Exploratory Graphics for Financial Time Series
Volatility
A. J. Lawrance
Department of Statistics, University of Warwick, Coventry, UK
Summary. This paper develops a framework for volatility graphics in financial
time series analysis which allows exploration of the time progression of volatility
and the dependence of volatility on past behaviour. It is particularly suitable for
identifying volatility structure to be incorporated in specific volatility models.
Plotting techniques are identified on the basis of a general time series volatility
model, and are illustrated on the FTSE100 financial time series. They are
statistically validated by bootstrapping and application to simulated volatile and
non-volatile series, generated by both conditionally heteroscedastic and
stochastic volatility models. An important point is that volatility can only be
properly visualized and analyzed for linearly uncorrelated or decorrelated series.
Keywords: Statistical graphics, Financial time series, Volatility, Smoothing, Bootstrap,
Heteroscedastic models, Stochastic volatility models
1. Introduction
Volatility has been an important topic in financial time series analysis and modelling
since the initial work of Engle (1982). Much effort has since been expended on
developing a variety of volatility models, with acronyms such as ARCH, GARCH, SV,
and their many variants. Two volumes which cover statistical aspects more than most
are Fan and Yao (2003) and Ruppert (2004), with the handbook, Anderson, Davies et
al. (2009) giving a comprehensive overview of financial time series. However, very
little appears to have been done in producing statistical graphics which quickly
identify volatility in time series data and guide the choice of model structure. This is
in spite of one motivating idea for Engle’s Garch model being to obtain volatility
bands which did not depend on moving averages of squared returns, so-called
historical volatility, Engle (2010). There are two general difficulties, firstly, volatility
is concerned with variance, particularly conditional variance, and this requires a
sample of values at a particular time, which is not usually available in a time series
context. Secondly, volatility can be visually confused with the effects of negative
autocorrelations and distorted by positive autocorrelations. The proposed graphics are
based on a general volatility model which suggests individual volatilities in the form
of absolutes; these can be smoothed, either temporally for volatility time progression
or in scatter plots for volatility dependence.
A.J. Lawrance, Department of Statistics, University of Warwick, Coventry, CV4 7LE, UK.
E-mail: A.J.Lawrance@warwick.ac.uk
2 A. J. Lawrance
2. A General Volatility Model
Denote a time series by the sequence of dependent random variables X 1 , X 2 ,  and
let X t be the current and previous values X t , X t 1 ,  . The proposed graphical
methods are motivated by the general structure of the autoregressive multiplicative
volatility model
X t   ( X t 1 )   ( X t 1 ) t
(1)
where  ( X
 t 1 ) denotes a stochastically varying conditional mean E ( X t | X t 1 )
giving the level of the series at time t ,  ( X t 1 ) similarly denotes the conditional
standard deviation sdev( X t | X
 t 1 ) at time t , being the volatility function, and
causing the variance of the innovations to change according to previous values of the
series. The basis of the innovations is the series 1 ,  2 ,... of identically and
independently distributed standardized random variables, not necessarily Gaussian,
such that  t 1 ,  t  2 ,.... are independent of X t .
Some aspects of the model (1) can be noted. It is predictive in the sense that the mean
and variance of the distribution of X t | X t 1 are available at time t  1. Models
encompassed by (1) include ARCH and GARCH models and many of their variants;
they have often been used for modelling financial return series, and also for currency
exchange and interest rate series.
An insight from the model (1) is that to study volatility, the levelled time series
X t   ( X t 1 ) must be used, from its equivalence to the innovations  ( X t 1 ) t . A
particular and easily verified property of these levelled values is that they are not
autocorrelated uncorrelated,
cov  X t   ( X t 1 ), X t  k   ( X t  k 1 )  0, k  1, 2,... .
(2)
This motivates the need for volatility to be analyzed under the condition of zero
autocorrelations, otherwise local behaviour due to autocorrelation may be taken as
volatility.
Of particular interest here is the non-parametric use of (1) as the basis for graphical
exploration of volatility in financial return series. Typically, such series are stationary,
with only local changes in level and with small and decreasing early-lag linear
autocorrelations. This suggests that  ( X t 1 ) can be taken as a linear function of its
first one or two previous returns with very small coefficients and which can easily be
estimated by linear autoregression using standard time series methodology. Then
changes in local level and the implied autocorrelations can be removed by subtraction,
as illustrated by (2). Modelling of the level function is thus a minimal issue for
financial return series and is only briefly mentioned.
Following unpublished work by Yau and Kohn, 2001, Ruppert (2004) takes a nonparametric approach with a model of the form (1), as does Franke, Neumann et al.
(2004), in a more theoretically based study giving results for kernel estimation and
bootstrapping.
For interest rates, and possibly for other types of financial series, volatility may be
riding on non-stationary changing level. In these situations, it will be necessary to
eliminate changing level, such as by differencing, Chan, Karoyli et al. (1992), Ruppert
(2004).
Volatility Graphics
3
3. Volatility Graphics for Return Series
Although volatility is conditional variance, in the usual time series analysis context
there is only a single value available at each time. One way round this problem is to
calculate variance moving along the series, although then there are questions of
determining extent and weights. Another way is the Riskmetrics approach, Kim and
Mina (2000), of calculating an exponentially moving average of squares. These ways
are undoubtedly useful, but both can be seen as pragmatic and to be missing any
structural basis and to ignore autocorrelations and their effects.
A more structural and intuitive way round these issues is to non-parametrically
explore the volatility function  t 1   ( X t 1 ) on the basis of the model (1), and the
proposal here is to construct individual volatilities defined as
ˆ t  X t  ˆt 1 )  , t  2,3,..., n  1
(3)
for a series of length n, where ˆ t 1  ˆ ( X
 t 1 ) is the estimated level function in model
  is a scaling term.
(1) and   E  t
This usually also needs estimating but there is
a sometimes useful Gaussian reference value of (2  )1 2 .
In the proposed procedure for estimating  , it is first taken as unity and then (3)

becomes unscaled individual volatilities { t } ; these are smoothed to give unscaled
smoothed individual volatilities  t . Use of these in model (1) leads to unscaled
residuals

 t   X t  ˆ t 1   t 1 , t  2,3,..., n  1
(4)
and then to standardized residuals {ˆt } .
 
The estimate ̂ suggested for  is the empirical version of E  t
which follows
from (4) by averaging the {| ˆt |} . This process is equivalent to determining  as
12
 n 1
2
ˆ  n   X t  ˆ t 1  t 1  n 1  ( X t  ˆt 1 ) /  t 1  .
(5)
t 2
 t 2


The unscaled individual volatilities { t } are divided by ̂ to give the individual
volatilities {ˆ t } envisioned by (3). Finally, dividing the unscaled smoothed

individual volatilities { t } by ̂ , gives the smoothed individual volatilities { t } , an
1
n 1
estimate of the volatility function.
The plot of the individual volatilities {ˆ t } against time with a superimposed smooth

version { t } constitute a volatility progression plot. The scaling means the plot may
be compared to the time series plot of volatility functions in volatile time series

models, such as the ARCH 1 model in Section 6. The smoothed volatilities { t } can
also be used to produce a non-parametric version of Engle’s Arch volatility bands plot,
Engle (2010). The return series { X t } is plotted with volatility bands formed as

{2 t } .
Both the previous plots involve smoothing and any of the main approaches may be
used, such as locally weighted least squares, nearest neighbours, splines, wavelets or
kernels. However, it should be noted that there are elements of subjectivity with each,
and in the degree of smoothing employed. The chosen method in this paper is the well
known lowess algorithm of Cleveland and Devlin (1988) which is widely
implemented in packages. Most smoothing procedures implicitly assume an additive
4 A. J. Lawrance
two-sided error structure, so when on the contrary scatters are multiplicative and of
positive values, an inverse-log smooth approach is taken in which the log of the
variable is smoothed and then the smooth is anti-logged.
The next plot is aimed at discerning the main features of volatility as a function of
previous values of X, usually X t 1 . The individual volatilities (3) are plotted against
X t 1 , yielding the volatility dependence plot
X
t
 ˆ ( X t 1 )  , X t 1  , t  1, 2,... .
(6)
The scatter of points is always very dispersed but with a concentration against its
lower zero boundary; considerable smoothing is needed to reveal the underlying
volatility function  ( x) . By the model (1), the scatter is multiplicative according to
|  t | error, and so the cloud of points cannot be central to its underlying curve. Thus,
as previously noted, it makes sense to use the inverse-log approach and first construct
the log volatility dependence plot
log X
t
 ˆ ( X t 1 )  log( ), X t 1  , t  1, 2,...
(7)
which should smooth with additive error to log{ ( x)} . This log smooth is then antilogged to give the preferred inverse log smooth which is overlaid on the original
scatter as the best depiction of  ( x).
Although the plots from (6) and (7) are based on the general autoregressive model (1),
they can detect volatility produced by other models, such as the stochastic volatility
model of Taylor (1982), to be discussed in Section 8.
The intuitive nonparametric approach here to volatility estimation has relied on the
introduction and lowess smoothing of individual volatilities. This makes it more
explicit and exploratory in nature than most other methods already mentioned, such as
the nearest neighbour and bootstrap approach of Franke, Neumann and Stockis (2004)
which provides the estimated volatility function. Further, the scatter of individual
volatilities is helpful in appreciating the basis of the estimated volatility curve and the
influence of outlying points. The choice of smoother is a matter of taste, rather than
integral to the approach.
There are several other computational approaches to volatility, as yet without
exploratory aspects or graphical interfaces. There is the implied volatility approach,
Beckers (1981), which invokes option pricing models to invert observed derivative
prices into the required volatility. With high frequency intra-day return data, there is
the realized volatility approach. A survey of this field may be found in Anderson and
Benzoni (2009). The Riskmetrics historical volatility approach uses individual
squares and exponential smoothing, although without modelling, scaling and
decorrelation.
4. Volatility Plotting and Refinements Using FTSE100 Data
The rest of the paper is concerned with the illustration, refinement and justification of
the suggested plots for exploratory volatility analysis, mainly the volatility
dependence plot, using FTSE100 adjusted closing price data from 4th January 2005 to
10th February 2011, sourced from Yahoo (2011), and various simulated data sets.
A plot of the FTSE100 index series I t , t  1, 2,...,1542 is given by the upper trace in
Figure 1, and the lower trace is the corresponding return series ( I t  I t 1 ) I t 1 . This is
Volatility Graphics
5
8000
FTSE Values
7000
6000
5000
4000
10%
Returns
5%
0%
-5%
-10%
01/01/2005
01/01/2006 01/01/2007
01/01/2008 01/01/2009
Daily Date
01/01/2010 01/01/2011
Figure 1. FTSE100 adjusted closing index values and returns from 4th January 2005 to
10th February 2011.
14
12
Volatility
10
8
6
4
2
0
01/01/2005
01/01/2006
01/01/2007
01/01/2008
Daily Date
01/01/2009
01/01/2010
01/01/2011
Figure 2. Volatility progression and its inverse-log lowess smooth for the FTSE100 series.
10
Return
5
0
-5
-10
01/01/2005
01/01/2006
01/01/2007
01/01/2008
Daily Date
01/01/2009
01/01/2010
01/01/2011
Figure 3. The FTSE100 return series with its inverse-log smooth Engle volatility bands.
a levelling of the index series and shows the percentage changes of the index, and that
strong oscillations in returns often occur after sharp drops in the index. They are
virtually uncorrelated linearly but with considerable quadratic autocorrelation. The
returns have a symmetric distribution, range over (8.37, 9.83)% , have a mean near
zero of 0.02, a standard deviation of 1.36 and a high kurtosis of 8.59. These results
correspond well to the so-called stylized facts of equity returns, Granger et al (2000).
6 A. J. Lawrance
10
Volatility
8
6
4
2
1
0
-6
-4
-2
0
Previous Return
2
4
6
Figure 4. Volatility dependence plot and its lowess smooth for the FTSE100 return series.
3
Log Volatility
2
1
0
-1
-2
-3
-6
-4
-2
0
Previous Return
2
4
5
6
Figure 5. Log Volatility dependence plot and its lowess smooth for the FTSE100 return series.
10
Volatility
8
6
4
2
1
0
-6
-4
-2
0
Previous Return
2
4
6
Figure 6. Volatility dependence plot and its inverse-log lowess smooth for the FTSE100 return series.
The volatility progression plot is illustrated in Figure 2; there is a rather slow gradual
increase in volatility up to mid 2008, then a high volatility period finishing with the
proposed split point of 3rd March 2009, and continuation at a lower level but still
higher than at the beginning of the period. As noted earlier, the smoothing of the
individual volatilities is by the inverse-log lowess method. The smoothing should be
regarded as descriptive, rather than predictive, since it is two-sided and not backwards;
the choice of lowess smoothing parameter is a matter of judgement. For the volatility
progressions in Figures 2 and 3 it was 0.03, meaning that 3% of the nearest time series
values were used at each time point.
Volatility Graphics
7
12
10
Volatility
8
6
4
2
1
0
-6
-4
-2
0
2
Previous Model-Residual
4
6
Figure 7. Volatility dependence plot and its inverse-log lowess smooth for residuals from FTSE100
return series.
10
Volatility
8
6
4
2
1
0
-6
-5
-4
-3
-2
-1
0
1
Previous Return
2
3
4
5
6
Figure 8. Volatility dependence plot and its inverse-log lowess smooth for the FTSE100 return series
from 5th March 2009 to 10th February 2011.
Figure 3 uses the smoothed volatilities to obtain an empirical version of Engle’s
volatility bands plot for the original returns; it gives a visually effective assessment of
the changing conditional variance of the volatile returns.
Volatility dependence on the previous return for the FTSE100 series is now explored
through the volatility dependence plot (6) and is seen in Figure 4 in its basic form. The
non-constant lowess smooth estimate of the volatility function is evidence of volatility
in the series, and moreover that it is not symmetric. Prominent features of the plot are
its compression of points with lowest volatility and the lowess smooth not being
central to the cloud of points, the effect of multiplicative error. However, the main
purpose of the plot is to identify that volatility is present; without volatility the lowess
smooth should be flat, and this is clearly not the case. Volatility increases more so
after negative returns than positive ones; thus a stylistic leverage effect of equity
return time series, Granger, Ding et al. (2000), is graphically evident.
In volatility dependence plots, such as Figure 4, the lowess parameter was set at 0.5,
and so chosen as the minimum needed to achieve a reasonably smooth curve. In
addition, to guard against edge-distortions of smoothers, extreme previous returns
well-separated from the main scatter, generally less than five points, were not
smoothed; this resulted in the range (6, 6) being used in the FTSE100 dependency
plots. Also, several of the largest volatilities, evident in Figure 2, have been omitted
from the plot in order to achieve better clarity. Such judgments are regarded as a
necessary part of exploratory analysis.
8 A. J. Lawrance
Figure 5 repeats the plot in Figure 4 using volatility on the log scale and avoids
compression of the scatter, as previously asserted.
The inverse of the log lowess smooth from Figure 5 is now used to replace the lowess
smooth in Figure 4. This results in Figure 6, the preferred version of the volatility
dependence plot.
The practical message from Figure 6 is a refinement of that from Figure 3, namely that
there is evidence for asymmetric volatility of the FTSE100 returns, with a tendency of
high volatility to follow low return when considered over the over the period 4th
January 2005 to 10th February 2011. Although this plot is fairly convincing, it is
natural to take the residuals from the model (1), as defined at (4), and examine them
for remaining volatility. In the case of the FTSE100 series, Figure 7 indicates clearly
that this is not the case.
Further similar analysis to that given by Figures 4-7 was under taken for prior- and
post- 4th March 2009, a proxy for the supposed end of recession. Essentially the same
results were obtained for the data up to the 4th March. From the 5th March onwards
very different results were obtained, as shown in the volatility dependence plot in
Figure 8; first, the range of returns is much smaller, within 5% and secondly, there
is now hardly any evidence of volatility, a previously very unusual situation in a
period of 488 trading days. This did, however, accord with current market perceptions
at the time.
5. Bootstrap Validation of Volatility Dependence Plots
There could be concern for the validity of volatility dependence plots in respect of
underlying statistical variation associated with them. Thus, there is a further need for
an uncertainty assessment of the lowess smoothed volatility function. While this
would be a complicated or intractable task to achieve formally, an exploratory
nonparametric bootstrap approach is a practical response. In this approach, the

FTSE100 smoothed volatilities { t } were used as the estimated volatility function in
model (1), justifiably assuming its conditional mean term to be zero, and the model
residuals were obtained. Then, as used by Atkinson (1981), Atkinson (1987) with
half-Normal plots and by Bowman and Azzalini (1977) with kernel smoothing, but
applicable more generally, the model was bootstrapped 19 times by using 19
resampled with-replacement versions of the FTSE100 residual series but keeping its
estimated volatility function; this leads to 19 series which are effectively independent
realizations of the same underlying process. For each of the realizations, a smoothed
volatility function was obtained, exactly as for the FTSE100 one. These functions are
effectively independent and identically distributed.
The suggested graphical
presentation is illustrated by Figure 9 which gives the estimated FTSE100 volatility
function together with its 19 bootstrap resampled versions. This sheaf of 19
replications is itself graphically informative, showing the changing level of
uncertainty associated with the estimated FTSE100 volatility function, progressively
increasing with distance from the centre but with higher uncertainty associated with
lower previous return values than upper ones. The 19 curves may also be thought of
as giving variability bands, Bowman and Azzalini (1977), in the form of a succession
of approximate point-wise confidence intervals. In each there is a 1 in 20 chance that
the observed volatility at a particular previous return value is higher than the
maximum of its simulated volatilities, and similarly for the lower tail probability.
Thus, each point of the band has a confidence interval coverage probability for the
FTSE100 volatility function of approximately 90%, but the overall coverage
Volatility Graphics
9
10
Volatility
8
6
4
2
1
0
-6
-4
-2
0
Previous Return
2
4
6
Figure 9. Volatility dependence plot and its inverse-log lowess smooth (dark line) for the FTSE100
return series and 19 bootstrap versions.
10
Volatility
8
6
4
2
1
0
-6
-4
-2
0
Previous Return
2
4
6
Figure 10. Volatility dependence plot and its inverse-log lowess smooth (dark line) for the FTSE100
return series and a sheaf of 19 volatility curves of the resampled model residuals.
probability is not easy to assess in a simple way. However, the main importance of
the graphics is to provide guidance as to what are extreme fluctuations.
Although the previous volatility curve estimate and its bootstrap uncertainty analysis
give quite compelling evidence that the FTSE100 series exhibits asymmetric
volatility, the same resampling approach can also be used to provide a graphicallybased informal test of the hypothesis that there is no volatility. Instead of creating 19
bootstrap resamplings using the estimated FTSE100 volatility function, a constant
volatility function is used, corresponding to the null hypothesis of no volatility,
together, with 19 resampled sets of FTSE100 residuals. A sheaf of 19 resampled
volatility functions is thus obtained, exactly as before. Figure 10 shows these together
with the originally estimated FTSE100 volatility function. It is quite clear that the null
hypothesis is untenable and additionally that the evidence comes mainly from the low
returns. The informal significance level of the point-wise tests follows as 10%
according to the previous arguments. Once again, graphical evidence from the sheaf
of 19 resamplings provides a good indication as to what can be considered extreme.
10 A. J. Lawrance
12
10
Volatility
8
6
4
2
1
0
-6
-4
-2
0
Previous Value
2
4
6
Figure 11. Volatility dependence plot and its inverse-log lowess smooth for 1542 simulated
AR1(-0.75) data, with analysis not including decorrelation.
12
10
Volatility
8
6
4
2
1
0
-6
-4
-2
0
Previous Value
2
4
6
Figure 12. Volatility dependence plot and its inverse-log lowess smooth for 1542 simulated
AR1(-0.75) data, with analysis including decorrelation.
6. Volatility Dependence Plots for Model-Generated Series
The previous plots were all concerned with the FTSE100 series and included empirical
justification of their validity using residuals and nonparametric bootstrap resampling.
This work is now broadened and extended to series generated by two standard time
series models, one linear and non-volatile and the other non-linear and volatile.
Considered first is the correlated but non-volatile first-order linear Gaussian
autoregressive model with correlation parameter of  0.75 and unit innovation
variance, the AR1( 0.75) model. Incidentally, this allows the importance of
decorrelation prior to volatility analysis, as stressed in Section 2, to be illustrated.
Figure 11 incorrectly indicates volatility since no decorrelation of the series was
applied. The series looked as if it might be volatile, due to the negative dependency
inducing many sign changes. In Figure 12 it has been decorrelated and the plot
correctly points to no volatility. For an AR1(0.75) series, not illustrated, the effects of
positive linear dependency are seen although volatility was less obvious in the series.
However, again the volatility dependence plots spuriously detect volatility when no
decorrelation has been applied and correctly suggest no volatility when applied
correctly with decorrelation. Figures 11 and 12 give further practical justification that
the volatility graphics can be reliably informative.
Volatility Graphics
11
12
10
Volatility
8
6
4
2
1
0
-6
-4
-2
0
Previous Value
2
4
6
Figure 13. Volatility dependence plot for 20 identically distributed ARCH1(0.1, 0.8) series
of 1532 values with their inverse-log lowess smooth volatility
functions and the true volatility function (dark line).
The second standard model to be considered is the non-correlated but volatile
ARCH 1( 0 , 1 ) model, introduced by Engle (1982), and specified here by the
equation
X t   t  t ,  t2   0  1 X t21
(8)
where  t is from an independent standardized Gaussian series of innovations. The
model was simulated with parameters ( 0  0.1, 1  0.8) to give 20 independent
sequences each with 1532 values. Figure 13 gives the estimated volatility functions of
the 20 series. Also shown is the true volatility function and a set of individual
volatilities. It can be seen that the volatility dependence graphics work satisfactorily
by capturing the true volatility function although there are some sampling extremes.
This type of display could be used in practice with estimated ARCH 1( 0 , 1 ) model
parameters to judge whether a historical volatility function was in broad agreement
with that of an ARCH 1 model, after allowance for uncertainty in the model’s volatility
function.
7. Asymmetric Volatility Models and the FTSE100 Series
In this section two popular but contrasting asymmetric volatility versions of the
standard GARCH model are considered for the FTSE100 series - the threshold
version TGARCH , Nelson(1991), and the exponential version EGARCH , Glosten,
Jaganathan et al. (1993). A definitive analysis is not attempted, rather an exploratory
graphical comparison between the often used and simplest cases TGARCH (1,1) and
EGARCH (1,1) is made. Such models are considered flexible and parsimonious in
applications.
The TGARCH (1,1) model is taken in the form
X t   t  t ,  t2   0  1 X t21   I ( X t 1 ) X t21  1 t21
(9)
where  is the asymmetry parameter and I ( X t 1 ) is a binary (0,1) indicator variable
taking value 1 when X t 1  0 and 0 otherwise. The EGARCH (1,1) model is
specified by the equation
12 A. J. Lawrance
10
Volatility
8
6
4
2
1
0
-6
-4
-2
0
Previous Return
2
4
6
Figure 14. Volatility dependence plot of fitted volatilities from the TGARCH (1,1) model for the
FTSE100 Series.
10
Volatility
8
6
4
2
1
0
-6
-4
-2
0
Previous Return
2
4
6
Figure 15. Volatility dependence plot of fitted volatilities from the EGARCH (1,1) model for the
FTSE100 series.
X t   t  t , log( t2 )   0  1
| X t 1 |
X
  t 1  1 log( t21 ) .
 t 1
 t 1
(10)
In both models  t is from an independent standardized Gaussian series of
innovations, although the Gaussian choice is not essential.
Fitting these models by maximum likelihood to the FTSE100 series gives
 t2  0.0142  0.0031X t21  0.1715 I ( X t 1 ) X t21  0.9083 t21 ,
log( t2 )  0.0926  0.1197
(11)
| X t 1 |
X
 0.1332 t 1  0.9864 log( t21 ) . (12)
 t 1
 t 1
The parameter estimation is without cause for concern but the residual behaviour is
somewhat unsatisfactory over the 2008-9 period of strong fluctuations. Estimates of
both asymmetry parameters are not near zero and both are very significantly different
from zero; their signs indicate higher volatility following negative returns. However,
from this information it is not clear which model should be chosen. An exploratory
graphical suggestion is to compare their volatility dependence plots with that of the
FTSE100 series shown in Figure 6. This requires for each model the fitted volatilities

 t2 calculated from (11) and (12) using the FTSE100 series. Figures 14 and 15 give

the results. Their lower boundaries are indications of the asymmetries since the  t21
Volatility Graphics
13
14
12
Volatility
10
8
6
4
2
0
-7
-5
-3
-1
0
1
Previous Value
3
5
7
Figure 16. Volatility dependence plot for 20 identically distributed SV1((0.9, 0.6538)
series of 1532 values with their inverse-log lowess smooth volatility functions and the
true volatility function (dark line).
of (11) and (12) are not involved with this effect. The tentative conclusion from
comparing Figures 14 and 15 with Figure 6 must be that the TGARCH (1,1) model is
preferred; it is more strongly asymmetric but neither model has the required strength
of asymmetry, a point for possible future investigation.
8. Stochastic Volatility Models
The graphical methods previously introduced were suggested on the basis of the
general multiplicative autoregressive volatility model (1) in which the volatility is
internally driven. The volatility dependency plot will now be empirically validated on
series produced by a so-called discrete stochastic volatility model in which the
volatility is driven by an external autoregressive process. The origins of such models
go back to a little-known 1972 working paper by B.Rosenberg, discovered by
Shephard (2005); the model was later independently re-introduced by Taylor (1982) as
the ‘product process’. A current overview of both discrete and continuous stochastic
volatility models is given by Shephard and Anderson (2009).
The stochastic volatility model from Taylor (1982) is considered here, and takes the
form
X t   (ht ) t ,  (ht )  exp(ht 2), ht   ht 1  t ,
(13)
denoted as SV 1(  , ). Here { t } is a series of independent and standardized
Gaussian distributed innovations,  () is a volatility link function of exponential
form, {ht } is a first-order linear autoregressive volatility driving process, with
autocorrelation parameter  , 0    1 and  is the standard deviation of its
innovations; these are based on the independent and standardized Gaussian-distributed
series {t }. This model is sometimes referred to as the log-normal SV model,
although it does not have a log-normal marginal distribution.
The SV 1 model has been simulated 20 times with   0.9,  0.6538 to give 20
sequences of 1532 values. Using these, Figure 16 includes one set of the individual
volatilities, 20 simulation estimates of the volatility functions and the implied
theoretical volatility function, to be given by equation (14). The shapes of the
simulation estimates confirm the model’s volatility, correctly pointing to its symmetry.
14 A. J. Lawrance
The SV 1 model is a two-equation time series model with two innovation variables  t
and t , and does not explicitly relate current value to past values as with singleequation models; therefore it can not be directly compared to models in the
autoregressive volatility family (1). It is instructive therefore to use (13) to relate X t
and X t 1 by eliminating ht and ht 1 between their two equations; then there is the
alternative representation
X t  | X t 1 | exp(t 2)   t  t1  .
(14)
This demonstrates that the SV 1 model has first-order autoregressive volatility
function | X t 1 | but that the innovations exp(t 2)  t  t1 are formed from the


two IID series of innovations and are dependent. Due to this structure it is not a
member of the autoregressive volatility family (1). Further, its volatility function is
not continuous at X t 1  0 , although a smooth approximation has been given by
lowess smoothing in Figure 16.
A useful feature of the representation (14) is afforded by its logarithmic-absolute value
transformation which brings it to be the linear model
log | X t |   log | X t 1 |  12 t  log |  t |   log |  t 1 |,
(15)
but with two innovation variables.
Thus, the logarithmic-absolute value
transformation eliminates volatility from the SV 1 model. This is a helpful feature in
data analysis since using the volatility dependence plot on both the untransformed and
transformed data could give evidence for the suitability or not of an SV 1 model.
9. A Theoretical Illustration of Volatility Decorrelation
As just seen in Section 8, in a multiple-equation time series model, the volatility
function may be implicit rather than explicit. Another such case is the first-order
autoregressive AR (1) model with ARCH (1) innovations. This is an autocorrelated
volatility model. Thus, suppose
X t   X t 1  et
(16)
where the innovations {et } are generated by the ARCH (1) model
et 
    e  .
0
2
1 t 1
(17)
t
The { t } are independent and identically distributed. A single-equation form of this
model is obtained by setting et  X t   X t 1 in (17) and becomes
X t   X t 1 
   (X
0
1
t 1

  X t 2 ) 2  t .
(18)
This is in the class of models (1) where there is a level term and the volatility is now a
function of the two immediately past values; hence the volatility dependence scatter
would be in three dimensions and need a smoothing surface. However, it is pertinent
to note that the volatility function of (18) involves the decorrelated X t 1 values, so
alternatively, the series could be decorrelated initially, and the decorrelated values
used as ARCH1 values in the volatility dependence plots (6) and (7). In effect, this is a
parametric version of the non-parametric decorrelation strategy.
Volatility Graphics
15
10. Discussion
The proposed graphics can be applied to many types of financial time series as well as
index returns, such as individual equity returns, portfolio returns, exchange rates and
interest rates, and to other time series areas requiring volatility analysis. The behaviour
of volatility dependence has been found to vary with the area of application. In
regard to interest rates, there is some evidence that differenced series point to
increased volatility after substantially increased rates.
There are a number of possible extensions to the proposed methods. Empirical
experience suggests that volatility dependence plots can also be univariately effective
with the previous two or three values. They could also be extended to jointly explore
a volatility surface in terms of the previous two values. The extent of practical gain
from more sophisticated non-parametric methods of volatility estimation could be
investigated. Higher order multiplicative autoregressive representations of stochastic
volatility models offer further model-checking developments. Other extensions, such
as generalizations to vector time series, should also be possible.
References
Anderson, T.G. and Benzoni, K. (2009) Realized volatility, in Handbook of Financial Time
Series, Ed. T.G. Anderson , R.A. Davis , J.-P. Kreiss and T. Mikosch, Heidelberg: SpringerVerlag.
Anderson, T.G., Davies, R.A., Kreiss, J.P. and Mikosch, T. (2009) Handbook of Financial
Time Series: Springer-Verlag.
Atkinson, A.C. (1981) Two graphical displays for outlying and influential observations in
regression, Biometrika, 68, 13-20.
Atkinson, A.C. (1987) Plots, Transformations and Regression, Oxford: Clarendon Press.
Beckers, S. (1981) Standard deviations implied in option prices as predictors of future stock
price variability, Journal of Banking and Finance, 5, 363-381.
Bowman, A.W. and Azzalini, A. (1977) Applied Smoothing Techniques for Data Analysis,
Oxford: Clarendon Press.
Chan, C., Karoyli, G.A., Longstaff, F.A. and Sanders, A.B. (1992) An empirical comparison
of alternative models for short term interest rates, Journal of Finance, XLVII, 1209-1277.
Cleveland, W.S. and Devlin, S.J. (1988) Locally-weighted regression: An approach to
regression analysis by local fitting, Journal of the American Statistical Association, 83, 596610.
Engle, R.F. (1982) Autoregressive conditional heteroscedasticity with estimates of the
variance of United Kingdom inflation, Econometrica, 50, 987-1008.
Engle, R.F. (2010) Robert Engle's FT Lectures on Volatility, parts 1-5:
http://readingthemarkets.blogspots.com/2010/09/robert-engles-ft-lectures.
Fan, J. and Yao, Q. (2003) Nonlinear Time Series, New York: Springer.
Franke, J., Neumann, M.H. and Stockis, J.-P. (2004) Bootstrapping nonparametric estimators
of the volatility function, Journal of Econometrics, 118, 189-218.
Glosten, L.R., Jaganathan, R. and Runkle, D. (1993) On the relation between expected value
and volatility of the normal excess return on stocks, Journal of Finance, 48, 1779-1801.
16 A. J. Lawrance
Granger, C.W., Ding, Z. and Spear, S. "Stylized facts on the temporal and distributional
properties of absolute returns: an update," Presented at Hong Kong International Workshop
on Statistics and Finance: An Interface, 2000, Eds., London: Imperial College Press.
Kim, J. and Mina, J. (2000) Riskgrades Technical Document, New York: Riskmetrics.
Ruppert, D. (2004) Statistics and Finance: An Introduction, New York: Springer.
Shephard, N. (2005) Stochastic Volatility: Selected Readings: Oxford University Press.
Shephard, N. and Anderson, T.G. (2009) Stochastic volatility: origins and overview, in
Handbook of Financial Time Series, Ed. T.G. Anderson , R.A. Davis , J.-P. Kreiss and T.
Mikosch, Heidelberg: Springer-Verlag, 233-254.
Taylor, S.J. (1982) Financial returns modelled by the product of two stochastic processes - a
study of daily sugar prices 1961-79, in Time Series Analysis: Theory and Practice, Ed. O.D.
Anderson, Amsterdam: North-Holland, 203-226.
Yahoo (2011) FTSE100 historical prices:
http://finance.yahoo.com/q/hp?s=%5EFTSE+Historical+Prices.