Complex Hurst and Fractional Analysis

for Stock Market Fluctuation

Presenter: Yufan Hu

Authors: Qing Zou, Yufan Hu, Jun Steed Huang

ICIMSA2015 May 26-28 Tokyo Japan

About the Authors

• Qing Zou is a student with Suqian College, who wrote 2 books on number theory.

• Yufan Hu is a master major math with Loughborough University.

• Jun Steed Huang is a distinguished professor.

Traditional Hurst Parameter

• Denoted by H, in honor of Harold Edwin Hurst and Ludwig Otto Hölder

• Meaning: Index of long-range dependence

• 0 1H≤ <

Meaning of Hurst Parameter

• : a time series with long-term switching between high and low values in adjacent pairs, e.g. High—Low or Low-High

• : a time series with long-term positive autocorrelation, e.g. High—High or Low-Low

• H=0.5 : a completely uncorrelated memory less series, e.g. Poisson point process

0 0 .5H≤ <

0 .5 1H< <

Defining Fractional Norm

• Fractional Norm:

, 1, 1,p

x l p r∀ ∈ > ≥

2 2

11 1 11 21 2 2

12 2

11 1

2

sup : ,( 1)

inf : ,(0 1)

yn n

i irrn i ir y

rri

yn ni

i iry i i

r

x y x

x x

x y x

−− = =≤−

=

−≤ = =

∈ >

= =

∈ < <

∑ ∑

∑∑ ∑

ℚ

ℚ

Defining Fractional Moment

• Fractional center moment:

Given as a discrete random variable,

exist, define order k center moment as

where

ξ ξE

−

=−= ∑

∞

=

− i

i

ikik

k Ei

kEEE )()1()(

0

ξξξξµ

∑∞

=

−−

=

0

)()1(i

ikiiEE

i

kξξ

1< k < 3

Defining Complex Hurst

• Complex Hurst Parameter CH

IDC: Index of Dispersion for Count

: mean of

m: the last point in the sample

CH =

1+ logIDC(m)

IDC(1)⋅ log

Level(1)

Level(m)

2

IDC =E(ξ − Eξ )k

µµ ξ

Level of Grouping

•Time span between two successive checks

1: 1 day 2: 2 days

5: 1 week 10: 2 weeks

20: 1 month 65: 3 months

130: 6 months 260: 1 year

Complex Hurst

•

• Real part: same as traditional H

• Imaginary part: Uncertainty of real part

−0.5 < Re(CH ), Im(CH ) <1.5

Forward, Backward & Irreversibility

(p1, p2,…, p700)

• Forward Hurst FH: (p1, p2,…, p700)

• Backward Hurst BH: (p700, p699,…, p1)

• Irreversibility Hurst IH:

earliest latest

2

FH BHIH

−=

IH ∈ (−0.5,1.5)

Irreversibility

The real part of IH is used to judge if

the stock market will repeat itself in

future, the lower the value, the better

chance it will repeat, the high the

value, unlikely it will.

Example of Real Part• MATLAB code:

http://www.mathworks.com/matlabcentral/fileexchange/49803-

calculate-complex-hurst/content/complexHurstV1.m

Example of Real & Imaginary Value

Company Re(FH)

F=2.5

Im(FH)

F=2.5

Re(BH)

F=1.5

Im(BH)

F=1.5

Re(IH)

ΔF=0.5

Im(IH)

ΔF=0.5

eBay 1.2529 -0.0477 0.0645 0.0196 0.5942 -0.0337

Microsoft 0.9756 0.0308 0.7929 -0.0103 0.0914 0.0206

Baidu 0.7681 0.0285 0.8119 -0.0088 -0.0219 0.0186

Vodafone 1.1049 0.0213 0.7720 -0.0070 0.1665 0.0142

Nokia 0.9958 0.0151 0.7891 -0.0059 0.1034 0.0105

Apple 1.0746 0.0137 0.7870 -0.0042 0.1438 0.0090

Samsung 1.2304 -0.0108 0.7128 0.0022 0.2588 -0.0065

Amazon 1.2035 0.0052 0.7332 -0.0015 0.2352 0.0034

Conclusion

• We have defined Complex Hurst and

related

• FH, BH, IH for stock market quote

estimation

• Tested their meaning of real &

imaginary parts

Thank you!