses the standard deviation of closing returns of January 2009 (3.56%), the estimated VaR turns out to be $0.0586.In contrast, if the risk manager uses the median of the VDPI (4.05%) the VaR will be $0.0666(13.76% higher).
The inability of the close-to-close or the open-to-open volatilities to assess the degree of uncertainty of an asset can also yield mispricing problems.For example, a trader can input a wrong volatility in an option pricing model.Suppose that an investor wish to price a call of VALE5 on March 1 st , 2007 with 6 months to maturity and exercise price R$66. 14Assume that the investor uses the Black & Scholes formula and estimate the volatility of VALE5 by the standard deviation of the returns of the last month (February 2007).The 6-month risk-free compound interest rate is 11.71% and the stock price is R$ 62.46.The call price provided by the Black & Scholes model is R$7.70,R$6.29, R$ 6.84, R$ 5,71 or R$8.88 if she uses the close-to-close volatility, the open-to-open volatility, the VDPI median, the VDPI 5% percentile or the VDPI 95% percentile, respectively. 11We did another t-test to verify if the open-to open volatility is greater than the close-to-close volatility.The alternative hypothesis for the stock i is: t With 5% signicance level, in 54 of 84 stocks the H 0 is rejected.With 10% signicance level, in 62 of 84 stocks the H 0 is rejected. 12Among others drawbacks, it is well-known that in a consumption-based model the riskiness of a payo depends on its covariance with the stochastic discount factor rather than its variance (see, for instance, Cochrane, 2005).Another problem with variance stems from the fact that it can fail to assess the downside risk (see Markowitz, 1991). 13For an amazing discussion of the important role of the volatility assessment, we refer to Poon and  Granger (2003). 14The Brazilian Real/US Dollar exchange rate was around 1.80 in 2009.
Policy makers are also interested in the evaluation of volatility.As point out by Poon  and Granger (2003), they rely on market estimates of volatility as a measure of the vulnerability of the economy.Thus, according to our results, policy makers who assess the volatility using close-to-close prices can underestimate the vulnerability while the one who uses open-to-open prices can superestimate it.
The impact of those issues are more pronounced in time of nancial crisis.Figure 3 shows the time series of the median (in the stock dimension) of the dierence between the VDPI 99% and 1% percentiles.The maximun distance of the VDPI 99% percentile to the VDPI 1% percentile occurs in the epicenter of the subprime crises (October 2008).The average of the median of the dierence between the VDPI 99% and 1% percentiles is 1.40% while in October 2008 it reaches the value of 3.36%.The poor performance of the close-to-close volatility in the assessement of daily risk points to the realized volatility models.However, the pratical use of these models is very costly. 15For some applications, such as risk regulation, clarity and parsimonious are fundamental.The VDPI can be very helpful in this case.For example, in order to taken into account the variability of the volatility perceived by investors, a regulator can adjust the capital requirement by a factor.This factor can be set as a mean value of the ratio between a high VDPI percentile (e.g.95%) and the close-to-close volatility.
Our ndings also imply that studies which use the close-to-close volatility as the dayto-day realized volatility may contain a bias, as this volatility tends to be on the left side of the VDPI.For example, works as Day and Lewis (1992), Lamoureux and Lastrapes (1993) and Christensen and Prabhala (1998), which investigate the relation between implied and realized volatility, may have dierent conclusions if the VDPI was taken into account.

Conclusion
The purpose of this paper to verify whether investors who trade daily but at dierent times have distinct perceptions about the risk of an asset.For this objective we propose a simple procedure to assess the volatility perceived by an investor who trades shares at random times of a day.The methodology consiste in calculating the realized daily volatility using prices that are obtained through a random drawing among all the negotiations occurred on each day.The set of volatilities estimated by this procedure is what we call the volatility distribution perceived by investors (VDPI).We nd that the dispersion of the VDPI can be very high.We show that for the same share the volatility perceived by an investor may be twice the volatility perceived by others in the same period.
It is common practice (for both market participants and academics) to calculate the daily volatility using opening or closing prices.Comparing these realized volatility with the VDPI allows us to assess whether they actually capture the volatility perceived by an investor.The results show that this practice can provide volatilities that do not represent the level of uncertainty perceived by an investor.The open-to-open and close-to-close volatilities are often far from the median of the VDPI.Therefore, an investor who measures the risk of her portfolio using these volatilities can often bias it.
The ndings of this study raise issues regarding the way risk management, portfolio allocation and pricing are traditionally performed in practice.In general, market professionals and regulators elect volatility calculated from the opening and closing prices as a metric for the dispersion of the daily returns of an asset.However, as shown in this study, these volatilities can be quite dierent from the volatility perceived by investors.This table presents the ratios of the VDPI percentiles presented in Table A. Each entry represents the median calculated over the sample period between the ratio of two VDPI percentiles.
, Hong and Wang (2000) among others present comparisons between the open-to-open and close-toclose volatilities.They show that the open-to-open returns are more volatile than close-toclose returns.Besides replicating this result our study extends these works since the opento-open and close-to-close volatilities are just samples of VDPI.The approach proposed here allows us to investigate several kinds of patterns of daily returns volatilities since VDPI encompasses all daily sampling of prices.
Now, let's see what can happen over the months of the sample (July 2006 until April 2009).Figure 2 illustrates the evolution of the open-to-open and close-to-close volatilities of Vale (VALE5).It also shows the 5% percentile, median and 95% percentile of Vale VDPI.We can observe that the open-to-open volatility is sometimes less than the 5% percentile (September 2006) and sometimes greater than the 95% percentile (March 2008,

Figure 1 :
Figure 1: Histogram of VALE5 VDPI This gure contains the histogram of Vale (VALE5) VDPI in January 2009.It also presents the open-to-open (labeled by the number 1) and the close-to-close (labeled by the number 2) volatilities.The histogram represents the frequency of daily volatilities calculated from 400,000 paths of prices of Vale in January 2009.Each path is obtained by a random process which draw an observed price in each day of the month.The bin width is 0.001.

Figure 2 :
Figure 2: VDPI percentiles of Vale This gure contains the time evolution of the open-to-open and close-to-close volatilities, the 5% percentile, the median, and the 95% percentile of VALE VDPI from July 2006 to April 2009.The VDPI for a month is obtained simulating 400,000 paths of daily random prices.From each path we calculate the realized volatility of the returns.

vol 1 vol 2
(i) = Med vol 1 (i;t) vol 2 (i;t) : t Table B in the Appendix presents the results.For example, the median of the ratio of the 99% percentile to 1% percentile of Natura (

4. 3
Open-to-open and close-to-close volatilities In this subsection we analyze the location of the open-to-open and close-to-close volatilities on the VDPI.We aim to answer questions such as: Is the open-to-open volatility almost close to VDPI median?Is the close-to-close volatility located in the left-side of the VDPI?More specically, our goal is to determine the frequency in which the open-to-open and the close-to-close volatilities are located in the 1%, 5%, 10%, 25%, 50%, 75%, 90%, 95% and 99% percentiles of the VDPI.As we have 77.03 stocks on average in each month and we have 34 months in the sample, there are 2,619 observations of the open-to-open and the close-to-close volatilities.

Figure 3 :
Figure 3: Dierence between the VDPI 99% and 1% percentiles This gure contains the time evolution of the median (in the stock dimension) of the dierence between the VDPI 99% and 1% percentiles.

Table A
in the Appendix presents the median of the VDPI percentiles of each stock calculated over the sample period.Each entry in Table A is given by vol(i) = Med vol(i; t); t where Med stands for the median operator.

Table 1
presents some descriptive statistics of Table B. It shows that the highest value of the ratio vol99 vol1 (i) is 1.86 (this value comes from AES Tietê (GETI4), see Table B in Appendix) and the lowest is 1.55 (i =Bradespar (BRAP4)).The highest median of the ratio

Table 2 :
Time evolution of the VDPI percentiles ratios.
presents all results of this procedure.Note that both the close-to-close and the open-to-open volatilities frequently are located in the tails of the VDPI.Moreover we can see that, at average, the open-to-open volatility is in the right side of the VDPI while the close-to-close volatility is in the left side.

Table 3 :
Location of the open-to-open and close-to-close volatilities.
This table presents the frequency in which the open-to-open and the close-to-close volatilities are located in each percentile of the VDPI.Each entry in the table represents the number of times that a specic volatility (open-to-open or close-to-close) lies on a specic VDPI percentile (1%, 5%, 10%, 25%, 50%, 75%, 90%, 95% and 99%).The analysis covers all the month and all stocks jointly.

Table A :
VDPI -Spatial dimensionThis table presents the medians of VDPI 1%, 5%, 10%, 25%, 50%, 75%, 90%, 95% and 99% percentiles of the VDPI for the 84 stocks in the sample.It also contains the median of the maximum and the minimun of the VDPI.The medians, the maximum and the minimun are calculated for each stock between July 2006 and April 2009.

Table B :
Ratios between percentiles of VDPI.