2012 novotny
TRANSCRIPT
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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