Multifractality in Asset Returns REStat 2002

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AdlaiFisher

UniversityofBritishColumbia-Vancouver

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The Review of Economics and StatisticsVOL LXXXIV NUMBER 3AUGUST 2002

MULTIFRACTALITY IN ASSET RETURNS THEORY AND EVIDENCE

Laurent Calvet and Adlai Fisher

AbstractmdashThis paper investigates the multifractal model of asset returns(MMAR) a class of continuous-tim e processes that incorporat e the thicktails and volatility persistence exhibited by many nancial time series Thesimplest version of the MMAR compounds a Brownian motion with amultifracta l time-deformation Prices follow a semi-martingale whichprecludes arbitrage in a standard two-asset economy Volatility has longmemory and the highest nite moments of returns can take any valuegreater than 2 The local variability of a sample path is highly heteroge-neous and is usefully characterized by the local Holder exponent at everyinstant In contrast with earlier processes this exponent takes a continuumof values in any time interval The MMAR predicts that the moments ofreturns vary as a power law of the time horizon We con rm this propertyfor Deutsche markUS dollar exchange rates and several equity seriesWe develop an estimation procedure and infer a parsimonious generatingmechanism for the exchange rate In Monte Carlo simulations the esti-mated multifractal process replicates the scaling properties of the data andcompares favorably with some alternative speci cations

I Introduction

THE multifractal model of asset returns (MMAR) is acontinuous-time process that captures the thick tails and

long-memory volatility persistence that are exhibited bymany nancial time series1 It is constructed by compound-ing a standard Brownian motion with a random time-deformation process which is speci ed to be multifractalThe time deformation produces clustering and long memoryin volatility and implies that the moments of returns vary asa power law of the time horizon We con rm this propertyempirically on the Deutsche markUS dollar exchange ratea US equity index and several individual stocks

The MMAR is characterized by a form of time-invariancecalled multiscaling which combines extreme returns withlong memory in volatility This speci cation improves ontraditional models with scaling features in several waysFirst the MMAR is consistent with economic equilibriumThe simplest version implies uncorrelated returns and semi-martingale prices thus precluding arbitrage in a standardtwo-asset economy The model also permits signi cant exibility in matching the data Returns have a nite vari-ance and their highest nite moment can take any valuegreater than 2 Consistent with many nancial series theunconditional distribution of returns displays thinner tails asthe time scale increases In contrast with earlier processeshowever the distribution need not converge to a Gaussian atlow frequencies and never converges to a Gaussian at highfrequencies The MMAR thus captures the distributionalnonlinearities that are exhibited by nancial data whileretaining the parsimony and tractability of scaling models

The time-deformation process is obtained as the limit ofa simple iterative procedure called a multiplicative cascadeThe construction begins with a uniform distribution ofvolatility at a suitably long time horizon and randomlyconcentrates volatility into progressively smaller time inter-vals The procedure follows the same rule at each stage ofthe cascade which provides parsimony and implies momentscaling By construction volatility clusters at all frequen-cies which is consistent with the intuition that economicfactors such as technological shocks business cycles earn-ings cycles and liquidity shocks have different time scales2

We anticipate that rational equilibrium models can generatethe MMAR either exogenously through multifractal shocksor endogenously due to market incompleteness or informa-tional cascades

The MMAR provides a fundamentally new class ofstochastic processes to nancial economists In particularthe multifractal model is a continuous diffusion that liesoutside the class of Ito processes Although these traditionalmodels locally vary as (dt)1 2 along their sample paths theMMAR generates the richer class (dt)a(t) where the localscale a(t) can take a continuum of values The relativeoccurrences of the local scales a(t) are conveniently sum-marized in a renormalized probability density called themultifractal spectrum Given a speci cation of the model

Received for publication June 12 2000 Revision accepted for publica-tion July 11 2001

Harvard University and University of British Columbia respectivel yWe are grateful to Benoit Mandelbrot for inspirationa l mentoring during

our years at Yale and beyond We also received helpful comments from TAndersen D Andrews D Backus J Campbell P Clark T Conley RDeo F Diebold J Geanakoplos W Goetzmann O Hart O Linton ALo E Maskin R Merton P C B Phillips M Richardson S Ross JScheinkman R Shiller C Sims J Stein J Stock B Zame S Zinanonymous referees and seminar participant s at numerous institutions including Yale Boston University British Columbia the Federal ReserveBoard Harvard Johns Hopkins MIT NYU Princeton Riverside theSanta Fe Institute the University of Colorado at Boulder the Universityof Miami the University of Utah the Berkeley Program in Finance andthe 1998 Summer Meeting of the Econometric Society

1 Long memory is convenientl y de ned by hyperbolicall y decliningautocorrelation s either for a process itself or functions of it This propertywas rst analyzed in the context of fractional integration of Brownianmotion by Mandelbrot (1965) and Mandelbrot and van Ness (1968) It hasbeen documented in squared and absolute returns for many nancial datasets (Taylor 1986 Ding Granger amp Engle 1993 Dacorogna et al 1993)Baillie (1996) provides a survey of long memory in economics 2 This idea is further elaborated by Calvet and Fisher (2001)

The Review of Economics and Statistics August 2002 84(3) 381ndash406copy 2002 by the President and Fellows of Harvard College and the Massachusetts Institute of Technology

we provide a general rule for calculating this function andderive its closed-form expressions in a number of examplesThe applied researcher can estimate the spectrum from themoments of the data and then infer the speci cation of themultifractal generating process

Our empirical work begins by examining the DeutschemarkUS dollar (DMUSD) exchange rate We use a high-frequency data set of approximately 15 million quotescollected over one year and a 23-year sample of dailyprices The exchange rate displays the moment-scalingproperty predicted by the model over a remarkable range oftime horizons We estimate the multifractal spectrum andinfer a generating mechanism that replicates DMUSD scal-ing Monte Carlo simulations then show that GARCH andFIGARCH are less likely to reproduce these results than theMMAR In addition we nd evidence of scaling in a USequity index and ve individual stocks3

Volatility modeling has received considerable attention in nance and the most common approaches currently includenumerous variants of the ARCHGARCH class (Engle1982 Bollerslev 1986) and stochastic volatility models(Wiggins 1987)4 Because early processes in this literaturehad dif culty capturing the outliers of nancial seriesresearchers have proposed conditional distributions of re-turns with thicker tails than the Gaussian In discrete timethese adaptations include the Student-t (Bollerslev 1987)and nonparametric speci cations (Engle amp Gonzalez-Rivera 1991) The problem of modeling thick tails is moreacute in continuous time and is typically addressed byincorporating an independent jump process Bates (19951996) thus nds that standard diffusions cannot producetails suf ciently fat to explain the implied volatility smile inoption prices and recommends the incorporation of jumpsAlthough a continuous diffusion the MMAR incorporatesenough bursts of extreme volatility to capture the fat tails of nancial series It also extends the characterization of vol-atility in continuous time by considering a multiplicity oflocal scales In particular the multifractal model can gen-erate local oscillations that are intermediate between Itodiffusion and discontinuous jumps

Although early processes from the ARCHGARCH liter-ature have weak persistence long memory in squared re-turns is a characteristic feature of FIGARCH (BaillieBollerslev amp Mikkelsen 1996) and the Long MemoryStochastic Volatility (LMSV) approach (Breidt Crato amp

DeLima 1998) The MMAR is reminiscent of the long-memory property of these models In addition the multi-fractal process has appealing temporal aggregation proper-ties and is parsimoniously consistent with the momentscaling of nancial data

Subsection IA discusses the relation between the MMARand earlier scaling models Section II de nes multifractalsand demonstrates their construction through a number ofsimple examples Section III formalizes the MMAR bycompounding a Brownian motion with a continuous time-deformation process Section IV shows that multifractalprocesses can take a continuum of local scales whosedistribution is conveniently characterized by the multifractalspectrum Section V extends the model to permit longmemory in returns which allows testing of the martingalehypothesis and may be useful in modeling economic serieswith persistence In section VI we verify the moment-scaling rule for DMUSD exchange rates and estimate thecorresponding multifractal spectrum We infer a data-generating process and show that simulated samples repli-cate the scaling features of the data Evidence of multifractalscaling is also found in a US equity index and veindividual stocks Section VII summarizes our results anddiscusses possible extensions

This paper simpli es the discussion and extends theresults of three earlier working papers (Mandelbrot Fisheramp Calvet 1997 Calvet Fisher amp Mandelbrot 1997 FisherCalvet amp Mandelbrot 1997) In the remainder of the textwe refer to the working papers as MFC CFM and FCMrespectively signifying the various permutations of theauthors All proofs are contained in the appendix

A Review of the Literature

The multifractal model combines several elements ofprevious research on nancial time series First the MMARgenerates fat tails in the unconditional distribution of re-turns and is thus reminiscent of the L-stable processes ofMandelbrot (1963)5 The MMAR improves on this earliermodel by generating returns with a nite variance as seemsto be empirically the case in most nancial series Further-more the L-stable model assumes that increments are inde-pendent through time and have thus the same variability atevery instant In contrast the MMAR helps model one ofthe main features of nancial markets uctuations in vol-atility

Second the multifractal model has long memory in theabsolute value of returns but the returns themselves have awhite spectrum Long memory is the characteristic featureof fractional Brownian motion (FBM) introduced by Kol-mogorov (1940) and Mandelbrot (1965) An FBM denotedBH(t) has continuous sample paths as well as Gaussian andpossibly dependent increments The FBM is an ordinary

3 The moment-scaling properties of nancial returns are also the objectof a growing physics literature (Galluccio et al 1997 Vandewalle ampAusloos 1998 Pasquini amp Serva 2000 Richards 2000) These contri-butions con rm that multiscaling is exhibited by many nancial timeseries and are thus complementary of the empirical work contained inFisher Calvet and Mandelbrot (1997) and further developed in this paperAlthough the physics literature focuses on these phenomenologica l regu-larities the MMAR is a parsimonious stochastic process that allows auni ed treatment of the theoretica l and empirical properties of the pricedynamics

4 See Ghysels Harvey and Renault (1996) for a recent survey of thestochastic volatility literature

5 Recent application s of the L-stable model to foreign exchange ratesinclude Koedijk and Kool (1992) and Phillips McFarland and McMahon(1996)

THE REVIEW OF ECONOMICS AND STATISTICS382

Brownian motion for H 5 1 2 is antipersistent when 0 H 1 2 and displays persistence and long memory when1 2 H 1 Granger and Joyeux (1980) and Hosking(1981) introduced ARFIMA a discrete-time counterpart ofthe FBM that helped the use of long memory in economicsFBM and ARFIMA do not disentangle volatility persistencefrom long memory in returns6 This has obvious limitationsin nancial applications and has led to the construction ofprocesses such as FIGARCH and LMSV Like these recentmodels the MMAR separates persistence in volatility frompersistence in returns but in a parsimonious continuous-time setting

The third essential component of the multifractal model isthe concept of trading time

De nition 1 Let B(t) be a stochastic process and u (t)an increasing function of t We call X(t) [ B[u (t)] acompound7 process The index t denotes clock time andu (t) is called the trading time or time-deformation process

When the directing process B is a martingale the tradingtime speeds up or slows down the process X(t) withoutin uencing its direction Compounding can thus separatethe direction and the size of price movements and has beenused in the literature to model the unobserved natural timescale of economic series (Mandelbrot amp Taylor 1967Clark 1973 Stock 1987 1988) More recently this methodhas been used to build models integrating seasonal factors(Dacorogna et al 1993 Muller et al 1995) and measuresof market activity (Ghysels Gourieroux amp Jasiak 1996)The MMAR also incorporates compounding and its pri-mary innovation is to specify the trading time u to bemultifractal Although the MMAR is not a structural modelof trade future work may de ne the trading time u to be afunction of observable data

Finally the MMAR generalizes the concept of scalingin the sense that a well-de ned rule relates returns overdifferent sampling intervals Mandelbrot (1963) sug-gested that the shape of the distribution of returns shouldbe the same when the time scale is changed or moreformally

De nition 2 A random process X(t) that satis es

$X~ct1 X~ctk 5d

$cHX~t1 cHX~tk

for some H 0 and all c k t1 tk $ 0 is calledself-af ne8 The number H is the self-af nity index orscaling exponent of the process X(t)

The Brownian motion the L-stable process and the FBMare the main examples of self-af ne processes in nanceEmpirical evidence suggests that many nancial series arenot self-af ne but instead have thinner tails and becomeless peaked in the bells when the sampling interval in-creases The MMAR captures this feature as well as ageneralized version of self-af nity exhibited by the dataWhile maintaining the simplicity of self-af ne processesthe MMAR is thus suf ciently exible to model the non-linearities fat tails and long-memory volatility persistenceexhibited by many nancial time series

II Multifractal Measures and Processes

The MMAR is constructed in section III by compoundinga Brownian motion B(t) with a random increasing func-tion u (t)

ln P~t 2 ln P~0 5 Bu ~t

The trading time u (t) will be speci ed as the cumulativedistribution function (cdf) of a multifractal measure m aconcept that we now present

A The Binomial Measure

Multifractal measures can be built by iterating a simpleprocedure called a multiplicative cascade We rst present oneof the simplest examples the binomial measure9 on [0 1]Consider the uniform probability measure m0 on the unitinterval and two positive numbers m0 and m1 adding up to1 In the rst step of the cascade we de ne a measure m1 byuniformly spreading the mass m0 on the left subinterval[0 12] and the mass m1 on the right subinterval [12 1] Thedensity of m1 is a step function as illustrated in gure 1a

In the second stage of the cascade we split the interval[0 12] into two subintervals of equal length The leftsubinterval [0 14] is allocated a fraction m0 of m1[0 12]whereas the right subinterval [14 12] receives a fractionm1 Applying a similar procedure to [12 1] we obtain ameasure m2 such that

m20 14 5 m0m0 m214 1 2 5 m0m1

m21 2 34 5 m1m0 m234 1 5 m1m1

Iteration of this procedure generates an in nite sequence ofmeasures (mk) that weakly converges to the binomial mea-sure m Figure 1b illustrates the density of the measure m4

obtained after k 5 4 steps of the recursionBecause m0 1 m1 5 1 each stage of the construction

preserves the mass of split dyadic intervals10 Consider the6 Taqqu (1975) establishes that BH(t) has long memory in the absolutevalue of increments when H 1 2

7 Processes of this type have also been called subordinated in the recenteconomics literature In mathematics subordinatio n differs from com-pounding and requires that u (t) have independent increments (Bochner1955 Feller 1968) The economics literature has evolved to describe anygeneric time-deformation process as a subordinato r

8 Self-af ne processes are sometimes called self-similar in the literature

9 The binomial is sometimes called the Bernoulli or Besicovitch mea-sure

10 A number t [0 1] is called dyadic if t 5 1 or t 5 h1221 1 1hk22k for a nite k and h1 hk 0 1 A dyadic interval hasdyadic endpoints

MULTIFRACTALITY IN ASSET RETURNS THEORY AND EVIDENCE 383

interval [t t 1 22k] where t 5 0 z h1 hk 5 yen i51

k h i22i

for some h1 hk 0 1 We denote by w0 and w1 therelative frequencies of 0rsquos and 1rsquos in (h1 hk) The measureof the dyadic interval then simpli es to m[t t 1 22k] 5m0

kw0m1kw1 This illustrates that the construction creates

large and increasing heterogeneity in the allocation ofmass As a result the binomial like many multifractalsis a continuous but singular probability measure that hasno density and no point mass

This construction is easily generalized For instance wecan uniformly split intervals into an arbitrary number b $ 2of cells at each stage of the cascade Subintervals indexedfrom left to right by b (0 b b 2 1) then receivefractions m0 mb21 of the measure We preserve mass

in the construction by imposing that these fractions alsocalled multipliers add up to 1 yen mb 5 1 This de nes theclass of multinomial measures which are discussed byMandelbrot (1989a)

A more signi cant extension randomizes the allocation ofmass between subintervals The multiplier of each cell isthen a discrete random variable Mb taking values m0m1 mb21 with probabilities p0 pb21 We pre-serve mass in the construction by imposing the additivityconstraint yen Mb 5 1 This modi ed algorithm generates arandom multifractal measure Figure 1c shows the randomdensity obtained after k 5 10 iterations with parametersb 5 2 p 5 p0 5 05 and m0 5 06 This density whichrepresents the ow of trading time begins to show the

FIGURE 1mdashCONSTRUCTION OF THE BINOMIAL MEASURE

In panels (a) and (b) the construction is deterministic with the fraction m0 5 06 of the mass always allocated to the left and fraction m1 5 04 always allocated to the right Panel (c) shows a randomized binomial measureafter k 5 10 stages The masses m0 and m1 each have equal probabilities of going to the left or right The nal panel shows the fractal character of ldquocutsrdquo of various sizes Each cut shows the set of instants at which the randommeasure in panel (c) exceeds a given level The clustering of these sets has a self-similar structure and the extreme bursts of volatility are intermittent as discussed in the appendix (subsection F)


properties that we desire in modeling nancial volatilityThe occasional bursts of trading time generate thick tails inthe compound price process and their clustering generatesvolatility persistence Because the reshuf ing of mass fol-lows the same rule at each stage of the cascade volatilityclustering is present at all time scales

B Multiplicative Measures

We can also consider nonnegative multipliers Mb (0 b b 2 1) with arbitrary distributions Assume forsimplicity that all multipliers are identically distributed(Mb

d5 M b) and that multipliers de ned at different

stages of the construction are independent The limitmultiplicative measure is called conservative when massis conserved exactly at each stage (yen Mb [ 1) andcanonical when it is preserved only on average ( (yenMb) 5 1 or equivalently M 5 1b) A canonicalmeasure can be conveniently generated by choosing in-dependent multipliers Mb within each stage of thecascade

The moments of multiplicative measures have interestingscaling properties To show this rst consider the generat-ing cascade of a conservative measure m Stage 1 uniformlysplits the unit interval into cells of length b21 and allocatesrandom masses M0 Mb21 to each cell Similarly themeasure of a b-adic cell of length Dt 5 b2k starting at t 50 z h1

hk 5 yen h ib2i is the product of k multipliers

m~Dt 5 Mh1Mh1h2 middot middot middot Mh1 hk (1)

Because multipliers de ned at different stages of the cas-cade are independent we infer that [m(Dt)q] 5 [ (Mq)]kor equivalently

m~Dtq 5 ~Dtt~q11 (2)

where t(q) 5 2logb (Mq) 2 1 The moment of anintervalrsquos measure is thus a power functions of the lengthDt This important scaling rule characterizes multifractals

Scaling relation (2) easily generalizes to a canonicalmeasure m which by de nition is generated by a cascadethat only conserves mass on average (yen Mb) 5 1 Themass of the unit interval is then a random variable V 5m[0 1] $ 0 More generally the measure of a b-adic cellsatis es

m~Dt 5 Mh1Mh1h2 middot middot middot Mh1 hkVh1hk (3)

where Vh1 hkhas the same distribution as V This directly

implies the scaling relationship

m~Dtq 5 ~Vq~Dtt~q11 (4)

which generalizes equation (2)The right tail of the measure m(Dt) is determined by the

way mass is preserved at each stage of the construction

When m is conservative the mass of the cell is boundedabove by the deterministic mass of the unit interval 0 m(Dt) m[0 1] 5 1 and has therefore nite moments ofevery order On the other hand consider a canonical mea-sure generated by independent multipliers Mb We assumefor simplicity that (Mq) ` for all q Guivarcrsquoh (1987)shows that the random mass V $ 0 of the unit interval thenhas a Paretian right tail

$V v C1v2qcr it as v 1`

where C1 0 and the critical moment qcrit is nite andlarger than 1 1 qcrit `11 By equation (3) the mass ofevery cell has the same critical moment qcrit as the randomvariable V The property qcrit 1 will prove particularlyimportant because it implies that returns have a nitevariance in the MMAR

The multiplicative measures constructed so far are grid-bound in the sense that the scaling rule of equation (4) holdsonly when t 5 0 z h1 hk and Dt 5 b2l l $ k Let denotethe set of couples (t Dt) satisfying scaling rule (4) hasinteresting topological properties that are summarized in Prop-erty 1 of the appendix (subsection A) Alternatively we canconsider grid-free random measures that satisfy scaling rule (4)for all admissible values of (t Dt) (Mandelbrot 1989a) Thisleads to the following

De nition 3 A random measure m de ned on [0 1] iscalled multifractal if it satis es

~mt t 1 Dtq 5 c~q~Dtt~q11

for all ~t Dt q

where is a subset of [0 1] 3 [0 1] is an interval andt(q) and c(q) are functions with domain Moreover[0 1] and satis es Property 1 which is de nedin the appendix

Maintaining the distinction between grid-bound andgrid-free measures would prove cumbersome and lead tounnecessary technicalities We will therefore neglect thedifference between the two classes in the remainder ofthis paper The interested reader can refer to Calvet andFisher (2001) for a detailed treatment of grid-free mul-tifractals

C Multifractal Processes

Multifractality is easily extended from measures to sto-chastic processes

De nition 4 A stochastic process X(t) is called multi-fractal if it has stationary increments and satis es

11 The cascade construction also implies that V satis es the invariancerelation yeni51

b MiVi

d5 V where M1 Mb V1 Vb are indepen-

dent copies of the random variables M and V


~ X~t q 5 c~qtt~q11 for all t q (5)

where and are intervals on the real line t(q) and c(q)are functions with domain Moreover and havepositive lengths and 0 [0 1] The function t(q) is called the scaling function of themultifractal process Setting q 5 0 in condition (5) we seethat all scaling functions have the same intercept t(0) 5 21In addition it is easy to show

Proposition 1 The scaling function t(q) is concaveWe will see that the distinction between linear and non-

linear scaling functions t(q) is particularly importantA self-af ne process X(t) is multifractal and has a

linear function t(q) as is now shown Denoting by H theself-af nity index we observe that the invariance conditionX(t) 5

d t HX(1) implies that ( X(t) q) 5 tHq ( X(1) q)Scaling rule (5) therefore holds with c(q) 5 ( X(1) q) andt(q) 5 Hq 2 1 In this special case the scaling functiont(q) is linear and fully determined by its index H Moregenerally linear scaling functions t(q) are determined by aunique parameter (their slope) and the corresponding pro-cesses are called uniscaling or unifractal

Uniscaling processes which may seem appealing for theirsimplicity do not satisfactorily model asset returns This isbecause most nancial data sets have thinner tails and becomeless peaked in the bell when the sampling intervals Dt in-creases In this paper we focus on multiscaling processeswhich have a nonlinear t(q) They provide a parsimoniousframework with strict moment conditions and enough exibil-ity to model a wide range of nancial prices

III The Multifractal Model of Asset Returns

We now formalize construction of the MMAR Considerthe price of a nancial asset P(t) on a bounded interval[0 T] and de ne the log-price process

X~t ln P~t 2 ln P~0

We model X(t) by compounding a Brownian motion with amultifractal trading time

Assumption 1 X(t) is a compound process

X~t Bu ~t

where B(t) is a Brownian motion and u (t) is a stochastictrading time

Assumption 2 Trading time u (t) is the cdf of a multi-fractal measure m de ned on [0 T]

Assumption 3 The processes B(t) and u (t) areindependent

This construction generates a large class of multifractalprocesses

We will show that the price process is a semi-martingalewhich implies the absence of arbitrage in simple economiesA straightforward generalization of this model allows thebroader class of fractional Brownians BH(t) in Assumption1 as developed in section V In Assumption 2 the multi-fractal measure m can be multinomial or multiplicativewhich implies a continuous trading time u (t) with nonde-creasing paths and stationary increments Assumption 3ensures that the unconditional distribution of returns issymmetric Weakening this assumption allows leverage ef-fects as in Nelson (1991) and Glosten Jagannathan andRunkle (1993) and is a promising direction for futureresearch

Under the previous assumptions

Theorem 1 The log-price X(t) is a multifractal processwith stationary increments and scaling function tX(q) [tu(q 2)

Trading time controls the tails of the process X(t) Asshown in the proof the q th moment of X exists if (and onlyif) the process u has a moment of order q 2 In particularif X(t) q is nite for some instant t then it is nite for allt We therefore drop the time index when discussing thecritical moment of the multifractal process

The tails of X(t) have different properties if the generat-ing measure is conservative or canonical This followsdirectly from the discussion of subsection IIA If m isconservative trading time is bounded and the process X(t)has nite moments of all (nonnegative) order Conservativemeasures thus generate ldquomildrdquo processes with relatively thintails Conversely the total mass u (T) [ m[0 T] of acanonical measure is a random variable with Paretian tailsIn particular there exists a critical exponent qcrit(u ) 1 fortrading time such that uq is nite when 0 q qcrit(u )and in nite when q $ qcrit(u )12 The log-price X(t) thenhas in nite moments and is accordingly called ldquowildrdquo Notehowever that X(t) always has nite variance sinceqcrit(X) 5 2qcrit(u ) 2 Overall the MMAR has enough exibility to accommodate a wide variety of tail behaviors

We can also analyze how the unconditional distributionof returns varies with the time horizon t Consider forinstance a conservative measure m such as a random bino-mial At the nal instant T the trading time u (T) isdeterministic implying that the random variable X(T) isnormally distributed As we move to a smaller horizon t theallocation of mass becomes increasingly heterogeneousas is apparent in gure 1 The tails of returns thus becomethicker at higher frequencies The mass of a dyadic cellcan be written as m[t t 1 22k] 5 m0

kw0m1k(12w0) where t 5

0 z h1 hk and w0 denotes the proportion of the

12 We also know that the scaling function tu(q) is negative when 0 q 1 and positive when 1 q qcri t(u )


multipliers Mh1 Mh1 hk

equal to m0 By the law oflarge numbers draws of w0 concentrate increasingly inthe neighborhood of 12 as k increases implying that thebell of the distribution becomes thicker The distributionof X(t) thus accumulates more mass in the tails and in thebell as the time horizon decreases while the middle ofthe distribution becomes thinner This property which isconsistent with empirical observations is further elabo-rated by Calvet and Fisher (forthcoming) In additionwhen the measure m is canonical the random variablesu (T) and X(T) have Paretian tails thus illustrating thatmultifractal returns need not converge to a Gaussian atlow frequency

The model also has an appealing autocorrelation structure

Theorem 2 The price P(t) is a semi-martingale (withrespect to its natural ltration) and the process X(t) is amartingale with nite variance and thus uncorrelated incre-ments

The model thus implies that asset returns have a whitespectrum a property that has been extensively discussed inthe market ef ciency literature13

The price P(t) is a semi-martingale14 which has importantconsequences for arbitrage15 Consider for instance the two-asset economy consisting of the multifractal security with priceP(t) and a riskless bond with constant rate of return r Follow-ing Harrison and Kreps (1979) we can analyze if arbitrage

pro ts can be made by frequently rebalancing a portfolio ofthese two securities Theorem 2 directly implies

Theorem 3 There are no arbitrage opportunities in thetwo-asset economy

This suggests that future research may seek to embed theMMAR in standard nancial models Because the price P(t) isa semi-martingale stochastic integration can be used to calcu-late the gains from trading multifractal assets which in futurework will greatly help to develop portfolio selection and optionpricing applications Further research will also seek to integratemultifractality into equilibrium theory We may thus obtain theMMAR in a general equilibrium model with exogenous mul-tifractal technological shocks in the spirit of Cox Ingersolland Ross (1985) Such a methodology is justi ed by themultifractality of many natural phenomena such as weatherpatterns and will help build new economic models of asset andcommodity prices Another line of research could also obtainmultifractality as an endogenous equilibrium property whichmight stem from the incompleteness of nancial markets(Calvet 2001) or informational cascades (Gennotte amp Leland1990 Bikhchandani Hirshleifer amp Welch 1992 Jacklin Klei-don amp P eiderer 1992 Bulow amp Klemperer 1994 Avery ampZemsky 1998)

Recent research focuses not only on predictability in returnsbut also on persistence in the size of price changes TheMMAR adds to this literature by proposing a continuous-timemodel with long memory in volatility Because the priceprocess is de ned only on a bounded time range the de nitionof long memory seems problematic We note however thatfor any stochastic process Z with stationary increments Z(aDt) [ Z(a 1 Dt) 2 Z(t) the autocovariance in levels

dZ~t q 5 Cov~ Z~a Dt q Z~a 1 t Dt q

13 See Campbell Lo and MacKinlay (1997) for a recent discussion ofthese concepts We also note that immediate extensions of the MMARcould add trends or other predictable components to the compoundprocess in order to t different nancial time series

14 Because the process X(t) 5 ln P(t) is a martingale Jensenrsquos inequal-ity implies that the price P(t) is a submartingal e but not a martingale Thisresult is of course not speci c to the MMAR

15 See Dothan (1990) for a discussion of semi-martingale s in the contextof nance

FIGURE 2mdashMMAR SIMULATION WITH RANDOM BINOMIAL TRADING TIME

This gure shows the rst differences of a simulation obtained by compounding a standard Brownian motion with a binomial trading time The simulation displays volatility clustering at all time scales andintermittent large uctuations


quanti es the dependence in the size of the processrsquos incre-ments It is well de ned when Z(a Dt) 2q is nite For a xed q we say that the process has long memory in the sizeof increments if the autocovariance in levels is hyperbolic int when tDt ` When the process Z is multifractal thisconcept does not depend on the particular choice of q16 Itis easy to show that when m is a multiplicative measure

Theorem 4 Trading time u (t) and log-price X(t) havelong memory in the size of increments

This result can be illustrated graphically Figure 2 showssimulated rst differences when u (t) is the cdf of arandomized binomial measure with parameter m0 5 06The simulated returns displays marked temporal heteroge-neity at all time scales and intermittent large uctuations

The MMAR is thus a exible continuous-time frameworkthat accommodates long memory in volatility a variety oftail behaviors and unpredictability in returns Furthermorethe multifractal model contains volatility persistence at alltime frequencies Table 1 compares the MMAR with exist-ing models of nancial time series

IV The Multifractal Spectrum

This section examines the geometric properties of samplepaths in the MMAR Although we previously focused onglobal properties such as moments and autocovariances wenow adopt a more local viewpoint and examine the regu-larity of realized paths around a given instant The analysisbuilds on a concept borrowed from real analysis the localHolder exponent On a given path the in nitesimal varia-tion in price around a date t is heuristically of the form17

ln P~t 1 dt 2 ln P~t Ct~dta~t

where a(t) and C t are respectively called the local Holderexponent and the prefactor at t As is apparent in thisde nition the exponent a(t) quanti es the scaling proper-ties of the process at a given point in time and is also calledthe local scale of the process at t

In continuous Ito diffusions the Holder exponent takesthe unique value a(t) 5 1 2 at every instant18 For thisreason traditional research obtains time variations in marketvolatility through changes in the prefactor Ct In contrastthe MMAR contains a continuum of local scales a(t) withinany nite time interval Thus multifractal processes are notcontinuous Ito diffusions and cannot be generated by stan-dard techniques Fractal geometry imposes that in theMMAR the instants t a(t) a with local scale lessthan a cluster in clock time thus accounting for the con-centration of outliers in our model The relative frequencyof the local exponents can be represented by a renormalizeddensity called the multifractal spectrum For a broad class ofmultifractals we calculate this spectrum by an applicationof large-deviation theory

A Local Scales

We rst introduce

De nition 5 Let g be a function de ned on the neigh-borhood of a given date t The number

a~t 5 Sup$b $ 0 g~t 1 Dt 2 g~t

5 O~ Dt b as Dt 0

is called the local Holder exponent or local scale of g at tThe Holder exponent thus describes the local scaling of apath at a point in time and lower values correspond tomore-abrupt variations The exponent a(t) is nonnegativewhen the function g is bounded around t as is always thecase in this paper The de nition readily extends to mea-sures on the real line At a given date t a measure simplyhas the local exponent of its cdf

We can easily compute Holder exponents for many func-tions and processes For instance the local scale of afunction is 0 at points of discontinuity and 1 at (nonsingu-lar) differentiable points Smooth functions thus have inte-gral exponents almost everywhere On the other hand theunique scale a(t) 5 1 2 is observed on the jagged samplepaths of a Brownian motion or of a continuous Ito diffusionSimilarly a fractional Brownian BH(t) is characterized by aunique exponent a(t) 5 H Thus the continuous processestypically used in nance each have a unique Holder expo-nent In contrast multifractal processes contain a continuumof local scales

The mathematics literature has developed a convenientrepresentation for the distribution of Holder exponents in amultifractal This representation called the multifractalspectrum is a function f(a) that we now describe From

16 Provided that Z(a Dt) 2q ` as is implicitly assumed in the restof the paper

17 The expression (dt)a(t) is an example of ldquononstandard in nitesimalrdquoas developed by Abraham Robinson

18 More precisely the set t a(t) THORN 1 2 of instants with a local scaledifferent from 12 has a Hausdorff-Besicovitc h measure (and therefore aLebesgue measure) equal to zero This set can thus be neglected in ouranalysis See Kahane (1997) for a recent survey of this topic

TABLE 1mdashCOMPARISON WITH OTHER MODELS

Models

PropertiesVolatility ClusteringMartingale Returns

CorrelatedReturns

MMAR FBM Long memoryContinuous time

FIGARCH ARFIMA Long memoryDiscrete time

GARCH ARMA Short memoryDiscrete time


De nition 5 the Holder exponent a(t) is the liminf of theratio

ln g~t Dt ln ~Dt as Dt 0

where consistent with previous notation g(t Dt) [ g(t 1Dt) 2 g(t) This suggests estimating the distribution of thelocal scale a(t) at a random instant For increasing k $ 1we partition [0 T] into bk subintervals [t i ti 1 Dt] wherelength Dt 5 b2kT and calculate for each subinterval thecoarse Holder exponent

ak~ti ln g~ti Dt ln Dt

This operation generates a set ak(t i) of bk observationsWe then divide the range of arsquos into small intervals of lengthDa and denote by Nk(a) the number of coarse exponentscontained in (a a 1 Da] It would then be natural tocalculate a histogram with the relative frequencies Nk(a)bk which converge as k ` to the probability that arandom instant t has Holder exponent a Using this methodhowever the histogram would degenerate into a spike andthus fail to distinguish the MMAR from traditional pro-cesses This is because multifractals typically have a dom-inant exponent a0 in the sense that a(t) 5 a0 at almostevery instant Mandelbrot (1974 1989a) instead suggested

De nition 6 The limit

f~a limln Nk~a

ln bk as k ` (6)

represents a renormalized probability distribution of localHolder exponents and is called the multifractal spectrum

For instance if b 5 3 and Nk(a) 5 2k the frequencyNk(a)bk 5 (23)k converges to zero as k ` while theratio ln Nk(a)ln bk 5 ln 2ln 3 is a positive constant Themultifractal spectrum thus helps to identify events thathappen many times in the construction but at a vanishingfrequency

Frisch and Parisi (1985) and Halsey et al (1986) in-terpreted f(a) as the fractal dimension of T(a) 5 t [0 T] a(t) 5 a the set of instants having local Holder exponent aFor various levels of the scale a Figure 1d illustrates thesubintervals with coarse exponent ak(ti) a When the num-ber of iterations k is suf ciently large these ldquocutsrdquo display aself-similar structure A more detailed discussion of this inter-pretation can be found in the appendix (subsection F)

B The Spectrum of Multiplicative Measures

We now use large-deviation theory to compute thespectrum of multiplicative measures First consider a con-servative measure m de ned on the unit interval [0 1] Afterk iterations we know the masses m[t t 1 Dt] 5Mh1

Mh1h2 Mh1 hk

in intervals of length Dt 5 b2k The

coarse exponents ak(t) 5 ln m[t t 1 Dt]ln Dt can thus berewritten as

ak~t 5 2~logb Mh1 1 middot middot middot 1 logb Mh1 hkk (7)

The multifractal spectrum is obtained by forming renormal-ized histograms Letting V i [ 2logb Mh1 h i

we caninterpret the coarse Holder exponents as draws of therandom variable

ak 51

ki51

k

V i (8)

The spectrum f(a) then directly depends on the asymptoticdistribution of ak

By the Strong Law of Large Numbers ak convergesalmost surely to19

a0 5 V1 5 2 logb M 1 (9)

As k ` almost all coarse exponents are contained in asmall neighborhood of a0 The standard histogram Nk(a)bk

thus collapses to a spike at a0 as anticipated in subsectionIVA The other coarse exponents nonetheless matter greatlyIn fact most of the mass concentrates on intervals withHolder exponents that are bounded away from a020 Infor-mation on these ldquorare eventsrdquo is presumably contained inthe tail of the random variable ak

Tail behavior is the object of large-deviation theory In1938 Cramer established the following theorem under con-ditions that were gradually weakened

Theorem 5 Let Xk denote a sequence of iid randomvariables Then as k `

1

kln

1

ki51

k

Xi a Infq

ln eq~a2X1

for any a X1

Proofs can be found in Billingsley (1979) and Durrett(1991) The theorem implies

Theorem 6 The multifractal spectrum f(a) is theLegendre transform

f~a 5 Infq

aq 2 t~q (10)

of the scaling function t(q)

19 The relation 2 logb M 1 follows from Jensenrsquos inequality andM 5 1b20 Let Tk denote the set of b-adic cells with local exponents greater than

(1 1 a0)2 When k is large Tk contains ldquoalmost allrdquo cells but its mass

t Tk

~Dtak~ t bk~Dt~a011 2 5 b2k~a021 2

vanishes as k goes to in nity


This result holds for both conservative and canonical mea-sures It provides the foundation of the empirical workdeveloped in section VI in which an estimation procedurefor the scaling function t(q) is obtained and the Legendretransform yields an estimate of the multifractal spectrumf(a)

The theorem allows us to derive explicit formulas for thespectrum in a number of useful examples To aid futurereference we denote by fu(a) the spectrum common to ameasure m and its cdf u Begin by considering a measuregenerated by a log normal multiplier M with distribution2logb M (l s2) Conservation of mass imposes that

M 5 1b or equivalently s2 5 2 ln b(l 2 1) It is easyto show that the scaling function t(q) [ 2logb ( Mq) 2 1has the closed-form expression t(q) 5 lq 2 1 2 q2s2(lnb) 2 We infer from Theorem 6 that the multifractal spec-trum is a quadratic function

fu~a 5 1 2 ~a 2 l24~l 2 1

parameterized by a unique number l 1 More generallytable 2 reports the spectrum when the random variable V isbinomial Poisson or Gamma (See CFM (1997) for de-tailed derivations) We note that the function fu(a) is verysensitive to the distribution of the multiplier which suggeststhat the MMAR has enough exibility to model a widerange of nancial prices In the empirical work this allowsus to identify a multiplicative measure from its estimatedspectrum

C Application to the MMAR

We now examine the spectrum of price processes gener-ated by the MMAR Denoting by fZ(a) the spectrum of aprocess Z(t) we show

Theorem 7 The price P(t) and the log-price X(t) haveidentical multifractal spectra fP(a) [ fX(a) [ fu(2a)

The log-price X(t) contains a continuum of local exponentsand thus cannot be generated by an Ito diffusion processLet a0(Z) denote the most probable exponent of a processZ Because a0(u) 1 the log-price has a local scalea0(X) [ a0(u ) 2 larger than 12 at almost every instantDespite their apparent irregularity the MMARrsquos samplepaths are almost everywhere smoother than the paths of a

Brownian motion Subsection IVB indicates that the vari-ability of the MMAR is in fact explained by the ldquorarerdquo localscales a a0(X) Although jump diffusions permit negli-gible sets to contribute to the total variation multifractalprocesses are notable for combining continuous paths withvariations dominated by rare events

Although the local scale is larger than 12 almost every-where Theorem 1 implies that the standard deviation of theprocess

$X~t 1 Dt 2 X~t2 5 cX~21 2 Dt

is of the order (Dt)1 2 Thus whereas most shocks are oforder (Dt)a0(X) the exponents a a0(X) appear frequentlyenough to alter the scaling properties of the variance Thiscontrasts with the textbook analysis that a standard devia-tion in (Dt)1 2 implies that most shocks are of the sameorder21 We expect these ndings to have interesting conse-quences for decision and equilibrium theory

V An Extension with Autocorrelated Returns

The multifractal model presented in section III is charac-terized by long memory in volatility but the absence ofcorrelation in returns Although there is little evidence offractional integration in stock returns (Lo 1991) longmemory has been identi ed in the rst differences of manyeconomic series22 including aggregate output (Adelman1965 Diebold amp Rudebusch 1989 Sowell 1992) theBeveridge (1921) wheat price index the US consumerprice index (Baillie Chung amp Tieslau 1996) and interestrates (Backus amp Zin 1993)23 This has led authors to modelthese series with the FBM or the discrete-time ARFIMA

21 Merton (1990 ch 3) provides an interesting discussion of multiplelocal scales and ldquorare eventsrdquo in nancial processes Assume that the pricevariation over a time interval Dt is a discrete random variable takingvalues e1 em with probability p1 pm and assume moreoverthat pi (Dt)q i e i (Dt)ri and r i 0 for all i Denote by I the eventsi such that piei

2 Dt When the variance of the process yen i51m piei

2 is of theorder Dt only events in I contribute to the variance If all events belongto I Merton establishes that only events of the order (Dt)1 2 matter TheMMAR shows that events outside I can play a crucial role in the statistica lproperties of the price process a property previously overlooked in theliterature

22 Maheswaran and Sims (1993) suggest potential application s in nancefor processes lying outside the class of semi-martingales

23 See Baillie (1996) for a review of this literature

TABLE 2mdashEXAMPLES OF MULTIFRACTAL SPECTRA

Distribution of V Multifractal Spectrum fu(a)

Normal (l s2) 1 2 (a 2 l)2[4(l 2 1)]

Binomial 2amax 2 a

amax 2 aminlog2

amax 2 a

amax 2 amin2

a 2 amin

amax 2 aminlog2

a 2 amin

amax 2 amin

Poisson (g) 1 2 gln b 1 a logb (gea)

Gamma (b g) 1 1 g logb (abg) 1 (g 2 ab)ln b

This table shows the multifractal spectrum of a multiplicative measure and its corresponding trading time when the random variable V 5 2logb M is respectively (1) a Gaussian density of mean l and variances2 (2) a binomial distribution taking discrete values amin and amax with equal probability (3) a discrete Poisson distribution p( x) 5 e2 ggxx and (4) a Gamma distribution with density p( x) 5 bgxg21e2 bxG(g)


speci cation We note however that these economic serieshave volatility patterns that seem closer to the multifractalmodel than to the fractional Brownian motion This suggestsusing the fractional Brownian motion of multifractal time

We model an economic series X(t) by replacing Assump-tion 1 in section III with

Assumption 1a X(t) is a compound process X(t) [BH[u (t)] where BH(t) is a fractional Brownian motion andu (t) is a stochastic trading time

In addition we maintain the multifractality of trading time(Assumption 2) and the independence of the processesBH(t) and u (t) (Assumption 3) Note that this coincideswith the earlier model if H 5 1 2 For other values of theindex H the increments of X(t) display either antipersistent(H 1 2) or positive autocorrelations and long memory(H 1 2) The more general model is fully developed byMFC CFM and FCM (1997)

The self-similarity of BH(t) implies

Theorem 8 The process X(t) is a multifractal processwith stationary increments scaling function tX(q) 5tu(Hq) and multifractal spectrum fX(a) 5 fu(aH)

The proof of these results is provided by MFC (1997) Weobserve that tX(1H) 5 tu(1) 5 0 which allows theestimation of the index H in the empirical work Thegeneralized construction has scaling properties analogous tothe model explored earlier and provides a useful additionaltool for empirical applications

VI Empirical Evidence

A Multifractal Moment Restrictions

Consider a price series P(t) on the time interval [0 T]and the log-price X(t) [ ln P(t) 2 ln P(0) Partitioning [0T] into integer N intervals of length Dt we de ne thesample sum or partition function

Sq~T Dti50

N21

X~iDt 1 Dt 2 X~iDt q (11)

When X(t) is multifractal the addends are identically dis-tributed and the scaling law (5) yields [Sq(T Dt)] 5Nc(q)(Dt)tX(q)11 when the q th moment exists This implies

ln Sq~T Dt 5 tX~q ln ~Dt 1 c~q (12)

where c(q) 5 ln c(q) 1 ln T For each admissible qequation (12) provides testable moment conditions describ-ing how the partition function varies with increment size DtVarious methods can be used to construct an estimator tX(q)from the sample moments of the data By equation (10) itsLegendre transform f(a) provides an estimate of the mul-tifractal spectrum and can then be mapped back into adistribution of the multipliers

The scaling function tX(q) is speci ed either parametri-cally or nonparametrically We can for instance choose aparametric family for the distribution of the multiplier MThe multifractal spectrum f(a) then belongs to a speci c

FIGURE 3mdashDMUSD DAILY DATA

The data is provided by Olsen and Associates and spans from June 1973 to December 1996 The outlined area labeled ldquoHF datardquo shows the one-year period from October 1992 to October 1993 that correspondsto the span of our high-frequency data Despite the apparent long cycles and clustering of volatility in the daily data the MMAR provides suf cient exibility to capture this behavior in a parsimonious stationarymodel Moreover both the daily data and the high-frequency data show similar scaling patterns


class of functions (table 2) a constraint that can be imposedin estimation On the other hand a nonparametric approachplaces fewer restrictions on the underlying process Becausetu(q) 5 2logb (Mq) 2 1 the sample moments provideall the nite moments of M and thus a great deal ofinformation on its underlying distribution24

This paper uses a very simple estimation procedureGiven a set of positive moments q and time scales Dt wecalculate the partition functions Sq(T Dt) of the data Thepartition functions are then plotted against Dt in logarithmicscales By equation (12) the multifractal model implies thatthese plots should be approximately linear when the q th

moment exists Regression estimates of the slopes thenprovide the corresponding scaling exponents tX(q) Thisprocedure reveals striking visual evidence of moment scal-ing in DMUSD data Simulation experiments are thenconducted to assess the joint performance of the multifractalmodel and the estimation methodology

B Deutsche MarkUS Dollar Exchange Rates

We begin by investigating the multifractality of theDeutsche markUS dollar (DMUSD) exchange rate Weuse two data sets provided by Olsen and Associates acurrency research and trading rm based in Zurich The rstdata set (ldquodailyrdquo) consists of a 24-year series of daily dataspanning June 1973 to December 1996 Olsen collects pricequotes from banks and other institutions through severalelectronic networks A price quote is converted to a singleprice observation by taking the geometric mean of theconcurrent bid and ask The reported price in the daily datais then calculated by linear interpolation of the price obser-vations closest to 1600 local UK time on each side25 Figure3 shows the daily data which exhibits volatility clustering atall time scales and intermittent large uctuations

The second data set (ldquohigh-frequencyrdquo) contains all bidask quotes and transmittal times collected over the one-yearperiod from October 31 1992 to September 1 1993 We

24 The distribution of M may not be uniquely determined by its mo-ments See Feller (1968) and Durrett (1991) for good discussions of theuniqueness problem in moments

25 An earlier working paper (FCM 1997) also uses noon buying ratesprovided by the Federal Reserve and nds no signi cant difference inreported results

FIGURE 4mdashDMUSD PARTITION FUNCTIONS FOR THE FULL SET OF TIMESCALES AND MOMENTS q NEAR 2

This gure identi es a scaling region from about 14 hours to at least six months the largest horizon for which the partition function was calculated As discussed in subsection VIC the change in scaling at highfrequencies is consistent with market frictions such as bid-ask spread discreteness of trading units and noncontinuou s trade Moments q near 2 were chosen to investigate the martingale hypothesi s for returns bythe equation t(q 5 1H) 5 0 We nd a at slope near q 5 188 implying H 5 053 or slight persistence Using simulation methods subsection VID investigates whether this evidence is suf cient to rejectthe martingale version of the MMAR


convert quotes to price observations using the same meth-odology as Olsen and obtain a round-the-clock data set of1472241 observations Olsen provides a ag for quotesthat are believed to be erroneous or not representative ofactual willingness to trade We eliminate these observationswhich constitute 036 of the data set Combining the dailydata and the high-frequency data allows us to calculatepartition functions over three orders of magnitude for Dt

The high-frequency data show strong patterns of dailyseasonality In continuous time seasonality is a smoothtransformation that does not affect local Holder exponentsBecause our data is discrete however we may expect

seasonality to introduce noise To reduce this effect we canwrite a seasonally modi ed version of the MMAR

ln P~t 2 ln P~0 5 BH$uSEAS~t

where the seasonal transformation SEAS(t) is a differentia-ble function of clock time In this paper we use a pre lterthat smoothes variation in average absolute returns over fteen-minute intervals of the week Except for thereduction in noise there are no systematic differences inreported results for ltered and un ltered data An earlierworking paper (FCM 1997) provides details on this and

FIGURE 5mdashDMUSD PARTITION FUNCTIONS IN THE SCALING REGION FOR MOMENTS 15 q 5

For each moment the rst solid line plotted from 14 hours to two weeks corresponds to the high-frequency data The second solid line ranges from Dt 5 1 day to six months and corresponds to thedaily data The lines are remarkably straight as predicted by the model and have nearly identical slopes Also their scaling is noticeably different from that of the Brownian motion which is shown bythe dotted lines


three other seasonal pre lters and nds small predictabledifferences in results depending on the deseasonalizingmethod

C Main Results

Figures 4 and 5 illustrate the partition functions of thetwo DMUSD data sets Values of Dt are chosen to increasemultiplicatively by a factor of 11 from minimum to maxi-mum Because we focus on the slopes tX(q) but not theintercepts plots for each q are renormalized by verticaldisplacement to begin at zero for the lowest value of Dt ineach graph This allows plots for many q to be presentedsimultaneously The daily and high-frequency plots arepresented in the same graph to highlight the similarity intheir slopes This is achieved by a second vertical displace-ment of the daily data that provides the best linear t underOLS restricting both lines to have the same slope

Figure 4 shows the full range of calculated Dt from fteen seconds to six months and ve values of q rangingfrom 175 to 225 This allows us to estimate the self-af nityindex H in the extended model presented in section VBecause tX(1H) 5 0 and the standard Brownian speci -cation H 5 1 2 has previous empirical support we expectto nd tX(q) 5 0 for a value of q near 2

We rst note the approximate linearity of the partitionfunctions beginning at Dt 5 14 hours and extending to thelargest increment used Dt 5 6 months In this range theslope is zero for a value of q slightly smaller than 2 and wereport H rsquo 053 which implies very slight persistence inthe DMUSD series It is not immediately clear whether thisresult is suf ciently close to H 5 1 2 to be consistent withthe martingale version of the MMAR but we will return tothis issue in the following section using simulation methods

The partition functions in gure 4 also show breaks inlinearity at high frequencies These are consistent withmicrostructure effects such as bid-ask spreads discretenessof quoting units and discontinuous trading In particularthese microstructure effects can be expected to induce anegative autocorrelation at high frequencies as is wellunderstood in the case of bid-ask bounce (Roll 1984)Negative autocorrelation effectively acts as an additionalsource of volatility as previously explored in the varianceratio literature (Campbell amp Mankiw 1987 Lo amp Mac-Kinlay 1988 Richardson amp Stock 1989 Faust 1992) Theresults in gure 4 are analogous to variance ratio testsexactly so if we focus on the moment q 5 2 As we moveto the left on the graph and sampling frequency increasesmicrostructure-induced negative autocorrelation increasesand the plots bend upwards corresponding to the increase invariability

Descriptive statistics help to con rm that high-frequencybreaks in linearity are caused by microstructure effects Thedeparture from linearity begins at a frequency of approxi-mately Dt 5 14 hours which is highlighted by the dottedline in gure 4 We rst note that the absolute change in the

DMUSD rate averages 014 pfennig26 over a time incre-ment of 14 hours Comparing this to the average spread of007 pfennig27 we observe that the spread covers a signif-icant proportion of average variation at this time horizon Itis thus sensible that microstructure effects begin to effectscaling properties at this frequency To further con rm thisintuition observe that for time scales between 36 minutesand 14 hours the partition function has an approximateslope of zero for the moment q 5 225 This implies thatH rsquo 044 1 2 which is consistent with the explanationthat microstructure-induced negative autocorrelation causesthe observed departure from scaling in this high-frequencyregion28

These high-frequency microstructure effects have severalpotential solutions First we could view the MMAR as anunderlying price process and overlay it with a structuralmodel of discrete trade and bid-ask spreads that could thenbe related to the observed Olsen quotes This would permitusing the data at all frequencies but at the cost of introduc-ing a new set of modeling issues A second alternativeconsists of pre ltering the data to remove high-frequencyautocorrelations This approach has recently been employedby Andersen Bollerslev Diebold and Ebens (2001) whouse an MA(1) lter to remove rst-order serial correlationfrom ve-minute returns on the thirty stocks tracked in theDow Jones Industrial Index Although this would certainlyremove high-frequency autocorrelations the effect of thisprocedure on scaling properties would require further inves-tigation For simplicity we choose to discard from furtheranalysis all values of Dt less than 14 hours29 and havethree orders of magnitude of sampling frequencies withwhich to test the scaling properties of the data

With attention now restricted to values of Dt between 14hours and six months gure 5 presents partition functionsfor a larger range of moments 15 q 5 Highermoments capture information in the tails of the distributionof returns and are thus generally more sensitive to devia-tions from scaling All of the plots are nonetheless remark-ably linear and the overlapping values from the two datasets appear to have almost the same slope Thus despite theapparent nonstationarity of the 24-year series such as longprice swings and long cycles of volatility the moment

26 One pfennig equals 001 DM27 The two most common spread sizes are 005 pfennig (3825) and

010 pfennig (5255) together comprising more than 90 of all ob-served spreads

28 To further con rm this explanation one could compare across assetsthe frequency at which departures from moment scaling begin If themicrostructur e explanation is correct one would expect scaling to extendto higher frequencie s when trading frictions (measured by variables suchas bid-ask spreads or the average time interval between trades) are lower

29 As an approximation the choice of Dt 5 14 hours as the high-frequency cutoff for our analysis is justi ed by gure 4 but the exactvalue was chosen by ad hoc rounding In the future there may be somegains to adopting a more formal test for the high-frequenc y cutoff valueThere is already a substantia l econometric literature (such as Andrews(1993)) that could be adapted to this purpose


restrictions imposed by the MMAR seem to hold over abroad range of sampling frequencies

Estimates of the slopes in gure 5 and additional mo-ments q are then used to obtain estimated scaling functionstX(q) for both data sets We note the increasing variabilityof the partition function plots with the time scale Dt whichcan be attributed to the shrinking number of addends in thepartition function at low frequencies This suggests aweighted least squares or generalized least squares ap-proach In practice however weighting the observationshas little effect on the results because the plots are verynearly linear Preferring simplicity we thus report in gure6 the estimated scaling functions from OLS regressions Theestimated scaling functions are strictly concave indicatingmultifractality and are fairly similar except for very largemoments

Theorem 6 suggests to estimate the multifractal spectrumfX(a) by taking the Legendre transform of tX(q) Followingthis logic gure 7 shows the estimated multifractal spec-trum of the daily data30 The estimated spectrum is concavein contrast to the degenerate spectra of Brownian motionand other unifractals Using the estimated spectrum we canrecover a generating mechanism for trading time based onthe canonical multiplicative cascades described in subsec-tion IIB

The spectrum of daily data is very nearly quadratic andsubsection IVB has shown that quadratic spectra are gener-ated by log normally distributed multipliers M We thusspecify 2logb M (l s2) giving trading time u (t) withmultifractal spectrum fu(a) 5 1 2 (a 2 l)2[4(l 2 1)]

The log-price process has most probable exponent a0 5lH and spectrum

fX~a 5 1 2~a 2 a0

2

4H~a0 2 H

Because H 5 053 the free parameter a0 is used to t theestimated spectrum We report a050589 which producesthe parabola shown in gure 7 Choosing a generatingconstruction with base b 5 231 this immediately impliesthat l 5 111 and s2 5 03232 It is also natural to considerthe martingale version of the MMAR with the restrictionH 5 1 2 For this case we estimate the single parametera050545

In both cases the estimated value of the most probablelocal Holder exponent a0 is greater than 12 On a set ofLebesgue measure 1 the estimated multifractal process istherefore more regular than a Brownian motion Howeverthe concavity of the spectrum also implies the existence oflower Holder exponents that correspond to more-irregularinstants of the price process These contribute dispropor-tionately to volatility

30 The estimated multifractal spectrum of the high-frequenc y data issimilar in many respects and is discussed by FCM (1997)

31 The base b of the multifractal generating process is not uniquelyidenti ed by the spectrum alone hence we assume the commonly usedvalue b 5 2 Calvet and Fisher (2001) develop a likelihood-base d lterunder which b can be estimated for the class of multinomial multifractals

32 Mandelbrot (1989a 1989b) shows that the partition function method-ology provides reasonable estimates of tX(q) only for moments q 1=a0(X)H 2 1 which is approximatel y equal to 566 in our estimatedprocess

FIGURE 6mdashESTIMATED DMUSD SCALING FUNCTIONS

For each partition function Sq(T Dt) we estimate the slope using OLS to obtain tX(q) The estimated scaling functions for both data sets are concave and have a similar shape until high moments are reachedThe contrast with Brownian motion is shown by the dotted line


D Monte Carlo Simulations

We now present simulation experiments that provide apreliminary assessment of the new model and the estimationprocedure Figure 8 shows the levels and log differences ofa random price path generated by the limit log normalMMAR estimated in subsection VIC33 The simulationshows a variety of large price changes apparent trendspersistent bursts of volatility and other characteristics foundin the DMUSD series

The following sections examine whether the inferredprocess captures the moment properties of the data Wesketch the simulation methodology and then provide asynthetic discussion of the numerical results

Methodology We use three types of tests to analyze theMMARrsquos performance First visual evidence is provided onthe moment properties of simulated data Figure 9a thusillustrates the partition functions corresponding to four sim-ulations of the MMAR For comparison we report in gure9b the partition functions of a GARCH(1 1) process withthe parameter estimates of Baillie and Bollerslev (1989)Figure 9c similarly considers the FIGARCH (1 d 0)speci cation of Baillie Bollerslev and Mikkelsen (1996)Each plot in gure 9 is based on a long sample of 100000observations This sample lengthmdashwhich exceeds the sam-ple size of 6118 daily DMUSD returnsmdashhas the advantageof reducing the noisiness of the partition functions

Second we consider small samples and examine distri-butional evidence on the linearity and slopes of the partitionfunctions The analysis focuses on four processes the ex-

33 The simulation of a multifractal path is discussed in the appendix(subsection I)

FIGURE 7mdashESTIMATED MULTIFRACTAL SPECTRUM OF DAILY DMUSD DATA

The estimated spectrum is obtained from the Legendre transform fX~a 5 Infqaq 2 t~q shown in this graph by the lower envelope of the dotted lines The shape is nearly quadratic with a tted parabola

shown by marked symbols A quadratic spectrum implies a log normal distribution for multipliers M in the multifractal-generating mechanism


tended MMAR (with arbitrary H) the martingale MMAR(H 5 1 2) FIGARCH and GARCH These models arerespectively indexed by m 1 4 For each modelm we simulate J 5 10000 paths with the same length T 56118 as the DMUSD data We denote each path by Y j

m 5Y jt

m t51T (1 j J) and focus the analysis on the

moments q Q 5 05 1 2 3 5 For each path andeach q an OLS regression provides a slope estimate t(qYj

m) and the corresponding sum of squared errors SSE(qYj

m) The distributions of these statistics appear unimodalwith smoothly declining tails Tables 3 and 4 report thepercentiles of these statistics

We summarize these ndings with several measures ofglobal t For a given model m 1 4 each pathYj

m generates a column vector of slope and SSE estimates34

h~Y jm 5 $t~q Y j

m ln SSE~q Y jmq Q9

It is convenient to denote the data by X 5 X t t51T and to

arrange the simulated paths in a J 3 T matrix Ym 5[Y1

m YJm]9 We also consider

H~X Ym 5 h~X 21

Jj51

J

h~Y jm

The function H is useful to test how a particular model tsthe moment properties of the data In particular we cande ne a global statistic G 5 H9WH for any positive-de nite matrix W The empirical work consider four differ-

ent weighting matrices Wm m 1 4 each ofwhich is obtained by inverting the simulated covariancematrix of moment conditions Wm 5 [yen j51

J H(Y jm

Ym) H(Y jm Ym)9J]21 The global statistics

Gmn~X

5 H~X Ym9WnH~X Ym m n $1 4

are indexed by the model m that generates the simulateddata Ym and the model n that generates the weighting matrixWn This gives a set of sixteen global statistics Assumingthat m is the true model we can estimate the cumulativedistribution function mn of each statistic from the setGmn(Yj

m)1 j J and then quantify the p-value 1 2mn[Gmn(X)] The global statistics Gmn(X) and their as-

sociated p-values are reported in table 5

Results The simulation results in gure 9 and tables 3through 5 con rm that the MMAR replicates the mainscaling features of the data The partition function plots in gure 9a are approximately linear and tend to follow theirtheoretically predicted slopes which are nearly identical tothe estimates from the DMUSD data Tables 3 and 4 permita more detailed assessment The extended MMAR is veryclose to the data in both its theoretically predicted slopes t0

and the mean slopes t Furthermore the estimated slopesfrom the DMUSD data are well within the central bells ofthe simulated slope distributions generated by the extendedMMAR

The martingale version is subtly different in both itstheoretically predicted and mean slopes For low moments

34 We use the logarithm of the SSE in calculating the global statisticsbecause table 4 shows that the SSE are heavily right-skewed

FIGURE 8mdashSIMULATED MULTIFRACTAL-GENERATING PROCESS FOR THE DMUSD DATA

We use the estimated values of H 5 053 and a0 5 0589 with the limit log normal construction of trading time The plots show volatility clustering at all time scales and occasional large uctuations


the estimated slopes from the DMUSD data are moretowards the upper tails of the simulated distributions gen-erated by the martingale MMAR In both cases but more sofor the martingale version there appears to be a slightdownward bias in the average simulated slope relative to itstheoretical value Future work may thus correct the bias inour estimation method by matching simulated moments(Ingram amp Lee 1991 Duf e amp Singleton 1993)

Table 4 analyzes the variability of the simulated partitionfunctions around their slopes The extended and martingaleversions of the MMAR yield nearly identical results Forlow moments the data falls well within the likely range ofthe SSE statistic for both models For high moments thepartition functions are typically more variable for the sim-ulated MMAR than for the data This at rst seems curiousbecause the model has been designed speci cally to producescaling It is consistent however with the nding of a slightdownward bias in the slopes of the estimated partition

functions Correcting this bias would give a slightly mildermultifractal process and thus reduce the variability of thepartition functions This presents a promising avenue forimproving estimation Overall these results suggest that theMMAR is successful in matching the main scaling featuresof the data

Another important question is whether other standardeconometric models possess scaling properties Figure 9bshows that GARCH(1 1) partition functions are fairlylinear but their apparent slope is similar to the predictedslope of Brownian motion rather than the data This issymptomatic of the fact that GARCH models are short-memory processes Over long time periods temporal clus-tering disappears and GARCH scales like a Brownian mo-tion Tables 3 and 4 con rm this visual evidence The SSEstatistics show that GARCH tends to be as linear as the databut for two of the ve moments the slopes from the dataare in the extreme tails of their distributions simulated underGARCH Because it contains long memory in volatilityFIGARCH can be expected to scale differently than Brown-ian motion at low frequencies This is con rmed in gure9c however the same plots suggest that simulatedFIGARCH partition functions are more irregular than thescaling plots generated by GARCH the MMAR and thedata Tables 3 and 4 again complement this visual evidenceThe simulated FIGARCH slopes improve over the GARCHslopes but are not as close to the data as the MMAR TheSSEs from the data are also far in the tails of their distri-butions generated under FIGARCH

The previous analysis has separately assessed ten mo-ment conditions that capture different scaling features of thedata We now consider the evidence provided by the globalstatistics which are quadratic functions of these ten momentconditions Each column of the results in table 5 is obtainedby a different weighting of the set of quadratic terms so thatwithin-column comparisons provide four separate views ofability to t scaling features of the data Each of theweighting matrices of course has different power against agiven model and asymptotic theory suggests that the most-powerful weighting matrix for each model is provided bythe inverse of its own covariance matrix of moment condi-tions Thus we expect the diagonal entries of the table toprovide the greatest power to reject each model and this isconsistent with our results Whether evaluated column-wiseor by the diagonal elements the results con rm that theMMAR is best able to replicate the scaling properties of thedata

These simulations are of course only a preliminary step toevaluating the usefulness of the multifractal model None-theless our results demonstrate that scaling properties con-tain important information for estimating and discriminatingbetween models A natural path for future work will be toincorporate this information in broader estimation and test-ing procedures As these techniques develop (for exampleCalvet and Fisher (2001)) and lead to more-rigorous eval-

FIGURE 9mdashSIMULATED PARTITION FUNCTIONS

Each panel shows estimated partition functions for a simulated sample of 100000 observations Thedata-generating process in (a) uses our estimates from the extended MMAR The data-generatingprocesses in (b) and (c) use speci cations from previously published research on daily DMUSDexchange rates Large simulated samples are used to reduce noise Dotted lines in each gure representthe scaling predicted for Brownian motion and dashed lines represent the scaling found in the data TheMMAR appears most likely to capture scaling in the DMUSD data The GARCH simulations tend toscale like Brownian motion FIGARCH simulations occasionally show scaling that is similar to the dataas in panel (c2) but in general tend to be much more irregular


uations it will be interesting to discover whether the prom-ise of these early simulation results is ful lled

E Equity Data

After observing multifractal properties in DMUSD ex-change rates it is natural to test the model on other nancialdata This section presents evidence of moment scaling in asample of ve major US stocks and one equity index35

The Center for Research in Security Prices (CRSP) pro-vides daily stock returns for 9190 trading days from July1962 to December 1998 We present results for the value-weighted NYSE-AMEX-NASDAQ index (ldquoCRSP Indexrdquo)and ve stocks Archer Daniels Midland (ADM) GeneralMotors (GM) Lockheed-Martin Motorola and United Air-lines (UAL) The individual stocks are issued by large

35 The multifractal model offers a exible framework that may beamenable to many types of nancial prices In particular equity data

requires additiona l consideratio n for the relationship between volatilityand mean returns Because this has not been incorporated in the version ofthe model presented in this paper the results in this section should beinterpreted as a model-free investigation of scaling

TABLE 3mdashMOMENT SCALING IN MODEL SIMULATIONS ESTIMATED SLOPES

A Predicted Slopes and Mean Slopes from Model Simulations and DMUSD Data

q

DMUSD Data i MMAR ii MMAR (H 5 12)iii FIGARCH

(1 d 0) iv GARCH (1 1)

t t0 t sim sd t0 t sim sd t sim sd t sim sd

05 20711 2071 2071 001 2073 2073 001 2074 001 2075 00110 20440 2044 2044 002 2049 2048 002 2049 003 2050 00220 0058 006 005 006 000 2001 006 2001 008 2001 00530 0500 049 047 012 043 041 010 042 016 047 00850 1208 119 116 027 116 110 024 114 035 136 017

B Probabilities that the Simulated Model Slopes are Less than the Slopes from DMUSD Data (tsim t)

05 05910 09812 09969 0999310 05465 09596 09787 0995920 05620 08867 08363 0918630 06032 08111 07209 0658750 05757 06771 05656 01754

C Percentiles of the Simulated Model SlopesP05 P1 P25 P5 P10 P25 P50 P75 P90 P95 P975 P99 P995

i MMAR

05 20745 20742 20738 20734 20729 20722 20714 20705 20698 20694 20690 20686 2068310 20506 20501 20493 20485 20475 20460 20443 20426 20412 20403 20396 20386 2037920 20106 20091 20069 20052 20028 0009 0049 0090 0126 0149 0170 0197 021530 0159 0192 0246 0284 0328 0399 0472 0549 0622 0666 0712 0758 079850 0343 0465 0601 0705 0821 0988 1162 1331 1491 1595 1691 1793 1871

ii MMAR (H 5 12)

05 20766 20762 20758 20754 20750 20743 20735 20727 20720 20716 20712 20708 2070510 20542 20536 20527 20519 20511 20497 20480 20465 20451 20443 20436 20428 2042320 20150 20136 20114 20098 20078 20045 20008 0029 0062 0083 0100 0124 013830 0143 0177 0211 0244 0283 0344 0410 0479 0542 0584 0621 0666 070750 0412 0502 0633 0714 0807 0954 1106 1258 1401 1490 1581 1684 1779

iii FIGARCH (1 d 0)

05 20776 20773 20768 20764 20760 20753 20744 20736 20729 20725 20721 20716 2071310 20566 20556 20546 20537 20527 20511 20494 20476 20460 20450 20442 20430 2042220 20244 20211 20174 20141 20111 20064 20016 0034 0084 0115 0147 0191 022130 20082 20003 0086 0151 0220 0320 0416 0514 0608 0680 0737 0815 085750 0033 0200 0412 0559 0703 0934 1155 1362 1560 1699 1807 1935 2031

iv GARCH (1 1)

05 20777 20774 20770 20766 20762 20755 20747 20740 20733 20729 20725 20721 2071810 20555 20548 20541 20534 20525 20512 20497 20482 20470 20461 20454 20447 2044220 20121 20110 20095 20080 20063 20036 20005 0025 0053 0071 0084 0101 011330 0281 0302 0322 0347 0375 0420 0470 0521 0571 0601 0629 0664 068550 0928 0978 1037 1094 1150 1250 1352 1461 1569 1641 1709 1790 1859

This table is based upon J 5 10000 simulated paths for each of the four models In panel A the column t0 is the theoretically predicted slope under the MMAR as calculated in subsection IVB The columnt shows the average slope over the J paths and sd the standard deviation Panel B provides the percentage of the simulated paths with slope values less than observed in the daily DMUSD data and panel C givespercentiles of the simulated distribution under each model


well-known corporations from various economic sectorsand have reported data for the full CRSP sample span36 Foreach series we convert the daily return data into a renor-malized log-price series X t and then apply the partitionfunction methodology described in subsection VIB37

Figure 10 shows results for the CRSP index and GMIn the rst two panels the full data sets are used with

increments Dt ranging from one day to approximatelyone year The partition functions for moments q 5 1 23 are approximately linear for both series with littlevariation around the apparent slope The slope for themoment q 5 2 is noticeably positive for the CRSP indexindicating persistence This characteristic is very atypicalof individual securities although short-horizon persis-tence has been observed previously in index returns andis typically attributed to asynchronous trading (Bou-doukh Richardson amp Whitelaw 1994) In contrast to theresults for low moments the partition functions fromboth series vary considerably for the moment q 5 5 This

36 Choosing stocks with full samples allows testing of the moment-scaling restrictions over a larger range of frequencies

37 The CRSP holding period returns r t 5 (P t 2 Pt21 1 d t)Pt21 includecash distributions dt We construct the series Xt t50

T by X0 5 0 X t 5Xt21 1 ln (1 1 rt)

TABLE 4mdashSUM OF SQUARED ERRORS IN MODEL SIMULATIONS

A Mean SSE from Simulated Models and DMUSD Data

q

DMUSD Data i MMARii MMAR(H 5 12)

iii FIGARCH(1 d 0)

iv GARCH(1 1)

SSE SSE sd SSE sd SSE sd SSE sd

05 0018 00200 00080 00213 000805 00241 000876 00214 00077910 00507 00604 00278 00637 00266 00832 00417 00586 0022920 0159 0330 0212 0311 0167 0635 0635 021 0093730 042 149 114 129 0864 298 295 0698 03950 276 101 713 984 601 173 137 491 354

B Probabilities that the Simulated Model SSE are Less than in the DMUSD Data

05 04640 03914 02541 0373310 04265 03553 01648 0419120 01084 01000 00204 0321130 00156 00197 00026 0190250 00155 00283 00034 02293

C Percentiles of the Simulated Model SSEP05 P1 P25 P5 P10 P25 P50 P75 P90 P95 P975 P99 P995

i MMAR

05 000703 000774 000889 00101 00116 00144 00185 00238 00298 00345 00394 00461 0052810 00206 00223 00253 00289 00331 0042 00546 00716 00938 0112 0131 0158 017320 00936 0103 0118 0133 0156 0203 0278 0392 0551 0695 087 111 13430 0351 0387 0457 0528 0625 0834 118 177 265 345 447 586 71850 223 253 298 344 411 565 818 123 184 234 287 362 425

ii MMAR (H 5 12)

05 000806 000882 000993 00111 00127 00157 00199 00253 00315 00364 00412 00469 0052710 00224 00246 00281 00318 00362 00455 00583 00759 00964 0112 013 0157 017520 00973 0106 0121 0138 0159 0203 0271 0369 0503 0617 0734 0902 10830 0332 0372 0437 05 0583 0769 106 152 221 281 355 465 55850 205 233 267 316 372 502 724 107 157 202 249 321 383

iii FIGARCH (1 d 0)

05 000873 000969 00112 00126 00144 00179 00229 00287 00355 00403 00447 00507 0056110 0026 00291 0034 00386 00448 00572 00745 00984 0129 0155 0185 0228 027620 0126 014 0165 0192 0227 031 0461 073 118 162 214 31 41830 0453 0527 064 0751 0931 134 211 36 575 774 103 143 19450 299 342 413 494 607 873 135 214 321 415 524 689 853

iv GARCH (1 1)

05 000804 000872 000995 00111 00128 00159 00202 00254 00315 00357 00397 00455 00510 00212 00232 00265 00299 00343 00427 00546 00697 00874 0101 0115 0135 014920 00725 00793 0091 0103 0116 0147 019 0249 0325 0383 0449 0548 062430 0214 0237 027 031 0356 0456 0608 0827 113 139 164 207 24650 125 139 159 183 214 284 395 583 85 111 139 184 232

This table is based upon the same simulated paths as table 3 In panel A the column SSE shows the average SSE over the J 5 10000 paths for each model and sd gives the standard deviation Panel B givesthe percentage of simulated paths with SSE less than in the daily DMUSD data and panel C provides percentiles of the simulated SSE under each model


suggests investigation of the tails of the data We nd thatthe behavior of the fth moment is dominated by vola-tility surrounding the stock market crash of October1987 This is demonstrated by the second two panels of gure 10 which show striking linearity after simplyremoving the crash day from both data sets

Because discarding outliers seems an unsatisfactory ap-proach to volatility modeling38 we reexamine the full datasets The partition functions Sq55(T Dt) for both seriesdrop considerably from Dt 5 2 days to Dt 5 3 days In theraw data the CRSP index falls 17 on the day of the crashbut rebounds more than 8 two days later GM loses 21in the crash recovering almost all its losses over the nexttwo days When Dt 5 3 days aggregation of these returnswithin a single interval contributes to the severe declines inthe q 5 5 partition functions39 As Dt grows the crash andthe two following days occasionally fall in separate inter-vals and the partition functions spike More often howeverthese three days lie within a single interval when Dt is largeMoreover when a spike does occur for this reason withlarge Dt its size is smaller because of the diminishing

38 Although discarding data may be justi ed in speci c circumstances our approach in this paper has been to build a stationary model that is exible enough to accomodate a wide range of changing economiccircumstances This includes both long-range structura l shifts and extremetail events such as the 1987 crash

39 When calculating the partition functions for Dt 5 2 days the crashday and the two following days belong to separate intervals each of whichmake large contributions to S5(T Dt 5 2 days) When Dt 5 3 dayshowever the crash and the two following days belong to a single interval Their returns partially cancel when aggregated explaining the decreasesin the partition functions S5(T Dt) at Dt 5 3 days

TABLE 5mdashGLOBAL TESTS FOR REPLICATION OF DMUSD SCALING

Model

WeightingW1 W2 W3 W4

Test Statistics and Simulated P-values

MMAR 757 1023 726 2755(05219) (04452) (05539) (03215)

MMAR (H 5 12) 1079 1246 1175 1969(02133) (02117) (02099) (02905)

FIGARCH (1 d 0) 2460 2929 2596 5879(01159) (01605) (00427) (02586)

GARCH (1 1) 2105 2295 2534 2704(00274) (00239) (00297) (00324)

The weighting matrices W1 W4 correspond to the inverse covariance matrix of the momentconditions under the four models MMAR MMAR (H 5 1 2) FIGARCH and GARCH The tenmoment conditions used are the expected slope and SSE statistics for the ve moments q 5 05 1 23 5 conditioned upon each model The expectations and covariances of the moment conditions areobtained by simulating 10000 paths under each model The statistics reported are from the dailyDMUSD data and the P-values show the probability of observing a larger statistic conditioning uponthe assumed model Asymptotic theory suggests that the diagonal entries will have most power againsteach model

FIGURE 10mdashPARTITION FUNCTIONS FOR CRSP INDEX AND GM

The data spans from 1962 to 1998 and the time increments labeled ldquodrdquo ldquowrdquo ldquomrdquo and ldquoyrdquo correspond to one day one week one month and one year respectively When the full data sets are used we observescaling in the rst three moments and for horizons Dt $ 3 days in the fth moments The drop in the fth moments between two and three days is caused by sharp rebounds for both series from the 1987 crashAfter removing the crash from the data the second two panels show striking linearity but change the slopes of the full-sample plots to increase and thus appear more Brownian

FIGURE 11mdashPARTITION FUNCTIONS FOR ADM LOCKHEED MOTOROLA AND UAL

Each of these show strong scaling properties despite the 1987 crash


in uence of the crash at low frequencies This discussionexplains why the partition functions S5(T Dt) show largevariability but primarily for low values of Dt Because ofthis variability the estimated slopes of the partition functionwould appear to be relatively imprecise

After removing the crash from both data sets the samepartition functions appear to give a more-precise t but ata cost Both slopes increase to appear more Brownian orldquomildrdquo suggesting that important information has been lostAdditionally removing the crash does not necessarily im-prove the t in the MMAR because the theoretically pre-dicted slopes constrain only the expectations of the partitionfunctions not their variability40 In fact the simulations inthe previous section indicate that multifractal paths oftenhave partition functions that vary considerable around theirexpected slope Thus removing the crash gives a falseimpression of improved model t and alters scaling prop-erties to imply a much milder process

The other four stocks in our sample scale remarkably welldespite the crash as shown in gure 11 Consistent with themartingale hypothesis for returns three of the four stockshave almost exactly at partition functions for q 5 2whereas ADM has a slight negative slope The differencebetween Brownian scaling and multiscaling becomesperceptible for q 5 3 and for the fth moment thisdifference is pronounced UAL appears the most variablewith lower slopes at higher moments and thus a widermultifractal spectrum

Although not exhaustive our empirical analysis indicatesthat moment scaling is a prominent feature of many nan-cial series Using DMUSD data we con rm this propertyacross three orders of magnitude of frequencies and 23years of daily returns A simple estimation procedure helpsto provide a speci cation of the multifractal model thatreproduces scaling patterns found in the data Finally ouranalysis of equity data shows that the partition functionplots summarize a great deal of information in a convenientform This new tool may thus be useful in uncoveringempirical regularities and building new nancial models

VII Conclusion

The multifractal model of asset returns (MMAR) is acontinuous-time stochastic process that incorporates theoutliers and volatility persistence exhibited by many nan-cial time series The model compounds a standard Brownianmotion with an independent multifractal time-deformationprocess that produces volatility clustering We show how toconstruct a class of candidate time deformations as the limitof a simple iterative procedure called a multiplicative cas-cade The cascade provides parsimonious modeling andresults in a generalized scaling rule that restricts return

moments to vary as power laws of the time increment Theprice process is a semi-martingale with uncorrelated returnsand thus precludes arbitrage in a standard two-asset setting

The MMAR offers a fundamentally new class of pro-cesses to both nance and mathematics Multifractal pro-cesses have continuous sample paths but lie outside theclass of Ito diffusions Whereas standard processes can becharacterized by a single local scale that describes the localgrowth rate of variation sample paths of multifractal pro-cesses contain a continuum of local Holder exponentswithin any time interval The distribution of these exponentsis conveniently quanti ed by a renormalized density themultifractal spectrum f(a) For a large class of multifractalprocesses the spectrum can be explicitly derived fromCramerrsquos large-deviation theory We demonstrate through anumber of examples the sensitivity of the multifractal spec-trum to the generating mechanism The applied researchermay thus relate an empirical estimate of the spectrum backto a particular construction of the process and is permittedconsiderable exibility in modeling different types of data

We nd evidence of multifractality in the moment-scalingbehavior of Deutsche markUS dollar exchange rates Overa range of observational frequencies from approximatelytwo hours to 180 days and over a range of time from 1973to 1996 moments of the data grow approximately like apower law We obtain an estimate of the multifractal spec-trum by a Legendre transform of the momentsrsquo growth ratesFrom the shape of the estimated spectrum we infer a lognormal distribution as the primitive of the generating mech-anism and estimate its parameters We simulate the processand con rm that the multifractal model replicates the mo-ment behavior found in the data We also demonstratescaling behavior in an equity index and ve major USstocks

Our results indicate several directions for future researchUsing our simulation results as a guide the moment-scalingfeatures of the data can be incorporated into broader esti-mation and testing procedures Risk analysis forecastingand option pricing are promising applications that are cur-rently being developed in other papers Further research willalso seek to derive the MMAR as an equilibrium process ofeconomies with fully rational agents In such frameworksmultifractality is expected to arise in equilibrium eitherexogenously for instance as a consequence of multifractaltechnological shocks or endogenously because of marketincompleteness or informational cascades The early empir-ical success of the MMAR thus offers new challenges ineconometrics nance and economic theory

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Andersen T and T Bollerslev ldquoHeterogeneou s Information Arrivals andReturn Volatility Dynamics Uncovering the Long-Run in HighFrequency Returnsrdquo Journal of Finance 523 (1997) 975ndash1005

40 The simulations in the previous section do however suggest that thevariability of partition function plots can usefully be incorporate d intoestimation


Andersen T T Bollerslev F Diebold and H Ebens ldquoThe Distribution ofStock Return Volatilityrdquo Journal of Financial Economics 611(2001) 43ndash76

Andrews D ldquoTests for Parameter Instability and Structural Change withUnknown Change Pointrdquo Econometrica 614 (1993) 821ndash856

Avery C and P Zemsky ldquoMultidimensiona l Uncertainty and HerdBehavior in Financial Marketsrdquo American Economic Review 884(1998) 724ndash748

Backus D K and S E Zin ldquoLong Memory In ation Uncertainty Evidence from the Term Structure of Interest Ratesrdquo Journal ofMoney Credit and Banking 253 (1993) 681ndash700

Baillie R T ldquoLong Memory Processes and Fractional Integration inEconometricsrdquo Journal of Econometrics 731 (1996) 5ndash59

Baillie R T and T Bollerslev ldquoThe Message in Daily Exchange RatesA Conditional Variance Talerdquo Journal of Business and EconomicStatistics 73 (1989) 297ndash305

Baillie R T T Bollerslev and H O Mikkelsen ldquoFractionally IntegratedGeneralized Autoregressiv e Conditional Heteroskedasticit yrdquo Jour-nal of Econometrics 741 (1996) 3ndash30

Baillie R T C F Chung and M A Tieslau ldquoAnalyzing In ation by theFractionally Integrated ARFIMA-GARCH Modelrdquo Journal of Ap-plied Econometrics 111 (1996) 23ndash40

Bates D S ldquoTesting Option Pricing Modelsrdquo NBER working paper no5129 (1995) ldquoJumps and Stochastic Volatility Exchange Rate Process Implicitin Deutsche Mark Optionsrdquo Review of Financial Studies 91(1996) 69ndash107

Beveridge W H ldquoWeather and Harvest Cyclesrdquo Economic Journal31124 (1921) 429ndash452

Bikhchandani S D Hirshleifer and I Welch ldquoA Theory of FadsFashion Custom and Cultural Change as Informationa l CascadesrdquoJournal of Political Economy 1005 (1992) 992ndash1027

Billingsley P Probability and Measure (New York John Wiley and Sons1979)

Bochner S Harmonic Analysis and the Theory of Probability (BerkeleyUniversity of California Press 1955)

Bollerslev T ldquoGeneralized Autoregressiv e Conditional Heteroskedastic -ityrdquo Journal of Econometrics 313 (1986) 307ndash327 ldquoA Conditional Heteroskedasti c Time Series Model for Specula-tive Prices and Rates of Returnrdquo this REVIEW 693 (1987) 542ndash547

Boudoukh J M Richardson and R Whitelaw ldquoA Tale of Three SchoolsInsights on Autocorrelation s of Short-Horizon Returnsrdquo Review ofFinancial Studies 73 (1994) 539ndash573

Breidt J N Crato and P DeLima ldquoOn the Detection and Estimation ofLong-Memory in Stochastic Volatilityrdquo Journal of Econometrics831ndash2 (1998) 325ndash348

Bulow J and P Klemperer ldquoRational Frenzies and Crashesrdquo Journal ofPolitical Economy 1021 (1994) 1ndash24

Calvet L ldquoIncomplete Markets and Volatilityrdquo Journal of EconomicTheory 982 (2001) 295ndash338

Calvet L and A Fisher ldquoForecasting Multifracta l Volatilityrdquo Journal ofEconometrics 1051 (2001) 27ndash58 ldquoCatastrophic Risk in Multifractal Returnsrdquo in E Altman and IVanderhoof (Eds) Integrated Risk and Return Management forInsurance Companies (Norwell MA Kluwer Academic Pressforthcoming)

Calvet L A Fisher and B B Mandelbrot ldquoLarge Deviation Theory andthe Distribution of Price Changesrdquo Cowles Foundation discussionpaper no 1165 Yale University available from the SSRN databaseat httpwwwssrncom (1997)

Campbell J and N G Mankiw ldquoAre Output Fluctuations TransitoryrdquoQuarterly Journal of Economics 1024 (1987) 857ndash880

Campbell J A Lo and A C MacKinlay The Econometrics of FinancialMarkets (Princeton Princeton University Press 1997)

Clark P K ldquoA Subordinated Stochastic Process Model with FiniteVariance for Speculative Pricesrdquo Econometrica 411 (1973) 135ndash156

Cox J C J E Ingersoll and S A Ross ldquoAn Intertemporal GeneralEquilibrium Model of Asset Pricesrdquo Econometrica 532 (1985)363ndash384

Dacorogna M M U A Muller R J Nagler R B Olsen and O VPictet ldquoA Geographica l Model for the Daily and Weekly Seasonal

Volatility in the Foreign Exchange Marketrdquo Journal of Interna-tional Money and Finance 124 (1993) 413ndash438

Diebold F X and G D Rudebusch ldquoLong Memory and Persistence inAggregate Outputrdquo Journal of Monetary Economics 242 (1989)189ndash209

Ding Z C W J Granger and R F Engle ldquoA Long Memory Property ofStock Returns and a New Modelrdquo Journal of Empirical Finance11 (1993) 83ndash106

Dothan M Prices in Financial Markets (Oxford Oxford UniversityPress 1990)

Duf e D and K J Singleton ldquoSimulated Moments Estimation ofMarkov Models of Asset Pricesrdquo Econometrica 614 (1993) 929ndash952

Durrett R Probability Theory and Examples (Paci c Grove CA Wads-worth amp Brooks 1991)

Engle R F ldquoAutoregressiv e Conditional Heteroscedasticit y with Esti-mates of the Variance of United Kingdom In ationrdquo Econometrica504 (1982) 987ndash1007

Engle R F and G Gonzalez-Rivera ldquoSemiparametric ARCH ModelsrdquoJournal of Business and Economic Statistics 94 (1991) 345ndash359

Faust J ldquoWhen Are Variance Ratio Tests for Serial Dependence Opti-malrdquo Econometrica 605 (1992) 1215ndash1226

Feller W An Introductio n to Probability Theory and Its Applications(New York John Wiley and Sons 1968)

Fisher A L Calvet and B B Mandelbrot ldquoMultifractality of DeutscheMarkUS Dollar Exchange Ratesrdquo Cowles Foundation discussionpaper no 1166 Yale University available from the SSRN databaseat httpwwwssrncom (1997)

Frisch U and G Parisi ldquoFully Developed Turbulence and Intermittencyrdquo(pp 84ndash88) M Ghil (Ed) Turbulence and Predictabilit y inGeophysica l Fluid Dynamics and Climate Dynamics (AmsterdamNorth-Holland 1985)

Galluccio S G Caldarelli M Marsili and Y C Zhang ldquoScaling inCurrency Exchangerdquo Physica A 2453ndash4 (1997) 423ndash436

Gennotte G and H Leland ldquoMarket Liquidity Hedging and CrashesrdquoAmerican Economic Review 805 (1990) 999ndash1021

Ghysels E C Gourieroux and J Jasiak ldquoTrading Patterns Time De-formation and Stochastic Volatility in Foreign Exchange MarketsrdquoCREST working paper no 9655 (1996)

Ghysels E A C Harvey and E Renault ldquoStochastic Volatilityrdquo (pp119ndash191) in G S Maddala and C R Rao (Eds) Handbook ofStatistics vol 14 (Amsterdam North-Holland 1996)

Glosten L R Jagannathan and D Runkle ldquoOn the Relation Between theExpected Value and the Volatility of the Nominal Excess Return onStocksrdquo Journal of Finance 485 (1993) 1779ndash1801

Granger C W J and R Joyeux ldquoAn Introduction to Long Memory TimeSeries Models and Fractional Differencing rdquo Journal of Time SeriesAnalysis 1 (1980) 15ndash29

Guivarcrsquoh Y ldquoRemarques sur les Solutions drsquoune Equation Fonctionnell eNon Lineaire de Benoit Mandelbrot rdquo Comptes Rendus (Paris)3051 (1987) 139

Halsey T C M H Jensen L P Kadanoff I Procaccia and B IShraiman ldquoFractal Measures and Their Singularities The Charac-terization of Strange Setsrdquo Physical Review Letters A 332 (1986)1141

Harrison J M and D Kreps ldquoMartingales and Arbitrage in MultiperiodSecurities Marketsrdquo Journal of Economic Theory 203 (1979)381ndash408

Hosking J R M ldquoFractional Differencingrdquo Biometrika 681 (1981)165ndash176

Ingram B F and B S Lee ldquoSimulation Estimation of Time SeriesModelsrdquo Journal of Econometrics 472ndash3 (1991) 197ndash250

Jacklin C A Kleidon and P P eiderer ldquoUnderestimation of PortfolioInsurance and the Crash of October 1987rdquo Review of FinancialStudies 51 (1992) 35ndash63

Kahane J P ldquoA Century of Interplay Between Taylor Series FourierSeries and Brownian Motionrdquo Bulletin of the London Mathemat-ical Society 293 (1997) 257ndash279

Koedijk K G and C J M Kool ldquoTail Estimates of East EuropeanExchange Ratesrdquo Journal of Business and Economic Statistics101 (1992) 83ndash96

Kolmogorov A N ldquoWienersche Spiralen und einige andere interessant eKurven im Hilbertschen raumrdquo Comptes Rendus (Doklady) delrsquoAcademie des Sciences de lrsquoURSS 26 (1940) 115ndash118


Lo A W ldquoLong Memory in Stock Market Pricesrdquo Econometrica 595(1991) 1279ndash1313

Lo A W and A C MacKinlay ldquoStock Market Prices Do Not FollowRandom Walks Evidence from a Simple Speci cation Testrdquo Re-view of Financial Studies 11 (1988) 41ndash66

Maheswaran S and C Sims ldquoEmpirical Implications of Arbitrage-FreeAsset Marketsrdquo (pp 301ndash316) in P C B Phillips (Ed) ModelsMethods and Applications of Econometrics Cambridge MA BasilBlackwell 1993)

Mandelbrot B B ldquoThe Variation of Certain Speculative Pricesrdquo Journalof Business 364 (1963) 394ndash419 ldquoUne Classe de Processus Stochastiques Homothetiques a SoirdquoComptes Rendus de lrsquoAcademie des Sciences de Paris 260 (March1965) 3274ndash3277 ldquoIntermittent Turbulence in Self-Similar Cascades Divergence ofHigh Moments and Dimension of the Carrierrdquo Journal of FluidMechanics 622 (1974) 331ndash358 ldquoMultifracta l Measures Especially for the Geophysicist rdquo Pureand Applied Geophysics 1311ndash2 (1989a) 5ndash42 ldquoExamples of Multinomial Multifracta l Measures that HaveNegative Latent Values for the Dimension f(a)rdquo (pp 3ndash29) in LPietronero (Ed) Fractalsrsquo Physical Origins and Properties (NewYork Plenum 1989b)

Mandelbrot B B A Fisher and L Calvet ldquoThe Multifracta l Model ofAsset Returnsrdquo Cowles Foundation discussion paper no 1164Yale University paper available from the SSRN database at httpwwwssrncom (1997)

Mandelbrot B B and J W van Ness ldquoFractional Brownian MotionFractional Noises and Application rdquo SIAM Review 104 (1968)422ndash437

Mandelbrot B B and H W Taylor ldquoOn the Distribution of Stock PriceDifferencesrdquo Operations Research 156 (1967) 1057ndash1062

Merton R C Continuous-Time Finance (Cambridge MA Blackwell1990)

Muller U A M M Dacorogna R D Dave O V Pictet R B Olsenand J R Ward ldquoFractals and Intrinsic Time A Challenge toEconometriciansrdquo discussion paper presented at the 1993 Interna-tional Conference of the Applied Econometrics Association (Lux-embourg) (1995)

Nelson D ldquoConditional Heteroskedasticit y in Asset Returns A NewApproachrdquo Econometrica 592 (1991) 347ndash370

Pasquini M and M Serva ldquoClustering of Volatility as a MultiscalePhenomenonrdquo European Physical Journal B 161 (2000) 195ndash201

Peyriere J ldquoMultifractal Measuresrdquo (pp 175ndash186) in J S Byrnes et al(Eds) Proceedings of the NATO ASI ldquoProbabilistic StochasticMethods in Analysis with Applicationsrdquo (Norwell MA KluwerAcademic Press 1992)

Phillips P C B J W McFarland and P C McMahon ldquoRobust Tests ofForward Exchange Market Ef ciency with Empirical Evidencefrom the 1920rsquosrdquo Journal of Applied Econometrics 111 (1996)1ndash22

Richards G ldquoThe Fractal Structure of Exchange Rates Measurement andForecastingrdquo Journal of Internationa l Financial Markets Institu-tions and Money 102 (2000) 163ndash180

Richardson M and J Stock ldquoDrawing Inferences from Statistics Basedon Multi-Year Asset Returnsrdquo Journal of Financial Economics252 (1989) 323ndash348

Roll R ldquoA Simple Implicit Measure of the Effective Bid-Ask Spread inan Ef cient Marketrdquo Journal of Finance 394 (1984) 1127ndash1140

Sowell F B ldquoModeling Long Run Behavior with the Fractional ARIMAModelrdquo Journal of Monetary Economics 292 (1992) 277ndash302

Stock J H ldquoMeasuring Business Cycle Timerdquo Journal of PoliticalEconomy 956 (1987) 1240ndash1261 ldquoEstimating Continuous-Time Processes Subject to Time Defor-mationrdquo Journal of the American Statistica l Association 83401(1988) 77ndash85

Taqqu M S ldquoWeak Convergence to Fractional Brownian Motion and tothe Rosenblatt Processrdquo Z Wahrscheinlichkeitstheori e verw Ge-biete 31 (1975) 287ndash302

Taylor S Modeling Financial Time Series (New York Wiley 1986)Vandewalle N and M Ausloos ldquoMulti-Af ne Analysis of Typical

Currency Exchange Ratesrdquo European Physical Journal B 42(1998) 257ndash261

Wiggins J B ldquoOption Values under Stochastic Volatility Theory andEmpirical Estimatesrdquo Journal of Financial Economics 192(1987) 351ndash372

APPENDIX

A Scaling Rule

This appendix analyzes the set de ned by a multiplicative measurewith parameter b $ 2 Consider a xed instant t [0 1] For all e 0 there exists a dyadic number tn such that tn 2 t e We can then nda number Dn 5 b2kn e for which (tn Dn) In the plane 2 thepoint (t 0) is thus the limit of the sequence (tn Dn) whichestablishes

Property 1 The closure of contains the set [0 1] 3 0The scaling relation (4) thus holds ldquoin the neighborhoo d of any instantrdquo

B Proof of Proposition 1

Consider two exponents q1 and q2 and two positive weights w1 andw2 adding up to 1 Holderrsquos inequality implies

~ X~t q ~ X~t q1w1 ~ X~t q2w2

where q 5 w1q1 1 w2q2 Taking logarithms and using equation (5) weobtain

ln c~q 1 t~q ln t w1t~q1 1 w2t~q2 ln t 1 w1 ln c~q1

1 w2 ln c~q2 (A1)

We divide by ln t 0 and let t go to zero

t~q $ w1t~q1 1 w2t~q2 (A2)

which establishes the concavity of t This proof also contains additiona linformation on multifractal processes Assuming that relation (5) holds fort [0 `) we divide inequality (A1) by ln t 0 and let t go to in nityWe obtain the reverse of inequality (A2) and conclude that t(q) is linearThus exact multiscaling can hold only for bounded time intervals

C Proof of Theorem 1

Because the trading time and the Brownian motion B(t) are indepen-dent conditioning on u (t) yields

$ X~t q u ~t 5 u 5 B~u q u ~t 5 u

5 u ~tq 2 B~1 q

and thus [ X(t) q] 5 [u (t)q 2] [ B(1) q] The process X(t) satis esthe multiscaling relation (5) with tX(q) [ tu(q 2) and cX(q) [cu(q 2) [ B(1) q]

D Proof of Theorem 2

Let t and 9t denote the natural ltrations of X(t) and X(t) u (t)For any t T u the independenc e of B and u implies

$X~t 1 T 9t u ~t 1 T 5 u 5 $B~u 9t

5 Bu ~t

because B(t) is a martingale We now infer that [X(t 1 T) t] 5X(t) which establishes that X(t) is a martingale and has thus uncorrelate dincrements The price P(t) is a smooth function of X(t) and therefore asemi-martingale which precludes arbitrage opportunitie s in the two-asseteconomy


E Proof of Theorem 4

Trading time

Consider a canonica l cascade after k $ 1 stages Consistent with thenotation of section II the interval [0 T] is partitioned into cells of lengthDt 5 b2kT and I1 5 [t1 t1 1 Dt] and I2 5 [t2 t2 1 Dt] denote twodistinct cells with lower endpoints of the form t1T 5 0h1 hk andt2T 5 0z1 zk Assume that the rst l $ 1 terms are equal in the b-adicexpansions of t1T and t2T so that z1 5 h1 zl 5 h l and z l11 THORNhl11 The distance t 5 t2 2 t1 satis es b2 l21 tT b2l and theproduct m(I1)qm(I2)q which is equal to

Vh1 hk

q Vz1 zk

q ~Mh1

2q Mh1 hl

2q

~Mh1 h l11

q Mh1 hk

q ~M z1 z l11

q Mz1 zk

q

has mean ( Vq)2[ M2q] l[ Mq]2(k2 l ) We conclude that

Cov m~I1q m~I2

q 5 ~ Vq2~ Mq2k$~ M2q~ Mq2l 2 1

5 C1~Dt2tu~q12b2ltu~2q22tu~q21 2 1

is bounded by two hyperbolic functions of t

Log-Price

Because B(t) and u (t) are independen t processes the conditiona lexpectation

$ X~0 Dt X~t Dt q u ~Dt 5 u1 u ~t 5 u2 u ~t 1 Dt 5 u3

(A3)

simpli es to

B~u1q B~u3 2 B~u2

q 5 u1q 2 u3 2 u2

q 2 B~1 q2

Taking expectations we infer that

X~0 Dt X~t Dt q 5 u ~0 Dtu ~t Dt q 2 B~1 q2

and therefore

dX~t q 5 du~t q 2 B~1 q2

F Interpretation of f( a ) as a Fractal Dimension

Fractal geometry considers irregular and winding structures that are notwell described by their Euclidean length For instance a geographe rmeasuring the length of a coastline will nd very different results as sheincreases the precision of her measurement In fact the structure of thecoastline is usually so intricate that the measured length diverges toin nity as the geographe rrsquos measurement scale goes to zero For thisreason we cannot use the Euclidean length to compare two differentcoastlines and it is natural to introduce a new concept of dimensionGiven a precision level e 0 we consider coverings of the coastline withballs of diameter e Let N(e) denote the smallest number of balls requiredfor such a covering The approximate length of the coastline is de ned byL(e) 5 eN(e) In many cases N(e) satis es a power law as e goes tozero N(e) e2D where D is a constant called the fractal dimensionFractal dimension helps to analyze the structure of a xed multifractal Forany a $ 0 we can de ne the set T(a) of instants with Holder exponenta As any subset of the real line T(a) has a fractal dimension D(a)which satis es 0 D(a) 1 It can be shown that for a large class ofmultifractals the dimension D(a) coincides with the multifracta l spec-trum f(a)

In the case of measures we can provide a heuristic interpretatio n of thisresult based on coarse Holder exponents Denoting by N(a Dt) thenumber of intervals [t t 1 Dt] required to cover T(a) we infer fromequation (1) that N(a Dt) (Dt)2f(a) We then rewrite the total mass

m[0 T] 5 yen m(Dt) yen (Dt)a(t) and rearrange it as a sum over Holderexponents

m0 T ~Dta2f~ada

The integral is dominated by the contribution of the Holder exponent a1that minimizes a 2 f(a) and therefore

m0 T ~Dta12f~a1

Because the total mass m[0 T] is positive we infer that f(a1) 5 a1 andf(a) a for all a When f is differentiable the coef cient a1 also satis esf9(a1) 5 1 The spectrum f(a) then lies under the 45-degree line withtangentia l contact at a 5 a1

G Large-Deviation Theory and the Multifractal Spectrum

This section sketches the proof of Theorem 6 and introduces theconcepts of latent and virtual Holder exponents 41 First consider a con-servative multiplicative measure m Application of large-deviatio n theory(LDT) begins with the histogram method of subsection IVA Subdivide therange of arsquos into intervals of length Da and denote by Nk(a) the numberof coarse Holder exponents in the interval (a a 1 Da] For large valuesof k we write

1

klogb

Nk~a

bk

1

klogb $a ak a 1 Da (A4)

This relation holds exactly for multinomial measures which have discretecoarse exponents ak but is postulated in more-general cases For any a a0 Cramerrsquos theorem implies that

k21 logb $ak a Infq

logb eq~a2V1 ln b (A5)

as k ` Using the de nition of the scaling function we simplify thelimit to Infq

aq 2 t~q 2 1 Combining this with equations (1) and

(A4) it follows42 that Theorem 6 holdsThese arguments easily extend to a canonica l measure m Given a

b-adic instant t the coarse exponent ak(t) 5 ln m[t t 1 Dt]ln Dt is thesum of a high-frequenc y component 2k21 logb Vh1 hk

and of thefamiliar low-frequency average

ak L~t 5 2logb Mh1 1 1 logb Mh1 h kk

The exponent ak(t) converges almost surely to a0 5 2 logb M and themultifractal spectrum is again the Legendre transform of the scalingfunction t(q)

Relation (A5) also shows that f(a) is the limit of

k21 logb $ak L~t a 1 1 if a a0

and


f(a) is therefore a hump-shaped function reaching a maximum at themost-probabl e exponent f(a) f(a0) 5 143 We have successivel yviewed the spectrum f(a) as

41 We refer the reader to Mandelbrot (1989b) Peyriere (1992) and CFM(1997) for more-detailed discussions

42 See CFM (1997) for a more-detailed proof43 It is easy to show that a0q 2 t(q) is minimal for q 5 0 The set

T(a0) has therefore fractal dimension f(a0) 5 2t(0) 5 1 and thuscarries all of the Lebesgue measure Moreover by the central limittheorem f(a) is locally quadratic around a0 as shown by CFM (1997)


(D1) the limit of a renormalized histogram of coarse Holder expo-nents

(D2) the fractal dimension of the set of instants with Holder exponenta and

(D3) the limit of k21 logb ak L(t) a 1 1 provided by LDT

The three de nitions coincide for multinomial measures and (D1) and(D2) agree for a large class of multifractals (Peyriere 1991) However(D1) and (D2) imply that f(a) $ 0 whereas (D3) imposes no suchrestriction When f(a) 0 the corresponding arsquos called latent are rarecoarse exponents which appear in few draws of the random measure andcontrol high and low moments (Mandelbrot 1989b) Similarly becausecanonica l measures allow M to be greater than 1 the low-frequencyaverage ak L(t) can be negative with positive probability (D3) thus de nesthe multifractal spectrum for negative or virtual values of a This topicfurther discussed by Mandelbrot (1989b) remains an active research areain mathematics

H Proof of Theorem 7

Given a process Z denote aZ(t) as its local scale at date t and TZ(a)as the set of instants with scale a At any date the in nitesimal variationof the log-price X(t 1 DT) 2 X(t) 5 B[u (t 1 Dt)] 2 B[u (t)] satis es

X~t 1 Dt 2 X~t u ~t 1 Dt 2 u ~t 1 2 Dt au~t 2

implying aX(t) [ au(t) 2 The sets TX(a) and Tu(2a) coincide and inparticular have identical fractal dimensions fX(a) [ fu(2a) Moreoverbecause the price P(t) is a differentiabl e function of X(t) the twoprocesses have identical local Holder exponents and spectra

I Simulation of Multifractal Paths

The construction of the simulated multifractal price paths used insubsection VID has two basic components First a nite-stage approxi-mation to a canonica l multifracta l measure is constructed as suggested insubsection IIB All simulations use the base b 5 2 so that if a simulationof length T is desired we choose the minimum integer number of stagesk such that 2k $ T At each stage in the construction we drawindependen t log normal multipliers with identical distribution s given bythe results in subsection VIC When k stages are completed the measuremk is used as a discrete approximation to the quadratic variation of amultifractal path Aggregating the increments of mk thus provides asimulated path from the trading time u (t) The second part of theconstruction involves compounding as suggested by Assumptions 1through 3 of section III For the martingale version of the MMAR wesimply calculate the standard deviation [mk(Dt)]1 2 over the discrete-timeinterval Dt and multiply by an independen t standard Gaussian To simu-late the extended MMAR we rst generate a discretized path from anFBM with parameter H taken from the estimates in subsection VICInterpolatio n provides values of the path BH[u (t)] at the simulated valuesfrom the path u (t)


The Review of Economics and StatisticsVOL LXXXIV NUMBER 3AUGUST 2002

MULTIFRACTALITY IN ASSET RETURNS THEORY AND EVIDENCE

Laurent Calvet and Adlai Fisher

AbstractmdashThis paper investigates the multifractal model of asset returns(MMAR) a class of continuous-tim e processes that incorporat e the thicktails and volatility persistence exhibited by many nancial time series Thesimplest version of the MMAR compounds a Brownian motion with amultifracta l time-deformation Prices follow a semi-martingale whichprecludes arbitrage in a standard two-asset economy Volatility has longmemory and the highest nite moments of returns can take any valuegreater than 2 The local variability of a sample path is highly heteroge-neous and is usefully characterized by the local Holder exponent at everyinstant In contrast with earlier processes this exponent takes a continuumof values in any time interval The MMAR predicts that the moments ofreturns vary as a power law of the time horizon We con rm this propertyfor Deutsche markUS dollar exchange rates and several equity seriesWe develop an estimation procedure and infer a parsimonious generatingmechanism for the exchange rate In Monte Carlo simulations the esti-mated multifractal process replicates the scaling properties of the data andcompares favorably with some alternative speci cations

I Introduction

THE multifractal model of asset returns (MMAR) is acontinuous-time process that captures the thick tails and

long-memory volatility persistence that are exhibited bymany nancial time series1 It is constructed by compound-ing a standard Brownian motion with a random time-deformation process which is speci ed to be multifractalThe time deformation produces clustering and long memoryin volatility and implies that the moments of returns vary asa power law of the time horizon We con rm this propertyempirically on the Deutsche markUS dollar exchange ratea US equity index and several individual stocks

The MMAR is characterized by a form of time-invariancecalled multiscaling which combines extreme returns withlong memory in volatility This speci cation improves ontraditional models with scaling features in several waysFirst the MMAR is consistent with economic equilibriumThe simplest version implies uncorrelated returns and semi-martingale prices thus precluding arbitrage in a standardtwo-asset economy The model also permits signi cant exibility in matching the data Returns have a nite vari-ance and their highest nite moment can take any valuegreater than 2 Consistent with many nancial series theunconditional distribution of returns displays thinner tails asthe time scale increases In contrast with earlier processeshowever the distribution need not converge to a Gaussian atlow frequencies and never converges to a Gaussian at highfrequencies The MMAR thus captures the distributionalnonlinearities that are exhibited by nancial data whileretaining the parsimony and tractability of scaling models

The time-deformation process is obtained as the limit ofa simple iterative procedure called a multiplicative cascadeThe construction begins with a uniform distribution ofvolatility at a suitably long time horizon and randomlyconcentrates volatility into progressively smaller time inter-vals The procedure follows the same rule at each stage ofthe cascade which provides parsimony and implies momentscaling By construction volatility clusters at all frequen-cies which is consistent with the intuition that economicfactors such as technological shocks business cycles earn-ings cycles and liquidity shocks have different time scales2

We anticipate that rational equilibrium models can generatethe MMAR either exogenously through multifractal shocksor endogenously due to market incompleteness or informa-tional cascades

The MMAR provides a fundamentally new class ofstochastic processes to nancial economists In particularthe multifractal model is a continuous diffusion that liesoutside the class of Ito processes Although these traditionalmodels locally vary as (dt)1 2 along their sample paths theMMAR generates the richer class (dt)a(t) where the localscale a(t) can take a continuum of values The relativeoccurrences of the local scales a(t) are conveniently sum-marized in a renormalized probability density called themultifractal spectrum Given a speci cation of the model

Received for publication June 12 2000 Revision accepted for publica-tion July 11 2001

Harvard University and University of British Columbia respectivel yWe are grateful to Benoit Mandelbrot for inspirationa l mentoring during

our years at Yale and beyond We also received helpful comments from TAndersen D Andrews D Backus J Campbell P Clark T Conley RDeo F Diebold J Geanakoplos W Goetzmann O Hart O Linton ALo E Maskin R Merton P C B Phillips M Richardson S Ross JScheinkman R Shiller C Sims J Stein J Stock B Zame S Zinanonymous referees and seminar participant s at numerous institutions including Yale Boston University British Columbia the Federal ReserveBoard Harvard Johns Hopkins MIT NYU Princeton Riverside theSanta Fe Institute the University of Colorado at Boulder the Universityof Miami the University of Utah the Berkeley Program in Finance andthe 1998 Summer Meeting of the Econometric Society

1 Long memory is convenientl y de ned by hyperbolicall y decliningautocorrelation s either for a process itself or functions of it This propertywas rst analyzed in the context of fractional integration of Brownianmotion by Mandelbrot (1965) and Mandelbrot and van Ness (1968) It hasbeen documented in squared and absolute returns for many nancial datasets (Taylor 1986 Ding Granger amp Engle 1993 Dacorogna et al 1993)Baillie (1996) provides a survey of long memory in economics 2 This idea is further elaborated by Calvet and Fisher (2001)

The Review of Economics and Statistics August 2002 84(3) 381ndash406copy 2002 by the President and Fellows of Harvard College and the Massachusetts Institute of Technology





















$X~ct1 X~ctk 5d

$cHX~t1 cHX~tk










m20 14 5 m0m0 m214 1 2 5 m0m1

m21 2 34 5 m1m0 m234 1 5 m1m1










k h i22i


































for all ~t Dt q







b MiVi













X~t ln P~t 2 ln P~0



X~t Bu ~t









































A Local Scales

We rst introduce



5 O~ Dt b as Dt 0








Models


CorrelatedReturns











f~a limln Nk~a

ln bk as k ` (6)






Mh1h2 Mh1 hk






ak 51

ki51

k

V i (8)



a0 5 V1 5 2 logb M 1 (9)





1

kln

1

ki51

k

Xi a Infq

ln eq~a2X1

for any a X1



f~a 5 Infq

aq 2 t~q (10)




t Tk

~Dtak~ t bk~Dt~a011 2 5 b2k~a021 2






fu~a 5 1 2 ~a 2 l24~l 2 1








$X~t 1 Dt 2 X~t2 5 cX~21 2 Dt











Normal (l s2) 1 2 (a 2 l)2[4(l 2 1)]

Binomial 2amax 2 a

amax 2 aminlog2

amax 2 a

amax 2 amin2

a 2 amin

amax 2 aminlog2

a 2 amin

amax 2 amin















Sq~T Dti50

N21





























C Main Results



















fX~a 5 1 2~a 2 a0

2

4H~a0 2 H




















m 5Y jt







h~Y jm 5 $t~q Y j

m ln SSE~q Y jmq Q9




H~X Ym 5 h~X 21

Jj51

J

h~Y jm



J H(Y jm


Gmn~X






















E Equity Data







q


(1 d 0) iv GARCH (1 1)


05 20711 2071 2071 001 2073 2073 001 2074 001 2075 00110 20440 2044 2044 002 2049 2048 002 2049 003 2050 00220 0058 006 005 006 000 2001 006 2001 008 2001 00530 0500 049 047 012 043 041 010 042 016 047 00850 1208 119 116 027 116 110 024 114 035 136 017


05 05910 09812 09969 0999310 05465 09596 09787 0995920 05620 08867 08363 0918630 06032 08111 07209 0658750 05757 06771 05656 01754


i MMAR

05 20745 20742 20738 20734 20729 20722 20714 20705 20698 20694 20690 20686 2068310 20506 20501 20493 20485 20475 20460 20443 20426 20412 20403 20396 20386 2037920 20106 20091 20069 20052 20028 0009 0049 0090 0126 0149 0170 0197 021530 0159 0192 0246 0284 0328 0399 0472 0549 0622 0666 0712 0758 079850 0343 0465 0601 0705 0821 0988 1162 1331 1491 1595 1691 1793 1871

ii MMAR (H 5 12)

05 20766 20762 20758 20754 20750 20743 20735 20727 20720 20716 20712 20708 2070510 20542 20536 20527 20519 20511 20497 20480 20465 20451 20443 20436 20428 2042320 20150 20136 20114 20098 20078 20045 20008 0029 0062 0083 0100 0124 013830 0143 0177 0211 0244 0283 0344 0410 0479 0542 0584 0621 0666 070750 0412 0502 0633 0714 0807 0954 1106 1258 1401 1490 1581 1684 1779

iii FIGARCH (1 d 0)

05 20776 20773 20768 20764 20760 20753 20744 20736 20729 20725 20721 20716 2071310 20566 20556 20546 20537 20527 20511 20494 20476 20460 20450 20442 20430 2042220 20244 20211 20174 20141 20111 20064 20016 0034 0084 0115 0147 0191 022130 20082 20003 0086 0151 0220 0320 0416 0514 0608 0680 0737 0815 085750 0033 0200 0412 0559 0703 0934 1155 1362 1560 1699 1807 1935 2031

iv GARCH (1 1)

05 20777 20774 20770 20766 20762 20755 20747 20740 20733 20729 20725 20721 2071810 20555 20548 20541 20534 20525 20512 20497 20482 20470 20461 20454 20447 2044220 20121 20110 20095 20080 20063 20036 20005 0025 0053 0071 0084 0101 011330 0281 0302 0322 0347 0375 0420 0470 0521 0571 0601 0629 0664 068550 0928 0978 1037 1094 1150 1250 1352 1461 1569 1641 1709 1790 1859








T by X0 5 0 X t 5Xt21 1 ln (1 1 rt)



q


iii FIGARCH(1 d 0)

iv GARCH(1 1)


05 0018 00200 00080 00213 000805 00241 000876 00214 00077910 00507 00604 00278 00637 00266 00832 00417 00586 0022920 0159 0330 0212 0311 0167 0635 0635 021 0093730 042 149 114 129 0864 298 295 0698 03950 276 101 713 984 601 173 137 491 354


05 04640 03914 02541 0373310 04265 03553 01648 0419120 01084 01000 00204 0321130 00156 00197 00026 0190250 00155 00283 00034 02293


i MMAR

05 000703 000774 000889 00101 00116 00144 00185 00238 00298 00345 00394 00461 0052810 00206 00223 00253 00289 00331 0042 00546 00716 00938 0112 0131 0158 017320 00936 0103 0118 0133 0156 0203 0278 0392 0551 0695 087 111 13430 0351 0387 0457 0528 0625 0834 118 177 265 345 447 586 71850 223 253 298 344 411 565 818 123 184 234 287 362 425

ii MMAR (H 5 12)

05 000806 000882 000993 00111 00127 00157 00199 00253 00315 00364 00412 00469 0052710 00224 00246 00281 00318 00362 00455 00583 00759 00964 0112 013 0157 017520 00973 0106 0121 0138 0159 0203 0271 0369 0503 0617 0734 0902 10830 0332 0372 0437 05 0583 0769 106 152 221 281 355 465 55850 205 233 267 316 372 502 724 107 157 202 249 321 383

iii FIGARCH (1 d 0)

05 000873 000969 00112 00126 00144 00179 00229 00287 00355 00403 00447 00507 0056110 0026 00291 0034 00386 00448 00572 00745 00984 0129 0155 0185 0228 027620 0126 014 0165 0192 0227 031 0461 073 118 162 214 31 41830 0453 0527 064 0751 0931 134 211 36 575 774 103 143 19450 299 342 413 494 607 873 135 214 321 415 524 689 853

iv GARCH (1 1)

05 000804 000872 000995 00111 00128 00159 00202 00254 00315 00357 00397 00455 00510 00212 00232 00265 00299 00343 00427 00546 00697 00874 0101 0115 0135 014920 00725 00793 0091 0103 0116 0147 019 0249 0325 0383 0449 0548 062430 0214 0237 027 031 0356 0456 0608 0827 113 139 164 207 24650 125 139 159 183 214 284 395 583 85 111 139 184 232








Model



MMAR 757 1023 726 2755(05219) (04452) (05539) (03215)

MMAR (H 5 12) 1079 1246 1175 1969(02133) (02117) (02099) (02905)

FIGARCH (1 d 0) 2460 2929 2596 5879(01159) (01605) (00427) (02586)

GARCH (1 1) 2105 2295 2534 2704(00274) (00239) (00297) (00324)











VII Conclusion






REFERENCES
















































































APPENDIX

A Scaling Rule





~ X~t q ~ X~t q1w1 ~ X~t q2w2



1 w2 ln c~q2 (A1)


t~q $ w1t~q1 1 w2t~q2 (A2)




$ X~t q u ~t 5 u 5 B~u q u ~t 5 u

5 u ~tq 2 B~1 q




$X~t 1 T 9t u ~t 1 T 5 u 5 $B~u 9t

5 Bu ~t




Trading time


Vh1 hk

q Vz1 zk

q ~Mh1

2q Mh1 hl

2q

~Mh1 h l11

q Mh1 hk

q ~M z1 z l11

q Mz1 zk

q


Cov m~I1q m~I2

q 5 ~ Vq2~ Mq2k$~ M2q~ Mq2l 2 1



Log-Price



(A3)

simpli es to

B~u1q B~u3 2 B~u2

q 5 u1q 2 u3 2 u2

q 2 B~1 q2



and therefore






m0 T ~Dta2f~ada


m0 T ~Dta12f~a1




1

klogb

Nk~a

bk

1



k21 logb $ak a Infq











and






































$X~ct1 X~ctk 5d

$cHX~t1 cHX~tk










m20 14 5 m0m0 m214 1 2 5 m0m1

m21 2 34 5 m1m0 m234 1 5 m1m1










k h i22i


































for all ~t Dt q







b MiVi













X~t ln P~t 2 ln P~0



X~t Bu ~t









































A Local Scales

We rst introduce



5 O~ Dt b as Dt 0








Models


CorrelatedReturns











f~a limln Nk~a

ln bk as k ` (6)






Mh1h2 Mh1 hk






ak 51

ki51

k

V i (8)



a0 5 V1 5 2 logb M 1 (9)





1

kln

1

ki51

k

Xi a Infq

ln eq~a2X1

for any a X1



f~a 5 Infq

aq 2 t~q (10)




t Tk

~Dtak~ t bk~Dt~a011 2 5 b2k~a021 2






fu~a 5 1 2 ~a 2 l24~l 2 1








$X~t 1 Dt 2 X~t2 5 cX~21 2 Dt











Normal (l s2) 1 2 (a 2 l)2[4(l 2 1)]

Binomial 2amax 2 a

amax 2 aminlog2

amax 2 a

amax 2 amin2

a 2 amin

amax 2 aminlog2

a 2 amin

amax 2 amin















Sq~T Dti50

N21





























C Main Results



















fX~a 5 1 2~a 2 a0

2

4H~a0 2 H




















m 5Y jt







h~Y jm 5 $t~q Y j

m ln SSE~q Y jmq Q9




H~X Ym 5 h~X 21

Jj51

J

h~Y jm



J H(Y jm


Gmn~X






















E Equity Data







q


(1 d 0) iv GARCH (1 1)


05 20711 2071 2071 001 2073 2073 001 2074 001 2075 00110 20440 2044 2044 002 2049 2048 002 2049 003 2050 00220 0058 006 005 006 000 2001 006 2001 008 2001 00530 0500 049 047 012 043 041 010 042 016 047 00850 1208 119 116 027 116 110 024 114 035 136 017


05 05910 09812 09969 0999310 05465 09596 09787 0995920 05620 08867 08363 0918630 06032 08111 07209 0658750 05757 06771 05656 01754


i MMAR

05 20745 20742 20738 20734 20729 20722 20714 20705 20698 20694 20690 20686 2068310 20506 20501 20493 20485 20475 20460 20443 20426 20412 20403 20396 20386 2037920 20106 20091 20069 20052 20028 0009 0049 0090 0126 0149 0170 0197 021530 0159 0192 0246 0284 0328 0399 0472 0549 0622 0666 0712 0758 079850 0343 0465 0601 0705 0821 0988 1162 1331 1491 1595 1691 1793 1871

ii MMAR (H 5 12)

05 20766 20762 20758 20754 20750 20743 20735 20727 20720 20716 20712 20708 2070510 20542 20536 20527 20519 20511 20497 20480 20465 20451 20443 20436 20428 2042320 20150 20136 20114 20098 20078 20045 20008 0029 0062 0083 0100 0124 013830 0143 0177 0211 0244 0283 0344 0410 0479 0542 0584 0621 0666 070750 0412 0502 0633 0714 0807 0954 1106 1258 1401 1490 1581 1684 1779

iii FIGARCH (1 d 0)

05 20776 20773 20768 20764 20760 20753 20744 20736 20729 20725 20721 20716 2071310 20566 20556 20546 20537 20527 20511 20494 20476 20460 20450 20442 20430 2042220 20244 20211 20174 20141 20111 20064 20016 0034 0084 0115 0147 0191 022130 20082 20003 0086 0151 0220 0320 0416 0514 0608 0680 0737 0815 085750 0033 0200 0412 0559 0703 0934 1155 1362 1560 1699 1807 1935 2031

iv GARCH (1 1)

05 20777 20774 20770 20766 20762 20755 20747 20740 20733 20729 20725 20721 2071810 20555 20548 20541 20534 20525 20512 20497 20482 20470 20461 20454 20447 2044220 20121 20110 20095 20080 20063 20036 20005 0025 0053 0071 0084 0101 011330 0281 0302 0322 0347 0375 0420 0470 0521 0571 0601 0629 0664 068550 0928 0978 1037 1094 1150 1250 1352 1461 1569 1641 1709 1790 1859








T by X0 5 0 X t 5Xt21 1 ln (1 1 rt)



q


iii FIGARCH(1 d 0)

iv GARCH(1 1)


05 0018 00200 00080 00213 000805 00241 000876 00214 00077910 00507 00604 00278 00637 00266 00832 00417 00586 0022920 0159 0330 0212 0311 0167 0635 0635 021 0093730 042 149 114 129 0864 298 295 0698 03950 276 101 713 984 601 173 137 491 354


05 04640 03914 02541 0373310 04265 03553 01648 0419120 01084 01000 00204 0321130 00156 00197 00026 0190250 00155 00283 00034 02293


i MMAR

05 000703 000774 000889 00101 00116 00144 00185 00238 00298 00345 00394 00461 0052810 00206 00223 00253 00289 00331 0042 00546 00716 00938 0112 0131 0158 017320 00936 0103 0118 0133 0156 0203 0278 0392 0551 0695 087 111 13430 0351 0387 0457 0528 0625 0834 118 177 265 345 447 586 71850 223 253 298 344 411 565 818 123 184 234 287 362 425

ii MMAR (H 5 12)

05 000806 000882 000993 00111 00127 00157 00199 00253 00315 00364 00412 00469 0052710 00224 00246 00281 00318 00362 00455 00583 00759 00964 0112 013 0157 017520 00973 0106 0121 0138 0159 0203 0271 0369 0503 0617 0734 0902 10830 0332 0372 0437 05 0583 0769 106 152 221 281 355 465 55850 205 233 267 316 372 502 724 107 157 202 249 321 383

iii FIGARCH (1 d 0)

05 000873 000969 00112 00126 00144 00179 00229 00287 00355 00403 00447 00507 0056110 0026 00291 0034 00386 00448 00572 00745 00984 0129 0155 0185 0228 027620 0126 014 0165 0192 0227 031 0461 073 118 162 214 31 41830 0453 0527 064 0751 0931 134 211 36 575 774 103 143 19450 299 342 413 494 607 873 135 214 321 415 524 689 853

iv GARCH (1 1)

05 000804 000872 000995 00111 00128 00159 00202 00254 00315 00357 00397 00455 00510 00212 00232 00265 00299 00343 00427 00546 00697 00874 0101 0115 0135 014920 00725 00793 0091 0103 0116 0147 019 0249 0325 0383 0449 0548 062430 0214 0237 027 031 0356 0456 0608 0827 113 139 164 207 24650 125 139 159 183 214 284 395 583 85 111 139 184 232








Model



MMAR 757 1023 726 2755(05219) (04452) (05539) (03215)

MMAR (H 5 12) 1079 1246 1175 1969(02133) (02117) (02099) (02905)

FIGARCH (1 d 0) 2460 2929 2596 5879(01159) (01605) (00427) (02586)

GARCH (1 1) 2105 2295 2534 2704(00274) (00239) (00297) (00324)











VII Conclusion






REFERENCES
















































































APPENDIX

A Scaling Rule





~ X~t q ~ X~t q1w1 ~ X~t q2w2



1 w2 ln c~q2 (A1)


t~q $ w1t~q1 1 w2t~q2 (A2)




$ X~t q u ~t 5 u 5 B~u q u ~t 5 u

5 u ~tq 2 B~1 q




$X~t 1 T 9t u ~t 1 T 5 u 5 $B~u 9t

5 Bu ~t




Trading time


Vh1 hk

q Vz1 zk

q ~Mh1

2q Mh1 hl

2q

~Mh1 h l11

q Mh1 hk

q ~M z1 z l11

q Mz1 zk

q


Cov m~I1q m~I2

q 5 ~ Vq2~ Mq2k$~ M2q~ Mq2l 2 1



Log-Price



(A3)

simpli es to

B~u1q B~u3 2 B~u2

q 5 u1q 2 u3 2 u2

q 2 B~1 q2



and therefore






m0 T ~Dta2f~ada


m0 T ~Dta12f~a1




1

klogb

Nk~a

bk

1



k21 logb $ak a Infq











and



















k h i22i


































for all ~t Dt q







b MiVi













X~t ln P~t 2 ln P~0



X~t Bu ~t









































A Local Scales

We rst introduce



5 O~ Dt b as Dt 0








Models


CorrelatedReturns











f~a limln Nk~a

ln bk as k ` (6)






Mh1h2 Mh1 hk






ak 51

ki51

k

V i (8)



a0 5 V1 5 2 logb M 1 (9)





1

kln

1

ki51

k

Xi a Infq

ln eq~a2X1

for any a X1



f~a 5 Infq

aq 2 t~q (10)




t Tk

~Dtak~ t bk~Dt~a011 2 5 b2k~a021 2






fu~a 5 1 2 ~a 2 l24~l 2 1








$X~t 1 Dt 2 X~t2 5 cX~21 2 Dt











Normal (l s2) 1 2 (a 2 l)2[4(l 2 1)]

Binomial 2amax 2 a

amax 2 aminlog2

amax 2 a

amax 2 amin2

a 2 amin

amax 2 aminlog2

a 2 amin

amax 2 amin















Sq~T Dti50

N21





























C Main Results



















fX~a 5 1 2~a 2 a0

2

4H~a0 2 H




















m 5Y jt







h~Y jm 5 $t~q Y j

m ln SSE~q Y jmq Q9




H~X Ym 5 h~X 21

Jj51

J

h~Y jm



J H(Y jm


Gmn~X






















E Equity Data







q


(1 d 0) iv GARCH (1 1)


05 20711 2071 2071 001 2073 2073 001 2074 001 2075 00110 20440 2044 2044 002 2049 2048 002 2049 003 2050 00220 0058 006 005 006 000 2001 006 2001 008 2001 00530 0500 049 047 012 043 041 010 042 016 047 00850 1208 119 116 027 116 110 024 114 035 136 017


05 05910 09812 09969 0999310 05465 09596 09787 0995920 05620 08867 08363 0918630 06032 08111 07209 0658750 05757 06771 05656 01754


i MMAR

05 20745 20742 20738 20734 20729 20722 20714 20705 20698 20694 20690 20686 2068310 20506 20501 20493 20485 20475 20460 20443 20426 20412 20403 20396 20386 2037920 20106 20091 20069 20052 20028 0009 0049 0090 0126 0149 0170 0197 021530 0159 0192 0246 0284 0328 0399 0472 0549 0622 0666 0712 0758 079850 0343 0465 0601 0705 0821 0988 1162 1331 1491 1595 1691 1793 1871

ii MMAR (H 5 12)

05 20766 20762 20758 20754 20750 20743 20735 20727 20720 20716 20712 20708 2070510 20542 20536 20527 20519 20511 20497 20480 20465 20451 20443 20436 20428 2042320 20150 20136 20114 20098 20078 20045 20008 0029 0062 0083 0100 0124 013830 0143 0177 0211 0244 0283 0344 0410 0479 0542 0584 0621 0666 070750 0412 0502 0633 0714 0807 0954 1106 1258 1401 1490 1581 1684 1779

iii FIGARCH (1 d 0)

05 20776 20773 20768 20764 20760 20753 20744 20736 20729 20725 20721 20716 2071310 20566 20556 20546 20537 20527 20511 20494 20476 20460 20450 20442 20430 2042220 20244 20211 20174 20141 20111 20064 20016 0034 0084 0115 0147 0191 022130 20082 20003 0086 0151 0220 0320 0416 0514 0608 0680 0737 0815 085750 0033 0200 0412 0559 0703 0934 1155 1362 1560 1699 1807 1935 2031

iv GARCH (1 1)

05 20777 20774 20770 20766 20762 20755 20747 20740 20733 20729 20725 20721 2071810 20555 20548 20541 20534 20525 20512 20497 20482 20470 20461 20454 20447 2044220 20121 20110 20095 20080 20063 20036 20005 0025 0053 0071 0084 0101 011330 0281 0302 0322 0347 0375 0420 0470 0521 0571 0601 0629 0664 068550 0928 0978 1037 1094 1150 1250 1352 1461 1569 1641 1709 1790 1859








T by X0 5 0 X t 5Xt21 1 ln (1 1 rt)



q


iii FIGARCH(1 d 0)

iv GARCH(1 1)


05 0018 00200 00080 00213 000805 00241 000876 00214 00077910 00507 00604 00278 00637 00266 00832 00417 00586 0022920 0159 0330 0212 0311 0167 0635 0635 021 0093730 042 149 114 129 0864 298 295 0698 03950 276 101 713 984 601 173 137 491 354


05 04640 03914 02541 0373310 04265 03553 01648 0419120 01084 01000 00204 0321130 00156 00197 00026 0190250 00155 00283 00034 02293


i MMAR

05 000703 000774 000889 00101 00116 00144 00185 00238 00298 00345 00394 00461 0052810 00206 00223 00253 00289 00331 0042 00546 00716 00938 0112 0131 0158 017320 00936 0103 0118 0133 0156 0203 0278 0392 0551 0695 087 111 13430 0351 0387 0457 0528 0625 0834 118 177 265 345 447 586 71850 223 253 298 344 411 565 818 123 184 234 287 362 425

ii MMAR (H 5 12)

05 000806 000882 000993 00111 00127 00157 00199 00253 00315 00364 00412 00469 0052710 00224 00246 00281 00318 00362 00455 00583 00759 00964 0112 013 0157 017520 00973 0106 0121 0138 0159 0203 0271 0369 0503 0617 0734 0902 10830 0332 0372 0437 05 0583 0769 106 152 221 281 355 465 55850 205 233 267 316 372 502 724 107 157 202 249 321 383

iii FIGARCH (1 d 0)

05 000873 000969 00112 00126 00144 00179 00229 00287 00355 00403 00447 00507 0056110 0026 00291 0034 00386 00448 00572 00745 00984 0129 0155 0185 0228 027620 0126 014 0165 0192 0227 031 0461 073 118 162 214 31 41830 0453 0527 064 0751 0931 134 211 36 575 774 103 143 19450 299 342 413 494 607 873 135 214 321 415 524 689 853

iv GARCH (1 1)

05 000804 000872 000995 00111 00128 00159 00202 00254 00315 00357 00397 00455 00510 00212 00232 00265 00299 00343 00427 00546 00697 00874 0101 0115 0135 014920 00725 00793 0091 0103 0116 0147 019 0249 0325 0383 0449 0548 062430 0214 0237 027 031 0356 0456 0608 0827 113 139 164 207 24650 125 139 159 183 214 284 395 583 85 111 139 184 232








Model



MMAR 757 1023 726 2755(05219) (04452) (05539) (03215)

MMAR (H 5 12) 1079 1246 1175 1969(02133) (02117) (02099) (02905)

FIGARCH (1 d 0) 2460 2929 2596 5879(01159) (01605) (00427) (02586)

GARCH (1 1) 2105 2295 2534 2704(00274) (00239) (00297) (00324)











VII Conclusion






REFERENCES
















































































APPENDIX

A Scaling Rule





~ X~t q ~ X~t q1w1 ~ X~t q2w2



1 w2 ln c~q2 (A1)


t~q $ w1t~q1 1 w2t~q2 (A2)




$ X~t q u ~t 5 u 5 B~u q u ~t 5 u

5 u ~tq 2 B~1 q




$X~t 1 T 9t u ~t 1 T 5 u 5 $B~u 9t

5 Bu ~t




Trading time


Vh1 hk

q Vz1 zk

q ~Mh1

2q Mh1 hl

2q

~Mh1 h l11

q Mh1 hk

q ~M z1 z l11

q Mz1 zk

q


Cov m~I1q m~I2

q 5 ~ Vq2~ Mq2k$~ M2q~ Mq2l 2 1



Log-Price



(A3)

simpli es to

B~u1q B~u3 2 B~u2

q 5 u1q 2 u3 2 u2

q 2 B~1 q2



and therefore






m0 T ~Dta2f~ada


m0 T ~Dta12f~a1




1

klogb

Nk~a

bk

1



k21 logb $ak a Infq











and










































for all ~t Dt q







b MiVi













X~t ln P~t 2 ln P~0



X~t Bu ~t









































A Local Scales

We rst introduce



5 O~ Dt b as Dt 0








Models


CorrelatedReturns











f~a limln Nk~a

ln bk as k ` (6)






Mh1h2 Mh1 hk






ak 51

ki51

k

V i (8)



a0 5 V1 5 2 logb M 1 (9)





1

kln

1

ki51

k

Xi a Infq

ln eq~a2X1

for any a X1



f~a 5 Infq

aq 2 t~q (10)




t Tk

~Dtak~ t bk~Dt~a011 2 5 b2k~a021 2






fu~a 5 1 2 ~a 2 l24~l 2 1








$X~t 1 Dt 2 X~t2 5 cX~21 2 Dt











Normal (l s2) 1 2 (a 2 l)2[4(l 2 1)]

Binomial 2amax 2 a

amax 2 aminlog2

amax 2 a

amax 2 amin2

a 2 amin

amax 2 aminlog2

a 2 amin

amax 2 amin















Sq~T Dti50

N21





























C Main Results



















fX~a 5 1 2~a 2 a0

2

4H~a0 2 H




















m 5Y jt







h~Y jm 5 $t~q Y j

m ln SSE~q Y jmq Q9




H~X Ym 5 h~X 21

Jj51

J

h~Y jm



J H(Y jm


Gmn~X






















E Equity Data







q


(1 d 0) iv GARCH (1 1)


05 20711 2071 2071 001 2073 2073 001 2074 001 2075 00110 20440 2044 2044 002 2049 2048 002 2049 003 2050 00220 0058 006 005 006 000 2001 006 2001 008 2001 00530 0500 049 047 012 043 041 010 042 016 047 00850 1208 119 116 027 116 110 024 114 035 136 017


05 05910 09812 09969 0999310 05465 09596 09787 0995920 05620 08867 08363 0918630 06032 08111 07209 0658750 05757 06771 05656 01754


i MMAR

05 20745 20742 20738 20734 20729 20722 20714 20705 20698 20694 20690 20686 2068310 20506 20501 20493 20485 20475 20460 20443 20426 20412 20403 20396 20386 2037920 20106 20091 20069 20052 20028 0009 0049 0090 0126 0149 0170 0197 021530 0159 0192 0246 0284 0328 0399 0472 0549 0622 0666 0712 0758 079850 0343 0465 0601 0705 0821 0988 1162 1331 1491 1595 1691 1793 1871

ii MMAR (H 5 12)

05 20766 20762 20758 20754 20750 20743 20735 20727 20720 20716 20712 20708 2070510 20542 20536 20527 20519 20511 20497 20480 20465 20451 20443 20436 20428 2042320 20150 20136 20114 20098 20078 20045 20008 0029 0062 0083 0100 0124 013830 0143 0177 0211 0244 0283 0344 0410 0479 0542 0584 0621 0666 070750 0412 0502 0633 0714 0807 0954 1106 1258 1401 1490 1581 1684 1779

iii FIGARCH (1 d 0)

05 20776 20773 20768 20764 20760 20753 20744 20736 20729 20725 20721 20716 2071310 20566 20556 20546 20537 20527 20511 20494 20476 20460 20450 20442 20430 2042220 20244 20211 20174 20141 20111 20064 20016 0034 0084 0115 0147 0191 022130 20082 20003 0086 0151 0220 0320 0416 0514 0608 0680 0737 0815 085750 0033 0200 0412 0559 0703 0934 1155 1362 1560 1699 1807 1935 2031

iv GARCH (1 1)

05 20777 20774 20770 20766 20762 20755 20747 20740 20733 20729 20725 20721 2071810 20555 20548 20541 20534 20525 20512 20497 20482 20470 20461 20454 20447 2044220 20121 20110 20095 20080 20063 20036 20005 0025 0053 0071 0084 0101 011330 0281 0302 0322 0347 0375 0420 0470 0521 0571 0601 0629 0664 068550 0928 0978 1037 1094 1150 1250 1352 1461 1569 1641 1709 1790 1859








T by X0 5 0 X t 5Xt21 1 ln (1 1 rt)



q


iii FIGARCH(1 d 0)

iv GARCH(1 1)


05 0018 00200 00080 00213 000805 00241 000876 00214 00077910 00507 00604 00278 00637 00266 00832 00417 00586 0022920 0159 0330 0212 0311 0167 0635 0635 021 0093730 042 149 114 129 0864 298 295 0698 03950 276 101 713 984 601 173 137 491 354


05 04640 03914 02541 0373310 04265 03553 01648 0419120 01084 01000 00204 0321130 00156 00197 00026 0190250 00155 00283 00034 02293


i MMAR

05 000703 000774 000889 00101 00116 00144 00185 00238 00298 00345 00394 00461 0052810 00206 00223 00253 00289 00331 0042 00546 00716 00938 0112 0131 0158 017320 00936 0103 0118 0133 0156 0203 0278 0392 0551 0695 087 111 13430 0351 0387 0457 0528 0625 0834 118 177 265 345 447 586 71850 223 253 298 344 411 565 818 123 184 234 287 362 425

ii MMAR (H 5 12)

05 000806 000882 000993 00111 00127 00157 00199 00253 00315 00364 00412 00469 0052710 00224 00246 00281 00318 00362 00455 00583 00759 00964 0112 013 0157 017520 00973 0106 0121 0138 0159 0203 0271 0369 0503 0617 0734 0902 10830 0332 0372 0437 05 0583 0769 106 152 221 281 355 465 55850 205 233 267 316 372 502 724 107 157 202 249 321 383

iii FIGARCH (1 d 0)

05 000873 000969 00112 00126 00144 00179 00229 00287 00355 00403 00447 00507 0056110 0026 00291 0034 00386 00448 00572 00745 00984 0129 0155 0185 0228 027620 0126 014 0165 0192 0227 031 0461 073 118 162 214 31 41830 0453 0527 064 0751 0931 134 211 36 575 774 103 143 19450 299 342 413 494 607 873 135 214 321 415 524 689 853

iv GARCH (1 1)

05 000804 000872 000995 00111 00128 00159 00202 00254 00315 00357 00397 00455 00510 00212 00232 00265 00299 00343 00427 00546 00697 00874 0101 0115 0135 014920 00725 00793 0091 0103 0116 0147 019 0249 0325 0383 0449 0548 062430 0214 0237 027 031 0356 0456 0608 0827 113 139 164 207 24650 125 139 159 183 214 284 395 583 85 111 139 184 232








Model



MMAR 757 1023 726 2755(05219) (04452) (05539) (03215)

MMAR (H 5 12) 1079 1246 1175 1969(02133) (02117) (02099) (02905)

FIGARCH (1 d 0) 2460 2929 2596 5879(01159) (01605) (00427) (02586)

GARCH (1 1) 2105 2295 2534 2704(00274) (00239) (00297) (00324)











VII Conclusion






REFERENCES
















































































APPENDIX

A Scaling Rule





~ X~t q ~ X~t q1w1 ~ X~t q2w2



1 w2 ln c~q2 (A1)


t~q $ w1t~q1 1 w2t~q2 (A2)




$ X~t q u ~t 5 u 5 B~u q u ~t 5 u

5 u ~tq 2 B~1 q




$X~t 1 T 9t u ~t 1 T 5 u 5 $B~u 9t

5 Bu ~t




Trading time


Vh1 hk

q Vz1 zk

q ~Mh1

2q Mh1 hl

2q

~Mh1 h l11

q Mh1 hk

q ~M z1 z l11

q Mz1 zk

q


Cov m~I1q m~I2

q 5 ~ Vq2~ Mq2k$~ M2q~ Mq2l 2 1



Log-Price



(A3)

simpli es to

B~u1q B~u3 2 B~u2

q 5 u1q 2 u3 2 u2

q 2 B~1 q2



and therefore






m0 T ~Dta2f~ada


m0 T ~Dta12f~a1




1

klogb

Nk~a

bk

1



k21 logb $ak a Infq











and



























X~t ln P~t 2 ln P~0



X~t Bu ~t









































A Local Scales

We rst introduce



5 O~ Dt b as Dt 0








Models


CorrelatedReturns











f~a limln Nk~a

ln bk as k ` (6)






Mh1h2 Mh1 hk






ak 51

ki51

k

V i (8)



a0 5 V1 5 2 logb M 1 (9)





1

kln

1

ki51

k

Xi a Infq

ln eq~a2X1

for any a X1



f~a 5 Infq

aq 2 t~q (10)




t Tk

~Dtak~ t bk~Dt~a011 2 5 b2k~a021 2






fu~a 5 1 2 ~a 2 l24~l 2 1








$X~t 1 Dt 2 X~t2 5 cX~21 2 Dt











Normal (l s2) 1 2 (a 2 l)2[4(l 2 1)]

Binomial 2amax 2 a

amax 2 aminlog2

amax 2 a

amax 2 amin2

a 2 amin

amax 2 aminlog2

a 2 amin

amax 2 amin















Sq~T Dti50

N21





























C Main Results



















fX~a 5 1 2~a 2 a0

2

4H~a0 2 H




















m 5Y jt







h~Y jm 5 $t~q Y j

m ln SSE~q Y jmq Q9




H~X Ym 5 h~X 21

Jj51

J

h~Y jm



J H(Y jm


Gmn~X






















E Equity Data







q


(1 d 0) iv GARCH (1 1)


05 20711 2071 2071 001 2073 2073 001 2074 001 2075 00110 20440 2044 2044 002 2049 2048 002 2049 003 2050 00220 0058 006 005 006 000 2001 006 2001 008 2001 00530 0500 049 047 012 043 041 010 042 016 047 00850 1208 119 116 027 116 110 024 114 035 136 017


05 05910 09812 09969 0999310 05465 09596 09787 0995920 05620 08867 08363 0918630 06032 08111 07209 0658750 05757 06771 05656 01754


i MMAR

05 20745 20742 20738 20734 20729 20722 20714 20705 20698 20694 20690 20686 2068310 20506 20501 20493 20485 20475 20460 20443 20426 20412 20403 20396 20386 2037920 20106 20091 20069 20052 20028 0009 0049 0090 0126 0149 0170 0197 021530 0159 0192 0246 0284 0328 0399 0472 0549 0622 0666 0712 0758 079850 0343 0465 0601 0705 0821 0988 1162 1331 1491 1595 1691 1793 1871

ii MMAR (H 5 12)

05 20766 20762 20758 20754 20750 20743 20735 20727 20720 20716 20712 20708 2070510 20542 20536 20527 20519 20511 20497 20480 20465 20451 20443 20436 20428 2042320 20150 20136 20114 20098 20078 20045 20008 0029 0062 0083 0100 0124 013830 0143 0177 0211 0244 0283 0344 0410 0479 0542 0584 0621 0666 070750 0412 0502 0633 0714 0807 0954 1106 1258 1401 1490 1581 1684 1779

iii FIGARCH (1 d 0)

05 20776 20773 20768 20764 20760 20753 20744 20736 20729 20725 20721 20716 2071310 20566 20556 20546 20537 20527 20511 20494 20476 20460 20450 20442 20430 2042220 20244 20211 20174 20141 20111 20064 20016 0034 0084 0115 0147 0191 022130 20082 20003 0086 0151 0220 0320 0416 0514 0608 0680 0737 0815 085750 0033 0200 0412 0559 0703 0934 1155 1362 1560 1699 1807 1935 2031

iv GARCH (1 1)

05 20777 20774 20770 20766 20762 20755 20747 20740 20733 20729 20725 20721 2071810 20555 20548 20541 20534 20525 20512 20497 20482 20470 20461 20454 20447 2044220 20121 20110 20095 20080 20063 20036 20005 0025 0053 0071 0084 0101 011330 0281 0302 0322 0347 0375 0420 0470 0521 0571 0601 0629 0664 068550 0928 0978 1037 1094 1150 1250 1352 1461 1569 1641 1709 1790 1859








T by X0 5 0 X t 5Xt21 1 ln (1 1 rt)



q


iii FIGARCH(1 d 0)

iv GARCH(1 1)


05 0018 00200 00080 00213 000805 00241 000876 00214 00077910 00507 00604 00278 00637 00266 00832 00417 00586 0022920 0159 0330 0212 0311 0167 0635 0635 021 0093730 042 149 114 129 0864 298 295 0698 03950 276 101 713 984 601 173 137 491 354


05 04640 03914 02541 0373310 04265 03553 01648 0419120 01084 01000 00204 0321130 00156 00197 00026 0190250 00155 00283 00034 02293


i MMAR

05 000703 000774 000889 00101 00116 00144 00185 00238 00298 00345 00394 00461 0052810 00206 00223 00253 00289 00331 0042 00546 00716 00938 0112 0131 0158 017320 00936 0103 0118 0133 0156 0203 0278 0392 0551 0695 087 111 13430 0351 0387 0457 0528 0625 0834 118 177 265 345 447 586 71850 223 253 298 344 411 565 818 123 184 234 287 362 425

ii MMAR (H 5 12)

05 000806 000882 000993 00111 00127 00157 00199 00253 00315 00364 00412 00469 0052710 00224 00246 00281 00318 00362 00455 00583 00759 00964 0112 013 0157 017520 00973 0106 0121 0138 0159 0203 0271 0369 0503 0617 0734 0902 10830 0332 0372 0437 05 0583 0769 106 152 221 281 355 465 55850 205 233 267 316 372 502 724 107 157 202 249 321 383

iii FIGARCH (1 d 0)

05 000873 000969 00112 00126 00144 00179 00229 00287 00355 00403 00447 00507 0056110 0026 00291 0034 00386 00448 00572 00745 00984 0129 0155 0185 0228 027620 0126 014 0165 0192 0227 031 0461 073 118 162 214 31 41830 0453 0527 064 0751 0931 134 211 36 575 774 103 143 19450 299 342 413 494 607 873 135 214 321 415 524 689 853

iv GARCH (1 1)

05 000804 000872 000995 00111 00128 00159 00202 00254 00315 00357 00397 00455 00510 00212 00232 00265 00299 00343 00427 00546 00697 00874 0101 0115 0135 014920 00725 00793 0091 0103 0116 0147 019 0249 0325 0383 0449 0548 062430 0214 0237 027 031 0356 0456 0608 0827 113 139 164 207 24650 125 139 159 183 214 284 395 583 85 111 139 184 232








Model



MMAR 757 1023 726 2755(05219) (04452) (05539) (03215)

MMAR (H 5 12) 1079 1246 1175 1969(02133) (02117) (02099) (02905)

FIGARCH (1 d 0) 2460 2929 2596 5879(01159) (01605) (00427) (02586)

GARCH (1 1) 2105 2295 2534 2704(00274) (00239) (00297) (00324)











VII Conclusion






REFERENCES
















































































APPENDIX

A Scaling Rule





~ X~t q ~ X~t q1w1 ~ X~t q2w2



1 w2 ln c~q2 (A1)


t~q $ w1t~q1 1 w2t~q2 (A2)




$ X~t q u ~t 5 u 5 B~u q u ~t 5 u

5 u ~tq 2 B~1 q




$X~t 1 T 9t u ~t 1 T 5 u 5 $B~u 9t

5 Bu ~t




Trading time


Vh1 hk

q Vz1 zk

q ~Mh1

2q Mh1 hl

2q

~Mh1 h l11

q Mh1 hk

q ~M z1 z l11

q Mz1 zk

q


Cov m~I1q m~I2

q 5 ~ Vq2~ Mq2k$~ M2q~ Mq2l 2 1



Log-Price



(A3)

simpli es to

B~u1q B~u3 2 B~u2

q 5 u1q 2 u3 2 u2

q 2 B~1 q2



and therefore






m0 T ~Dta2f~ada


m0 T ~Dta12f~a1




1

klogb

Nk~a

bk

1



k21 logb $ak a Infq











and












































A Local Scales

We rst introduce



5 O~ Dt b as Dt 0








Models


CorrelatedReturns











f~a limln Nk~a

ln bk as k ` (6)






Mh1h2 Mh1 hk






ak 51

ki51

k

V i (8)



a0 5 V1 5 2 logb M 1 (9)





1

kln

1

ki51

k

Xi a Infq

ln eq~a2X1

for any a X1



f~a 5 Infq

aq 2 t~q (10)




t Tk

~Dtak~ t bk~Dt~a011 2 5 b2k~a021 2






fu~a 5 1 2 ~a 2 l24~l 2 1








$X~t 1 Dt 2 X~t2 5 cX~21 2 Dt











Normal (l s2) 1 2 (a 2 l)2[4(l 2 1)]

Binomial 2amax 2 a

amax 2 aminlog2

amax 2 a

amax 2 amin2

a 2 amin

amax 2 aminlog2

a 2 amin

amax 2 amin















Sq~T Dti50

N21





























C Main Results



















fX~a 5 1 2~a 2 a0

2

4H~a0 2 H




















m 5Y jt







h~Y jm 5 $t~q Y j

m ln SSE~q Y jmq Q9




H~X Ym 5 h~X 21

Jj51

J

h~Y jm



J H(Y jm


Gmn~X






















E Equity Data







q


(1 d 0) iv GARCH (1 1)


05 20711 2071 2071 001 2073 2073 001 2074 001 2075 00110 20440 2044 2044 002 2049 2048 002 2049 003 2050 00220 0058 006 005 006 000 2001 006 2001 008 2001 00530 0500 049 047 012 043 041 010 042 016 047 00850 1208 119 116 027 116 110 024 114 035 136 017


05 05910 09812 09969 0999310 05465 09596 09787 0995920 05620 08867 08363 0918630 06032 08111 07209 0658750 05757 06771 05656 01754


i MMAR

05 20745 20742 20738 20734 20729 20722 20714 20705 20698 20694 20690 20686 2068310 20506 20501 20493 20485 20475 20460 20443 20426 20412 20403 20396 20386 2037920 20106 20091 20069 20052 20028 0009 0049 0090 0126 0149 0170 0197 021530 0159 0192 0246 0284 0328 0399 0472 0549 0622 0666 0712 0758 079850 0343 0465 0601 0705 0821 0988 1162 1331 1491 1595 1691 1793 1871

ii MMAR (H 5 12)

05 20766 20762 20758 20754 20750 20743 20735 20727 20720 20716 20712 20708 2070510 20542 20536 20527 20519 20511 20497 20480 20465 20451 20443 20436 20428 2042320 20150 20136 20114 20098 20078 20045 20008 0029 0062 0083 0100 0124 013830 0143 0177 0211 0244 0283 0344 0410 0479 0542 0584 0621 0666 070750 0412 0502 0633 0714 0807 0954 1106 1258 1401 1490 1581 1684 1779

iii FIGARCH (1 d 0)

05 20776 20773 20768 20764 20760 20753 20744 20736 20729 20725 20721 20716 2071310 20566 20556 20546 20537 20527 20511 20494 20476 20460 20450 20442 20430 2042220 20244 20211 20174 20141 20111 20064 20016 0034 0084 0115 0147 0191 022130 20082 20003 0086 0151 0220 0320 0416 0514 0608 0680 0737 0815 085750 0033 0200 0412 0559 0703 0934 1155 1362 1560 1699 1807 1935 2031

iv GARCH (1 1)

05 20777 20774 20770 20766 20762 20755 20747 20740 20733 20729 20725 20721 2071810 20555 20548 20541 20534 20525 20512 20497 20482 20470 20461 20454 20447 2044220 20121 20110 20095 20080 20063 20036 20005 0025 0053 0071 0084 0101 011330 0281 0302 0322 0347 0375 0420 0470 0521 0571 0601 0629 0664 068550 0928 0978 1037 1094 1150 1250 1352 1461 1569 1641 1709 1790 1859








T by X0 5 0 X t 5Xt21 1 ln (1 1 rt)



q


iii FIGARCH(1 d 0)

iv GARCH(1 1)


05 0018 00200 00080 00213 000805 00241 000876 00214 00077910 00507 00604 00278 00637 00266 00832 00417 00586 0022920 0159 0330 0212 0311 0167 0635 0635 021 0093730 042 149 114 129 0864 298 295 0698 03950 276 101 713 984 601 173 137 491 354


05 04640 03914 02541 0373310 04265 03553 01648 0419120 01084 01000 00204 0321130 00156 00197 00026 0190250 00155 00283 00034 02293


i MMAR

05 000703 000774 000889 00101 00116 00144 00185 00238 00298 00345 00394 00461 0052810 00206 00223 00253 00289 00331 0042 00546 00716 00938 0112 0131 0158 017320 00936 0103 0118 0133 0156 0203 0278 0392 0551 0695 087 111 13430 0351 0387 0457 0528 0625 0834 118 177 265 345 447 586 71850 223 253 298 344 411 565 818 123 184 234 287 362 425

ii MMAR (H 5 12)

05 000806 000882 000993 00111 00127 00157 00199 00253 00315 00364 00412 00469 0052710 00224 00246 00281 00318 00362 00455 00583 00759 00964 0112 013 0157 017520 00973 0106 0121 0138 0159 0203 0271 0369 0503 0617 0734 0902 10830 0332 0372 0437 05 0583 0769 106 152 221 281 355 465 55850 205 233 267 316 372 502 724 107 157 202 249 321 383

iii FIGARCH (1 d 0)

05 000873 000969 00112 00126 00144 00179 00229 00287 00355 00403 00447 00507 0056110 0026 00291 0034 00386 00448 00572 00745 00984 0129 0155 0185 0228 027620 0126 014 0165 0192 0227 031 0461 073 118 162 214 31 41830 0453 0527 064 0751 0931 134 211 36 575 774 103 143 19450 299 342 413 494 607 873 135 214 321 415 524 689 853

iv GARCH (1 1)

05 000804 000872 000995 00111 00128 00159 00202 00254 00315 00357 00397 00455 00510 00212 00232 00265 00299 00343 00427 00546 00697 00874 0101 0115 0135 014920 00725 00793 0091 0103 0116 0147 019 0249 0325 0383 0449 0548 062430 0214 0237 027 031 0356 0456 0608 0827 113 139 164 207 24650 125 139 159 183 214 284 395 583 85 111 139 184 232








Model



MMAR 757 1023 726 2755(05219) (04452) (05539) (03215)

MMAR (H 5 12) 1079 1246 1175 1969(02133) (02117) (02099) (02905)

FIGARCH (1 d 0) 2460 2929 2596 5879(01159) (01605) (00427) (02586)

GARCH (1 1) 2105 2295 2534 2704(00274) (00239) (00297) (00324)











VII Conclusion






REFERENCES
















































































APPENDIX

A Scaling Rule





~ X~t q ~ X~t q1w1 ~ X~t q2w2



1 w2 ln c~q2 (A1)


t~q $ w1t~q1 1 w2t~q2 (A2)




$ X~t q u ~t 5 u 5 B~u q u ~t 5 u

5 u ~tq 2 B~1 q




$X~t 1 T 9t u ~t 1 T 5 u 5 $B~u 9t

5 Bu ~t




Trading time


Vh1 hk

q Vz1 zk

q ~Mh1

2q Mh1 hl

2q

~Mh1 h l11

q Mh1 hk

q ~M z1 z l11

q Mz1 zk

q


Cov m~I1q m~I2

q 5 ~ Vq2~ Mq2k$~ M2q~ Mq2l 2 1



Log-Price



(A3)

simpli es to

B~u1q B~u3 2 B~u2

q 5 u1q 2 u3 2 u2

q 2 B~1 q2



and therefore






m0 T ~Dta2f~ada


m0 T ~Dta12f~a1




1

klogb

Nk~a

bk

1



k21 logb $ak a Infq











and
























f~a limln Nk~a

ln bk as k ` (6)






Mh1h2 Mh1 hk






ak 51

ki51

k

V i (8)



a0 5 V1 5 2 logb M 1 (9)





1

kln

1

ki51

k

Xi a Infq

ln eq~a2X1

for any a X1



f~a 5 Infq

aq 2 t~q (10)




t Tk

~Dtak~ t bk~Dt~a011 2 5 b2k~a021 2






fu~a 5 1 2 ~a 2 l24~l 2 1








$X~t 1 Dt 2 X~t2 5 cX~21 2 Dt











Normal (l s2) 1 2 (a 2 l)2[4(l 2 1)]

Binomial 2amax 2 a

amax 2 aminlog2

amax 2 a

amax 2 amin2

a 2 amin

amax 2 aminlog2

a 2 amin

amax 2 amin















Sq~T Dti50

N21





























C Main Results



















fX~a 5 1 2~a 2 a0

2

4H~a0 2 H




















m 5Y jt







h~Y jm 5 $t~q Y j

m ln SSE~q Y jmq Q9




H~X Ym 5 h~X 21

Jj51

J

h~Y jm



J H(Y jm


Gmn~X






















E Equity Data







q


(1 d 0) iv GARCH (1 1)


05 20711 2071 2071 001 2073 2073 001 2074 001 2075 00110 20440 2044 2044 002 2049 2048 002 2049 003 2050 00220 0058 006 005 006 000 2001 006 2001 008 2001 00530 0500 049 047 012 043 041 010 042 016 047 00850 1208 119 116 027 116 110 024 114 035 136 017


05 05910 09812 09969 0999310 05465 09596 09787 0995920 05620 08867 08363 0918630 06032 08111 07209 0658750 05757 06771 05656 01754


i MMAR

05 20745 20742 20738 20734 20729 20722 20714 20705 20698 20694 20690 20686 2068310 20506 20501 20493 20485 20475 20460 20443 20426 20412 20403 20396 20386 2037920 20106 20091 20069 20052 20028 0009 0049 0090 0126 0149 0170 0197 021530 0159 0192 0246 0284 0328 0399 0472 0549 0622 0666 0712 0758 079850 0343 0465 0601 0705 0821 0988 1162 1331 1491 1595 1691 1793 1871

ii MMAR (H 5 12)

05 20766 20762 20758 20754 20750 20743 20735 20727 20720 20716 20712 20708 2070510 20542 20536 20527 20519 20511 20497 20480 20465 20451 20443 20436 20428 2042320 20150 20136 20114 20098 20078 20045 20008 0029 0062 0083 0100 0124 013830 0143 0177 0211 0244 0283 0344 0410 0479 0542 0584 0621 0666 070750 0412 0502 0633 0714 0807 0954 1106 1258 1401 1490 1581 1684 1779

iii FIGARCH (1 d 0)

05 20776 20773 20768 20764 20760 20753 20744 20736 20729 20725 20721 20716 2071310 20566 20556 20546 20537 20527 20511 20494 20476 20460 20450 20442 20430 2042220 20244 20211 20174 20141 20111 20064 20016 0034 0084 0115 0147 0191 022130 20082 20003 0086 0151 0220 0320 0416 0514 0608 0680 0737 0815 085750 0033 0200 0412 0559 0703 0934 1155 1362 1560 1699 1807 1935 2031

iv GARCH (1 1)

05 20777 20774 20770 20766 20762 20755 20747 20740 20733 20729 20725 20721 2071810 20555 20548 20541 20534 20525 20512 20497 20482 20470 20461 20454 20447 2044220 20121 20110 20095 20080 20063 20036 20005 0025 0053 0071 0084 0101 011330 0281 0302 0322 0347 0375 0420 0470 0521 0571 0601 0629 0664 068550 0928 0978 1037 1094 1150 1250 1352 1461 1569 1641 1709 1790 1859








T by X0 5 0 X t 5Xt21 1 ln (1 1 rt)



q


iii FIGARCH(1 d 0)

iv GARCH(1 1)


05 0018 00200 00080 00213 000805 00241 000876 00214 00077910 00507 00604 00278 00637 00266 00832 00417 00586 0022920 0159 0330 0212 0311 0167 0635 0635 021 0093730 042 149 114 129 0864 298 295 0698 03950 276 101 713 984 601 173 137 491 354


05 04640 03914 02541 0373310 04265 03553 01648 0419120 01084 01000 00204 0321130 00156 00197 00026 0190250 00155 00283 00034 02293


i MMAR

05 000703 000774 000889 00101 00116 00144 00185 00238 00298 00345 00394 00461 0052810 00206 00223 00253 00289 00331 0042 00546 00716 00938 0112 0131 0158 017320 00936 0103 0118 0133 0156 0203 0278 0392 0551 0695 087 111 13430 0351 0387 0457 0528 0625 0834 118 177 265 345 447 586 71850 223 253 298 344 411 565 818 123 184 234 287 362 425

ii MMAR (H 5 12)

05 000806 000882 000993 00111 00127 00157 00199 00253 00315 00364 00412 00469 0052710 00224 00246 00281 00318 00362 00455 00583 00759 00964 0112 013 0157 017520 00973 0106 0121 0138 0159 0203 0271 0369 0503 0617 0734 0902 10830 0332 0372 0437 05 0583 0769 106 152 221 281 355 465 55850 205 233 267 316 372 502 724 107 157 202 249 321 383

iii FIGARCH (1 d 0)

05 000873 000969 00112 00126 00144 00179 00229 00287 00355 00403 00447 00507 0056110 0026 00291 0034 00386 00448 00572 00745 00984 0129 0155 0185 0228 027620 0126 014 0165 0192 0227 031 0461 073 118 162 214 31 41830 0453 0527 064 0751 0931 134 211 36 575 774 103 143 19450 299 342 413 494 607 873 135 214 321 415 524 689 853

iv GARCH (1 1)

05 000804 000872 000995 00111 00128 00159 00202 00254 00315 00357 00397 00455 00510 00212 00232 00265 00299 00343 00427 00546 00697 00874 0101 0115 0135 014920 00725 00793 0091 0103 0116 0147 019 0249 0325 0383 0449 0548 062430 0214 0237 027 031 0356 0456 0608 0827 113 139 164 207 24650 125 139 159 183 214 284 395 583 85 111 139 184 232








Model



MMAR 757 1023 726 2755(05219) (04452) (05539) (03215)

MMAR (H 5 12) 1079 1246 1175 1969(02133) (02117) (02099) (02905)

FIGARCH (1 d 0) 2460 2929 2596 5879(01159) (01605) (00427) (02586)

GARCH (1 1) 2105 2295 2534 2704(00274) (00239) (00297) (00324)











VII Conclusion






REFERENCES
















































































APPENDIX

A Scaling Rule





~ X~t q ~ X~t q1w1 ~ X~t q2w2



1 w2 ln c~q2 (A1)


t~q $ w1t~q1 1 w2t~q2 (A2)




$ X~t q u ~t 5 u 5 B~u q u ~t 5 u

5 u ~tq 2 B~1 q




$X~t 1 T 9t u ~t 1 T 5 u 5 $B~u 9t

5 Bu ~t




Trading time


Vh1 hk

q Vz1 zk

q ~Mh1

2q Mh1 hl

2q

~Mh1 h l11

q Mh1 hk

q ~M z1 z l11

q Mz1 zk

q


Cov m~I1q m~I2

q 5 ~ Vq2~ Mq2k$~ M2q~ Mq2l 2 1



Log-Price



(A3)

simpli es to

B~u1q B~u3 2 B~u2

q 5 u1q 2 u3 2 u2

q 2 B~1 q2



and therefore






m0 T ~Dta2f~ada


m0 T ~Dta12f~a1




1

klogb

Nk~a

bk

1



k21 logb $ak a Infq











and





















fu~a 5 1 2 ~a 2 l24~l 2 1








$X~t 1 Dt 2 X~t2 5 cX~21 2 Dt











Normal (l s2) 1 2 (a 2 l)2[4(l 2 1)]

Binomial 2amax 2 a

amax 2 aminlog2

amax 2 a

amax 2 amin2

a 2 amin

amax 2 aminlog2

a 2 amin

amax 2 amin















Sq~T Dti50

N21





























C Main Results



















fX~a 5 1 2~a 2 a0

2

4H~a0 2 H




















m 5Y jt







h~Y jm 5 $t~q Y j

m ln SSE~q Y jmq Q9




H~X Ym 5 h~X 21

Jj51

J

h~Y jm



J H(Y jm


Gmn~X






















E Equity Data







q


(1 d 0) iv GARCH (1 1)


05 20711 2071 2071 001 2073 2073 001 2074 001 2075 00110 20440 2044 2044 002 2049 2048 002 2049 003 2050 00220 0058 006 005 006 000 2001 006 2001 008 2001 00530 0500 049 047 012 043 041 010 042 016 047 00850 1208 119 116 027 116 110 024 114 035 136 017


05 05910 09812 09969 0999310 05465 09596 09787 0995920 05620 08867 08363 0918630 06032 08111 07209 0658750 05757 06771 05656 01754


i MMAR

05 20745 20742 20738 20734 20729 20722 20714 20705 20698 20694 20690 20686 2068310 20506 20501 20493 20485 20475 20460 20443 20426 20412 20403 20396 20386 2037920 20106 20091 20069 20052 20028 0009 0049 0090 0126 0149 0170 0197 021530 0159 0192 0246 0284 0328 0399 0472 0549 0622 0666 0712 0758 079850 0343 0465 0601 0705 0821 0988 1162 1331 1491 1595 1691 1793 1871

ii MMAR (H 5 12)

05 20766 20762 20758 20754 20750 20743 20735 20727 20720 20716 20712 20708 2070510 20542 20536 20527 20519 20511 20497 20480 20465 20451 20443 20436 20428 2042320 20150 20136 20114 20098 20078 20045 20008 0029 0062 0083 0100 0124 013830 0143 0177 0211 0244 0283 0344 0410 0479 0542 0584 0621 0666 070750 0412 0502 0633 0714 0807 0954 1106 1258 1401 1490 1581 1684 1779

iii FIGARCH (1 d 0)

05 20776 20773 20768 20764 20760 20753 20744 20736 20729 20725 20721 20716 2071310 20566 20556 20546 20537 20527 20511 20494 20476 20460 20450 20442 20430 2042220 20244 20211 20174 20141 20111 20064 20016 0034 0084 0115 0147 0191 022130 20082 20003 0086 0151 0220 0320 0416 0514 0608 0680 0737 0815 085750 0033 0200 0412 0559 0703 0934 1155 1362 1560 1699 1807 1935 2031

iv GARCH (1 1)

05 20777 20774 20770 20766 20762 20755 20747 20740 20733 20729 20725 20721 2071810 20555 20548 20541 20534 20525 20512 20497 20482 20470 20461 20454 20447 2044220 20121 20110 20095 20080 20063 20036 20005 0025 0053 0071 0084 0101 011330 0281 0302 0322 0347 0375 0420 0470 0521 0571 0601 0629 0664 068550 0928 0978 1037 1094 1150 1250 1352 1461 1569 1641 1709 1790 1859








T by X0 5 0 X t 5Xt21 1 ln (1 1 rt)



q


iii FIGARCH(1 d 0)

iv GARCH(1 1)


05 0018 00200 00080 00213 000805 00241 000876 00214 00077910 00507 00604 00278 00637 00266 00832 00417 00586 0022920 0159 0330 0212 0311 0167 0635 0635 021 0093730 042 149 114 129 0864 298 295 0698 03950 276 101 713 984 601 173 137 491 354


05 04640 03914 02541 0373310 04265 03553 01648 0419120 01084 01000 00204 0321130 00156 00197 00026 0190250 00155 00283 00034 02293


i MMAR

05 000703 000774 000889 00101 00116 00144 00185 00238 00298 00345 00394 00461 0052810 00206 00223 00253 00289 00331 0042 00546 00716 00938 0112 0131 0158 017320 00936 0103 0118 0133 0156 0203 0278 0392 0551 0695 087 111 13430 0351 0387 0457 0528 0625 0834 118 177 265 345 447 586 71850 223 253 298 344 411 565 818 123 184 234 287 362 425

ii MMAR (H 5 12)

05 000806 000882 000993 00111 00127 00157 00199 00253 00315 00364 00412 00469 0052710 00224 00246 00281 00318 00362 00455 00583 00759 00964 0112 013 0157 017520 00973 0106 0121 0138 0159 0203 0271 0369 0503 0617 0734 0902 10830 0332 0372 0437 05 0583 0769 106 152 221 281 355 465 55850 205 233 267 316 372 502 724 107 157 202 249 321 383

iii FIGARCH (1 d 0)

05 000873 000969 00112 00126 00144 00179 00229 00287 00355 00403 00447 00507 0056110 0026 00291 0034 00386 00448 00572 00745 00984 0129 0155 0185 0228 027620 0126 014 0165 0192 0227 031 0461 073 118 162 214 31 41830 0453 0527 064 0751 0931 134 211 36 575 774 103 143 19450 299 342 413 494 607 873 135 214 321 415 524 689 853

iv GARCH (1 1)

05 000804 000872 000995 00111 00128 00159 00202 00254 00315 00357 00397 00455 00510 00212 00232 00265 00299 00343 00427 00546 00697 00874 0101 0115 0135 014920 00725 00793 0091 0103 0116 0147 019 0249 0325 0383 0449 0548 062430 0214 0237 027 031 0356 0456 0608 0827 113 139 164 207 24650 125 139 159 183 214 284 395 583 85 111 139 184 232








Model



MMAR 757 1023 726 2755(05219) (04452) (05539) (03215)

MMAR (H 5 12) 1079 1246 1175 1969(02133) (02117) (02099) (02905)

FIGARCH (1 d 0) 2460 2929 2596 5879(01159) (01605) (00427) (02586)

GARCH (1 1) 2105 2295 2534 2704(00274) (00239) (00297) (00324)











VII Conclusion






REFERENCES
















































































APPENDIX

A Scaling Rule





~ X~t q ~ X~t q1w1 ~ X~t q2w2



1 w2 ln c~q2 (A1)


t~q $ w1t~q1 1 w2t~q2 (A2)




$ X~t q u ~t 5 u 5 B~u q u ~t 5 u

5 u ~tq 2 B~1 q




$X~t 1 T 9t u ~t 1 T 5 u 5 $B~u 9t

5 Bu ~t




Trading time


Vh1 hk

q Vz1 zk

q ~Mh1

2q Mh1 hl

2q

~Mh1 h l11

q Mh1 hk

q ~M z1 z l11

q Mz1 zk

q


Cov m~I1q m~I2

q 5 ~ Vq2~ Mq2k$~ M2q~ Mq2l 2 1



Log-Price



(A3)

simpli es to

B~u1q B~u3 2 B~u2

q 5 u1q 2 u3 2 u2

q 2 B~1 q2



and therefore






m0 T ~Dta2f~ada


m0 T ~Dta12f~a1




1

klogb

Nk~a

bk

1



k21 logb $ak a Infq











and




























Sq~T Dti50

N21





























C Main Results



















fX~a 5 1 2~a 2 a0

2

4H~a0 2 H




















m 5Y jt







h~Y jm 5 $t~q Y j

m ln SSE~q Y jmq Q9




H~X Ym 5 h~X 21

Jj51

J

h~Y jm



J H(Y jm


Gmn~X






















E Equity Data







q


(1 d 0) iv GARCH (1 1)


05 20711 2071 2071 001 2073 2073 001 2074 001 2075 00110 20440 2044 2044 002 2049 2048 002 2049 003 2050 00220 0058 006 005 006 000 2001 006 2001 008 2001 00530 0500 049 047 012 043 041 010 042 016 047 00850 1208 119 116 027 116 110 024 114 035 136 017


05 05910 09812 09969 0999310 05465 09596 09787 0995920 05620 08867 08363 0918630 06032 08111 07209 0658750 05757 06771 05656 01754


i MMAR

05 20745 20742 20738 20734 20729 20722 20714 20705 20698 20694 20690 20686 2068310 20506 20501 20493 20485 20475 20460 20443 20426 20412 20403 20396 20386 2037920 20106 20091 20069 20052 20028 0009 0049 0090 0126 0149 0170 0197 021530 0159 0192 0246 0284 0328 0399 0472 0549 0622 0666 0712 0758 079850 0343 0465 0601 0705 0821 0988 1162 1331 1491 1595 1691 1793 1871

ii MMAR (H 5 12)

05 20766 20762 20758 20754 20750 20743 20735 20727 20720 20716 20712 20708 2070510 20542 20536 20527 20519 20511 20497 20480 20465 20451 20443 20436 20428 2042320 20150 20136 20114 20098 20078 20045 20008 0029 0062 0083 0100 0124 013830 0143 0177 0211 0244 0283 0344 0410 0479 0542 0584 0621 0666 070750 0412 0502 0633 0714 0807 0954 1106 1258 1401 1490 1581 1684 1779

iii FIGARCH (1 d 0)

05 20776 20773 20768 20764 20760 20753 20744 20736 20729 20725 20721 20716 2071310 20566 20556 20546 20537 20527 20511 20494 20476 20460 20450 20442 20430 2042220 20244 20211 20174 20141 20111 20064 20016 0034 0084 0115 0147 0191 022130 20082 20003 0086 0151 0220 0320 0416 0514 0608 0680 0737 0815 085750 0033 0200 0412 0559 0703 0934 1155 1362 1560 1699 1807 1935 2031

iv GARCH (1 1)

05 20777 20774 20770 20766 20762 20755 20747 20740 20733 20729 20725 20721 2071810 20555 20548 20541 20534 20525 20512 20497 20482 20470 20461 20454 20447 2044220 20121 20110 20095 20080 20063 20036 20005 0025 0053 0071 0084 0101 011330 0281 0302 0322 0347 0375 0420 0470 0521 0571 0601 0629 0664 068550 0928 0978 1037 1094 1150 1250 1352 1461 1569 1641 1709 1790 1859








T by X0 5 0 X t 5Xt21 1 ln (1 1 rt)



q


iii FIGARCH(1 d 0)

iv GARCH(1 1)


05 0018 00200 00080 00213 000805 00241 000876 00214 00077910 00507 00604 00278 00637 00266 00832 00417 00586 0022920 0159 0330 0212 0311 0167 0635 0635 021 0093730 042 149 114 129 0864 298 295 0698 03950 276 101 713 984 601 173 137 491 354


05 04640 03914 02541 0373310 04265 03553 01648 0419120 01084 01000 00204 0321130 00156 00197 00026 0190250 00155 00283 00034 02293


i MMAR

05 000703 000774 000889 00101 00116 00144 00185 00238 00298 00345 00394 00461 0052810 00206 00223 00253 00289 00331 0042 00546 00716 00938 0112 0131 0158 017320 00936 0103 0118 0133 0156 0203 0278 0392 0551 0695 087 111 13430 0351 0387 0457 0528 0625 0834 118 177 265 345 447 586 71850 223 253 298 344 411 565 818 123 184 234 287 362 425

ii MMAR (H 5 12)

05 000806 000882 000993 00111 00127 00157 00199 00253 00315 00364 00412 00469 0052710 00224 00246 00281 00318 00362 00455 00583 00759 00964 0112 013 0157 017520 00973 0106 0121 0138 0159 0203 0271 0369 0503 0617 0734 0902 10830 0332 0372 0437 05 0583 0769 106 152 221 281 355 465 55850 205 233 267 316 372 502 724 107 157 202 249 321 383

iii FIGARCH (1 d 0)

05 000873 000969 00112 00126 00144 00179 00229 00287 00355 00403 00447 00507 0056110 0026 00291 0034 00386 00448 00572 00745 00984 0129 0155 0185 0228 027620 0126 014 0165 0192 0227 031 0461 073 118 162 214 31 41830 0453 0527 064 0751 0931 134 211 36 575 774 103 143 19450 299 342 413 494 607 873 135 214 321 415 524 689 853

iv GARCH (1 1)

05 000804 000872 000995 00111 00128 00159 00202 00254 00315 00357 00397 00455 00510 00212 00232 00265 00299 00343 00427 00546 00697 00874 0101 0115 0135 014920 00725 00793 0091 0103 0116 0147 019 0249 0325 0383 0449 0548 062430 0214 0237 027 031 0356 0456 0608 0827 113 139 164 207 24650 125 139 159 183 214 284 395 583 85 111 139 184 232








Model



MMAR 757 1023 726 2755(05219) (04452) (05539) (03215)

MMAR (H 5 12) 1079 1246 1175 1969(02133) (02117) (02099) (02905)

FIGARCH (1 d 0) 2460 2929 2596 5879(01159) (01605) (00427) (02586)

GARCH (1 1) 2105 2295 2534 2704(00274) (00239) (00297) (00324)











VII Conclusion






REFERENCES
















































































APPENDIX

A Scaling Rule





~ X~t q ~ X~t q1w1 ~ X~t q2w2



1 w2 ln c~q2 (A1)


t~q $ w1t~q1 1 w2t~q2 (A2)




$ X~t q u ~t 5 u 5 B~u q u ~t 5 u

5 u ~tq 2 B~1 q




$X~t 1 T 9t u ~t 1 T 5 u 5 $B~u 9t

5 Bu ~t




Trading time


Vh1 hk

q Vz1 zk

q ~Mh1

2q Mh1 hl

2q

~Mh1 h l11

q Mh1 hk

q ~M z1 z l11

q Mz1 zk

q


Cov m~I1q m~I2

q 5 ~ Vq2~ Mq2k$~ M2q~ Mq2l 2 1



Log-Price



(A3)

simpli es to

B~u1q B~u3 2 B~u2

q 5 u1q 2 u3 2 u2

q 2 B~1 q2



and therefore






m0 T ~Dta2f~ada


m0 T ~Dta12f~a1




1

klogb

Nk~a

bk

1



k21 logb $ak a Infq











and






































C Main Results



















fX~a 5 1 2~a 2 a0

2

4H~a0 2 H




















m 5Y jt







h~Y jm 5 $t~q Y j

m ln SSE~q Y jmq Q9




H~X Ym 5 h~X 21

Jj51

J

h~Y jm



J H(Y jm


Gmn~X






















E Equity Data







q


(1 d 0) iv GARCH (1 1)


05 20711 2071 2071 001 2073 2073 001 2074 001 2075 00110 20440 2044 2044 002 2049 2048 002 2049 003 2050 00220 0058 006 005 006 000 2001 006 2001 008 2001 00530 0500 049 047 012 043 041 010 042 016 047 00850 1208 119 116 027 116 110 024 114 035 136 017


05 05910 09812 09969 0999310 05465 09596 09787 0995920 05620 08867 08363 0918630 06032 08111 07209 0658750 05757 06771 05656 01754


i MMAR

05 20745 20742 20738 20734 20729 20722 20714 20705 20698 20694 20690 20686 2068310 20506 20501 20493 20485 20475 20460 20443 20426 20412 20403 20396 20386 2037920 20106 20091 20069 20052 20028 0009 0049 0090 0126 0149 0170 0197 021530 0159 0192 0246 0284 0328 0399 0472 0549 0622 0666 0712 0758 079850 0343 0465 0601 0705 0821 0988 1162 1331 1491 1595 1691 1793 1871

ii MMAR (H 5 12)

05 20766 20762 20758 20754 20750 20743 20735 20727 20720 20716 20712 20708 2070510 20542 20536 20527 20519 20511 20497 20480 20465 20451 20443 20436 20428 2042320 20150 20136 20114 20098 20078 20045 20008 0029 0062 0083 0100 0124 013830 0143 0177 0211 0244 0283 0344 0410 0479 0542 0584 0621 0666 070750 0412 0502 0633 0714 0807 0954 1106 1258 1401 1490 1581 1684 1779

iii FIGARCH (1 d 0)

05 20776 20773 20768 20764 20760 20753 20744 20736 20729 20725 20721 20716 2071310 20566 20556 20546 20537 20527 20511 20494 20476 20460 20450 20442 20430 2042220 20244 20211 20174 20141 20111 20064 20016 0034 0084 0115 0147 0191 022130 20082 20003 0086 0151 0220 0320 0416 0514 0608 0680 0737 0815 085750 0033 0200 0412 0559 0703 0934 1155 1362 1560 1699 1807 1935 2031

iv GARCH (1 1)

05 20777 20774 20770 20766 20762 20755 20747 20740 20733 20729 20725 20721 2071810 20555 20548 20541 20534 20525 20512 20497 20482 20470 20461 20454 20447 2044220 20121 20110 20095 20080 20063 20036 20005 0025 0053 0071 0084 0101 011330 0281 0302 0322 0347 0375 0420 0470 0521 0571 0601 0629 0664 068550 0928 0978 1037 1094 1150 1250 1352 1461 1569 1641 1709 1790 1859








T by X0 5 0 X t 5Xt21 1 ln (1 1 rt)



q


iii FIGARCH(1 d 0)

iv GARCH(1 1)


05 0018 00200 00080 00213 000805 00241 000876 00214 00077910 00507 00604 00278 00637 00266 00832 00417 00586 0022920 0159 0330 0212 0311 0167 0635 0635 021 0093730 042 149 114 129 0864 298 295 0698 03950 276 101 713 984 601 173 137 491 354


05 04640 03914 02541 0373310 04265 03553 01648 0419120 01084 01000 00204 0321130 00156 00197 00026 0190250 00155 00283 00034 02293


i MMAR

05 000703 000774 000889 00101 00116 00144 00185 00238 00298 00345 00394 00461 0052810 00206 00223 00253 00289 00331 0042 00546 00716 00938 0112 0131 0158 017320 00936 0103 0118 0133 0156 0203 0278 0392 0551 0695 087 111 13430 0351 0387 0457 0528 0625 0834 118 177 265 345 447 586 71850 223 253 298 344 411 565 818 123 184 234 287 362 425

ii MMAR (H 5 12)

05 000806 000882 000993 00111 00127 00157 00199 00253 00315 00364 00412 00469 0052710 00224 00246 00281 00318 00362 00455 00583 00759 00964 0112 013 0157 017520 00973 0106 0121 0138 0159 0203 0271 0369 0503 0617 0734 0902 10830 0332 0372 0437 05 0583 0769 106 152 221 281 355 465 55850 205 233 267 316 372 502 724 107 157 202 249 321 383

iii FIGARCH (1 d 0)

05 000873 000969 00112 00126 00144 00179 00229 00287 00355 00403 00447 00507 0056110 0026 00291 0034 00386 00448 00572 00745 00984 0129 0155 0185 0228 027620 0126 014 0165 0192 0227 031 0461 073 118 162 214 31 41830 0453 0527 064 0751 0931 134 211 36 575 774 103 143 19450 299 342 413 494 607 873 135 214 321 415 524 689 853

iv GARCH (1 1)

05 000804 000872 000995 00111 00128 00159 00202 00254 00315 00357 00397 00455 00510 00212 00232 00265 00299 00343 00427 00546 00697 00874 0101 0115 0135 014920 00725 00793 0091 0103 0116 0147 019 0249 0325 0383 0449 0548 062430 0214 0237 027 031 0356 0456 0608 0827 113 139 164 207 24650 125 139 159 183 214 284 395 583 85 111 139 184 232








Model



MMAR 757 1023 726 2755(05219) (04452) (05539) (03215)

MMAR (H 5 12) 1079 1246 1175 1969(02133) (02117) (02099) (02905)

FIGARCH (1 d 0) 2460 2929 2596 5879(01159) (01605) (00427) (02586)

GARCH (1 1) 2105 2295 2534 2704(00274) (00239) (00297) (00324)











VII Conclusion






REFERENCES
















































































APPENDIX

A Scaling Rule





~ X~t q ~ X~t q1w1 ~ X~t q2w2



1 w2 ln c~q2 (A1)


t~q $ w1t~q1 1 w2t~q2 (A2)




$ X~t q u ~t 5 u 5 B~u q u ~t 5 u

5 u ~tq 2 B~1 q




$X~t 1 T 9t u ~t 1 T 5 u 5 $B~u 9t

5 Bu ~t




Trading time


Vh1 hk

q Vz1 zk

q ~Mh1

2q Mh1 hl

2q

~Mh1 h l11

q Mh1 hk

q ~M z1 z l11

q Mz1 zk

q


Cov m~I1q m~I2

q 5 ~ Vq2~ Mq2k$~ M2q~ Mq2l 2 1



Log-Price



(A3)

simpli es to

B~u1q B~u3 2 B~u2

q 5 u1q 2 u3 2 u2

q 2 B~1 q2



and therefore






m0 T ~Dta2f~ada


m0 T ~Dta12f~a1




1

klogb

Nk~a

bk

1



k21 logb $ak a Infq











and























fX~a 5 1 2~a 2 a0

2

4H~a0 2 H




















m 5Y jt







h~Y jm 5 $t~q Y j

m ln SSE~q Y jmq Q9




H~X Ym 5 h~X 21

Jj51

J

h~Y jm



J H(Y jm


Gmn~X






















E Equity Data







q


(1 d 0) iv GARCH (1 1)


05 20711 2071 2071 001 2073 2073 001 2074 001 2075 00110 20440 2044 2044 002 2049 2048 002 2049 003 2050 00220 0058 006 005 006 000 2001 006 2001 008 2001 00530 0500 049 047 012 043 041 010 042 016 047 00850 1208 119 116 027 116 110 024 114 035 136 017


05 05910 09812 09969 0999310 05465 09596 09787 0995920 05620 08867 08363 0918630 06032 08111 07209 0658750 05757 06771 05656 01754


i MMAR

05 20745 20742 20738 20734 20729 20722 20714 20705 20698 20694 20690 20686 2068310 20506 20501 20493 20485 20475 20460 20443 20426 20412 20403 20396 20386 2037920 20106 20091 20069 20052 20028 0009 0049 0090 0126 0149 0170 0197 021530 0159 0192 0246 0284 0328 0399 0472 0549 0622 0666 0712 0758 079850 0343 0465 0601 0705 0821 0988 1162 1331 1491 1595 1691 1793 1871

ii MMAR (H 5 12)

05 20766 20762 20758 20754 20750 20743 20735 20727 20720 20716 20712 20708 2070510 20542 20536 20527 20519 20511 20497 20480 20465 20451 20443 20436 20428 2042320 20150 20136 20114 20098 20078 20045 20008 0029 0062 0083 0100 0124 013830 0143 0177 0211 0244 0283 0344 0410 0479 0542 0584 0621 0666 070750 0412 0502 0633 0714 0807 0954 1106 1258 1401 1490 1581 1684 1779

iii FIGARCH (1 d 0)

05 20776 20773 20768 20764 20760 20753 20744 20736 20729 20725 20721 20716 2071310 20566 20556 20546 20537 20527 20511 20494 20476 20460 20450 20442 20430 2042220 20244 20211 20174 20141 20111 20064 20016 0034 0084 0115 0147 0191 022130 20082 20003 0086 0151 0220 0320 0416 0514 0608 0680 0737 0815 085750 0033 0200 0412 0559 0703 0934 1155 1362 1560 1699 1807 1935 2031

iv GARCH (1 1)

05 20777 20774 20770 20766 20762 20755 20747 20740 20733 20729 20725 20721 2071810 20555 20548 20541 20534 20525 20512 20497 20482 20470 20461 20454 20447 2044220 20121 20110 20095 20080 20063 20036 20005 0025 0053 0071 0084 0101 011330 0281 0302 0322 0347 0375 0420 0470 0521 0571 0601 0629 0664 068550 0928 0978 1037 1094 1150 1250 1352 1461 1569 1641 1709 1790 1859








T by X0 5 0 X t 5Xt21 1 ln (1 1 rt)



q


iii FIGARCH(1 d 0)

iv GARCH(1 1)


05 0018 00200 00080 00213 000805 00241 000876 00214 00077910 00507 00604 00278 00637 00266 00832 00417 00586 0022920 0159 0330 0212 0311 0167 0635 0635 021 0093730 042 149 114 129 0864 298 295 0698 03950 276 101 713 984 601 173 137 491 354


05 04640 03914 02541 0373310 04265 03553 01648 0419120 01084 01000 00204 0321130 00156 00197 00026 0190250 00155 00283 00034 02293


i MMAR

05 000703 000774 000889 00101 00116 00144 00185 00238 00298 00345 00394 00461 0052810 00206 00223 00253 00289 00331 0042 00546 00716 00938 0112 0131 0158 017320 00936 0103 0118 0133 0156 0203 0278 0392 0551 0695 087 111 13430 0351 0387 0457 0528 0625 0834 118 177 265 345 447 586 71850 223 253 298 344 411 565 818 123 184 234 287 362 425

ii MMAR (H 5 12)

05 000806 000882 000993 00111 00127 00157 00199 00253 00315 00364 00412 00469 0052710 00224 00246 00281 00318 00362 00455 00583 00759 00964 0112 013 0157 017520 00973 0106 0121 0138 0159 0203 0271 0369 0503 0617 0734 0902 10830 0332 0372 0437 05 0583 0769 106 152 221 281 355 465 55850 205 233 267 316 372 502 724 107 157 202 249 321 383

iii FIGARCH (1 d 0)

05 000873 000969 00112 00126 00144 00179 00229 00287 00355 00403 00447 00507 0056110 0026 00291 0034 00386 00448 00572 00745 00984 0129 0155 0185 0228 027620 0126 014 0165 0192 0227 031 0461 073 118 162 214 31 41830 0453 0527 064 0751 0931 134 211 36 575 774 103 143 19450 299 342 413 494 607 873 135 214 321 415 524 689 853

iv GARCH (1 1)

05 000804 000872 000995 00111 00128 00159 00202 00254 00315 00357 00397 00455 00510 00212 00232 00265 00299 00343 00427 00546 00697 00874 0101 0115 0135 014920 00725 00793 0091 0103 0116 0147 019 0249 0325 0383 0449 0548 062430 0214 0237 027 031 0356 0456 0608 0827 113 139 164 207 24650 125 139 159 183 214 284 395 583 85 111 139 184 232








Model



MMAR 757 1023 726 2755(05219) (04452) (05539) (03215)

MMAR (H 5 12) 1079 1246 1175 1969(02133) (02117) (02099) (02905)

FIGARCH (1 d 0) 2460 2929 2596 5879(01159) (01605) (00427) (02586)

GARCH (1 1) 2105 2295 2534 2704(00274) (00239) (00297) (00324)











VII Conclusion






REFERENCES
















































































APPENDIX

A Scaling Rule





~ X~t q ~ X~t q1w1 ~ X~t q2w2



1 w2 ln c~q2 (A1)


t~q $ w1t~q1 1 w2t~q2 (A2)




$ X~t q u ~t 5 u 5 B~u q u ~t 5 u

5 u ~tq 2 B~1 q




$X~t 1 T 9t u ~t 1 T 5 u 5 $B~u 9t

5 Bu ~t




Trading time


Vh1 hk

q Vz1 zk

q ~Mh1

2q Mh1 hl

2q

~Mh1 h l11

q Mh1 hk

q ~M z1 z l11

q Mz1 zk

q


Cov m~I1q m~I2

q 5 ~ Vq2~ Mq2k$~ M2q~ Mq2l 2 1



Log-Price



(A3)

simpli es to

B~u1q B~u3 2 B~u2

q 5 u1q 2 u3 2 u2

q 2 B~1 q2



and therefore






m0 T ~Dta2f~ada


m0 T ~Dta12f~a1




1

klogb

Nk~a

bk

1



k21 logb $ak a Infq











and





























m 5Y jt







h~Y jm 5 $t~q Y j

m ln SSE~q Y jmq Q9




H~X Ym 5 h~X 21

Jj51

J

h~Y jm



J H(Y jm


Gmn~X






















E Equity Data







q


(1 d 0) iv GARCH (1 1)


05 20711 2071 2071 001 2073 2073 001 2074 001 2075 00110 20440 2044 2044 002 2049 2048 002 2049 003 2050 00220 0058 006 005 006 000 2001 006 2001 008 2001 00530 0500 049 047 012 043 041 010 042 016 047 00850 1208 119 116 027 116 110 024 114 035 136 017


05 05910 09812 09969 0999310 05465 09596 09787 0995920 05620 08867 08363 0918630 06032 08111 07209 0658750 05757 06771 05656 01754


i MMAR

05 20745 20742 20738 20734 20729 20722 20714 20705 20698 20694 20690 20686 2068310 20506 20501 20493 20485 20475 20460 20443 20426 20412 20403 20396 20386 2037920 20106 20091 20069 20052 20028 0009 0049 0090 0126 0149 0170 0197 021530 0159 0192 0246 0284 0328 0399 0472 0549 0622 0666 0712 0758 079850 0343 0465 0601 0705 0821 0988 1162 1331 1491 1595 1691 1793 1871

ii MMAR (H 5 12)

05 20766 20762 20758 20754 20750 20743 20735 20727 20720 20716 20712 20708 2070510 20542 20536 20527 20519 20511 20497 20480 20465 20451 20443 20436 20428 2042320 20150 20136 20114 20098 20078 20045 20008 0029 0062 0083 0100 0124 013830 0143 0177 0211 0244 0283 0344 0410 0479 0542 0584 0621 0666 070750 0412 0502 0633 0714 0807 0954 1106 1258 1401 1490 1581 1684 1779

iii FIGARCH (1 d 0)

05 20776 20773 20768 20764 20760 20753 20744 20736 20729 20725 20721 20716 2071310 20566 20556 20546 20537 20527 20511 20494 20476 20460 20450 20442 20430 2042220 20244 20211 20174 20141 20111 20064 20016 0034 0084 0115 0147 0191 022130 20082 20003 0086 0151 0220 0320 0416 0514 0608 0680 0737 0815 085750 0033 0200 0412 0559 0703 0934 1155 1362 1560 1699 1807 1935 2031

iv GARCH (1 1)

05 20777 20774 20770 20766 20762 20755 20747 20740 20733 20729 20725 20721 2071810 20555 20548 20541 20534 20525 20512 20497 20482 20470 20461 20454 20447 2044220 20121 20110 20095 20080 20063 20036 20005 0025 0053 0071 0084 0101 011330 0281 0302 0322 0347 0375 0420 0470 0521 0571 0601 0629 0664 068550 0928 0978 1037 1094 1150 1250 1352 1461 1569 1641 1709 1790 1859








T by X0 5 0 X t 5Xt21 1 ln (1 1 rt)



q


iii FIGARCH(1 d 0)

iv GARCH(1 1)


05 0018 00200 00080 00213 000805 00241 000876 00214 00077910 00507 00604 00278 00637 00266 00832 00417 00586 0022920 0159 0330 0212 0311 0167 0635 0635 021 0093730 042 149 114 129 0864 298 295 0698 03950 276 101 713 984 601 173 137 491 354


05 04640 03914 02541 0373310 04265 03553 01648 0419120 01084 01000 00204 0321130 00156 00197 00026 0190250 00155 00283 00034 02293


i MMAR

05 000703 000774 000889 00101 00116 00144 00185 00238 00298 00345 00394 00461 0052810 00206 00223 00253 00289 00331 0042 00546 00716 00938 0112 0131 0158 017320 00936 0103 0118 0133 0156 0203 0278 0392 0551 0695 087 111 13430 0351 0387 0457 0528 0625 0834 118 177 265 345 447 586 71850 223 253 298 344 411 565 818 123 184 234 287 362 425

ii MMAR (H 5 12)

05 000806 000882 000993 00111 00127 00157 00199 00253 00315 00364 00412 00469 0052710 00224 00246 00281 00318 00362 00455 00583 00759 00964 0112 013 0157 017520 00973 0106 0121 0138 0159 0203 0271 0369 0503 0617 0734 0902 10830 0332 0372 0437 05 0583 0769 106 152 221 281 355 465 55850 205 233 267 316 372 502 724 107 157 202 249 321 383

iii FIGARCH (1 d 0)

05 000873 000969 00112 00126 00144 00179 00229 00287 00355 00403 00447 00507 0056110 0026 00291 0034 00386 00448 00572 00745 00984 0129 0155 0185 0228 027620 0126 014 0165 0192 0227 031 0461 073 118 162 214 31 41830 0453 0527 064 0751 0931 134 211 36 575 774 103 143 19450 299 342 413 494 607 873 135 214 321 415 524 689 853

iv GARCH (1 1)

05 000804 000872 000995 00111 00128 00159 00202 00254 00315 00357 00397 00455 00510 00212 00232 00265 00299 00343 00427 00546 00697 00874 0101 0115 0135 014920 00725 00793 0091 0103 0116 0147 019 0249 0325 0383 0449 0548 062430 0214 0237 027 031 0356 0456 0608 0827 113 139 164 207 24650 125 139 159 183 214 284 395 583 85 111 139 184 232








Model



MMAR 757 1023 726 2755(05219) (04452) (05539) (03215)

MMAR (H 5 12) 1079 1246 1175 1969(02133) (02117) (02099) (02905)

FIGARCH (1 d 0) 2460 2929 2596 5879(01159) (01605) (00427) (02586)

GARCH (1 1) 2105 2295 2534 2704(00274) (00239) (00297) (00324)











VII Conclusion






REFERENCES
















































































APPENDIX

A Scaling Rule





~ X~t q ~ X~t q1w1 ~ X~t q2w2



1 w2 ln c~q2 (A1)


t~q $ w1t~q1 1 w2t~q2 (A2)




$ X~t q u ~t 5 u 5 B~u q u ~t 5 u

5 u ~tq 2 B~1 q




$X~t 1 T 9t u ~t 1 T 5 u 5 $B~u 9t

5 Bu ~t




Trading time


Vh1 hk

q Vz1 zk

q ~Mh1

2q Mh1 hl

2q

~Mh1 h l11

q Mh1 hk

q ~M z1 z l11

q Mz1 zk

q


Cov m~I1q m~I2

q 5 ~ Vq2~ Mq2k$~ M2q~ Mq2l 2 1



Log-Price



(A3)

simpli es to

B~u1q B~u3 2 B~u2

q 5 u1q 2 u3 2 u2

q 2 B~1 q2



and therefore






m0 T ~Dta2f~ada


m0 T ~Dta12f~a1




1

klogb

Nk~a

bk

1



k21 logb $ak a Infq











and




























E Equity Data







q


(1 d 0) iv GARCH (1 1)


05 20711 2071 2071 001 2073 2073 001 2074 001 2075 00110 20440 2044 2044 002 2049 2048 002 2049 003 2050 00220 0058 006 005 006 000 2001 006 2001 008 2001 00530 0500 049 047 012 043 041 010 042 016 047 00850 1208 119 116 027 116 110 024 114 035 136 017


05 05910 09812 09969 0999310 05465 09596 09787 0995920 05620 08867 08363 0918630 06032 08111 07209 0658750 05757 06771 05656 01754


i MMAR

05 20745 20742 20738 20734 20729 20722 20714 20705 20698 20694 20690 20686 2068310 20506 20501 20493 20485 20475 20460 20443 20426 20412 20403 20396 20386 2037920 20106 20091 20069 20052 20028 0009 0049 0090 0126 0149 0170 0197 021530 0159 0192 0246 0284 0328 0399 0472 0549 0622 0666 0712 0758 079850 0343 0465 0601 0705 0821 0988 1162 1331 1491 1595 1691 1793 1871

ii MMAR (H 5 12)

05 20766 20762 20758 20754 20750 20743 20735 20727 20720 20716 20712 20708 2070510 20542 20536 20527 20519 20511 20497 20480 20465 20451 20443 20436 20428 2042320 20150 20136 20114 20098 20078 20045 20008 0029 0062 0083 0100 0124 013830 0143 0177 0211 0244 0283 0344 0410 0479 0542 0584 0621 0666 070750 0412 0502 0633 0714 0807 0954 1106 1258 1401 1490 1581 1684 1779

iii FIGARCH (1 d 0)

05 20776 20773 20768 20764 20760 20753 20744 20736 20729 20725 20721 20716 2071310 20566 20556 20546 20537 20527 20511 20494 20476 20460 20450 20442 20430 2042220 20244 20211 20174 20141 20111 20064 20016 0034 0084 0115 0147 0191 022130 20082 20003 0086 0151 0220 0320 0416 0514 0608 0680 0737 0815 085750 0033 0200 0412 0559 0703 0934 1155 1362 1560 1699 1807 1935 2031

iv GARCH (1 1)

05 20777 20774 20770 20766 20762 20755 20747 20740 20733 20729 20725 20721 2071810 20555 20548 20541 20534 20525 20512 20497 20482 20470 20461 20454 20447 2044220 20121 20110 20095 20080 20063 20036 20005 0025 0053 0071 0084 0101 011330 0281 0302 0322 0347 0375 0420 0470 0521 0571 0601 0629 0664 068550 0928 0978 1037 1094 1150 1250 1352 1461 1569 1641 1709 1790 1859








T by X0 5 0 X t 5Xt21 1 ln (1 1 rt)



q


iii FIGARCH(1 d 0)

iv GARCH(1 1)


05 0018 00200 00080 00213 000805 00241 000876 00214 00077910 00507 00604 00278 00637 00266 00832 00417 00586 0022920 0159 0330 0212 0311 0167 0635 0635 021 0093730 042 149 114 129 0864 298 295 0698 03950 276 101 713 984 601 173 137 491 354


05 04640 03914 02541 0373310 04265 03553 01648 0419120 01084 01000 00204 0321130 00156 00197 00026 0190250 00155 00283 00034 02293


i MMAR

05 000703 000774 000889 00101 00116 00144 00185 00238 00298 00345 00394 00461 0052810 00206 00223 00253 00289 00331 0042 00546 00716 00938 0112 0131 0158 017320 00936 0103 0118 0133 0156 0203 0278 0392 0551 0695 087 111 13430 0351 0387 0457 0528 0625 0834 118 177 265 345 447 586 71850 223 253 298 344 411 565 818 123 184 234 287 362 425

ii MMAR (H 5 12)

05 000806 000882 000993 00111 00127 00157 00199 00253 00315 00364 00412 00469 0052710 00224 00246 00281 00318 00362 00455 00583 00759 00964 0112 013 0157 017520 00973 0106 0121 0138 0159 0203 0271 0369 0503 0617 0734 0902 10830 0332 0372 0437 05 0583 0769 106 152 221 281 355 465 55850 205 233 267 316 372 502 724 107 157 202 249 321 383

iii FIGARCH (1 d 0)

05 000873 000969 00112 00126 00144 00179 00229 00287 00355 00403 00447 00507 0056110 0026 00291 0034 00386 00448 00572 00745 00984 0129 0155 0185 0228 027620 0126 014 0165 0192 0227 031 0461 073 118 162 214 31 41830 0453 0527 064 0751 0931 134 211 36 575 774 103 143 19450 299 342 413 494 607 873 135 214 321 415 524 689 853

iv GARCH (1 1)

05 000804 000872 000995 00111 00128 00159 00202 00254 00315 00357 00397 00455 00510 00212 00232 00265 00299 00343 00427 00546 00697 00874 0101 0115 0135 014920 00725 00793 0091 0103 0116 0147 019 0249 0325 0383 0449 0548 062430 0214 0237 027 031 0356 0456 0608 0827 113 139 164 207 24650 125 139 159 183 214 284 395 583 85 111 139 184 232








Model



MMAR 757 1023 726 2755(05219) (04452) (05539) (03215)

MMAR (H 5 12) 1079 1246 1175 1969(02133) (02117) (02099) (02905)

FIGARCH (1 d 0) 2460 2929 2596 5879(01159) (01605) (00427) (02586)

GARCH (1 1) 2105 2295 2534 2704(00274) (00239) (00297) (00324)











VII Conclusion






REFERENCES
















































































APPENDIX

A Scaling Rule





~ X~t q ~ X~t q1w1 ~ X~t q2w2



1 w2 ln c~q2 (A1)


t~q $ w1t~q1 1 w2t~q2 (A2)




$ X~t q u ~t 5 u 5 B~u q u ~t 5 u

5 u ~tq 2 B~1 q




$X~t 1 T 9t u ~t 1 T 5 u 5 $B~u 9t

5 Bu ~t




Trading time


Vh1 hk

q Vz1 zk

q ~Mh1

2q Mh1 hl

2q

~Mh1 h l11

q Mh1 hk

q ~M z1 z l11

q Mz1 zk

q


Cov m~I1q m~I2

q 5 ~ Vq2~ Mq2k$~ M2q~ Mq2l 2 1



Log-Price



(A3)

simpli es to

B~u1q B~u3 2 B~u2

q 5 u1q 2 u3 2 u2

q 2 B~1 q2



and therefore






m0 T ~Dta2f~ada


m0 T ~Dta12f~a1




1

klogb

Nk~a

bk

1



k21 logb $ak a Infq











and























T by X0 5 0 X t 5Xt21 1 ln (1 1 rt)



q


iii FIGARCH(1 d 0)

iv GARCH(1 1)


05 0018 00200 00080 00213 000805 00241 000876 00214 00077910 00507 00604 00278 00637 00266 00832 00417 00586 0022920 0159 0330 0212 0311 0167 0635 0635 021 0093730 042 149 114 129 0864 298 295 0698 03950 276 101 713 984 601 173 137 491 354


05 04640 03914 02541 0373310 04265 03553 01648 0419120 01084 01000 00204 0321130 00156 00197 00026 0190250 00155 00283 00034 02293


i MMAR

05 000703 000774 000889 00101 00116 00144 00185 00238 00298 00345 00394 00461 0052810 00206 00223 00253 00289 00331 0042 00546 00716 00938 0112 0131 0158 017320 00936 0103 0118 0133 0156 0203 0278 0392 0551 0695 087 111 13430 0351 0387 0457 0528 0625 0834 118 177 265 345 447 586 71850 223 253 298 344 411 565 818 123 184 234 287 362 425

ii MMAR (H 5 12)

05 000806 000882 000993 00111 00127 00157 00199 00253 00315 00364 00412 00469 0052710 00224 00246 00281 00318 00362 00455 00583 00759 00964 0112 013 0157 017520 00973 0106 0121 0138 0159 0203 0271 0369 0503 0617 0734 0902 10830 0332 0372 0437 05 0583 0769 106 152 221 281 355 465 55850 205 233 267 316 372 502 724 107 157 202 249 321 383

iii FIGARCH (1 d 0)

05 000873 000969 00112 00126 00144 00179 00229 00287 00355 00403 00447 00507 0056110 0026 00291 0034 00386 00448 00572 00745 00984 0129 0155 0185 0228 027620 0126 014 0165 0192 0227 031 0461 073 118 162 214 31 41830 0453 0527 064 0751 0931 134 211 36 575 774 103 143 19450 299 342 413 494 607 873 135 214 321 415 524 689 853

iv GARCH (1 1)

05 000804 000872 000995 00111 00128 00159 00202 00254 00315 00357 00397 00455 00510 00212 00232 00265 00299 00343 00427 00546 00697 00874 0101 0115 0135 014920 00725 00793 0091 0103 0116 0147 019 0249 0325 0383 0449 0548 062430 0214 0237 027 031 0356 0456 0608 0827 113 139 164 207 24650 125 139 159 183 214 284 395 583 85 111 139 184 232








Model



MMAR 757 1023 726 2755(05219) (04452) (05539) (03215)

MMAR (H 5 12) 1079 1246 1175 1969(02133) (02117) (02099) (02905)

FIGARCH (1 d 0) 2460 2929 2596 5879(01159) (01605) (00427) (02586)

GARCH (1 1) 2105 2295 2534 2704(00274) (00239) (00297) (00324)











VII Conclusion






REFERENCES
















































































APPENDIX

A Scaling Rule





~ X~t q ~ X~t q1w1 ~ X~t q2w2



1 w2 ln c~q2 (A1)


t~q $ w1t~q1 1 w2t~q2 (A2)




$ X~t q u ~t 5 u 5 B~u q u ~t 5 u

5 u ~tq 2 B~1 q




$X~t 1 T 9t u ~t 1 T 5 u 5 $B~u 9t

5 Bu ~t




Trading time


Vh1 hk

q Vz1 zk

q ~Mh1

2q Mh1 hl

2q

~Mh1 h l11

q Mh1 hk

q ~M z1 z l11

q Mz1 zk

q


Cov m~I1q m~I2

q 5 ~ Vq2~ Mq2k$~ M2q~ Mq2l 2 1



Log-Price



(A3)

simpli es to

B~u1q B~u3 2 B~u2

q 5 u1q 2 u3 2 u2

q 2 B~1 q2



and therefore






m0 T ~Dta2f~ada


m0 T ~Dta12f~a1




1

klogb

Nk~a

bk

1



k21 logb $ak a Infq











and























Model



MMAR 757 1023 726 2755(05219) (04452) (05539) (03215)

MMAR (H 5 12) 1079 1246 1175 1969(02133) (02117) (02099) (02905)

FIGARCH (1 d 0) 2460 2929 2596 5879(01159) (01605) (00427) (02586)

GARCH (1 1) 2105 2295 2534 2704(00274) (00239) (00297) (00324)











VII Conclusion






REFERENCES
















































































APPENDIX

A Scaling Rule





~ X~t q ~ X~t q1w1 ~ X~t q2w2



1 w2 ln c~q2 (A1)


t~q $ w1t~q1 1 w2t~q2 (A2)




$ X~t q u ~t 5 u 5 B~u q u ~t 5 u

5 u ~tq 2 B~1 q




$X~t 1 T 9t u ~t 1 T 5 u 5 $B~u 9t

5 Bu ~t




Trading time


Vh1 hk

q Vz1 zk

q ~Mh1

2q Mh1 hl

2q

~Mh1 h l11

q Mh1 hk

q ~M z1 z l11

q Mz1 zk

q


Cov m~I1q m~I2

q 5 ~ Vq2~ Mq2k$~ M2q~ Mq2l 2 1



Log-Price



(A3)

simpli es to

B~u1q B~u3 2 B~u2

q 5 u1q 2 u3 2 u2

q 2 B~1 q2



and therefore






m0 T ~Dta2f~ada


m0 T ~Dta12f~a1




1

klogb

Nk~a

bk

1



k21 logb $ak a Infq











and






















VII Conclusion






REFERENCES
















































































APPENDIX

A Scaling Rule





~ X~t q ~ X~t q1w1 ~ X~t q2w2



1 w2 ln c~q2 (A1)


t~q $ w1t~q1 1 w2t~q2 (A2)




$ X~t q u ~t 5 u 5 B~u q u ~t 5 u

5 u ~tq 2 B~1 q




$X~t 1 T 9t u ~t 1 T 5 u 5 $B~u 9t

5 Bu ~t




Trading time


Vh1 hk

q Vz1 zk

q ~Mh1

2q Mh1 hl

2q

~Mh1 h l11

q Mh1 hk

q ~M z1 z l11

q Mz1 zk

q


Cov m~I1q m~I2

q 5 ~ Vq2~ Mq2k$~ M2q~ Mq2l 2 1



Log-Price



(A3)

simpli es to

B~u1q B~u3 2 B~u2

q 5 u1q 2 u3 2 u2

q 2 B~1 q2



and therefore






m0 T ~Dta2f~ada


m0 T ~Dta12f~a1




1

klogb

Nk~a

bk

1



k21 logb $ak a Infq











and





























































































APPENDIX

A Scaling Rule





~ X~t q ~ X~t q1w1 ~ X~t q2w2



1 w2 ln c~q2 (A1)


t~q $ w1t~q1 1 w2t~q2 (A2)




$ X~t q u ~t 5 u 5 B~u q u ~t 5 u

5 u ~tq 2 B~1 q




$X~t 1 T 9t u ~t 1 T 5 u 5 $B~u 9t

5 Bu ~t




Trading time


Vh1 hk

q Vz1 zk

q ~Mh1

2q Mh1 hl

2q

~Mh1 h l11

q Mh1 hk

q ~M z1 z l11

q Mz1 zk

q


Cov m~I1q m~I2

q 5 ~ Vq2~ Mq2k$~ M2q~ Mq2l 2 1



Log-Price



(A3)

simpli es to

B~u1q B~u3 2 B~u2

q 5 u1q 2 u3 2 u2

q 2 B~1 q2



and therefore






m0 T ~Dta2f~ada


m0 T ~Dta12f~a1




1

klogb

Nk~a

bk

1



k21 logb $ak a Infq











and



















Trading time


Vh1 hk

q Vz1 zk

q ~Mh1

2q Mh1 hl

2q

~Mh1 h l11

q Mh1 hk

q ~M z1 z l11

q Mz1 zk

q


Cov m~I1q m~I2

q 5 ~ Vq2~ Mq2k$~ M2q~ Mq2l 2 1



Log-Price



(A3)

simpli es to

B~u1q B~u3 2 B~u2

q 5 u1q 2 u3 2 u2

q 2 B~1 q2



and therefore






m0 T ~Dta2f~ada


m0 T ~Dta12f~a1




1

klogb

Nk~a

bk

1



k21 logb $ak a Infq











and


















Multifractality in Asset Returns REStat 2002

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Transcript of Multifractality in Asset Returns REStat 2002