Table Of ContentAcknowledgments
Our interest in fractal modeling was spurred during our graduate years
at Yale by conversations with Benoˆıt Mandelbrot, the father of fractal
geometry. This interaction evolved into the collaborative development of
the Multifractal Model of Asset Returns (MMAR), which was originally
circulated in three Cowles Foundation Discussion Papers (Calvet, Fisher,
and Mandelbrot, 1997). The core of this work was eventually published in
the Review of Economics and Statistics in 2002.
In order to develop practical applications such as volatility forecast-
ing and pricing, we then independently developed a Markov model with
multifrequency characteristics, which we called the Markov-switching mul-
tifractal (MSM). We first presented MSM, both in discrete and continuous
time, at the 1999 National Bureau of Economic Research Summer Insti-
tute organized by Francis Diebold and Kenneth West, and published this
work in the corresponding special issue of Journal of Econometrics. MSM
stimulated a flow of subsequent research on the econometric and pricing
applications of multifrequency volatility risk, which we are now presenting
in this book.
Manyofthechaptersdrawheavilyfromearlierpublicationsinacademic
journals.Thepresentationofdiscrete-timeMSMinChapter3follows“How
to Forecast Long-Run Volatility: Regime-Switching and the Estimation of
Multifractal Processes,”presented in 2002 at the CIRANO Conference on
Extremal Events in Finance, organized by Ren´e Garcia and Eric Renault,
andpublishedintheJournalofFinancialEconometrics.MultivariateMSM
in Chapter 4 was presented in September 2003 at the Penn-IGIER confer-
ence organized by Francis Diebold and Carlo Favero, and appeared in the
Journal of Econometrics underthetitle“VolatilityComovement:AMulti-
frequencyApproach.”ThisarticleisjointwithSamuelThompson,aformer
Harvard colleague now at Arrowstreet Capital, L.P.
Chapters 6 and 8 follow the Review of Economics and Statistics article
“Multifractality in Asset Returns: Theory and Evidence.” The discussion
of continuous-time MSM in Chapter 7 is a thoroughly revised version of
theJournal of Econometrics publication“ForecastingMultifractalVolatil-
ity.” Chapter 9 is based on “Multifrequency News and Stock Returns,”
which was first presented at Wharton in March 2003 and appeared in the
JournalofFinancialEconomics.Chapter10borrowsfrom“Multifrequency
Jump-Diffusions: An Equilibrium Approach,” which was prepared for the
2005 NSF/CEME Conference in Honor of G´erard Debreu organized by
Chris Shannon at UC Berkeley and was published in Journal of Mathe-
matical Economics. We have liberally added and subtracted from these
earlier papers in order to produce a coherent book.
x Acknowledgments
Wewouldliketothankmanycolleaguesandfriendswhohavecontributed
to the development of the multifrequency research presented in this book.
We are especially grateful to John Campbell and Francis Diebold for their
continuedsupportoverthepastdecade.Duringourgraduateyears,thefac-
ulty of Yale University gave us insightful feedback, and John Geanakoplos,
Peter Phillips, Robert Shiller and Christopher Sims in particular provided
uswithinvaluablehelp.WethankSamuelB.Thompsonforthefruitfulcol-
laborationonwhichChapter4isbased.Wealsoacknowledgethegenerous
help of Karim Abadir, Andrew Abel, Torben Andersen, Donald Andrews,
TimBollerslev,MichaelBrandt,AndreaBuraschi,BernardCornet,Walter
Distaso, Darrell Duffie, Robert Engle, Francesco Franzoni, Ren´e Garcia,
William Goetzmann, James Hamilton, Lars Hansen, Oliver Hart, Jean
Jacod,GuidoKuersteiner,GuyLaroque,OliverLinton,EricMaskin,Nour
Meddahi, Andrew Metrick, Marcelo Moreira, Jacques Olivier, Jack Porter,
Stephen Ross, Jose Scheinkman, Neil Shephard, Bruno Solnik, Robert
Stambaugh, Jeremy Stein, James Stock, Jessica Wachter, Kenneth West,
Amir Yaron, Paolo Zaffaroni, Stanley Zin, and our current and former
colleagues at Harvard, HEC Paris, Imperial College London, New York
University, and the University of British Columbia.
We thank Karen Maloney, our editor at Elsevier, who encouraged us
to write this book and displayed considerable enthusiasm throughout the
durationofthisproject.WearealsogratefultoRoxanaBobocandMelinda
Ritchie for their patient assistance.
Last but not least, we thank our families for their love and kind support
while this work was under way. This book is dedicated to them.
Foreword
Thestudyoffinancialmarketshasbecomeoneofthemostactiveandpro-
ductive empirical endeavors in the social sciences. A cynic, or a trained
economist,mightsaythatthevolumeoffinancialresearchreflectsthehigh
price that market participants are willing to pay for it, but there are also
deepintellectualreasonsfortheinterestinfinancialmarketdata.
First,thesedataareabundantatallfrequenciesfromtick-by-tickdataat
one extreme, to century-by-century data at the other, and across multiple
correlated assets. Such data abundance is atypical in the social sciences,
anditcompensatestosomedegreeforthefactthatthedataaregenerated
by the interactions of investors rather than by controlled experiments.
Second, the uncertainty that financial econometricians face when esti-
mating their models is the same uncertainty that investors face when they
trade in asset markets and determine market prices. As Campbell, Lo,
and MacKinlay (1997) express it, “The random fluctuations that require
the use of statistical theory to estimate and test financial models are inti-
mately related to the uncertainty on which those models are based.” This
gives financial modelling a special relevance to market participants. Intel-
lectually, it poses the challenge of building stochastic equilibrium models
in which the second and higher moments of random shocks determine the
first moments of asset returns.
Asfinancialeconometricianshaveexploredthepropertiesofassetreturn
dataoverthepast30years,theyhaveuncoveredsomefascinatingregulari-
ties.First,measuredathighfrequencies,thedistributionofassetreturnsis
veryfarfromnormal;thereisexcesskurtosis,astrongtendencyforreturns
tostayeitherveryclosetothemeanorfarfromit.Investorswereforcefully
reminded of the non-normality of stock returns when stock prices crashed
in October 1987, although the basic facts were already well established in
the academic literature.1 This non-normality diminishes when returns are
measured over longer time intervals.
Second, conditioning information can be used to predict both first and
second moments of asset returns. The predictability in first moments is
subtle, as one would expect given the fundamental insight of the efficient
marketshypothesis,thatcompetitionamonginvestorsensuresmodesttrad-
ingprofitssothatmostpricevariationisdrivenbythearrivalofnews.The
predictability in second moments, on the other hand, is obvious even to a
1I vividly remember delivering a technical lecture on excess kurtosis at Princeton
duringthemorningofOctober19,1987.Astudentatthebackoftheroomputuphis
handandsaid“ProfessorCampbell,areyouawarethattheDowisdown200points?”
Iwasnot.
xii Foreword
casualobserver.Avastliteratureonthisphenomenonhasdocumentedthat
conditionalvolatilitycanmovesuddenlybutalsovariespersistently,sothat
time-varying volatility is apparent both within the trading day and across
decades. Movements in conditional volatility often appear to be correlated
with asset returns themselves.
A direct way to model changing second moments is to write down a
processforthevolatilityofreturns,conditionalonpastreturnsandpossibly
otherinformationavailabletoaneconometrician.Intheearly1980’sRobert
Engle followed this direct approach when he proposed the ARCH model.
Thisseminalcontributionwasrecognizedin2003bytheawardtoEngleof
the Nobel Memorial Prize in Economic Sciences.
While the ARCH framework is a natural way to model changing volatil-
ity, it does not offer an integrated explanation of return phenomena at dif-
ferent frequencies. It explains high-frequency non-normality by exogenous
non-normal shocks, and persistence of volatility using a slowly decaying
(fractionally integrated) volatility process. Each of these features must be
chosen separately to match different aspects of financial market data.
An alternative approach, which has attracted interest in recent years, is
to write down a process for a latent state variable that is not observable
to the econometrician.2 The simplest version of this “stochastic volatil-
ity” approach assumes that, conditional on the state variable, returns are
normally distributed with a volatility governed by the state variable, and
that the state variable follows a smooth autoregressive process. Since the
state variable is unobserved, returns are not in general normal conditional
on observed information, so in principle a smoothly evolving stochastic
volatility model can explain a wide variety of asset pricing phenomena. In
practice, however, empirical analyses of stochastic volatility models find
that additional jumps are needed to fit the data: jumps in volatility, to
accommodate sudden changes in volatility, and jumps in prices, to gen-
erate the extreme non-normality observed in high-frequency return data.
Correlation between these jumps causes sudden movements in asset prices
to predict subsequent volatility (Duffie, Pan, and Singleton, 2000, Eraker;
Johannes, and Polson, 2003).
Once stochastic volatility is allowed to jump, it becomes natural to con-
sider a discrete-state regime-switching model of the sort introduced to the
economics literature by Hamilton (1988, 1989). Regime-switching models
are particularly appealing because they are tractable both for econome-
tricians and for financial economists solving asset pricing models.3 Given
2For surveys, see Ghysels, Harvey, and Renault (1996) or Andersen and Benzoni
(2008).
3Mehra and Prescott (1985) used a regime-switching model in their seminal paper
on the equity premium puzzle. Garcia, Meddahi, and T´edongap (2008) use a regime-
switchingmodeltoanalyzeseveralmodelsthathavebeenpopularintherecentliterature
onconsumption-basedassetpricing.
Foreword xiii
the richness of financial data, however, one would like to allow for many
possible states in volatility. Until recently, this requirement has barred the
useofregime-switchingmodelsinvolatilitymodelling,becausethenumber
of parameters in regime-switching models increases with the square of the
numberofstates,sothemodelsbecomeunusableveryquicklyasnewstates
are added.
TheresearchthatLaurentCalvetandAdlaiFisherpresentinthisvolume
is exciting because it breaks this barrier, and does so in a way that pro-
videsaunifiedexplanationofmanyofthestylizedfactsofassetpricing.The
Markov-SwitchingMultifractal(MSM)modelassumesthatvolatilityisthe
productofalargenumberofdiscretevariables,eachofwhichcanrandomly
switchtoanewvaluedrawnfromacommondistribution.Thevariablesare
ordered by their switching probability, which increases smoothly from low-
frequency to high-frequency volatility components. Volatility jumps when
a regime switch occurs, and the change in volatility can be extremely per-
sistent if the switch affects a low-frequency component of volatility. The
modelhasonlyfourparametersevenifithasmanymorevolatilitycompo-
nentsandanenormousnumberofstates.Thisparsimonymakesthemodel
a strong performer in forecasting volatility out of sample.
BecausetheMSMiseasytoembedinageneralequilibriumassetpricing
framework, Calvet and Fisher are able to calculate the effects on prices
of jumps in the volatility of fundamentals when investors are risk-averse.
Theyfindthatexogenousjumpsinfundamentalvolatilitycauseendogenous
jumps in asset prices, providing an attractive economic explanation of the
jump correlations documented by Eraker, Johannes, and Polson (2003)
among others.
The appeal of the multifractal approach becomes clear when one con-
sidersitsvariousapplicationstogether.Calvet,Fisher,andtheircoauthors
have published a series of papers on multifractal volatility, but the impor-
tance of the work is much easier to see when it is presented in a
unified manner in this volume. Multifractal volatility modelling is a major
advance, and this book is a milestone in the modern literature on financial
econometrics.
John Y. Campbell
Harvard University
June 2008
Credits and Copyright Exceptions
Chapter 3: The material in this chapter appeared in an earlier form in
Journal of Financial Econometrics, 2, L. E. Calvet and A. J. Fisher, ‘How
to Forecast Long-Run Volatility: Regime-Switching and the Estimation of
MultifractalProcesses,’pp.49–83,Copyright(cid:2)c 2004,withpermissionfrom
Oxford University Press.
Chapter 4: The material in this chapter appeared in an earlier form
in Journal of Econometrics, 131, L. E. Calvet, A. J. Fisher, and
S. Thompson, ‘Volatility Comovement: A Multifrequency Approach,’
pp. 179–215, Copyright (cid:2)c 2006, with permission from Elsevier.
Chapter 6: Adapted from L. E. Calvet and A. J. Fisher, ‘Multifractality
in Asset Returns: Theory and Evidence,’ The Review of Economics and
Statistics, 84:3 (August, 2002), pp. 381–406. Copyright (cid:2)c 2002 by the
President and Fellows of Harvard College and the Massachusetts Institute
of Technology.
Chapter 7: Thoroughly revised version from Journal of Econometrics,
105, L. E. Calvet and A. J. Fisher. ‘Forecasting Multifractal Volatility,’
pp. 27–58, Copyright (cid:2)c 2001, with permission from Elsevier.
Chapter 8: Adapted from L. E. Calvet and A. J. Fisher, ‘Multifractality
in Asset Returns: Theory and Evidence,’ The Review of Economics and
Statistics, 84:3 (August, 2002), pp. 381–406. Copyright (cid:2)c 2002 by the
President and Fellows of Harvard College and the Massachusetts Institute
of Technology.
Chapter 9: The material in this chapter appeared in an earlier form in
Journal of Financial Economics, 86, L. E. Calvet and A. J. Fisher. ‘Multi-
frequency News and Stock Returns,’ pp. 178–212, Copyright (cid:2)c 2007, with
permission from Elsevier.
Chapter 10: The material in this chapter appeared in an earlier form in
Journal of Mathematical Economics, 44, L. E. Calvet and A. J. Fisher.
‘MultifrequencyJump-Diffusions:AnEquilibriumApproach,’pp.207–226,
Copyright (cid:2)c 2008, with permission from Elsevier.
1
Introduction
Financial markets are uniquely complicated systems, combining the
interactions of thousands of individuals and institutions and generating
at every instant the prices to buy and sell claims to future uncertain cash
flows. The timing of dividends, coupons, and other payoffs varies widely
acrossassets,andvaluationsarecorrespondinglydrivenbynewsasdiverse
asshort-runweatherforecasts(Roll,1984a)ortechnologicalbreakthroughs
that may take decades to come to fruition (Greenwood and Jovanovic,
1999; Pastor and Veronesi, 2008).Market participants adopt a variety of
trading strategies and investment horizons. High-frequency speculators,
arbitrageurs, and day traders attempt to exploit opportunities over the
veryshortrun,whileinsurancecompanies,pensionfundsand401(k)partic-
ipants have investment objectives spanning several decades. Furthermore,
the recent diffusion of algorithmic trading techniques implies that even
long-run investors now routinely engage in sophisticated high-frequency
transactions.
Thecomplexityofassetmarketsismatchedbytherichdynamicproper-
tiesofthereturndatathattheyproduce,andquantifyingriskhasbeenfor
overacenturyoneoftheleadingtopicsofinvestigationinfinance.Forsome
researchers, the pure scientific challenge of understanding this intricate
environmentissufficientmotivation.Atthesametime,thepotentialpecu-
niary rewards to improved modeling of financial data are substantial.The
fieldsofportfoliomanagementandassetpricingrequireanaccurateviewof
thestatisticalpropertiesofassetreturns.Riskmanagersmustquantifythe
exposure of trading positions and portfolios of contingent claims. Option
values are largely determined by market forecasts of future volatility.
The models we develop in this book capture empirically relevant and
seemingly disparate features of financial returns in a single, parsimonious
framework.
1.1 Empirical Properties of Financial Returns
One well-known property of financial volatility is its persistence. When
returns are substantially positive or negative on a given day, further
2 1. Introduction
large movements are likely to follow.1 In 1982, Robert Engle concisely
modeledtime-varyingvolatilityinhisseminalpublicationonautoregressive
conditional heteroskedasticity (ARCH), which laid the foundation for his
2003 Nobel Memorial Prize in Economic Sciences.2 Subsequent research
showedthatvolatilityclusteringcanremainsubstantialoververylonghori-
zons(Ding,Granger,andEngle,1993),andthatanaccuraterepresentation
mayrequirethepossibilityofextremechanges,orjumps,involatility(e.g.,
Duffie,Pan,andSingleton,2000).Underconditionswherevolatilityisboth
highly persistent and highly variable, we should expect volatility fluctua-
tions to have substantial valuation and risk management implications.
Asset prices themselves can change by large amounts in short periods
of time, a phenomenon often described as tail risk. In continuous-time set-
tings, such extreme events can be modeled as jumps (Press, 1967; Merton,
1976), or as sudden bursts of volatility (Mandelbrot and Taylor, 1967;
Rosenberg, 1972; Clark, 1973). In discrete time, postulating a thick-tailed
conditional distribution of returns can achieve a similar effect (Boller-
slev, 1987). Extreme returns have been a pervasive feature of financial
markets throughout their history, and recent turbulence (e.g., Greenspan,
2007) suggests that tail risk will continue to be important in the new
century.
A deeper understanding of financial returns can be obtained by inves-
tigating their characteristics at various frequencies. For instance, the per-
sistence and variability of financial volatility can be apparent whether one
observes returns at intradaily, daily, weekly, monthly, yearly, or decennial
intervals.Thatis,ascasualobservationsuggests,therearevolatiledecades
and quiet decades, volatile years and quiet years, and so on. Intuition
suggests that these features are important for volatility forecasting, and
thatimprovedfilteringcanbeusedtodistinguishbetweencyclesofdifferent
durations.
The multifrequency nature of volatility is consistent with the intuition
thateconomicshockshavehighlyheterogeneousdegreesofpersistence.For
example,liquidityshockstendtobesuddenandtransitory,buttheirdura-
tion is quite random. The liquidity crisis that started in the fall of 2007 is
still in progress as we are writing these lines in April 2008. Volatility also
1Earlysuggestionsofvolatilitypersistenceandcorroborativeempiricalevidencefor
variousfinancialseriesappearinOsborne(1962),Mandelbrot(1963),Fama(1965,1970),
Beaver (1968), Kassouf (1969), Praetz (1969), Fisher and Lorie (1970), Fielitz (1971),
Black and Scholes (1972), Rosenberg (1972), Officer (1973), Hsu, Miller, and Wichern
(1974),Black(1976),LataneandRendleman(1976),andSchmalenseeandTrippi(1978).
Additionalevidenceisprovidedby,amongothers,Akgiray(1989),BaillieandBollerslev
(1989),Bollerslev(1987),Chou(1988),DieboldandNerlove(1989),French,Schwert,and
Stambaugh(1987),McCurdyandMorgan(1987),Milhoj(1987),PoterbaandSummers
(1986),Schwert(1989),andTaylor(1982).
2SeeDiebold(2004)foradiscussionofEngle’scontributions.
1.1 Empirical Properties of Financial Returns 3
variesoverhorizonsofafewyears,forinstanceinconjunctionwithearnings
andbusinesscycles(e.g.,Schwert,1989;HamiltonandLin,1996).Atlonger
horizons,slowvariationsinmacroeconomicvolatility,oruncertaintyabout
oilreserves,technology,andglobalsecuritycanaffectvolatilityoveragen-
eration or longer (e.g., Bansal and Yaron, 2004; Lettau, Ludvigson, and
Wachter, 2004).
This heterogeneity is accompanied by important nonlinearities. For
instance, the unconditional distribution of returns varies nonlinearly as
the frequency of observation changes (e.g., Campbell, Lo, and MacKinlay,
1997, Chapter 1). At short horizons, returns tend to be either close to the
mean or to take large values. By contrast, for longer horizons the bell and
the tails of the return distribution become thinner, while the intermediate
regionsgainmass.Thesenonlinearitiesarealsoapparentinthebehaviorof
returnmoments,asdocumentedbytheexpandingliteratureonpowervari-
ation.Inmanyfinancialseries,themomentsoftheabsolutevalueofreturns
vary as a power function of the frequency of observation. Additionally, the
growth rate of the qth moment is a nonlinear and strictly concave function
of q, a feature consistent with the nonlinear variations of the return distri-
bution with the sampling horizon (e.g., Andersen et al., 2001; Barndorff-
NielsenandShephard,2003;CalvetandFisher,2002a;Calvet,Fisher,and
Mandelbrot, 1997; Galluccio et al., 1997; Ghashghaie et al., 1996; Pasquini
and Serva, 1999, 2000; Richards, 2000; Vandewalle and Ausloos, 1998;
Vassilicos, Demos, and Tata, 1993).
Inequitymarkets,theunconditionaldistributionofreturnsisnegatively
skewed,sincelargenegativereturnsaremorefrequentlyobservedthanlarge
positivereturns.Moreover,aspointedoutbyFischerBlack(1976),volatil-
ity is typically higher after a stock market fall than after a stock market
rise, so stock returns are negatively correlated with future volatility. Black
hypothesized that this effect could be caused by financial leverage, which
risesafterthemarketvalueofafirmdeclinestherebytendingtoexacerbate
risk in the residual equity claim. The financial leverage channel is gener-
ally regarded as too small, however, to fully account for the skewness of
equity index returns (e.g., Schwert, 1989; Aydemir, Gallmeyer, and Holli-
field, 2006). An alternative explanation focuses on feedback between news
aboutfuturevolatilityandcurrentprices(Abel,1988;Barsky,1989;French,
Schwert,andStambaugh,1987;Pindyck,1984).Whenmarketparticipants
reviseupwardtheirforecastsoffuturedividendvolatility,theytendtoprice
downthestock,creatinganegativecorrelationbetweencurrentreturnsand
futurevolatility.Intuitionsuggeststhatheterogeneoushorizonscanplayan
importantroleinthiscontext.High-frequencyvolatilityshockscancapture
the dynamics of typical variations, while lower-frequency movements can
generate extreme returns through the feedback channel. Multifrequency
shocksmaythereforebehelpfultounderstandingtheskewnessandkurtosis
of asset returns.
