Table Of ContentDirect evidence for inversion formula in multifractal financial volatility measure
Zhi-Qiang Jiang1,2 and Wei-Xing Zhou1,2,3,4,∗
1School of Business, East China University of Science and Technology, Shanghai 200237, China
2School of Science, East China University of Science and Technology, Shanghai 200237, China
3Research Center for Econophysics, East China University of Science and Technology, Shanghai 200237, China
4Research Center of Systems Engineering, East China University of Science and Technology, Shanghai 200237, China
(Dated: December 8, 2008)
The inversion formula for conservative multifractal measures was unveiled mathematically a
decade ago, which is however not well tested in real complex systems. In this Letter, we pro-
pose to verify the inversion formula using high-frequency turbulent financial data. We construct
conservativevolatilitymeasurebasedonminutelyS&P500 indexfrom 1982to1999anditsinverse
8 measure of exit time. Both the direct and inverse measures exhibit nice multifractal nature, whose
0 scaling ranges are not irrelevant. Empirical investigation shows that the inversion formula holds in
0 financial markets.
2
PACSnumbers: 89.75.Da,89.65.Gh,05.45.Df
n
a
J
Inrecentyears,theconceptofinversestatisticshasat- the inversion formula was given for conservational mul-
3 tracted much attention in turbulence [1, 2] as well as in tifractal measures[19, 20]. It is noteworthy pointing out
2
financial markets [3] based on time series analysis. The that, the inversion formula given by Roux and Jensen
directstructurefunctionconcernswiththestatisticalmo- is obtained based on a special case of conservative mea-
]
T ments of a physical quantity g measured over a distance sures,althoughthey verifiedthe inversionformulainthe
S s such that Sq(s) = hgk(s)qi. The multifractal nature directandinversestructurefunctionsofshellmodels[14].
. of direct structure functions has been well documented It is thus natural that Xu et al. proposed to test the in-
n
in turbulence [4, 5, 6], as well as in finance [7, 8, 9], version formula in the energy dissipation rate (a kind of
i
f which is characterized by S (s) ∼ sζ(q) with a nonlinear conservative measure) rather than in the structure func-
- q
q scaling function ζ(q). In contract, the inverse structure tions and they did found sound evidence in favor of the
[ function is related to the exit distance, where the physi- proof [18].
cal quantity fluctuation exceeds a prescribed value, such Theinversionformulawastheoreticallyestablishedfor
1
v that Tp(g) = hsp(g)i. One can intuitively expect that bothdiscontinuousandcontinuousmultifractalmeasures
4 there is a power law scaling stating that T (g) ∼ gφ(p), byRiediandMandelbrot[19,20]. Letµbeaprobability
p
9 where φ(p) is also a nonlinear function. Furthermore, if measureon[0,1]whoseintegralfunctionM(a)=µ([0,a])
4 s ∼ g1/h, Schmitt has shown that there is an inversion isright-continuousandnondecreasing. Sincethemeasure
.3 formulabetweenthe twotypes ofscalingexponents such is self-similar, we have µ = ni=0miµ(wi−1(·)), where
1 that ζ(q) = −p and φ(p) = −q [10]. A similar intuitive wi’sarethe similaritymaps withscalecontractionratios
0 derivationfortheinversionformulaisgivenforLaplacian ri ∈(0,1)and ni=1mi =1wiPthmi >0. Themultifrac-
8 random walks [11]. tal spectrum f(α) of measure µ can be obtained via the
0 P
Legendre transform of τ, which is defined by
: Thepower-lawscalingininversestructurefunctionwas
v
observed in the signals of two dimensional turbulence n
Xi [12, 13], in the synthetic velocity data of the GOY shell mqiri−τ =1 . (1)
r model [1, 14], and in the temperature and longitudinal Xi=1
a andtransversevelocitydataingrid-generatedturbulence The inverse measure of µ∗ can be defined as follows,
[15]. However, this scaling behavior was not observed
inf{a:M(a)>b} if b<1
in other three dimensional turbulent flows from different µ∗ =M∗(b)= , (2)
1 if b=1
experiments [2, 16, 17]. The inversion formula for direct (cid:26)
and inverse structure functions is verified for synthetic where M∗(b) is the inverse function of M(a). Since µ
turbulence data of shell models [14] but not for wind- is self-similar, its inverse measure µ∗ is also self-similar
tunnel turbulence data, which covera range of Reynolds with ratios r∗ = m and probabilities m∗ = r , whose
i i i i
numbers Re=400−1000 [16]. multifractal spectrum f∗(α∗) is the Legendre transform
It is arguedthat [18], the absence of inversionformula of θ, which is defined implicitly by
betweenthescalingexponentsofdirectandinversestruc- n
turefunctionsisduetothefactsthatthevelocityfluctu- (m∗)p(r∗)−θ =1 . (3)
i i
ationisnotaconservativequantitywhileastrictproofof
i=1
X
The inversion formula follows immediately that
τ(q) = −p
. (4)
∗Electronicaddress: [email protected] θ(p) = −q
(cid:26)
2
Equivalently, we have where µ = max{µ : n = 1,··· ,N}. This trick ap-
max n
plies for the calculation of inverse partition functions as
τ(q)=−θ−1(−q) (5)
well.
or Figure 1 plots [χq(s)]1/q−1 as a function of box size s
for different orders. Sound power laws are observed for
θ(p)=−τ−1(−p) . (6) each partition function such that
These two equivalent relations are testable. Follow- χ (s)∼sτ(q) , (11)
q
ing this line, the inversion formula was verified in [18]
with high-Reynolds turbulence data collected at the S1 in which the scaling range spans about three orders of
ONERA wind tunnel [6], which is however the only ev- magnitude. The scaling exponent τ(q) can be estimated
idence. Due to the well documented analogues between through a power-law fit to the data in the scaling range.
turbulent flows and financial markets [21], in this letter, We will see that τ(q) is a nonlinear function, confirm-
weproposetotesttheinversionformulainfinancialmar- ing the presence of multifractality in the direct volatility
kets using high-frequency historicaldata of the S&P 500 measure.
index. Our data consist of 18-yearminutely prices span-
ning from 1 January, 1982 to 31 December, 1999 with a 2
10
totalof1.7milliondatapoints. Theminutelyreturnr(t)
is calculated as follows,
−1
10
r(t)=ln[I(t)/I(t−1)] , (7) 1)
−
q
where {I(t):t=1,··· ,T} is the time series of minutely 1/( 10−4
)
S&P 500 index. s
(q q = −3
Wefirstconstructthedirectvolatilitymeasureandin- χ −7 q = −1
vestigate its multifractal nature. The absolute return is 10 q = 0
q = 2
utilized as a proxy for volatility such that v(t) = |r(t)|. q = 4
According to the partition function method for multi- 10−10 q = 6
0 2 4 6
fractal analysis, the series is firstly covered by N boxes 10 10 10 10
s
with identical size s = T/N. The sizes of the boxes are
chosen such that the number of boxes of each size is an
integer to cover the whole time series. On each box, we FIG. 1: (color online.) Dependence of χq(s)1/q−1 as a func-
tion of box size s for q = −3, q = −1, q = 0, q = 4, and
construct the direct measure as
q=6. Thecurveshavebeen translated vertically byafactor
ns
1 of 0.001, 0.01, 0.1, 10, and 100 in turn for better visibility.
µ (s)= v(t), (8)
n The solid lines are power-law fitsin thescaling range.
V
t=(nX−1)s+1
where V = T v(t) and n = 1,··· ,N. By construc- Wenowinvestigatethescalingbehaviorofinversepar-
t=1
tition function of exit times. For each threshold ∆v, a
tion, this volatility measure µ is conservative. The pres-
P sequence of exit times s (∆v) can be determined succes-
ence of multifractality in µ has been confirmed based on j
sively from j =1 to j =J by
the multiplier method utilizing the same data set of the
minutelyS&P500index[22]. Alternatively,thevolatility
j t
measure µ of Chinese stocks and indexes exhibits multi-
s =inf t: ν(t)dt>j∆v , (12)
k
fractalbehaviorbasedonthepartitionfunctionapproach
k=1 (cid:26) Z0 (cid:27)
[23]. X
For order q, the direct partition function χq(s) can be where ν(t) = v(t) for t ∈ [t,t+1). The inverse measure
estimated using is defined as the normalized exit time
N µ∗(∆v)=s /T , (13)
χ (s)= [µ (s)]q . (9) j j
q n
n=1 and the inverse partition function can be determined as
X
When µ ≪ 1 and q ≫ 1, the estimation of the parti- follows
tion function χ will be very difficult since the value is so
J
small that it is “out of the memory”. To overcome this χ∗(∆v)= µ∗(∆v) p , (14)
p j
problem, we can calculate the logarithm of the partition
j=1
functionlnχ (s)ratherthanthepartitionfunctionitself. X(cid:2) (cid:3)
q
A simple manipulation results in the following formula where
N µ (s) q 1 T
n
lnχ (s)=ln +qlnµ , (10) J = ν(t)dt . (15)
q max
µ ∆v
n=1(cid:20) max (cid:21) " Z0 #
X
3
Figure2showsthedependenceofχ∗(∆v)1/(p−1) onthe Figure3showstheempiricalPDFsσf(xσ)ofthenormal-
p
threshold values ∆v for different values of p. Power-law ized exit times x=s/σ for different thresholds ∆v. The
scaling can be observed PDFsatdifferentthresholdscannotbesuperposedbythe
simplenormalizationprocedure. AssketchedinFig.3(a),
χ∗(s)∼∆vθ(p) . (16) the PDFs are strongly asymmetric and not log-normal,
p
whichdiffersremarkablyfromthesituationofenergydis-
wherethescalingrangecoversaboutthreeordersofmag- sipation in three-dimensional fully developed turbulence
nitude. The straightlines are the best fits to the data in showingroughlylog-normaldistribution[18]. Moreinter-
the scaling range, whose slopes correspond to the expo- estingly, the probability density functions show plateaus
nents θ(p)/(p−1). We note that the two scaling ranges on the left tails, which ensures the existence of any neg-
(s1,s2) and (∆v1,∆v2) for direct and inverse partition ative moments. Figure 3(b) shows that the right tail
functions are related by relaxes exponentially for small thresholds or faster for
large thresholds. This relaxation behavior is different
∆v =s×vmean , (17) from those exit times extracted from financial return se-
ries exhibiting a power-law tail [3, 27, 28, 29, 30, 31],
where v = 2.69× 10−4. Specifically, we find that
mean
∆v ≈ s × v and ∆v ≈ s × v . This puts
forw1ard s1oundmeevaindence upo2n the2detemrmeainnation of the 1 v /2
(a) mean
scaling laws.
0.8 v
mean
2v
2 mean
10 0.6
)
x
(
ρ
1) 10−2 0.4
−
p
1/( −6 0.2
) 10
v
∆
*χ(p10−10 ppp === −−031 100 −2 10−1 x 100 101
p = 2
p = 4
0
10−14 p = 6 10 (b)
−4 −2 0 2
10 10 10 10
∆v 10−1
−2
FIG. 2: (color online.) Dependence of χ∗p(∆v)1/p−1 on the x)10
(
thresholds ∆v. The same translated processes are also per- ρ
−3
formed on the data points of p = −3, p= −1, p = 0, p = 4, 10
and p=6. The solid lines are thebest fits tothe data.
−4
10
A subtle issue concerning negative moments arises, −5
10
which is related to the probability density functions 0 2 4 6 8 10 12 14
x
(PDFs) of volatility and exit time respectively. The
volatilityofS&P500indexislog-normallydistributedin
the center followed by a power-law tail for large volatil- FIG. 3: (color online.) Empirical probability density func-
tion ρ(x) of the normalized exit time x = s/σ for different
ities while the right tail seems truncated [24, 25]. In
addition, due to the construction of the volatility mea- thresholds ∆v=vmean/2 (◦), vmean ((cid:3)), and 2vmean (⋄).
sure, µ(s) > 0 for large s in the scaling range shown in
Fig. 1. Therefore, negative moments can be estimated Thepower-lawexponentsτ(q)fordirectpartitionfunc-
numerically. Taking into account the statistical signifi- tionsareplottedasopencirclesinFig.4,whiletheexpo-
canceoftheestimationofpartitionfunctions[17,26],we nentsθ(q)areillustratedastriangles. Bothτ(q)andθ(p)
focus on q ∈[−4,8]. arenonlinear,indicatingthatthe time seriesofvolatility
The PDFs of exit time defined in this letter have not and exit time possess multifractal properties. The func-
beeninvestigatedbefore. Letusdenotebyf(s)thePDF tion −θ−1(−q) is determined numerically from the θ(p)
of exit times for a fixed threshold ∆v. For comparison, curve,whichisplottedinFig.4asadashedline. Wecan
we normalize the exit time by their standard deviation find that the values of −θ−1(−q) are in excellent agree-
σ(∆v) for each given ∆v. Then the PDF of the normal- ment with the values of τ(p), which provides strong evi-
ized exit times x=s/σ can be determined by dence supporting the inversionformula in Eq. (5). Simi-
larly,the−τ−1(−p)curvenumericallyobtainedfromthe
ρ(x)=σf(xσ) . (18) τ(p) function is depicted as a solid line, which coincides
4
remarkably with the θ(q) curve. In other words, the in- volatilityandexittimeseriesbasedonthepartitionfunc-
version formula Eq. (6) also holds as expected. We note tionmethod. Ourinvestigationconfirmsthatbothdirect
that the differences between the comparing curves are and inverse partition functions exhibit nice multifractal
well within the error bars. properties. The two scaling ranges are consistent with
each other. Furthermore, we found that the function
(−q) 10 t−hτe−θ1((p−)p)cuerxvteraacnteddtnhuemfeurniccatilolynfr−oθm−τ1((−q)qo)vdeerltaeprmswinietdh
1
− from the θ(p) curve collapses on the τ(q) curve, which
τ 6
− verifies the inversion formula. We also investigated for
p), 2 the first time the empirical distributions of exit time of
(
θ financial volatility at different thresholds. The PDFs of
p), −2 θ(p) exittimearenontrivial,whichareneitherlog-normalnor
−
1( −τ−1(−p) power laws observed in other systems.
−
θ −6 τ(q)
−
), −θ−1(−q)
τ(q−1−06 −4 −2 0 2 4 6 8 10 Acknowledgments
p, q
We are grateful to Gao-Feng Gu for discussion. This
FIG. 4: Testing theinversion formula in financial volatility.
work was partly supported by the National Natural
Science Foundation of China (Grant No. 70501011),
In summary, we have attempted to test the inver- the Fok Ying Tong Education Foundation (Grant No.
sion formula for conservative multifractal measures us- 101086), the Shanghai Rising-Star Program (Grant No.
ing high-frequency volatility data of the S&P 500 in- 06QA14015), and the Program for New Century Excel-
dex. We have performed multifractal analysis on both lent Talents in University (Grant No. NCET-07-0288).
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