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•

•

•

•

•

•

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p

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αT

α±

T

α±

T

pRV p/p

T (t)

pRV p/p

T (t)

F

RS/BP

t

RS/BP t

RRV/BPV Day

F RRV/BPV Day

F

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jT (t) = |r(t)|

< |r(t)| >T ,

< |r(t)| >T T

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< |r(t)| >T = 1

N T

T −1i=0

|r(t − i)| ,

N T

r−α

α ≈ 4

rnT (t) =

r(t)− < r(t) >T

σT (t) ,

< r(t) >T σT (t)

L2

T

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α

T

T

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T

T

T

T

T = 12 24 100 1000 2000 5000

T = 12 24

T = 100

jT (t)

∝ s−α(f )T

P ( jT > s) ∼ s−α

(f )

T ,

α(f )T T

T

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α(f )T α

α 4

α ≤

3

α

α

ln P ( j > s) ∝ −α ln s .

ln P ( j > s) = α0 − α ln s + ν ,

α0 ν

j s j

α α

R2

s j

α

α

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α

α

N (N + 1) /2 N

U i =

i∈Samplei

ranki − ni (ni + 1)

2 ,

ni i ranki i

U = U 1 + U 2

z = U − mU

σU ,

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mU = n1n2

2 σU =

n1n2(n1+n2+1)12

z ∼ N (0, 1)

K

KW = (N −

1) K i=1 ni (ri − r)2

K i=1

j∈Samplei (r j − r)2 ,

ni i r = (N + 1) /2

ri =

j∈Samplei r j

/ni N K

KW ni

KW ∼ χ2K −1 KW

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PX

BUX

WIG

DAX 0

− 0 . 0

5 %

0 . 0

5 %

− 0 . 0

5 %

0

0 . 0

5 %

− 0 . 0

5 %

0

0 . 0

5 % −

0 . 0

5 %

0

0 . 0

5 %

9:00 10:00 11:00 12:00 13:00 14:00 15:00 16:00 17:00

PX

BUX

WIG

DAX 0 %

0 . 2 5

%

0 %

0 . 2

5 %

0 %

0 . 2

5 %

0 %

0 . 2

5 %

0 . 5

%

9:00 10:00 11:00 12:00 13:00 14:00 15:00 16:00 17:00

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PX

BUX

WIG

DAX 0

2 5 0

0

2 5 0

0

2 5 0

0

2 5 0

5 0 0

9:00 10:00 11:00 12:00 13:00 14:00 15:00 16:00 17:00

2.5

97.5

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f N µ σ S K

r(t)

f

N

µ

σ

S

K

p

2.5 97.5

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∆t

σ∆t ∝ √ ∆t .

p

T = 12 24

100 1000 2000 5000

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− 1 0

− 1

l n ( P ( x > s ) ) , P X ,

5 m i n

1 1.5 2 2.5 3 3.5ln(s)

− 1 0

− 1

l n ( P ( x > s ) ) , P X ,

1 0 m i n

1 1.5 2 2.5 3 3.5ln(s)

− 1 0

− 1

l n ( P ( x > s ) ) , P X ,

1 5 m i n

1 1.5 2 2.5 3 3.5ln(s)

− 1 0

− 1

l n ( P ( x > s ) ) , P X ,

3 0 m i n

1 1.5 2 2.5 3 3.5ln(s)

T

T = 12

T = 24

T = 100

T = 1000

T = 2000

T = 5000

T

T = 5000

α

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αT

αT (σα) t αT (σα) t αT (σα) t αT (σα)

a>

a<

a<

a<

a>

a<

a<

a>

a<

a> ≈ ≈

a> ≈ ≈

a> ≈

b>

a>

a>

a>

≈

b>

a>

c< ≈

a>

a>

a>

a>

a>

a>

a>

a<

a>

a>

T = 5000 T = 2000

T = 1000

t

t

a

b

c

a

99%

b

95%

c

90%

α

R2

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5%

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b

a

a

a

a

a

a

a

a

a a a a a a a

a

a

a

a

a

a

a

a

a

a

c

a

a

a

a

a

c

a

a

a

a

a

T = 5000

α

+

α+T

α−T

−

a

b

c

a

99%

b

95%

c

90%

H 0 α(+)T =

α(−)T

+

α(+)T α

(−)T

a

b

c

−

α(+)T α

(−)T

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α

α

α

H 0

α(+)T = α

(−)T

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F D

↑

c< ≈ ≈ ≈ ≈ ≈

↑

a<

c<

a<

c<

c<

b<

↓

b< ≈

b<

c< ≈ ≈

↓

b<

a<

a<

b<

c<

a<

↑ ≈ ≈ ≈ ≈ ≈ ≈

↑

c< ≈ <b

c< ≈

b<

↓

b< ≈ ≈ ≈ ≈ ≈

↓

a< ≈

a<

b<

b<

a<

↑ ≈ ≈ ≈ ≈ ≈ ≈

↑ ≈ ≈

c< ≈ ≈ ≈

↓

c

<

≈

≈

≈

≈

≈

↓ ≈ ≈

c<

b< ≈

b<

↑ ≈ ≈ ≈ ≈ ≈ ≈

↑ ≈ ≈

b<

c< ≈

b<

↓ ≈ ≈ ≈ ≈ ≈ ≈

↓ ≈ ≈

b< ≈ ≈ ≈

α±

F

D

↑

↓

a

99%

b

95%

c

90%

a b c

≈

α

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α)

α

∆ = αUp − αDown

5

5

±σ

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F P D

↑ ≈

c< ≈ ≈ ≈ ≈

B ↑

a

<

b

<

a

<

b

<

b

<

a

<

↓

b< ≈

c<

c< ≈

c<

↓

b<

a<

a<

c<

b<

a<

↑ ≈ ≈ ≈ ≈ ≈ ≈

C ↑

c< ≈

b< ≈ ≈

a<

↓ ≈ ≈ ≈ ≈ ≈ ≈

↓ ≈

≈

c<

≈

≈

c<

↑ ≈ ≈ ≈ ≈ ≈ ≈

B ↑

b<

≈

b<

b<

≈

b<

↓

b

<

≈

≈

≈

≈

≈

↓

b< ≈

a<

c< ≈

a<

↑ ≈ ≈ ≈ ≈ ≈ ≈

C ↑ ≈ ≈

b< ≈ ≈ ≈

↓ ≈ ≈ ≈ ≈ ≈ ≈

↓

c<

b<

b< ≈

c<

b<

↑ ≈ ≈ ≈ ≈ ≈ ≈

B ↑ ≈ ≈ ≈ ≈ ≈ ≈

↓ ≈ ≈ ≈

c< ≈ ≈

↓

c<

≈

b<

b<

≈

c<

↑ ≈ ≈ ≈ ≈ ≈ ≈

C ↑ ≈ ≈

c< ≈ ≈ ≈

↓ ≈ ≈ ≈ ≈ ≈ ≈

↓ ≈ ≈ ≈ ≈ ≈ ≈

↑ ≈ ≈ ≈ ≈ ≈ ≈

B ↑

c< ≈

c< ≈ ≈ ≈

↓ ≈ ≈ ≈ ≈ ≈ ≈

↓ ≈ ≈

c< ≈ ≈ ≈

↑ ≈ ≈ ≈ ≈ ≈ ≈

C ↑ ≈ ≈ ≈ ≈ ≈

c<

↓ ≈ ≈ ≈ ≈ ≈ ≈

↓ ≈ ≈ ≈ ≈ ≈ ≈

α±

F

D

↑

↓

P

B

C

a

99% b

95% c

90%

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D

↑

b

b

↓ b

(↑ − ↓)

c

↑

↓

(↑ − ↓)

↑

b

c

c

↓

(↑ − ↓)

c

↑

c

c

b

↓

b

(↑ − ↓)

b

α±

D

↑ ↓

(↑ − ↓)

a

99%

b

95%

c

90%

α

α

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0 .

0 0 1

0 .

0 0 2

0 .

0 0 3

04/Q1 05/Q1 06/Q1 07/Q1 08/Q1 09/Q1 10/Q1

DAX: 5−minutes

0 .

0 0 1

0 .

0 0 2

0 .

0 0 3

04/Q1 05/Q1 06/Q1 07/Q1 08/Q1 09/Q1 10/Q1

BUX: 5−minutes

0 .

0 0 1

0 .

0 0 2

0 .

0 0 3

04/Q1 05/Q1 06/Q1 07/Q1 08/Q1 09/Q1 10/Q1

WIG: 5−minutes

0 .

0 0 1

0 .

0 0

2

0 .

0 0 3

04/Q1 05/Q1 06/Q1 07/Q1 08/Q1 09/Q1 10/Q1

DAX: 5−minutes

5

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0

5

1 0

1 5

2 0

2005/Q1 2006/Q1 2007/Q1 2008/Q1 2009/Q1 2010/Q1

PX: 5−minutes

0

5

1 0

1 5

2 0

2005/Q1 2006/Q1 2007/Q1 2008/Q1 2009/Q1 2010/Q1

BUX: 5−minutes

0

5

1 0

1 5

2 0

2005/Q1 2006/Q1 2007/Q1 2008/Q1 2009/Q1 2010/Q1

WIG: 5−minutes

0

5

1 0

1 5

2 0

2005/Q1 2006/Q1 2007/Q1 2008/Q1 2009/Q1 2010/Q1

DAX: 5−minutes

Price jump index; Up

0

5

1 0

1 5

2 0

2005/Q1 2006/Q1 2007/Q1 2008/Q1 2009/Q1 2010/Q1

PX: 5−minutes

0

5

1 0

1 5

2 0

2005/Q1 2006/Q1 2007/Q1 2008/Q1 2009/Q1 2010/Q1

BUX: 5−minutes

0

5

1 0

1 5

2 0

2005/Q1 2006/Q1 2007/Q1 2008/Q1 2009/Q1 2010/Q1

WIG: 5−minutes

0

5

1 0

1 5

2 0

2005/Q1 2006/Q1 2007/Q1 2008/Q1 2009/Q1 2010/Q1

DAX: 5−minutes

Price jump index; Down

±σ

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α

U

T = 5000

t

χ2n1+n2−2 n1

n2

t

t

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4.244a −0.646 −3.193a

−0.237 0.580 −0.176 0.641 6.896a

−4.350a

−1.195 −0.694 13.062a

−1.155 1.266 −0.070 −0.908 −0.550 1.462 2.268b −4.153a 0.836 0.577 −2.581b 2.162c

−3.002b −0.159 2.765b

−1.011 −1.169 2.202c

−1.167 1.115 0.343 2.353c 1.505 −3.269b

−2.039c 0.172 1.697 0.050 −1.165 1.611 2.157c −4.643a 0.389 −0.254 −3.468b 3.476a

t

α

t

χ2n1+n2−2

n1

n2

a

99% b

95% c

90%

t

t

30

10 30 t

10 30

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α±

T

α±

T (σα)

T = 5000

+

−

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α±

T

α(σα)

T = 5000

−

+

z

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α±

z

H 0 a b c

a

99% b

95% c

90%

α+

α−

a

c

c

a

a

a

a

α±

H 0

χ23

a

99%

b

95%

c

90%

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a

b

b

a

c

a

a

a

a

c

α±

H 0

χ23

a

99%

b

95%

c

90%

χ23

1 4

1

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a

b

c

b

c

b

c

a

c

c

c

a

a

a

a

c

b

b

c

b

b

c

b

b

b

b

a

b

b

b

c

b

c

c

a

α±

U D

α

α

a

99%

b

95%

c

90%

z

5 10

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c

c

c

c

c

b

c

b

a

b

b

c

b

a

b

b

c

b

b

c

a

a

b

b

c

a

a

a

a

c

c

b

b

c

b c b c b

c

c

c

b

b

a

b

α±

U

D

α

α

a

99%

b

95%

c

90%

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S t

dS t = µtdt + σtdW t + Y tdJ t ,

µt σt dW t

Y tdJ t σtdW t

Y tdJ t

p

t t

t

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|rτ | τ

(100 − X )

|rτ − rτ S | τ

(100 − X )

< X >S

τ

X/2 (100 − X/2)

B(r) = 12T

exp− |r|

T

,

T r

T

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σ2T = 2T 2

p

p

pRV pT (t) =

t

τ =t−T +1

|rτ | p1/p

,

T p

p

p

jT (t) t

jT,t = |rt|

< |rt| >T ,

< |rt| >T = 1T

T −1i=0 rt−i T

jT,t

P ( jT > s) ∝ s−α(f )T ,

αT

T s

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α

X t X t =N i=1 W i,t + V 2,t

W 1,t

W i,t

i

V N,t

N

W i,t

N = 2

W 1,t =

L−1l=0

h1,lX t−l N W 2,t

L−1l=0 h1,l

V 1,t−2l N

V 1,t =L−1l=0

g1,lX t−l

N V 2,t =L−1l=0

g1,l V 1,t−2l

N ,

hl

gl

N = 2 X 2 =W 1

2 +W 2

2 +V 2

2 X L2

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σ2t =

1

T − 1

t−1τ =t−T

(rτ − < rτ >T )2

< rτ >T = 1T

T −1i=0 rt−i

ˆσ2t =

1

T − 2

t−1τ =t−T +2

|rτ | |rτ −1| .

RS/BP t = σ2

t /ˆσ2t R

S/BP t ≥

1 T

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RV Day =Day

(rτ )2 ,

BP V Day = π

2 Day

|rτ | |rτ −1| ,

π/2

RRV/BPV Day = RV Day/BPV Day

Lt = rt/ˆσt

ξ

maxτ ∈An |Lτ | − C nS n

→ ξ ,

An n C n = (2lnn)1/2

c −lnπ+ln(lnn)

2c(2lnn)1/2 S n = 1

c(2lnn)1/2 c =

√ 2/

√ π

ξ P (ξ ≤x) = exp (e−x)

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ξ CV ξ ξ CV

ξ

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| p(t) − pk(t)| < 3σk(t) + γ ,

pk(t) ±k t

σk(t)

γ = 0.005 k = 5

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rt = log (Rt/Rt−1) Rt

t

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D a y i n S a m p l e

0

5 0

1 0 0 1 5 0 2 0 0 2 5 0 3 0 0 3 5 0

S t o

c k

2

4

6

8

1 0

1 2

1 4

1 6

- 0 . 4

- 0 . 3

- 0 . 2

- 0 . 1 0

0 . 1

0 . 2

0 . 3

0 . 4

- 3

1 0

×

M e a n

D a y i n S a m p l e

0

5 0

1 0 0 1 5 0 2 0 0 2 5

0 3 0 0 3 5 0

S t o

c k

2

4

6 8

1 0

1 2

1 4

1 6

0

0 . 0

0 2

0 . 0

0 4

0 . 0

0 6

0 . 0

0 8

0 . 0

1 S t a n d a r d

d e v i a t i o n

D a y i n S a m p l e

0

5 0

1 0 0 1 5 0 2 0 0 2 5 0 3 0 0 3

5 0

S t o

c k

2

4

6

8

1 0

1 2

1 4

1 6

- 3 - 2 - 1 0 1 2 3 4 S k e w n e s s

D a y i n S a m p l e

0

5 0

1 0 0 1 5 0 2 0 0 2 5

0 3 0 0 3 5 0

S t o

c k

2

4

6

8

1 0

1 2

1 4

1 6

1 0

2 0

3 0

4 0

5 0

6 0

K u r t o s i s

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J B = N 6

S 2 + (K −3)2

4

S K N

χ22

χ22 95%

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APPL

XOM

BAC

C

CVXGE

GOOG

HPQ

IBM

JNJ

KO

MSFT

PFE

PG

SPYT

WFC

S t o c k

0 100 200 300 400Day is Sample

95%

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p

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•

•

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F

F

S 2C

S 2No−C

∼ F (N C −1,N No−C −1) ,

S 2 C

No

−C N C N No−C

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90 95 99

90

95

99

5/95 2.5/97.5

0.5/99.5

T

ln B(r) = ln 1

2T − 1

T r .

T

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D a y

i n S a m p

l e

0

5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0 3 5 0

S t o

c k s

2 4 6 8

1 0

1 2

1 4

1 6 0

2 0

4 0 6 0 8 0

1 0 0

1 2 0

1 4 0

1 6 0

1 8 0

2 0 0

2 2 0

2 4 0

A b s ol u t er e t ur n s , t h r e sh ol d

a t 9 0 % C en t i l e

D a y

i n S a m p

l e

0

5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0 3 5 0

S t o

c k s

2 4 6 8

1 0

1 2 1 4

1 6 0

2 0

4 0

6 0

8 0

1 0 0

1 2 0

1 4 0

1 6 0

1 8 0

A b s

ol u t er e t ur n s , t h r e sh ol d a t 9 5 % C en t i l e

D a y

i n S a m p

l e

0

5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0 3 5 0

S t o

c k s

2 4 6 8

1 0

1 2

1 4

1 6 0

2 0

4 0

6 0

8 0

1 0 0

A b s ol u t er e t ur n s , t h r e sh ol d a t 9 9 % C en t i l e

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D a y i n S a m p l e

0

5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0 3 5 0

S t o

c k

s

2 4

6 8

1 0

1 2

1 4

1 6 0

2 0

4 0

6 0

8 0

1 0 0

1 2 0

1 4 0

1 6 0

1 8 0

2 0 0

2 2 0

2 4 0

C e n t e r e d r e t u r n s ,

t h r e s h o l d a t 5 / 9 5 % C

e n t i l e

D a y i n S a m p l e

0

5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0 3 5 0

S t o

c k

s

2 4

6

8 1 0

1 2

1 4

1 6 0

2 0

4 0

6 0

8 0

1 0 0

1 2 0

1 4 0

1 6 0

1 8 0

C e n t e r e d r e t u r n s ,

t h r e s h o l d a t 2 . 5

% / 9 7 . 5

% C

e n t i l e

D a y i n S a m p l e

0

5 0 1 0

0 1 5 0 2 0 0 2 5 0 3 0 0 3 5 0

S t o

c k

s

2 4

6 8

1 0

1 2

1 4

1 6 0

2 0

4 0

6 0

8 0

1 0 0

C e n t e r e d r e t u r n s ,

t h r e s h o l d a t 0 . 5

% / 9 9

. 5 % C

e n t i l e

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D a y

i n S a m p

l e

0

5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0 3 5 0

S t o

c k

2 4 6 8

1 0

1 2

1 4

1 6 0

2 0

4 0 6 0 8 0

1 0 0

1 2 0

1 4 0

1 6 0

1 8 0

2 0 0

2 2 0

2 4 0

R e t ur n s , 5 % / 9 5 %

D a y

i n S a m p

l e

0

5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0 3 5 0

S t o

c k

2 4 6 8

1 0

1 2 1 4

1 6 0

2 0

4 0

6 0

8 0

1 0 0

1 2 0

1 4 0

1 6 0

1 8 0

R e

t ur n s ,2 . 5 % / 9 7 . 5 %

D a y

i n S a m p

l e

0

5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0 3 5 0

S t o

c k

2 4 6 8

1 0

1 2

1 4

1 6 0

2 0

4 0

6 0

8 0

1 0 0

R e t ur n s , 0 . 5 % / 9 9 . 5 %

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D a y i n S a m p

l e

0 50 100 150 200

250 300 350

S t o c k

24

68

1012

1416

0

5

10

15

20

25

30

35

40

45

Temperature

ln B(r) = ln 1

2T − 1

T r + ν ,

ν

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p

p = 1 p = 4

p = 4 p = 1

pRV p/p

T (t) ≡ pRV pT (t)/pRV p

T (t) ,

pRV pT (t)

p = 4 p = 1

T

T = 60 T = 120

T

z

z

z T = 60

z −3.332

99% z

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pRV p/p

T (t)

T = 60

T = 120

z

H 0

z

z

p

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pRV p/p

T (t)

T = 60

T = 120

z

H 0

z

pRV p/p

T (t)

α

T = 60 T = 120

α

α

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z

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T = 60

T = 120

z

H 0

z

F

F

F

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F

F

F F 225,172

F 29,368

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L = 8

W 1

2 / X 2

W 2

2 / X 2

W 1

2 / X 2

W 2

2 / X 2

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D a y i n S a

m p l e

0

5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0 3 5 0

S t o

c k

2 4

6 8

1 0

1 2

1 4

1 6 0 1 2 3 4 5 6

- 6

1 0 ×

W 1

D a y i n S a m p l e

0

5 0 1 0 0 1 5 0 2 0 0 2 5

0 3 0 0 3 5 0

S t o

c k

2 4

6 8

1 0

1 2

1 4

1 6 0 5

1 0

1 5

2 0

2 5

3 0

3 5

- 6

1 0 ×

W 2

˜ W i

2

/ X 2

˜ W i

i =

1 , 2

X 2

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RS/BP t

T = 60

T = 120

z H 0

z

RS/BP t = σ2

t /ˆσ2t

T = 60 T = 120 t

σt

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RS/BP t

T = 60

T = 120

z H 0

z

T

p

RS/BP t

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RRV/BPV Day

RRV/BPV Day

T = 60 T = 120

z H 0

z

RRV/BPV Day = RV Day/BPV Day

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F RRV/BPV Day

RRV/BPV Day

F RRV/BPV Day

F

F 225,172

F 29,368

z

F

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T = 60

T = 120

z

H 0

z

T = 270

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F

F

F

F 225,172 F 29,368

T = 60 T = 120

95%

z

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F

F

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z

z

z

z

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L1

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Y t

p11 = P r (1|1) = P r

Y = 1|Y = 1

Y = 1

p22 = P r (0|0) = P r

Y = 0|Y = 0

Y = 0

p12 = P r (1|0) = P r

Y = 1|Y = 0

Y = 0

p21 = P r (0|1) = P r

Y = 0|Y = 1

Y = 1

p11

p22 p12 p21

P r (1|0) P r (0|1)

P r = P r (0|0) + P r (1|1) = p11 + p22.

P r = P r (0|1) + P r (1|0) = p21 + p12.

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I 1

Σ

I 2

n11 n12 n1•

n21 n22 n2•

Σ n•1 n•2 n

I 1 I 2

n n11

n12 I 2

I 1 n21 I 1

I 2 n22

I 1 I 2

n1•

I 1

I 1 I 2

H 0 n1• = n•1 H 0 n12 = n21

χ21 = (n12−n21)

2

n12+n21 n12 + n21 ≥ 8 n12 + n21

n · n•i · nj•

n · n•i · nj•

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P r (1|0) P r (0|1)

I 1 I 2

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420

pt − pt−1 = F t ,

F t

pt − pt−1 = σZ t ,

Z t = Z ∼ N (0, 1) σ

U

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pt − pt−1 = σtZ t ,

σt

σt =

σhigh t ∈ [0, α · Day)

σlow t ∈ [α · Day,β · Day)

σhigh t ∈ [β · Day,Day]

,

σlow < σhigh α β

σlow σhigh

U

pt − pt−1 = σtZ t ,

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σt =

3σhigh t ∈ [0, α · Day)

2σhigh t ∈ [α · Day,β · Day)

1σhigh t ∈ [β · Day,γ · Day)

σlow t ∈ [γ · Day,δ · Day)

1σhigh t ∈ [δ · Day, · Day)

2σhigh t ∈ [ · Day,φ · Day)

3σhigh t ∈ [φ · Day,Day]

,

σlow < σhigh α β γ δ φ

U

pt − pt−1 = σtZ t ,

σt =

3σhigh − (3σhigh−σlow)α·Day (t) t ∈ [0, α · Day)

σlow t ∈ [α · Day,β · Day)

σlow + (3σhigh−σlow)

(1−β)·Day (t − β · Day) t ∈ [β · Day, Day]

,

σlow < σhigh 3 · σhigh

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pt − pt−1 = σZ t ,

σ = 0.0004

pt − pt−1 = σtZ t ,

σt =

σhigh t ∈ [0,Day/4)

σlow t ∈ [Day/4, 3Day/4)

σhigh t ∈ [3Day/4,Day]

,

σlow = 0.0001 σhigh = 0.0004

pt − pt−1 = σtZ t ,

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σt =

3σhigh t ∈ [0, 45min)

2σhigh t ∈ [45min, 90min)

1σhigh t ∈ [90min, 135min)

σlow t ∈ [135min, 285min)

1σhigh t ∈ [285min, 330min)

2σhigh t ∈ [330min, 375min)

3σhigh t ∈ [375min, 420min]

.

45 Day/9

σlow = 0.0001 σhigh = 0.0002

σhigh

σt =

3σhigh − (3σhigh−σlow)135min (t) t ∈ [0, 135min)

σlow t ∈ [135min, 285min)

σlow + (3σhigh−σlow)

135min (t

−285) t

∈ [285min, 420min]

,

σlow < σhigh σlow = 0.0001 σhigh = 0.0002

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pt − pt−1 = F t = σtZ t + J · jt ,

σtZ t J · jt

jt λ j

jt =

0 p j

1 1 − p j

,

p j = e−λj J

J = ±J param

J

n

P ( n) = e−λ (λ)n

n! .

P ( ) = e−λ

P ( ) = 1 − e−λ

J = ±const

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J = 5 jump = 5/N Day

J = 7 jump = 5/N Day

J = 9 · jump = 5/N Day

jump = 0.0004 N day

0 < a < b

J ∼ ±U (5σ jump , 9σ jump) λ j = 5/N Day

J ∼ U (5σ jump , 9σ jump) λ j = 15/N Day

jump = 0.0004 N day

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Z RJ,TP

n = 60

Z RJ,TP n = 120

Z RJ,TP n = 60

Z RJ,TP n = 120

ξ n = 60

ξ n = 120

JORatio n = 60

JORatio n = 120

J ORatio n = 60

J ORatio n = 120

pji > 4

n = 120

pji > 4

n = 420

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p0 = 100

n = 60 n = 120

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ξ n = 120

n = 60

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∆S =

σ∆X + ∆J ∆S = S t − S t−∆t

∆t ∆S

∆t X J β

∆X = (∆t)1/2 X 1 ∆J = (∆t)1/β J 1

X

J β

J

∆S = ∆J

∆S = σ (∆t)1/2 X 1 ∆t α (∆t)γ

α > 0 γ ∈ (0, 1/2) ∆S > α (∆t)1/2 ∆S

J

σ = σ (t)

J

J

J

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RV t =n

i=2 r2i

BV t = µ−21

n−1n−2

ni=3 |ri| |ri−1| µα = E (|Z |α) Z ∼ N (0, 1)

µα = 2α/2Γ α+12 /

√ π

´ σ4

T P = nµ−3

4/3

n

−1

n − 3

ni=4 |ri|

4/3

|ri−1|4/3

|ri−2|4/3

→ ˆ σ

4

,

QP = nµ−41

n − 1

n − 4

ni=5

|ri| |ri−1| |ri−2| |ri−3| →ˆ

σ4 .

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Z RJ,TP

Z RJ,TP = RJ

π2

2+ π − 5

1n

max

1, TP

BV 2

,

RJ = (RV − BV ) /RV

Φ−1 (α)

α

RV j =

ji= j−n+2

r2i ,

BV j = µ−21

n − 1

n−

2 j

i= j−n+3

|ri| |ri−1| .

n j

j

n = 60

n = 120

Z t−1 ≤ Φ−1 (α) Z t > Φ−1 (α)

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n = 60 n = 120

L (i) = ri

σ (i) ,

σ2 (i) = 1n−2

i−1 j=i−n+2 |ri| |ri−1|

maxi |L (i)| − C nS n

→ ξ ,

ξ P (ξ ≤ x) = exp (−e−x)

C n = (2logn)1/2

c −log π+log(log n)

2c(2logn)1/2 S n = 1

c(2logn)1/2 c =

√ 2/

√ π ξ

ξ CV ti

n15−min = 156 n5−min = 270

n = 60 n = 120

SwV = 2n

i=2

(Ri − ri) ,

Ri = P i−P i−1

P i P i = exp ( pi) ri = pi − pi−1

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J ORatio = nBV √ ΩSwV

1 −

RV

SwV

,

RV BV

z ∼ N (0, 1)

JOt−1 ≤ Φ−1(α) JOt Φ−1(α)

n = 60 n = 120

n = 60 n = 120

pjii = |ri|

1n

i j=i−n+1 |ri|

,

n

P ( pji > s) ∼ s−α

s s = 4

n = 120 n = 420

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B∗∗∗

B∗∗∗ B∗∗∗

B∗∗∗ B∗∗∗ A∗∗∗

B∗∗∗ B∗∗∗ B∗∗∗ B∗∗∗

B∗∗∗ B∗∗∗ = B∗∗∗ =

B∗∗ A∗∗∗ A∗∗∗ A∗∗∗ A∗∗∗ A∗∗∗

A∗∗ A∗∗∗ A∗∗∗ A∗∗∗ A∗∗∗ A∗∗∗ A∗∗∗

B∗∗∗ B∗∗∗ A∗∗∗ A∗∗∗ A∗∗∗ A∗∗∗ B∗∗∗ B∗∗∗

B∗∗∗ B∗∗∗ A∗∗∗ A∗∗∗ A∗∗∗ A∗∗∗ B∗∗∗ B∗∗∗ A∗∗∗

B∗∗∗ B∗∗∗ A∗∗∗ A∗∗∗ A∗∗∗ A∗∗∗ B∗∗∗ B∗∗∗ B∗∗∗ B∗∗∗

B∗∗∗ B∗∗∗ A∗∗∗ A∗∗∗ A∗∗∗ A∗∗∗ B∗∗∗ B∗∗∗ B∗∗∗ B∗∗∗ A∗∗∗

B∗∗∗ A∗∗∗ A∗∗∗ A∗∗∗ A∗∗∗ A∗∗∗ B∗∗∗ B∗∗∗ A∗∗∗ A∗∗∗ A∗∗∗ A∗∗∗

B∗∗∗ A∗∗∗ A∗∗∗ A∗∗∗ A∗∗∗ A∗∗∗ B∗∗∗ B∗∗∗ A∗∗∗ A∗∗∗ A∗∗∗ A∗∗∗ A∗∗

B∗∗∗

B∗∗∗ B∗∗∗

B∗∗∗ B∗∗∗ A∗∗∗

B∗∗∗

B∗∗∗

B∗∗∗

B∗∗∗

B∗∗∗ B∗∗∗ B∗∗∗ B∗∗∗ =

B∗∗∗ B∗∗∗ A∗∗∗ A∗∗∗ A∗∗∗ A∗∗∗

B∗∗∗ B∗∗∗ A∗∗∗ = A∗∗∗ A∗∗∗ B∗∗∗

B∗∗∗ B∗∗∗ A∗∗∗ A∗∗∗ A∗∗∗ A∗∗∗ = A∗∗∗

B∗∗∗ B∗∗∗ A∗∗∗ A∗∗∗ A∗∗∗ A∗∗∗ A∗∗∗ A∗∗∗ A∗∗∗

B∗∗∗ B∗∗∗ A∗∗∗ B∗∗∗ A∗∗∗ A∗∗∗ B∗∗∗ B∗∗ B∗∗∗ B∗∗∗

B∗∗∗ B∗∗∗ A∗∗∗ A∗∗∗ A∗∗∗ A∗∗∗ B∗∗∗ A∗∗∗ B∗∗∗ B∗∗∗ A∗∗∗

B∗∗∗ B∗∗∗ B∗∗∗ B∗∗∗ A∗∗∗ A∗∗∗ B∗∗∗ B∗∗∗ B∗∗∗ B∗∗∗ B∗∗∗ B∗∗∗

B∗∗∗ B∗∗∗ A∗∗∗ = A∗∗∗ A∗∗∗ B∗∗∗ = B∗∗∗ B∗∗∗ A∗∗ B∗∗∗ A∗∗∗

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B∗∗∗

B∗∗∗ B∗∗∗

B∗∗∗ B∗∗∗ =

B∗∗∗ B∗∗∗ B∗∗∗ B∗∗∗

B∗∗∗ B∗∗∗ B∗∗∗ B∗∗∗ B∗∗

B∗∗∗ B∗∗∗ A∗∗∗ A∗∗∗ A∗∗∗ A∗∗∗

B∗∗∗ B∗∗∗ A∗∗∗ A∗∗∗ A∗∗∗ A∗∗∗ B∗∗∗

B∗∗∗ B∗∗∗ A∗∗∗ A∗∗∗ A∗∗∗ A∗∗∗ B∗∗∗ B∗∗∗

B∗∗∗ B∗∗∗ A∗∗∗ A∗∗∗ A∗∗∗ A∗∗∗ B∗∗∗ A∗∗ A∗∗∗

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ξ N = 120