Volatility conditional on price trends

0.21 MB·English

by  Gilles Zumbach

Volatility conditional on price trends
Gilles Zumbach 1
5 December, 2004
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Abstract
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] The influence of the past price behaviour on the realized volatility is investigated in the present
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e article. The results show that trending (drifting) prices lead to increased (decreased) realized
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volatility. This “volatility induced by trend” constitutes a new stylized fact. The past price be-
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haviour ismeasured by aproduct of 2non overlapping returns, ofthe form r·L[r] where L isthe
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t lag operator. The effect is studied empirically using USD/CHFforeign exchange data, in a large
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m rangeoftimehorizons. AsetofARCHbasedprocessesaremodifiedinordertoincludethetrend
- effect, andtheirforecasting performances arecompared. Forabetterforecast, itisshownthatthe
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main factor is the shape of the memory kernel (i.e. power law), and the following factor is the
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Keywords: trendeffect,ARCHprocess, volatility forecast, longmemory.
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1ConsultinginFinancialResearch
CheminCharlesBaudouin8
1228Saconnexd’Arve
Switzerland
e-mail:[email protected]
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1 Introduction
The extensive study of the volatility of financial time series starts 20 years ago with the seminal
paper of Engle [?]. After a rapid improvement with the GARCH(1,1) process [?, ?], the quan-
titative results obtained since then have not been much better, despite the very large number of
studieswithvariousvolatilityprocesses. Eventhoughwehavetodayamuchbetterunderstanding
of the financial markets volatility, particularly in the high-frequency domain (see e.g. in [?] and
references therein), it remains difficult to translate this knowledge into better processes or better
volatility forecasts.
In order to overcome the apparent limitation of the classical processes, Zumbach, Pictet and Ma-
sutti[?]launched astudyusinggeneticprogramming(GP).Theirworkfocused onimprovingthe
efficiencyoftheGPinordertoturnitintoapracticaltooltoinvestigatefinancialtimeseries,with
an application to volatility forecast. One important advantage of the GP is that it is not biased
by our a priori bais and knowledge, as the program searches in the whole space of models (yet
notveryefficiently). Indeed, in[?]theGPveryquickly rediscovered inessence theGARCH(1,1)
model,andthen,withmoretime,wasabletoobtainbettersolutions. Theanalysisofthesolutions
discovered by the program showed that the new terms leading to the improvement werw of the
form of a product of returns at two different time horizons. This can be expressed in a sum of 2
terms,onewithareturnsquare,asinmostvolatility processes, andonewithtwononoverlapping
returns. This term is like r[d t ](t)·r[d t′](t−d t ), namely at time t, a product of a return at time
r r r
horizon d t with a return at another time horizon d t′, lagged by d t so as to not overlap the first
r r r
return. Inshort,wedenotegenerically suchtermsasr·L[r],whereL[r]denotes thelaggedreturn.
Notice that such terms are even in the return, namely under the change r→−r, the term r·L[r]
doesnotchange itssign.
Thisnewtermcanbeinterpretedasameasureofthepastpricemoves,namelywhetherthemarket
is trending or drifting. The action of the “trend term” r·L[r] can be understood as follows. If
both returns have the same sign (both positive or both negative), the market is trending, and this
may induce the market participants to change their positions because of the price move. The
trading of their positions increases volatility in the subsequent period. If the market is drifting,
the two returns have different signs, and the unchanged price makes the market participants to
keep their positions. This decreases the volatility in the subsequent period. This behaviour of
the market participants creates a positive correlation with the realized volatility in the following
period. Indeed, the correlation of the r·L[r] term with the realized volatility, namely with the
volatility computed after t, is positive (see below). This “volatility induced by trend” is a new
stylized fact for financial time series. Notice that if a return differs by the sign but has the same
magnitude, its contribution to the historical volatility is identical. Therefore, the trend term is
measuring therecentpricebehaviour, andnotthehistorical volatility.
Thegoalofthispaperistwofolds. First,wewouldliketoinvestigate thetrendeffectinempirical
data. The main questions are its magnitude, and the time horizons of the return, lagged return
and realized volatility where the trend effect is important. Second, we want to include the effect
in ARCH like processes. The addition of such a term in a process is easy, it is enough to add
one or several r·L[r] terms in the equations. In this context, the point isnot to create yet another
ARCHlike process, but to investigate the respective importance ofthe manyingredients that can
enter into a volatility process. For example, what are the respective importance of the shape of
thememorykernel(rectangular, exponential, powerlaw),theshorttermmeanreversion,thetrend
effect,orofthevariousclassesofmarketparticipants. Thepointhereistomeasuretheimportance
ofthesedifferent stylizedfacts.
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Thepaperisorganized essentially alongthelineoftheprevious paragraph. First,wedescribe the
empiricaldata,andcomputethecorrelation betweenther·L[r]termandtherealizedvolatility. In
section 4, we extend several processes with trend term(s), while respecting the basic idea of the
process(oneorseveralcomponents,powerlaw,etc...). Theprocessesarecomparedinsec.5with
respecttotheirforecasting performances, inlightoftheproperties oftheprocesses.
Inmanyrespects,thiscontributionisanextensionofsomeresultspresentedin[?]. Inparticular, it
willfollow thesamenotation andprocesses definitions. Thispaper isself-contained, but inorder
no to repeat extensively this reference, some sections are reduced to the essential. More details
can be found in [?], for example a discussion of the relative merit of the log-likelihood or of the
volatility forecast, inrelationtoagivenquadratic process.
2 The data set
The data set used for this article is derived from high frequency tick-by-tick data for the foreign
exchangeUSD/CHF.Essentially,fortheempiricalanalysis,thedatasetisaregulartimeseriesfor
USD/CHF,sampledevery3minutesinbusiness time. First,thehighfrequency dataisfilteredfor
theincoherenteffect[?]: averyshortexponentialmovingaverageistakenonthepricesinorderto
attenuate the tick-by-tick incoherent price formation noise. Second, the price is sampled every 3
minutesinbusinesstime. Moreprecisely, weusethedynamicbusinesstimescaleasdevelopedin
[?]. Similarly tothe familiar daily business timescale, the dynamic business timescale contracts
periodsoflowactivity(night,week-end)andexpandsperiodsofhighactivity. Theactivitypattern
during the week is related to the measured volatility, averaged on a moving sample of 6 months.
Holidaysanddaylightsavingtimearetakenintoaccount. Thehomogeneous timeseriesusedfor
the empirical analysis is computed from the high frequency filtered tick-by-tick USD/CHF data,
thatissampledusingalinearinterpolation, every3minutesindynamictimescale. Asthismarket
is open essentially 24 hours per day, 5 days per week, the sampling time interval corresponds to
anaverage of2’8” (=5/7 3’)during the market opening hours. Theauthor isgrateful toOlsen &
Associates, inZu¨rich,Switzerland,toprovidethedata.
3 Empirical analysis
Thehistoricalreturnsr[d t ]andlaggedhistoricalreturnsL[d t ;r[d t′]]arecomputedbysimpleprice
r r r
difference fromthesampledlogarithmicprices. Therealizedvolatility iscomputedwith
s 2[d ts ,d tr =d ts /32]= 1 (cid:229) r2[d tr](t′) (1)
n
t+d tr≤t′≤t+d ts
andwherenisthenumberoftermsinthesum. Therealizedvolatility measuresthefluctuation of
the prices aftert, in the interval from t tot+d ts , The time horizon d ts , over which the volatility
iscomputed, isthemainvolatility parameter. Thereturnsaretakenattimehorizons d tr =d ts /32,
namely they grow with d ts . Other choices can be made, with a minor influence on the empirical
results. Thesumineq.1iscomputedwithallpointsonthe3’samplinggrid.
Withthethreetimeseriesr,L[r]ands ,wecomputethecorrelation
r [d tr,d tr′,d ts ]=r (r[d tr]·L[d tr;r[d tr′]],s [d ts ]) (2)
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wherer (x,y) ontherighthand sidedenotes theusual linear correlation between thetimeseries x
andy.
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Figure 1: Correlation between the trend term r·L[r] and the realized volatility. The fixed volatility time
horizon is d ts =1h 36. The horizontal axis is the (historical) return time interval d tr, the vertical axis is
the(historical)laggedreturntime intervald t′. Theaxis divisionscorrespondtothe logarithmicof thetime
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intervals. Themainaveragephysicaltimeintervalscorrespondingtothelabelsare1hour(n=12),8hour
(n=24),1day(n=31),1week(n=40),and1month(n=47). DatacourtesyofOlsenandAssociates,
Zurich.
The analysis of the empirical correlation is difficult to visualize, as it is a function of three time
intervals. The2dimensional resultspresented belowarecutinthis3dimensional space, showing
the level of correlation with colors. The figure 1 is a cut at fixed volatility time interval, at the
shortestvolatility estimated ts =32·3′ =96′ (inbusinesstime). Thetwoaxescorrespond respec-
tively to the return time interval and lagged return time interval. The main structures emerging
fromthisfigureare:
• Thecorrelationisessentiallypositive. Thisisconsistentwiththeintuitiveexplanationgiven
in the introduction, as trends will make the market participants to modify their positions,
increasing the subsequent volatility. Moreover, the levelofcorrelation isranging from 3to
8%,corresponding toaneffectofmediumimportance.
• We observe 4 pockets with higher correlations, roughly along the diagonal or above. They
are located approximately at positions 10 to 20, 30, 40 and 50, corresponding respectively
to time intervals intra-day, 1 day, 1 week and 1 month. These time intervals are indeed
precisely the expected time horizons for the main groups of traders, in agreement with the
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findingof[?]. Themaximaat1monthisverywelldefined,andwithd t =d t′. Ontheother
r r
corner, theintra-daymaximaisfairlysoft,andclearlyabovethediagonal. Themaximaat1
day showssimilar characteristics, whereas themaxima at1weekisalong the diagonal and
weaker.
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Figure 2: Correlation between the trend term r·L[r] and the realized volatility. The fixed volatility time
horizonisd ts =1day.Theaxesandcolorscodingareasforfig.1.
The figure 2 is a cut at fixed volatility time interval, with d ts = 1 day. For this daily volatility,
the 4 maxima can be seen, essentially along the diagonal d t =d t′. The intra-day maximum is
r r
very weak, the daily and monthly maxima are very clear, while the weekly maximum is weaker.
Thisshowsthatthevolatilityatthedailytimehorizonisnotinfluencedbyintra-daytrends,butby
trendsatdailyhorizonorlonger. Bothpreviousfiguresindicatethatthemajorcorrelationisalong
thediagonal d t =d t′,orslightlyabove.
r r
The figure 3 is a cut along the plane d tr = d tr′, and with the volatility time horizon d ts on the
vertical axis. The labels on the vertical volatility axis d ts correspond to the same time intervals
on the horizontal axis d t . The maximum at the daily and monthly time horizons are very clear,
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andtheweaker intra-day andweeklymaximumcanalsobeseen. Thisfigureshowsthattrends at
agiventimehorizon influence thevolatility uptoahorizon immediately larger, andthendecline.
Forexample, atrend measured byconsecutive dailyreturn influences strongly thevolatility upto
3to4days, butmuchlessatoneweekandabove. Indeed, thisbehaviour iswhatcanbeexpected
fromaportfoliomanagerworkingonadailyhorizonandadjustingitsportfoliobasedontheprice
trendsordriftsofthelastfewdays.
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Figure 3: Correlationbetweenthe trendtermr·L[r] and therealizedvolatility. The horizontalaxis is the
timehorizonforbothhistoricalreturnsd tr =d tr′. Thevolatilitytimehorizond ts isgivenontheverticalaxis.
Theaxesdivisionsandcolorscodingareasforfig.1.
4 Modelisation in ARCH-like processes: the ARTCH family
The influence of trends on the subsequent volatility is fairly easy to incorporate in ARCH-like
processes. Essentially, we must add one or several terms of the form r·L[r] in the conditional
volatility. For a given ARCH-like model, there is in general one corresponding minimal model
incorporating theinfluenceoftrends. Wecallgenerically thisnewfamilyofmodelsby’ARTCH’,
forAutoRegressiveTrendConditionalHeteroskedastic. Letusemphasizethatourgoallaysnotin
writing new ARCHprocesses, but inquantifying various effects that can be included in amodel.
Toaddresssuchquestions, webuildanetofprocessesofincreasingcomplexity. Alltheprocesses
are then estimated by minimizing the 1 day volatility forecast error on the USD/CHF data. The
comparison of their optimal forecasting performances measures their relative adequation to the
data,andtheimportance ofthevariouseffects(atthechosentimehorizon).
Ourgeneral strategy istousequadratic ARCH-likeprocesses togenerate volatility forecasts. For
quadratic processes, conditional averagescanbecomputedanalytically, andoneobtainsvolatility
forecaststhatdependontheprocessparameters. Thepropertiesoftheforecastderivedirectlyfrom
theproperties andparameters oftheunderlying process. Thisapproach givesasimpleframework
in which data generating processes and volatility forecasts are closely related. For forecasting
purposes only, other direct approaches can be pursued. Poon and Granger [?] wrote an extensive
reviewonvolatility forecast.
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Albeittheabove strategy isappealing, thedifficulty ofitspractical implementation should not be
underestimated. Foreachprocess, theneededconditional averagesmustbecomputed, andimple-
mented numerically. Moreover, as we are estimating the parameters by minimizing the forecast
error, the derivatives with respect to the parameters should also be implemented. The theoretical
setting isfollowing closely [?],andonly thesalient points aregivenhereforcompleteness. First,
wesetthebasiccommonequations, andthendescribe thevarious ARTCHprocesses.
4.1 The basicstructure ofthe processes
Weareconsidering processes forthepricewiththefollowingstructure
x(t+d t) = x(t)+r(t+d t) (3)
r(t+d t) = s (t+d t)e (t+d t) (4)
eff
s 2 (t+d t) = s˜2[W (t),J ,d t]. (5)
eff
Therandom variables e arei.i.d. withE[e (t)]=0andE[e 2(t)]]=1. Thetimeindexesarechosen
toemphasize thats 2 (t+d t)isaforecast forthevolatility attimet+d t. Theforecast function s˜
eff
isbasedontheinformation setW (t)attimet,anddepends onasetofparametersJ .
Intheprocesses below, theright handside ineq. 5contains termsoftheformr·L[r], whichhave
no definite sign. Asa consequence, the volatility square could become negative. In practice, this
never occurs at the optimal values of the parameters, but it could happen during the parameters
estimation. Topreventasquarerootofanegativevalue,therighthandsideincludesalowervalue
threshold. ImplicitinalltheARTCHequations,amaxfunctionmax(s 2 ,s 2 )isincluded,where
eff min
s 2 is given in the equations below. The minimal value for the volatility square s 2 has value
eff min
s 2 =10−10.
min
This possible negative variance and the related max function are likely to be an artifact of the
present ARCH setting, which includes only price and volatility. A more complete framework
should include also the tick rate. It is likely that the “volatility induced by trend” stylized fact is
related to trading decisions that influence the tick rate and/or the new orders rate, which in turn
influencesthevolatility.
4.2 The GARTCH(1,1)process
TheGARTCH(1,1)process equations are
s 2(t) = µs 2(t−d t)+(1−µ)r2(t) (6)
1 1
s 2eff(t+d t) = s 2+(1−w¥ ) s 21(t)−s 2 +q r[ld t](t)r[ld t](t−ld t) (7)
(cid:0) (cid:1)
with the 4 parameters s ,w¥ ,µ,q , and the integer lag parameters l. For q = 0, these equations
reduce to the usual GARCH(1,1) process. The rational for the equations is the following. The
“internal”states measuresthevolatilityatthetimehorizont =−d t/ln(µ),whichisthevolatility
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computed or perceived by a group of market participants. The equation 7 models their trading
pattern which depends on the difference with the (expected) mean volatility, and on the recent
trend/driftoftheprice. The“parameter”lcanbechosenaprioriwiththeguidanceoftheempirical
analysis above. Itcanalsobestudiedsystematically asdoneinsec.5.3.
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4.3 The I-GARTCH(1)process
Forw¥ =0,theGARTCH(1,1)equations reducetothelinearI-GARTCH(1)process
s 2(t) = µs 2(t−d t)+(1−µ)r2(t) (8)
1 1
s 2 (t+d t) = s 2(t)+q r[ld t](t)r[ld t](t−ld t) (9)
eff 1
withtwoparametersµandq . Aslightvariationconsists inusingthedefinition
s 2 (t+d t)=µs 2 (t)+(1−µ)r2(t)+q r[ld t](t)r[ld t](t−ld t) (10)
eff eff
In practice, both definitions give very similar results. The empirical results on the 1 day forecast
accuracyhavebeencomputedwiththesecond definition.
We have also included in the study the RiskMetric volatility. This process corresponds to the I-
GARCHprocess withafixedparameter µ. Asweare working withhourly data (inbusiness time
scale), wetakeµ=0.931/24.
4.4 The Long Memory volatilityprocesses
Thelongmemoryprocess hasbeenintroduced in[?]. Itincorporates inaminimalwaythepower
law decay of the lagged correlation for the absolute value of the return, or of the square return.
This model is structureless with respect to the time horizons, namely it has a uniform structure
betweenaloweranduppercut-off. Therefore, itdoesnotinclude thespecificmarketcomponents
asobservedandmodeledin[?,?]. Intheempiricalanalysisbelow,wehaveusedthe“microscopic”
version of the long memory process (see [?]). The inclusion of a trend term in the long memory
process should preserve this simple and uniform structure. Therefore, the idea is to include one
r·L[r]termforeachpartialvolatility s ,andtohaveweightsgivenbyasimplepowerlaw.
k
Thelongmemorymodelsarebuiltwithasetof(historical)volatilitiescomputedoverasetoftime
horizons increasing asageometricseries:
t = l t k=1,···,n
k k 0
µ = µ1/lk =exp(−d t/t ) (11)
k 0 k
s 2(t) = µ s 2(t−d t)+(1−µ )r2(t).
k k k k
The time horizons t correspond to the characteristic times of the EMA at which the historical
k
volatility is measured. The time structure l of the process is a geometric series l =r k−1, with
k k
the progression ofthe series chosen tober =2inthis work. Thebase timescale t corresponds
0
to the shortest time scale at which a volatility is measured, and is one of the process parameters.
Theempiricalstudies inthisworkhavebeendonewithhourlydataandwithn=12components,
corresponding to an upper cut-off of 6 months. The effective volatility s for the long memory
eff
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(LM)affine(Aff)processes LM-Aff-ARTCH(n),withncomponents andtrends, is
n n
s 2eff(t+d t) = s 2+(1−w¥ )(cid:229) c k s 2k(t)−s 2 +(cid:229) q k r[lk](t)r[lk](t−lkd t)
k=1 (cid:0) (cid:1) k=1
n n
= (cid:229) wks 2k(t)+w¥ s 2+(cid:229) q k r[lk](t)r[lk](t−lkd t) (12)
k=1 k=1
c = cr −kl =c 1 l with 1/c= (cid:229) n r −(k−1)l
k
(cid:18)l (cid:19)
k k=1
wk = (1−w¥ )c k
l ′
1
q = q r −(k−1)l ′ =q (13)
k 0 0
(cid:18)l (cid:19)
k
The“normalization constant” cischosen sothat(cid:229) c k =1and(cid:229) wk+w¥ =1. The“meanterms”
introduce two constants s and w¥ . For w¥ =0, the linear model is obtained. The “trend terms”
dependonthetwoconstants q andl ′.
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In the comparative study, the long memory models allow us to compare the processes with short
memory(exponential) andlongmemory(powerlaw). Weinclude 4versions ofthelongmemory
process, namely one linear model LM-Lin-AR(T)CH (linear, similar to I-GAR(T)CH, but with
long memory) and one affine model LM-Aff-AR(T)CH (affine, similar to GAR(T)CH, but with
longmemory).
4.5 The Market Component volatilityprocesses
In[?,?],thecorrelation betweenhistorical andrealized volatility iscomputed forarange oftime
horizons, and a similar computation was done for the change in historical volatility versus the
realized volatility. These correlations show clearly the characteristic time horizons of the market
participants, essentially at intra-day, daily, weekly and monthly time intervals. The observed
heterogeneity ofthevolatility correlations canbereproduced withaprocess that incorporates the
same characteristic time horizons. As noted in the empirical section, the trend effect shows very
similarcharacteristic timeintervals. Inordertoincludethetrendeffectinthemarketcomponents
process, it is enough to add a trend term for each volatility component. The market component
ARCHprocess ispresented indetailin[?],wegivehereforcompleteness thedefinition.
For the market component model, instead of measuring the (historical) volatilities with a simple
Exponential Moving Average (EMA), which has an exponential kernel, we use an MA operator
whichhasamorerectangular-like kernel. TheMAoperatorisdefinedby[?]:
MA[t ,m;z](t) = 1 (cid:229)m EMA (t) (14)
j
m
j=1
EMA (t) = µEMA (t−d t)+(1−µ)z(t)
1 1
EMA (t) = µEMA (t−d t)+(1−µ)EMA (t)
j j j−1
µ = exp(−d t(m+1)/t )
The coefficient µ is computed from the time horizon t , so that the memory length of the MA
operator is t . The parameter m control the shape of the kernel, and we take m=8 which gives a
fairlyrectangular kernel.
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TheMkt-Aff-ARTCHprocessequations, withthetrendterms,are:
s 2(t) = MA[t ,m;r2](t) (15)
k k
n n
s 2eff(t+d t) = s 2+(1−w¥ )(cid:229) c k s 2k(t)−s 2 +(cid:229) q k r[lkd t](t)r[lkd t](t−lkd t) (16)
k=1 (cid:0) (cid:1) k=1
withtheconstraint
n
(cid:229) c =1. (17)
k
k=1
Thelinearversionoftheprocess Mkt-Lin-ARTCHisobtainedbytakingw¥ =0.
4.6 Other processes
Forcompleteness,weincludeinthestudyafewprocesseswhichdonothaveanobviousextension
to incorporate the trend effect. Thefirstone is the“permanent forecast”. Thehistorical volatility
s 2 [d ts ],withequalweightonatimeintervald ts ,is
hist
s 2hist[d ts ](t)= n1 (cid:229) r2[d tr](t′). (18)
t−d ts +d tr≤t′≤t
As all the points are equally weighted from t−d ts to t, the memory kernel corresponds to a
rectangular moving average. The “permanent forecast” uses s for the volatility forecast over
hist
anytimehorizon.
The I-GARCH(2) process is a natural extension of GARCH(1,1), where the mean volatility s is
replaced byanexponential movingaverage:
s 2(t) = µ s 2(t−d t)+(1−µ )r2(t) (19)
1 1 1 1
s 2(t) = µ s 2(t−d t)+(1−µ )r2(t)
2 2 2 2
s 2 (t+d t) = ws (t)2+(1−w)s (t)2(t).
eff 2 1
This process is linear (in s 2 and r2), with two components. We name it I-GARCH(2) as it is a
naturalextension ofI-GARCH(1).
5 Volatility forecast using the processes
5.1 The data set
For all the empirical results presented in this section, the data set is a regular time series for
USD/CHF, sampled hourly in business time. This series is obtained by aggregation by a factor
28 of the homogeneous time series used in the empirical analysis. The resulting sampling time
intervalof1h24mcorresponds to7/5ofonehour,andissuchthatinaverage,120pointsperweek
10