Neglected chaos in international stock markets:Bayesian analysis of the joint return volatility dynamical system

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Accepted Manuscript

Neglected chaos in international stock markets: Bayesian analysis of the

joint return-volatility dynamical system

Mike G. Tsionas, Panayotis G. Michaelides

PII:

S0378-4371(17)30364-3

DOI:

http://dx.doi.org/10.1016/j.physa.2017.04.060

Reference:

PHYSA 18162

To appear in:

Physica A

Received date : 22 December 2016

Revised date :

5 April 2017

Please cite this article as: M.G. Tsionas. M.G. Tsionas, P.G. Michaelides. P.G. Michaelides,

Neglected chaos in international stock markets: Bayesian analysis of the joint return-volatility

dynamical system,

Physica A

(2017), http://dx.doi.org/10.1016/j.physa.2017.04.060

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HIGHLIGHTS



We use a novel Bayesian inference procedure for the Lyapunov exponent



We apply the new technique to daily stock data for a group of six countries



The interaction between returns and volatility is taken into consideration jointly



We employ Sequential Monte Carlo techniques

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*Manuscript

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(5)

{xt;t= 1, ..., T}

yt=f(yt−L, yt−2L, ..., yt−mL) +ut, ut|σt∼N(0, σ2t), t= 1, ..., T,

σ2

t m L

F: ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ yt−L

yt−2L

yt−mL ⎤ ⎥ ⎥ ⎥ ⎥ ⎦→ ⎡ ⎢ ⎢ ⎢ ⎢ ⎣

yt=f(yt−L, yt−2L, ..., yt−mL) +εt

yt−L

yt−(m−1)L

⎤ ⎥ ⎥ ⎥ ⎥ ⎦.

y0 △y0 t △y(y0, t)

λ= lim τ→∞τ

−1_ln|△y(y0, τ)|

|∆y0|

,

J χ

Jt(x) = df t_(χ) dχ Jt= ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ∂f ∂yt−L

∂f ∂yt−2L · · ·

∂f ∂yt−(m−1)L

∂f ∂yt−mL

1 0 · · · 0 0

0 1 · · · 0 0

0 0 0 1 0

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ .

yt=f(xt) +ut,

xt= [xt−L, ..., xt−mL]∈ ℜd_.

λ= lim M→∞

1 2Mlogν

∗_,

ν∗ _T′

MTM

TM=

M−_'1

t=1

JM−1,

M ≤T J

(6)

yt= G

(

g=1

αgϕ(x′

tβg) +ut,

ϕ(z) = 1

1+exp(−z), z ∈ ℜ βg ∈ ℜL, ∀g = 1, ..., G

α1≤...≤αG.

T′ MTM

logσ2

t =µ+ρlogσ2t−1+εt, εt∼iidN

)

0, ω2*_.

yt=+G1_g₌₁αgϕ(x′

tβg++l2l=1γglσt−lL22 ) +ut, ut|σt∼N(0, σt2), t= 1, ..., T, logσ2

t =+Gg=1δgϕ(x′tζg++l2l=1ψglσt−lL22 ) +εt, εt∼N(0, ω2), t= 1, ..., T,

L2

α1≤...≤αG δ1≤...≤δG.

yt=+G1_g₌₁αgϕ(x′

tβg+

+l2

l=1γglσt−lL22 ) +σtξt1, ξt1∼N(0,1), t= 1, ..., T,

logσ2

t =

+G

g=1δgϕ(x′tζg+

+l2

l=1ψglσt−lL22 ) +ωξt2, ξt2∼N(0,1), t= 1, ..., T.

σt

ξt

ξ∗= [ξ∗

1, ξ∗2]′∼iidN(0,1) ξt1=ξ∗1 ξt2=ξ2∗.

ξ∗

2 = 0

λ(ξ∗) = lim n→∞n

−1

n−₍1

i=0

log||∂Φ(χi)/∂χi||,

χ=,yt−1, ....yt−m1L1, σt−2 1, ..., σ2t−m2L2

-Φ(χ) ||.||

λ(ξ∗) = lim M→∞

1 2Mlogν

(7)

ν∗ _T′ MTM

TM=

M−_'1

t=1

JM−1,

M≤T M=T2/3 _Ji₌_∂Φ(χ

i)/∂χi

Ξ∗ ξ∗

λ= min ξ∗_∈Ξ∗λ(ξ

∗_).

λ >0 λ

ξ∗

S = ξ∗

1 ξ2∗

S2

λ(ξ∗) ξ∗

1 ξ∗2

ξ∗

1 ξ2∗

L(θ;Y) = (2πω2₎−T /2.

ℜT

+

/T

t=1(2π)−1/2(σ2t)−1exp

0

−[yt−! G1

g=1αgϕ(x′tβg+

!l2

l=1γglσt2−lm2)] 2

2σ2

t

1

·,

exp

0

−(logσ 2

t−

!G

g=1δgϕ(x′tζg+

!l2

l=1ψglσ2t−lm2)) 2

2ω2

1

dσ2_,

θ=,α′_,_β′_,_γ′_,_δ′_,ψ′_{, ω}-′

⊆ ℜp _σ2_{= [σ}2

1, ..., σT2] Y= [y1, ..., yT] λ≥0

p(θ)∝ω−1.

L, L2l1, l2

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G1=G2=G

p(θ|Y)∝ L(θ;Y)·p(θ).

L G

M(Y) =

2

ℜpL

(θ;Y)·p(θ)dθ.

λ

G L n

M(Y) G= 1 L= 1 n

G L m n

λ σt=σ

(9)

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yT∼p(yt|xt), st∼p(st|st−1),

st

Yt p(st|Yt)

3

s(ti), i= 1, ..., .N

4 3

w(ti), i= 1, ..., N

4 ₊_N

i=1w (i)

t = 1

p(st+1|Yt) =

2

p(st+1|st)p(st|Yt)dst≃

N

(

i=1

p(st+1|s(ti))wt(i),

p(st+1|Yt)∝p(yt+1|st+1)p(st+1|Yt)≃p(yt+1|st+1)

N

(

i=1

p(st+1|s(ti))w

(i)

t .

3

s(ti), wt(i), i= 1, . . . , N

4 3

s(_ti₊₁) , w_t(₊₁i) , i= 1, . . . , N4 θ∈Θ∈ ℜk

p(θ|Yt)≃

N

(

i=1

w(ti)N(θ|aθ

(i)

t + (1−a)¯θt, b2Vt),

¯

θt=+Ni=1w

(i)

t θ

(i)

t Vt=

+N i=1w

(i)

t (θ

(i)

t −θt)(θ¯

(i)

t −θt)¯ ′ a b

δ∈(0,1) a= (1−b2₎1/2 _b2_{= 1}₋_[(3δ₋_1)/2δ]2_.

ˆ

pN(Y|θ)

p(Y|θ)

θ(j) _Lj_{= ˆ}_pN(Y_|θ(j)₎

θc _q(θc_|_θ(j)₎ _Lc_{= ˆ}_pN(Y_|_θc₎ _θ(j+1)₌_θc

A= min

0

1, p(θ c_)Lc

p(θ(j)_Lj

q(θ(j)_|_θc q(θc_|θ(j)₎

1

,

5

θ(j+1)_{, L}j+16₌5_θ(j)_{, L}j6

p(Y|θ)

p(st|st−1)

st=g(st−1, ut)

p(yt|st−1) =

2

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p(ut|st−1;yt) =p(yt|st−1, ut)p(ut|st−1)/p(yt|st−1).

p(yt|st−1) p(ut|st−1;yt)

g(yt|st−1)

p(yt|st−1) p(yt|st−1) =N(E(yt|st−1),V(yt|st−1))

p(ut|yt, st−1)∝p(yt|st−1, ut)p(ut) p(ut|yt, st−1)

M uˆm

t m= 1, . . . , M

l(ut) =log[p(yt|st−1, ut)p(ut)]

l(ut)≃l(ˆumt ) +12(ut−uˆ

m

t )′∇2l(ˆumt)(ut−uˆmt).

g(ut|xt, st−1) =

M

(

m=1

λm(2π)−d/2_|_Σm|−1/2_exp51

2(ut−uˆmt )′Σ−m1(ut−uˆmt

6

,

Σm=−,∇2_l(ˆ_um t )

-−1

λm∝exp{l(um

t)}

+M

m=1= 1 sit.

st

p(yt|st)

p(si_t|_si t−1)

p(yt|si t)

t= 0, . . . , T−1 sk

t ∼p(st|Y1:t) πkt k= 1, ..., N

k= 1, . . . , N ωk

t|t+1=g(yt+1|skt)πtk, πkt|t+1=ωt|tk+1/

+N i=1ωt|ti +1

k= 1, . . . , N ˜sk t∼

+N

i=1πt|ti +1δist(dst)

k= 1, . . . , N uk

t+1∼g(ut+1|s˜kt, yt+1) skt+1=h(skt;ukt+1)

k= 1, . . . , N

ωk

t+1=

p(yt+1|skt+1)p(ukt+1)

g(yt+1|skt)g(ukt+1|s˜kt, yt+1)

, πk t+1=

ωk

t+1

+N i=1ωit+1

.

p(Y1:T) =

T

'

t=1

7₍N

i=1

ωi

t−1|t

8 7

N−1

N ( i=1 ωi t 8 .

p(st|Y1:t, θ)

p(st|Y1:t, θ)∝g(yt|xt, θ)

2

f(st|st−1, θ)p(st−1|y1:t−1, θ)dst−1,

p(st−1|y1:t−1, θ) t−1 t−1

5

si

t−1, i= 1, . . . , N

6 5

wi

t−1, i= 1, . . . .N

6

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ˆ

p(st−1|y1:t−1, θ)∝

N

(

i=1

wit−1f(st|sit−1, θ).

si

t−1 st

ωt=w

i

t−1g(yt|st, θ)f(s|sit−1, θ)

ξi

tq(st|sit−1, yt, θ)

,

q(st|si

t−1, yt, θ)

ξi

tq(st|sit−1, yt, θ)≃cg(yt|st, θ)f(st|sit−1, θ),

c

θc=θ(s)+λz+λ2

2∇logp(θ (s)

|Y1:T),

z∼N(0, I)

λ

a(θc|θ(s)_{) = min}0_1, p(Y1:T|θc)q(θ(s)|θc) p(Y1:T|θ(s))q(θc|θ(s))

1

.

∇logp(Y1:T|θ) =E[∇logp(s1:T, Y1:T|θ)|Y1:T, θ],

s1:T p(s1:T|Y1:T, θ)

ˆ

p(s1:T|Y1:T, θ) αit−1=∇logp(si1:t−1, Y1:t−1|θ)

κi t−1

i

αi

t=aκt−i1+∇logg(yt|sti, θ) +∇logf(sit|sit−1, θ).

∇logp(Y1:t|θ)

αit−1∼N(mit−1, Vt−1).

n

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αi

t−1 αt−1

mi

t−1=δαit−1+ (1−δ)

N

(

i=1

wi

t−1αit−1,

δ∈(0,1) αi

t

mi

t=δmκt−i1+ (1−δ)

N

(

i=1

wi

t−1mit−1+∇logg(yt|sit, θ) +∇logf(sit|sκt−i1, θ),

∇log ˆp(Y1:t|θ) =

N

(

i=1

wi

tmit.

δ= 0.95

p(Y1:T|θ) =p(y1|θ)

T

'

t=2

p(yt|Y1:t−1, θ),

p(yt|Y1:t−1, θ) =

2

g(yt|st)

2

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λ

−0.010 0 0.01 0.02 0.03 0.04 50

100 150 200

λ

density

US

before crisis after crisis

−0.010 0 0.01 0.02 0.03 100

200 300 400

λ

density

UK

−5 0 5 10 15 x 10−3 0

200 400 600

λ

density

Switzwerland

−5 0 5 10 15 20 x 10−3 0

500 1000 1500

λ

density

Netherlands

−0.010 0 0.01 0.02 0.03 50

100 150

λ

density

France

−0.010 0 0.01 0.02 200

400 600 800

λ

density

Germany

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λ(ξ∗) 0.02 0.02 0.02 0.02 0.02 0.02 0.04 0.04 0.04 0.04 0.06 0.06 0.06 0.06 0.08 0.08 0.08 0.08 ξ₁ ξ 2

U.S, before crisis (returns)

−2 0 2

−3 −2 −1 0 1 2 3 _0.020001 0.020001 0.020001 0.040001 0.040001 0.040001 0.06

0.060.08 0.06

0.08

ξ₁

ξ 2

U.S, after crisis (returns)

−2 0 2

−3 −2 −1 0 1 2 3 0.02 0.02 0.02 0.02 0.02 0.02 0.04 0.04 0.06 0.06 0.08 0.08 ξ₁ ξ 2

U.S, before crisis (log volatility)

−2 0 2

−3 −2 −1 0 1 2 3 0.020002 0.020002 0.020002 0.040002 0.040002 0.040002 0.060001 0.060001 0.060001 0.080001 0.080001 ξ₁ ξ 2

U.S, after crisis (log volatility)

−2 0 2

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λ(ξ∗) 0.020001 0.020001 0.020001 0.020001 0.040001 0.040001 0.060001 0.060001 0.08 0.08

ξ₁*

ξ 2

*

U.K, before crisis (returns)

−2 0 2

−3 −2 −1 0 1 2 3 0.020001 0.020001 0.020001 0.020001 0.04 0.04

0.06_0.08 0.06

0.08

ξ₁*

ξ 2

*

U.K, after crisis (returns)

−2 0 2

−3 −2 −1 0 1 2 3 0.020002 0.020002 0.020002 0.040002 0.040002 0.040002 0.060001 0.060001 0.080001 _0.080001

ξ₁*

ξ 2

*

U.K, before crisis (log volatility)

−2 0 2

−3 −2 −1 0 1 2 3 0.020002 0.020002 0.020002 0.020002 0.040001 0.040001 0.060001 0.060001 0.08 0.08

ξ₁*

ξ 2

*

U.K, after crisis (log volatility)

−2 0 2

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G L n m

0 2 4 6 8 10

0 5 10 15 20

G

normalized marginal likelihood

0 2 4 6 8 10

0 5 10 15 20

L

0 200 400 600 800 1000

0 5 10 15 20

n

0 5 10 15 20

0 2 4 6 8

embedding dimension, m

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λ

−0.050 0 0.05 0.1 0.15 20

40 60

λ

density

US

−0.050 0 0.05 0.1 0.15 20

40 60

λ

density

UK

−0.040 −0.02 0 0.02 0.04 50

100 150 200

λ

density

Switzerland

−0.010 0 0.01 0.02 50

100 150 200

λ

density

Netherlands

−5 0 5 10

x 10−3 0

100 200 300 400

λ

density

France

−0.030 −0.02 −0.01 0 0.01 50

100 150

λ

density

Germany

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λ

−0.040 −0.03 −0.02 −0.01 0 0.01 50

100

λ

density

US

−0.030 −0.02 −0.01 0 0.01 100

200 300 400

λ

density

UK

−10 −5 0 5 x 10−3 0

500 1000 1500

λ

density

Switzerland

−0.030 −0.02 −0.01 0 0.01 0.02 20

40 60 80

λ

density

Netherlands

−6 −4 −2 0 2 x 10−3 0

500 1000 1500 2000

λ

density

France

−0.030 −0.02 −0.01 0 0.01 50

100 150 200

λ

density

Germany