Transcript Document

Multifractality. Theory and Evidence
An Application to the Romanian Stock Market
MSc Student: Cristina-Camelia Paduraru
Supervisor: PhD Professor Moisa Altar
Presentation contents
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Motivation
Review of the Literature
Basics of Multifractal Modeling
Methodology to Detect Multifractality
Data
Main Results
Conclusions
Bibliography
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Motivation
• The major discrepancies between the Bachelier model and actual
financial data:
- long memory in the absolute values of returns
- long tails relative to the Gaussian.
• The Multifractal Model of Asset Returns (MMAR) – Mandelbrot,
Calvet, Fisher (1997) – accounts for these empirical regularities of
financial time series and adds scale consistency.
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Literature Review
• Mandelbrot, Calvet, Fisher (1997)
– the MMAR is developed
– the focus is on the scaling property: the moments of the returns
scale as a power law of the time horizon.
• Calvet, Fisher, Mandelbrot (1997)
– the focus is on the local properties of the multifractal processes.
• Fisher, Calvet, Mandelbrot (1997)
– an empirical investigation of the MMAR
– evidence of multifractality in Deutschemark/US Dollar currency
exchange rates
• Calvet and Fisher (2002, 2008)
– simplified version of the MMAR.
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MMAR incorporates:
• fat (long) tails - Mandelbrot (1963), but the MMAR does not
necessarily imply infinite variance.
• long dependence - fractional Brownian motion (FBM), Mandelbrot
and Van Ness (1968). MMAR displays long dependence in the
absolute value of price increments, while price increments
themselves can be uncorrelated.
• the concept of trading time - Mandelbrot and Taylor (1967):
explicit modeling of the relationship between unobserved natural
time-scale of the returns process, and clock time.
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Multifractal processes bridge the gap between Itô
and Jump diffusions
• Itô diffusions - increments that grow locally at the rate (dt)1/2
throughout their sample paths.
• FBM - local growth rates of order (dt)H , where H invariant over time
(the Hurst exponent).
• Multifractals - a multiplicity of local growth rates for increments:
(dt)α(t), where α(t) represents the Hölder exponent.
• Jump diffusions have α(t) = 0.
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Construction of the MMAR
Consider the price of a financial asset P(t) on [0, T] and the log-price
process:
X (t )  ln P(t )  ln P(0)
• 1.
X (t )  BH [ (t )]
• 2.
θ(t) - the cumulative distribution function of a multifractal
measure μ defined on [0, T].
• 3.
BH(t) and θ(t) are independent.
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Under Conditions 1 – 3:
• X(t) - multifractal process with stationary increments; the moments
of returns scale as a power law of the frequency of observation:

 X (t )
q
 c
X
(q) t
 X ( q ) 1
as t→0.
• The scaling function τX(q) - concave
- has intercept τX(0) = -1
 X (q)   ( Hq)
Concavity of the scaling function => multifractality.
Unifractal processes – linear scaling functions fully determined by H.
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Hölder Exponent
• Let g be a function defined on the neighborhood of a given date t.
The number
 (t )  Sup{  0 :| g (t  t )  g (t ) |  (| t |  ) as t  0}
,
is called the Hölder exponent of g at t.
• Describes the local variability of the function at a point in time.
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Multifractal Spectrum
• Describes the distribution of local Hölder exponents in a multifractal
process.
• The multifractal spectrum f(α) is the Legendre transform
f ( )  Min [q   (q)]
q
of the scaling function τ(q).
• Between the spectrum of the log-price process and the spectrum of
the trading time we have:
f X ( )  f  (

H
).
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Testing for Multifractality
• log-price series
X (t )  ln P(t )  ln P(0)
• Partitioning [0, T] into integer N intervals of length Δt, we define the
partition function:
N 1
S q (T , t )   | X (i t  t )  X (it ) | q
i 0
• X(t) – multifractal => the addends are identically distributed; the
scaling law yields:
[Sq (T , t )]  Nc X (q)(t ) X (q)1
, when the qth moment exists.
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Taking logs:
log[Sq (T , t )] X (q) log(t )  c* (q),
where c * (q)  logc X (q)  logT .
• We plot logSq(Δt) vs log(Δt) for various values of q and Δt.
• Linearity of those functions => scaling.
• OLS estimations of the partition functions => τX(q), the scaling
function.
• Of particular interest: the value of q where
ˆ(q)  0
• This value of q identifies H:
1
 X ( )0
H
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The Scaling Function
• Calvet, Fisher, Mandelbrot (1997) shows that the scaling function
- is concave
- has intercept τX(0) = -1
and
 X (q)   ( Hq)
• From the scaling function we estimate the multifractal spectrum
through the Legendre transform:
f ( )  Min [q   (q)]
q
f X ( )  f  (

H
).
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The Data
• High frequency data sets – all transaction prices with transaction
time during the period Jan, 2007 – May, 2009 for four Romanian
securities listed at the Bucharest Stock Exchange: SIF2, BRD, SNP
and TEL.
• We have 328,555 transactions for SIF2
179,617 transactions for SNP
178,562 transactions for BRD
68,289 transactions for TEL.
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Plots of the Partition Functions SIF2
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Plots of the Partition Functions BRD
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Plots of the Partition Functions SNP
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Plots of the Partition Functions TEL
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Plots of the Scaling Functions
Scaling Function SIF2
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6
5
τ(q)
4
3
2
1
0
-1
-2
0
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18 19 20
q
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Plots of the Scaling Functions
Scaling Function BRD
7
6
5
τ(q)
4
3
2
1
0
-1
-2
0
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18 19 20
q
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Plots of the Scaling Functions
Scaling Function SNP
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5
4
τ(q)
3
2
1
0
-1
-2
0
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18 19 20
q
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Plots of the Scaling Functions
Scaling Function TEL
5
4
τ(q)
3
2
1
0
0
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18 19 20
-1
-2
q
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Plots of the Multifractal Spectrum
Multifractal Spectrum SIF2
1
0.9
0.8
0.7
f(α)
0.6
0.5
0.4
0.3
0.2
0.1
0
0.3
0.33
0.36
0.39
0.42
0.45
0.48
0.51
0.54
0.57
0.6
α
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Plots of the Multifractal Spectrum
Multifractal Spectrum BRD
1
0.9
0.8
0.7
f(α)
0.6
0.5
0.4
0.3
0.2
0.1
0
0.25
0.3
0.35
0.4
0.45
0.5
0.55
0.6
α
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Plots of the Multifractal Spectrum
Multifractal Spectrum SNP
1
0.9
0.8
0.7
f(α)
0.6
0.5
0.4
0.3
0.2
0.1
0
0.25
0.3
0.35
0.4
α
0.45
0.5
0.55
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Plots of the Multifractal Spectrum
Multifractal Spectrum TEL
1
0.9
0.8
0.7
f(α)
0.6
0.5
0.4
0.3
0.2
0.1
0
0.15
0.2
0.25
0.3
0.35
α
0.4
0.45
0.5
0.55
0.6
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Main Results
• The partition functions – approximately linear => scaling in the
moments of returns
• The scaling functions – concave => evidence for multifractality.
• Of particular interest: the value of q where
• This value of q identifies H:
X(
ˆ(q)  0
1
)0
H
• All scaling functions have intercept -1.
• Each of the scaling functions is asymptotically linear, with a slope
approximately equal to αmin. The minimum α corresponds to the
most irregular instants on the price path, and thus the riskiest events
for investors.
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Main Results
• The multifractal spectrum is also concave and its maximum is
approximately 1 in all of the four cases.
• The estimated multifractal spectrum: approximately quadratic => the
limit lognormal multifractal measure for modeling the trading time.
• Calvet, Fisher, and Mandelbrot (1997)
(   ) 2
f ( ) 1 
.
2
2 logb
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Main Results
• We find:
Hˆ
ˆ 0
ˆ  0 ˆ
H
ˆ
ˆ 2
SIF2
0.56
0.57
1.02
0.13
BRD
0.56
0.58
1.04
0.27
SNP
0.53
0.54
1.02
0.13
TEL
0.52
0.58
1.12
0.79
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Conclusions
• We found evidence of multifractal scaling in 4 Romanian securities
prices.
• Using a methodology based on scaling function and multifractal
spectrum => we recovered the MMAR components.
• The estimated multifractal spectrum: approximately quadratic.
• We found slight persistence in the analyzed data.
• The scaling property holds from 4 days to one year. No intraday
scaling! =>
We can model our series with multifractal processes at large
time scales.
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Bibliography
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Calvet, L.E., A.J. Fisher, and B.B. Mandelbrot (1997) “Large Deviations and the
Distribution of Price Changes”, Cowles Foundation Discussion Paper No. 1165;
Sauder School of Business Working Paper
Calvet, L.E. and A.J. Fisher (1999), “A Multifractal Model of Assets Returns”, New
York University Working Paper No. FIN-99-072
Calvet, L.E. and A.J. Fisher (2002), “Multifractality in Asset Returns: Theory and
Evidence”, Review of Economics and Statistics 84, 381-406
Calvet, L.E. and A.J. Fisher (2008), “Multifractal Volatility: Theory, Forecasting, and
Pricing”, Elsevier
Fama, E.F. (1963), “Mandelbrot and the Stable Paretian Hypothesis”, Journal of
Business 36, 420-429
Fillol, J. (2003) "Multifractality: Theory and Evidence an Application to the French
Stock Market", Economics Bulletin 3, 1−12
Fisher, A.J., L.E. Calvet, and B.B. Mandelbrot (1997), “Multifractality of Deutschemark
/ US Dollar Exchange Rates”, Cowles Foundation Discussion Paper No. 1166;
Sauder School of Business Working Paper
Mandelbrot, B.B. (1963), “The Variation of Certain Speculative Prices”, Journal of
Business 36, 394-419
Mandelbrot, B.B. (1967), “The Variation of the Prices of Cotton, Wheat, and Railroad
Stocks, and of some Financial Rates”, Journal of Business 40, 393-413
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Bibliography 2
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Mandelbrot, B.B., and H.M. Taylor (1967), “On the Distribution of Stock Price
Differences”, Operations Research 15, 1057-1062
Mandelbrot, B.B., and J.W. Van Ness (1968), “Fractional Brownian Motions,
Fractional Noises and Applications”, SIAM (Society for Industrial and Applied
Mathematics) Review 10, 422-437
Mandelbrot , B.B. (1972), “Possible refinement of the lognormal hypothesis
concerning the distribution of energy dissipation in intermittent turbulence”, Statistical
Models and Turbulence 12, 333-351
Mandelbrot, B.B., A.J. Fisher, and L.E. Calvet (1997), “A Multifractal Model of Asset
Returns”, Cowles Foundation Discussion Paper No. 1164; Sauder School of Business
Working Paper
Mandelbrot, B.B. (2001), “Scaling in Financial Prices: Tails and Dependence”,
Quantitative Finance 1, 113-123
Mandelbrot, B.B. (2001), “Scaling in Financial Prices: Multifractals and the Star
Equation”, Quantitative Finance 1, 124-130
Mandelbrot, B.B. (2001), “Scaling in Financial Prices: Cartoon Brownian Motions in
Multifractal Time”, Quantitative Finance 1, 427-440
Mandelbrot, B.B. (2001), “Scaling in Financial Prices: Multifractal Concentration”,
Quantitative Finance 1, 641-649
Mandelbrot, B.B., and Richard L. Hudson (2004), “The (mis) Behavior of Markets”,
Basic Books
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