1. INTRODUCTION.
Almost anyone will agree that volatility is the lifeblood for trading. Volatility measurements are essential for risk a trader takes when trading. The phrase “the higher he rewards you want, the higher the risk you have to take” is illustrative for this general knowing. There has been a growing interest in describing and forecasting volatility the last ten to fifteen years.
So giving us the chances to trade, the understanding and use of volatility also gives many problems and even a definition is not general accepted. A nice definition is the next:
The degree to which the price of a market asset rises or falls within
a defined period.
The setup of the article is as follows: first we describe how volatility is measured and discuss some new and modern insights of the characteristics of volatility. In chapter three we discuss volatility data for the FDAX future. In four we give the results for autocorrelation of the data and try to fit the data into some known distribution and what it means. In chapter five we compare intraday volatility of the Fridays to all the data. In six some conclusions are drawn.
2. CHARACTERISTICS AND MEASURES OF VOLATILITY
Measurements
Volatility is measured alongside price changes. The most used measure of volatility is the standard deviation of price returns. The formula is well known and you can find it in every spreadsheet package:
σ = √(∑ (xi - µ)² ∕ (n – 1) ) (1)
i=0,N
Where :
xi = return for period i
µ = average return for the total period
Another measure is the absolute or relative difference of the Highest point and Lowest point of a given timeframe e.g., the difference between the High and Low of the day. The advantage of this measurement is fourfold.
- First it measures intra period changes of prices so less information is lost.
- Secondly and more importantly when trends in prices occur these won’t be reflected in High Low differences. This is important because we are only interested in price changes and not in their directions.
- Thirdly, high minus low data are only positive (> zero) so some standard distributions (as the Gamma fit) can be used to fit the data which would otherwise not be possible.
- Another reason to use intraday data for futures is the limited lifetime for these contracts and the difference in value between two adjacent contracts. Because H –L data only measures the intraday movement and not the difference between two days we don’t have to concern about the connection these contracts.
So we have:
σ = H - L (2).
H = Highest point of the day
L = Lowest point of the day.
Another measure which has become very popular lately amongst derivatives traders is the so called implied volatility but is not treated here further.
Characteristics of Volatility
Some characteristics of volatility are:
Extreme values
Autocorrelation
Heteroskedasticity.
Extreme events
Compared to a normal distribution the distribution of observed price movements contains too many observations in the tails. What is meant by this can be seen in fig.1
where a fat tail distribution (thick line) is sketched over the well known bell shaped normal distribution: to the right and to left more observations are made.
As an example see fig. 2 where you can see a normal distribution fitted onto the S&P 500.
Figure 2. Fat tailed (thick line) S&P 500 rate of return distribution compared to a normal distribution (dashed line). Copyright 1998 by Shiki Levy,
Fat tails1 are considered undesirable because of the additional riks they introduce. The study of fat tails is subject of what is now known as Extreme Value Theory (EVT). EVT became very important after the boom and the following crahes of the eighties and the nineties on the stockmarkets.
Very short outlined, the idea behind EVT is as follows. We don’t know the precise distribution of volatilty and price returns on the financial markets. Extreme losses are a part of the total datadistribution and are considered as a distribution within this total distribution (or parent distribution). It can be shown that such an extreme loss distribution within the parent takes a specific form which can be modelled exactly2.
The point here is is that when we use our total observations of volatility, data lying in the center (around the mean or median) are more weighed in a distribution curve than data in the tails. So to understand and predict extreme events, and these are just occuring in the tails, we have to make estimations how such a tail looks like without giving data in the center too much weight.
Autocorrelation
On the markets periods of large price movements alternate with periods of relative tranquility. But there is more about this. Volatility (and price returns) tend to cluster together over time. This clustering of movements, especially in the case of extreme events, is due to what is known as autocorrelation and heteroskedasticity .
Autocorrelation, sometimes called serial dependency, can be described in the same way as correlation between two variables. You know that price movements of stocks can be highly correlated in a sense that when stock A goes up, stock B also does3.
Autocorrelation measures the correlation for the same variable, e.g. the volatility, with itself. The autocorrelation function in a time series describes the correlation between the process at different time points in time and can be plotted in an autocorrelogram. See fig. 3
Figure 3. An example of an autocorrelogram.
An autocorrelation plot can be used to see if data are random or in contrary correlated in time. If random, such autocorrelations should be near zero or between 95% confidence level drawn in the plot as two horizontal lines. The plot shows that the volatility in time is not random, but has a high degree of autocorrelation between adjacent and near-adjacent observations.
Such a relation may be expected between nearest observations but is the extension in time which is of interest. Therefore as an extension to autocorrelation, partial autocorrelograms are used in which correlations are computed between lags k and k+1 but not within a lag4.
Autocorrelation functions are used in forecasting time series analysis such as AR (Auto Regression) and ARIMA (Auto Regressive Induced Moving Average) models.
Heteroskedasticity
Volatility is not only autocorrelated in time, also the variance is varying in time. This is called heteroskedasticity5 in contrast to homoskedasticity, where the error term has a constant value.
3. VOLATILITY DATA OF THE FDAX
Daily High Low data of the FDAX have been gathered over three years starting from June 2003 till June 2006 and computed with formula (2). The results are analyzed for their statistics, distribution and autocorrelation. These data are further denoted as H-L Total. The descriptive statistics can be found in table I.
Table I. Descriptive analysis for the H - L Total data.
The histogram can be seen in fig. 4 and the Q–Q plot in fig 5. When data are normally distributed the Q–Q plot would perfectly fit on the line with angle of 450 . As can be seen the H – L data don’t fit on this line so clearly the H – L Total data are distributed non normally. This is especially the case on the right and left side of the plot indicating a departure from normality in the tails of the data. Later we will see how we can fit these data in another than the normal distribution.
Figure 4. Histogram of the H – L Total data of the FDAX
The H-L data are divided in two sub categories: one containing all the High minus Low data of the Fridays (Friday) and one containing all the other days (H-L Min Friday). Their description can be found in respectively table II and table III. Q-Q plots of these data sets (fig 6.) also show a clear departure from normality.
Figure 6. Q–Q plots of the H – L Min Friday (left) and the Friday data (right)
Some statistical tests have been performed to see if there are differences between these sets of data.
Autocorrelation
A time series chart of the H-L Total data is given in fig. 7. The accompanied chart of the DAX index itself can be seen in fig. 8. Notice how a strong upward trend of the DAX can be seen during this period, a trend which is not seen on the H-L Total chart. This absence of a trend is important because we are only interested in intraday volatility not a as a result of existing underlying direction. When there is an upward trend in the prices you will expect that the distribution of volatility measured by formula (1) will show more observations to right than to the left. A downward trend will show itself in more observations to the left.
Spikes in the H-L Total chart represent days of extreme intraday movements (above 100 to 200 FDAX points). An autocorrelogram of the data is given in fig. 3. The plot suggests a strong autocorrelation over time. A plot of the partial autocorrelation in fig. 9 confirms this : during the first eight lags a fairly strong autocorrelation exists for high minus low data for the analyzed period.
This can be used as input for AR and ARIMA models because it is known that the order (p) of the AR part of the model will be somewhere for the k-th lag where the partial autocorrelation coefficient becomes practically zero. So for our data suggesting an AR parameter p of 8 or 9. 6
While a Q-Q plot of all three data distributions clearly shows a non normality (fig. 5 and 6) the question arises if the data can be fitted in some other known distributions. Fittings have been performed with XLSTAT software from Addinsoft (website: www.xlstat.com).
The procedure for these fittings is as follows. First the data are statistically described by so called moments of the data. These moments include the mean, the standard deviation, skewness which is a measure of lack of symmetry around central part of distribution and kurtosis. Kurtosis is a measure of whether the data are peaked or flat relative to a normal distribution. Formulas exist for skewness and kurtosis7.
When these moments are known, a best fit will be chosen for a known distribution curve for which we know the theoretical moments. When this is done hypothesis tests are performed. Such are used to test whether some kind of null hypothesis is true against some alternative hypothesis.
An example may clarify this. When fitting the H – L Total data into a Gamma distribution the null hypothesis H0 could be: the data follow a Gamma distribution and the alternative hypothesis Ha: the data don’t follow a Gamma distribution. A confidence interval is normally set at 95%.
Let’s return to the intraday movements of the FDAX. The intraday H – L data are plotted in so called probability density functions (PDF) in fig. 10a to c. As you can see all three data sets fit good to so called Fisher-Tippett and General Extreme Value (GEV) distributions8, both clearly showing a departure from a normal distribution.
Both distributions are characterized by µ, which is a measure for the location of the data, comparable to the mean and ß, a measure for the spread of the data.
Very interesting on all three data sets are the skewness to the right: you can see it as right tails of the distributions. This means the existence of more extreme observations of high minus low intraday data on the FDAX during 2003 till 2006.
In table IV a comparison of the fitted parameters for three data sets is given. Also the risk to reject the null hypothesis H0 while it is true is given, H0 being the hypothesis if the data is following the relevant distribution.
While in all cases the null hypothesis is accepted and the data do follow the relevant distribution the risk to reject the null hypothesis is fairly high but it is for this moment the best we can get.
A natural next step would be to compute so called Cumulative Density functions (CDF). The cumulative distribution function is the probability that the variable takes a value less than or equal to H – L, where H - L on the horizontal axis. Since the vertical axis is a probability, it must fall between zero and one. It increases from zero to one as we go from left to right on the horizontal axis.
There is a direct relation between a pdf and a cdf for a distribution but it asks some math knowledge to understand this9. In fig.11 you can see the cumulative density functions for the H – L Total data.
Cumulative density functions are used to measure the risk a fund manager, or you, may expect on his portfolio. Look at fig. 11 for the FDAX. Based on these probability distributions you may expect H – L data equal to 100 points or less to occur in 90% (90 percentile) of the days (vertical axis). Said otherwise, in 1 of 10 days an intraday movement exceeding 100 points may be expected. This is a higher risk than when a normal distribution of movements would have been used.
The widely used Value at Risk (VaR) analysis is based on this: generally made under the assumption of normality of returns (volatility) the 95-th percentile of the distribution provides an estimate how much money can be lost. But from the above it is clear that the risks involved may be greater when non normality or returns or volatility is assumed.
5. FRIDAY DATA COMPARED TO THE TOTAL H – L DATA.
As is known that on Friday often important
We already saw in table IV a possible difference in the H – L data for the Fridays compared to the total data when we discussed the fitted parameters for these distributions.
Various tests exist for comparing sets of data. These tests can be divided into two groups: parametric and non parametric. The parametric are straightforward and relatively easily performed. They compare the mean and or the variance10 of the two data sets and based under the assumption that if a difference between data sets exists, these differences are normally distributed. Also sometimes normality of the underlying distribution is required.
The best known parametric tests are the Fisher’s F-test which tests for a possible difference between variances of two sample distributions and a T-test. In Fishers’ F-test the tested variable is the coefficient of the two variances. The T-test is a test for measuring difference of the means.
For non parametric tests the assumption of normality is not needed but they are typically less powerful and less flexible in terms of types of conclusions that they can provide. They test whether data are consistent with a specific distribution.
In many cases we can still use the normal distribution-based test if we only make sure that the size of our samples is large enough. The latter option is based on an extremely important principle which is largely responsible for the popularity of tests that are based on the normal function. Namely, as the sample size increases, the shape of the sampling distribution approaches a normal shape, even if the distribution of the variable in question is not normal.
Parametric tests
Fisher’s F-tests T-tests have been performed for the Friday data against the H – L minus Friday data and the Friday data against the H – L Total data.
The results for the T–tests are shown in table 5 and 7. Clearly there is no significant difference between the means, of the three distributions. So no difference in the means of the H – L of Friday and the rest of the week can be distinguished.
This is not the case with the variance, the spread around the mean. This is shown in table 6 and 8. For both data sets, the H-L Total and the H-L minus Friday data do significant differ in their variances to the Friday data. Intraday movements on Friday are significant smaller than those on other days.
6. CONCLUSIONS.
The conclusions which can be drawn from above are as follows.
Also statistical methods were used with software which is also now available for small investors and traders. A wide range of software can now be obtained and used.
Despite a very strong upward trend of the DAX Index the last years the intraday High minus Low did not show a clear trend. This is in contrast to what is sometimes heard or said between intraday traders. No clear trend could be distinguished for the intraday volatility giving still enough possibilities for this type of trading on the FDAX. There is no reason to assume that this will different on other popular future markets.
We concentrated on High minus Low data of the FDAX as a measure for volatility and tried to fit the data into known distributions. Importantly the intraday movements do not follow a normal distribution but exhibit a fat right tail and skewness to the right.
The non normality of these intraday movements has consequences for risk analysis as done in VaR.
The observation of bigger intraday movements (spikes see fig. 3) than could be expected when normally distributed has consequences for risk management. Fitting data in a Fisher-Tippett or GEV distribution and looking at the relevant cumulative distribution gives a much better insight in the possible intraday movements of the FDAX and thus in the risk you may expect, but also in the opportunities this gives.
Finally we found a small but significant difference in the intraday volatility of the FDAX on Fridays and the rest of the week. The difference not being in the kind of distribution or the mean, but consists in a difference in the spread. The intraday movements of the FDAX on Fridays seem to be smaller than in the rest of the week.
NOTES
1. Fat tails must be discerned from long tails known from Pareto distributions. Long tails have been very useful in understanding of various micro-economical phenomena such as describing the existence of niche markets. An example: the reason why an internet book seller as Amazone.com can make money by keeping in stock many low volume titles while a local bookstore only has the big volume sellers. This is called the "More of Many" principle.
2. More precisely said: in the limiting case of N observations goes to ∞, extreme value distributions within a parent distribution are represented by only three limiting possibilities. This is called the Fisher-Tippett theorem. The three possibilities are the Gumbel, the Frechet and the Weibull distributions and these are known exactly.
3. Correlation between the Mini Dow Jones future (YM, CBOT) and the FDAX future sampled on 15 minutes can be shown to be around 90% the last two years. Data C. Giuntoni
4. Even when prices of financial assets are considered as purely random walks some autocorrelation may be expected between nearest observations due to the nature of the calculations.
In physics symmetry relations between nearest molecules are known as short term order, only effecting a few hundred nearest molecules in contrast to long term order which involves relations between millions of molecules throughout the material.
5. In the
6. AR en ARIMA analyses have been performed on the H-L Total data indeed suggesting a fairly good fit for p (AR) = 8, q = 3 and d (MA) = 1, without seasonal influences. Because we are strictly not interested in forecasting the volatility these analyses have been omitted in this article.
7. Formulas for skewness and kurtosis:
Where Ỹ is the mean and s is the standard deviation.
8. Fisher-Tippett and General Extreme Value distributions are closely related: sometimes the Fisher-Tippett is formulated as a special case of the GEV.
9. When f(t) = P(X) being the probability function than the cumulative function F(x) is given by ∫x f(t)dt , the area under the curve of the pdf.
10. Variance being the cubic of the standard deviation