A Bayesian Analysis of a Jump-Diffusion Model with double exponential distribution

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A Bayesian Analysis of a Jump-Diffusion
Model with double exponential distribution
Senniang Chen
Dr. Yu, Major Advisor
Department of Statistics, Iowa State University
September 8, 2011
1
Outline
Introduction
The model, MCMC method
Simulation results
Empirical results
Conclusion
2
Introduction
 Three stylized behaviors: stochastic volatility, leverage
effect, and jumps
 Continuous-time model
 AJD (Affine Jump-Diffusion) model
Let Yt =log(St) is the log price. νt is the stochastic volatility
 dY  dt   dB y  dJ y

t
t
t
t


d



(



)
dt



dB

t

t t
 t
where(Byt , Bνt )are two standard Brownian motions with Corr(Byt , Bνt )=ρdt
3
Jumps in the return
 Compound Poisson process
 Finite activity jumps
 Advantage: capture the three stylized behaviors
 Disadvantage: fail to capture many small movements
 Levy jump process proposed by Li, Wells, and Yu (2008)
 Infinity activity jumps
 Advantage: capture many small movements
 Double exponential jumps proposed by Kou(2002)
 Finite activity jumps
 Advantage: more flexible than compound Poisson process
Li, H., Wells , M. T., and Yu , C. L., 2008, A Bayesian Analysis of Return Dynamics with Levy
Jumps, The Review of Financial Studies,21:2345-2378
Kou, S.G., 2002,A Jump-Diffusion Model for Option Pricing, Management Science, 48:1086-1101
4
Goals of the project
Propose a AJD model with double exponential
jumps
Develop a MCMC method
model parameters
latent variables
Whether our model can capture the many
small movements in the real data
5
AJD Model with Double Exponential Jumps
First-order Euler discretized version of the model
 Y  Y      y  J y
t
 t 1
t t 1 t 1

 t 1   t  (   t )    t 
t 1

y
y



~ N(0,1), 
~ N(0,1), Corr( ,  )  
t 1
t 1
t 1 t 1
y
 I( I
 1)   I(I
 1)(  ),
t 1
t 1
t 1
t 1
t 1
 1, w.p. p1
  ~ exp(  )

 t 1
I
  0, w.p. p 0
 
t 1
)

~
exp(

1, w.p. p

 t 1
1

J
6
Eraker, B., Johannes M., and Polson N., 2003, The impact of jumps in equity index volatility and returns, Journal of Finance, 58:1269-1300
The model
Observations {Yt}Tt=0
Model parameters
Θ={μ, κ, θ, ρ, σν,η+, η-, p1 , p-1 , p0}
Latent variables {Xt}Tt=0
volatility {νt}Tt=0
 Jump time {It}Tt=0, positive jump size {ξ+t }Tt=0 and negative
jump size {ξ-t}Tt=0
 Stochastic
7
Challenge of Likelihood
Challenge: high dimensional latent variables
p( | Y)   p(, X | Y)dX
Bayes’ rule
p(Θ,X|Y)∝ p(Y|X,Θ)× p(X| Θ)× p(Θ)
The model is Markov
p(, X | Y)  t 0 p( y t 1  y t | X t , )  t 0 p(X t 1  X t | )  p()
T 1
T 1
8
Markov Chain Monte Carlo Method
Clifford-Hammersley theorem
p(Θ,X|Y)↔ p(Θ|X,Y) and p(X|Θ,Y)
Given the initial values Θ(0) and X (0)
Θ(1) ~ p(Θ|X (0),Y)
X(1) ~ p(X|Θ (1),Y)
… …
{Θ(g) , X(g) }Gg=0
The posterior Chain will converge to p(Θ,X|Y)
1 G
ˆ
 (, X̂)  g 1 ((g ) , X (g ) )
G
9
MCMC Method in Our Case
 Joint distribution
p(, , I,  | Y)  t 0 p( y t 1  y t |  t , J t , )  t 0 p(( t 1   t ), I t ,  t | )  p()
T 1
T 1
 Posterior distributions of parameters and latent variables
 Given Θ(0) , ν(0), ξ(0) and I(0), we can draw
Θ(1) ~ p(Θ| ν (0) , ξ(0), I(0),Y)
ν(1) ~ p(V|Θ (1), ξ(0), I(0), Y)
I(1) ~ p(J|Θ (1), ν(1), ξ(0), Y)
ξ(1) ~ p(J|Θ (1), ν(1), I(1), Y)
… …
{Θ(g) , ν(g) , I(g) , ξ(g)}Gg=0
ˆ , ˆ , Î, ˆ )  1 G ((g ) ,  ( g ) , I( g ) , ( g ) )
 (
G g 1
10
Priors of the Model Parameters
μ~N(f,F2)
θ~N (g,G2)I(θ>0)
κ ~N (k,K2)I(κ >0)
η+~ IG(α+, β+), and η- ~IG(α-, β-), where IG
represents inverse gamma distribution
(p1,p0,p-1) ~ Dirichlet(d1,d0,d-1)
11
Priors of the Model Parameters
According to Jacquier, Polson, and Rossi (1994)
,  
are reparameterized as
 ,  ,
where
   
,

2
2
    (1   )
and the jo int priors are
1

)
  |  ~ N(0,
2 .

 ~ Gamma ( w , W )
Jacquier, E., Polson, N., and Rossi, P., 1994, Bayesian Analysis of Stochastic Volatility Models, Journal of
Business & Economic Statistics,12:371-389
12
Joint Distribution
1
1


p(, V, I, , Y)  
exp{ 
   
21   
  1 
T 1
t 0


T 1
t 0

2

t 1

 2ty1 t 1 }
t
I ( I t 1 1)
1
p
2
2
y 2
t 1
p
I ( I t 1  0 )
0
p
I ( I t 1  1)
1
 t 1  t 1
1
exp{    }p()
 



 Y  Y       y  J y
t
 t 1
t t 1 t 1 ,
AJDDE : 
 t 1   t  (   t )     t  
t 1

y
y
where 
~ N(0,1),  ~ N(0,1), Corr ( ,  )   and
t 1
t 1
t 1 t 1
y
J
 I( I
 1)   I(I
 1)(  ),
t 1
t 1
t 1
t 1
t 1
 1, w.p. p1
  ~ exp(  )

 t 1
I
  0, w.p. p0
 
t 1

1, w.p. p
 t 1 ~ exp(  )
1

13
Joint Distribution
p(, V, I, , Y)  t 0
1
T 1
  t 1  
2
exp{ 


2
1
y 2

y









2


t 1
t 1
t 1 t 1 }
2
21   
T 1 I ( I 1) I ( I  0 ) I ( I  1)
 t 1  t 1
1
t 1
t 1
t 1
exp{


}
p
p
p
p ( )

1
0

1
 


t 0



   f 2 
   k 2 
   f 2     1

p()  exp 
exp 
exp{  }
 exp 
 
2
2 
2
2F 
2F 


 2K 

 

   1
exp{


d1 d 0 d 1
}
p
1 p 0 p 1


  1 
1
exp{ } 
   

2

w 1
exp{
W
}

14
Posterior Distribution of μ
2




1


f
T 1
y
2
y

p( | Θ  , V, I, , Y)  exp 
[( t 1 )  2t 1 t 1 ] 

2 t  0
2
2
(
1


)
2
F


Define C t 1  y t 1  y t  J ty1 ; D t 1   t 1   t     t ; then,

D t 1

 
C


 f  
1
1
T 1 
 1
T 1 t 1
2
 exp  [

]


 2  
2 t  0
2

2 t  0
 t 2F
(1   )
t
F
 2(1   )

 

 

 |  ~ N( s
1
T 1
, 1 ), s 

w w
(1  2 ) t 0
C t 1 
D t 1
t


f
1
1
T 1 
,
w



F2
(1  2 ) t 0  t F2
15
Posterior Distribution of θ

 
   g 2 

2 
1
T 1

y

p( |   , V, I, , Y)  exp 
 2t 1 t 1   t 1   exp 
2 t 0
2 
2
(
1


)
2
G




define C t 1  y t 1  y t    J ty1 ; D t 1   t 1   t  t ; then,
2

1
1  2  1
g  
T 1  
T 1   C t 1  D t 1
  
 exp  

  2 
2 t 0 2
2 
2 t 0
2
   t 2G 
  t 
G  
 (1   )
 2(1   )
Thus  |  ~ N( s
, 1 )  I(0,),
w w
2
1
g
1
1
T 1   C t 1  D t 1
T 1  
s



,
w




(1   2 ) t 0
 2  t 
G2
(1   2 ) t 0  2  t G 2
16
General method of Sampling from a
Truncated Distribution
For a typical truncated distribution x ~ f(x)×I(a<x<b), we
can do sampling following these two steps below.
Draw u~ U[0,1]
Compute x=F-1(F(a)+(F(b)-F(a))×u), where F(∙) is the
cdf of f(x)
17
Sampling from a Truncated Distribution
b
x~f(x)×I(a<x<b), fx(x)=f(x)/c , a<x<b and c= a f ( x )dx .
Suppose u~ U[0,1] and ξ=F-1( F(a)+(F(b)-F(a))*u).
Then
P (  x )
 P(F(a )  (F(b)  F(a ))  u  F( x ))
 P((F(b)  F(a ))  u  F( x )  F(a ))
F( x )  F(a )
)
F(b)  F(a )
F( x )  F(a ) F( x )  F(a )


F(b)  F(a )
c
1
1
f x ( x )  F' ( x )  f ( x )
c
c
 P(u 
18
Posterior Distribution of κ

 
   k 2 

2 
1
T 1

y

p(  |    , V, J, Y)  exp 
 2t 1 t 1   t 1   exp 
2 t 0
2 
2
(
1


)
2
K




define C t 1  y t 1  y t    J ty1 ; D t 1   t 1   t ; then,
2

1
1  2  1
k  
T 1    t  
T 1   C t 1  D t 1
  
   t   2 
 exp  

2 t 0
2
2 
2 t 0
2

  t
2K 
  t
K  
 2(1   )
 (1   )
Thus  |  ~ N( s
, 1 )  I(0,)
w w
1
k
1
1
T 1   C t 1  D t 1
T 1    t  


s





,
w




t
(1   2 ) t 0
 2  t
K2
(1   2 ) t 0  2  t
K2
2
19
Posterior Distribution of (ρ,σν)
p (   ,  |   (   , ) , J , V , Y ) 

1

1  2

 2  1
exp 

   
T



exp{
1
2 1  2
t 0 ( vt 1 ) 2  2vt 1}
T 1
1

w 1
 W
exp 





a 2

 * 1
(


) 


 1 
 * 1 
b
b

 
exp 
exp 
 2
 



 

 



2
b


  ~ IG ( * , 1 * )




 |  ~ N ( a ,  )
b
b

1 T 1 2
a2
where b  2   t 0 C ; a   t 0 C t 1D t 1 ;   w  T / 2;   W   t 0 D t 1 
2
2b
y t 1  y t    J ty1
   t   (   t ) 
C t 1 
; D t 1  t 1
;
t
t
T 1
2
t 1
T 1
We draw  from IG( * , 1





  2

     2


 
*
*
), then  is sampled from N ( a ,
b

*

b
)
20
Posterior Distribution of (η+,η-)

  t 1 
1
p( |  5 , J, V, Y)  t 0  exp     

  
 
 
T 
Thus, 

 
T 1




follows IG(T   ,





t 0 t 1 )
T 1
  t 1 
1
p( |  5 , J, V, Y)  t 0  exp     

  
 
T 1



 T    1
Thus, 

  
exp   
  
T 1 

1
 t 0  t 1   

exp  (

)

 

 




 1
  

   1
   1
  
exp   
  
T 1 


 t 0  t 1  

exp  (

)

 







follows IG(T   ,





t 0 t 1 )
T 1
21
Posterior Distribution of (p1,p-1,p0)
p(p1 , p 1 , p 0 |  6 , V, J, Y)  p1d1 p d0 0 p d11 t 0 p1I ( I t1 1) p 0I ( I t1 0) p I(1I t1  1)
T 1
I ( I t 1 1) d 0   I ( I t 1  0 ) d 1   I ( I t 1  1)

t 0
 p1
p 0 t 0
p 1 t0
d1 
T 1
T 1
T 1
Thus (p1 , p 1 , p 0 ) ~ Dirichlet (D1 , D 1 , D 0 )
D1  d1  Tt 01 I(I t 1  1), D 0  d 0  Tt 01 I(I t 1  0), D 1  d 1  Tt 01 I(I t 1  1)
To sample (p1 , p 1 , p 0 ), we use following steps:

simulate( x1 , x 1 , x 0 ) seperately from Gamma (D1 , C), Gamma (D 1 , C),
and Gamma (D 0 , C), where C is any positive number .

p1 
x0
x1
x 1
, p 1 
, p0 
x1  x 1  x 0
x1  x 1  x 0
x1  x 1  x 0
Devroye, Luc 1986, Non-Uniform Random Variate Generation, Page 594
22
Posterior Distribution of Jump Times It+1

 

1
T 1
I ( I t 1 1) I ( I t 1  0 ) I ( I t 1  1)

y
y 2 
p(I t 1 | , V, Y)  exp 

2





p
p0
p 1


t

1
t

1
t

1
1
2
t 0
 2(1   )

 p1 exp u1
; i 1

S
 p exp u 
1
p(I t 1  i | )   1
; i  1
S

 p 0 exp u 0  ; i  0

S
1
2


where u1  
C

2

,
where
C

y

y





1
t

1
1
t

1
t
t 1 ;
2
2(1   )
1
2

where u 0  
C

2

0
t 1 , where C 0  y t 1  y t  ;
2(1   2 )
1
2


where u 1  
C

2

,
where
C

y

y






1
t

1
1
t

1
t
t 1 ;
2
2(1   )
where S  p1 exp u1 p 0 exp u 0  p 1 exp u 1






23
Posterior Distribution of Jump Sizes

p (

t 1

1
T 1
| , V, Y )  exp 
 2 t 1 ty1   ty1
2 t 0
 2(1   )
p (

t 1

1
T 1

y
y
| , V, Y )  exp 

2





t

1
t

1
t
1
2
t 0
 2(1   )
The posterior distribution of



ξ t+1

2
  t 1 

  exp   

  

  t 1 

  exp   

  
are following :

 t 1 and  t 1
+
ξ t+1and
2


 D t 1  t  1   2
C t 1 





1

s
1

N
(
,
),
s

,
w

; when I t 1  1

2
2
w
w
 t 1 |  ~ 
t 1 
t 1 

  t 1 
exp   , when I t 1  1


 








 D t 1  t  1   2
C t 1 





1

s , 1 ), s 

N
(
,
w

; when I t 1  1

w
w
 t 1 |  ~ 
 t  1  2
 t  1  2

  t 1 
exp   , when I t 1  1


 

24
C t 1  y t 1  y t   and D t 1   t 1   t     t 




Posterior Distribution of Stochastic Volatility νt

For 0  t  1  T, the posterior of  t 1 is
  
  2y     
t 1 t 1
t 1
p( t 1 | , Y, J) ~ exp 
2
2(1   )

where 
y
t 1
2
 
y t 1  y t    J ty1
    (   t ) 

,  t 1  t 1 t
t
  t 
Jo int distribution : p(, V, I, , Y)  t 0
T 1
  
  y 2  2y     
1
t 2 t 2
t 2
exp  t  2

2
2(1   )
  t 1

1
  t 1  
2
exp{ 
 
2



  
1
y 2
 2
y





2


t 1
t 1
t 1 t 1 }
2
2 1 

T 1 I ( I 1) I ( I  0 ) I ( I  1)
 t 1  t 1
1
t 1
t 1
t 1
exp{


}
p
p
p
p ( )

1
0

1
 


t 0



25
Posterior Distribution of Stochastic Volatility νt

For 0  t  1  T, the posterior of  t 1 is
  
  2y     
t 1 t 1
t 1
p( t 1 | , Y, J) ~ exp 
2
2(1   )

where 
y
t 1
2
 
  
  y 2  2y     
1
t 2 t 2
t 2
exp  t  2

2
2(1   )
  t 1

2


y t 1  y t    J ty1
    (   t ) 

,  t 1  t 1 t
t
  t 
1
and the sec ond exp onential part.
0

For t  1  0, the posterior of  0 depends on

For t  1  T, the posterior of  T depends on the first exp onential part.
26
Metropolis-Hastings Method
To sample θ from f(θ), given θ(t) at tth iteration,
θ(t+1) is generated as following:
 Sample θ* from g(θ|θ(t))
R(θ*, θ(t))= g(θ(t)|θ*)/g(θ*|θ(t)) × f(θ*)/f(θ(t))

( t 1)
 * , with prob. (* , ( t ) )
*
(t)
*
(t)
  (t)
,

(

,

)

min{
R
(

,

),1}
*
(t)
 , with prob. 1  ( ,  )
If θ*= θ(t)+N(0,σ2), σ2 >0, then we call it normal
random walk sampling
27
Simulation Study Setup
 To examine whether MCMC method can accurately
estimate model parameters and latent variables
 50 sample paths with length T=1000
 True parameters μ=0.05, θ=0.9, κ=0.25, ρ=0.25, σν=0.4, p0=0.6, p1=0.1, p-1=0.3,η+=10, η-=30
 10K iterations for each paths, the estimates of the
parameters are the means of the last 5K iterations.
28
Hyperparameters
The hyperparameters are listed below
μ ~ N(0,1)
θ ~ N(0,1)I(θ >0)
κ ~ N(0,1)I(κ >0)
(p1, p0, p-1) ~ dirichlet(0.5, 0.1, 0.4)
η+~IG(1,1) η- ~ IG(1,3)
(ρ, σν) is transformed to (ων, φν)
1


)
  |  ~ N(0,
, 
2 .

2
2
   (1   )  ~ Gamma (1,120)
 
  
29
Initial Values
From the model, J y  (Yt 1  Yt )     t  y
t 1
t 1
E( J
y
)  (Yt 1  Yt )  
t 1
change of y
50
change of y
0
I (t 0)
-50
-100
 1, when ( y t 1  y t )  5

  1, when ( y t 1  y t )  10

0, otherwise

-150
-200
0
200
400
600
800
1000
time
Figure 1 Plots of the changes of y
30
Initial Values
(η+)(0) is the mean of the initial positive jump sizes and
(η-)(0) is the mean of the initial negative jump sizes
The initials of jump intensities and sizes are estimated as

I((I )  1)

(0)
t 0
 p1 
T 1

T
(0)

I
(
(
I

t )  1)
(0)
t 0
p 1 
T 1

T
(0)
 ( 0) t 0 I((I t )  0)
 p0 
T 1

T
(0)
t
( t ) ( 0)
( t ) ( 0)
y t 1  y t   ( 0) , when I t  1

 ( 0)
exp
(

) , otherwise



 y t 1  y t   ( 0) , when I t  1

 ( 0)
exp
(

) , otherwise



31
Simulation Results of 50 sample paths
μ
κ
ρ
θ
συ
p1
p-1
p0
η+
η-
True
0.05
0.25
-0.4
0.9
0.25
0.1
0.6
0.3
10
30
Monte Carlo Mean
0.0406
0.257
-0.297
1.194
0.297
0.0932
0.613
0.293
10.515
30.678
RMSE
0.0357
0.0646
0.1711
0.469
0.0913
0.0177
0.0290
0.0185
1.713
2.155
Monte Carlo STD
0.0345
0.0642
0.1370
0.366
0.0727
0.0163
0.0171
0.0257
1.634
2.046
Table 1 This table reports simulation results on the accuracy of MCMC estimators of
model parameters Θ={μ, κ, ρ, θ, συ, p1 , p-1 , p0, η+, η- } for 50 sample paths
32
Simulation Results of one sample path
μ
κ
ρ
θ
συ
p1
p-1
p0
η+
η-
True
0.05
0.25
-0.4
0.9
0.25
0.1
0.6
0.3
10
30
Initial
0.0181
0.0784
-0.3906
2.0525
0.3024
0.0560
0.7300
0.2150
18.7409
38.7680
Mean
0.0566
0.2146
-0.1454
0.8452
0.2404
0.0856
0.5988
0.3157
13.1253
27.9566
RMSE
0.0392
0.0511
0.2871
0.0716
0.0204
0.0177
0.0173
0.0224
3.4746
2.6295
Std
0.0387
0.0368
0.1326
0.0556
0.0180
0.0103
0.0173
0.0159
1.5186
1.6551
Table 2 This table reports simulation results on the accuracy of MCMC estimators of model
parameters Θ={μ, κ, ρ, θ, συ, p1 , p-1 , p0, η+, η-} for the one sample path.
33
chain plot of , initial=0.0784, mean=0.2146,
std=0.0368, RMSE=0.0511
chain plots of , initial=0.0181, mean=0.0566,
std=0.0287, RMSE=0.0392
chain plot of , initial=-.03906, mean=-0.1454,
std=0.1326, RMSE=0.2871
0.8
0.3
2.5
0.6

0.1
0.4
-0.2
1.5

2
0
0
2000
4000
6000
iteration time
8000
10000
-0.4
0.5
x
0
-0.6
0
2000
chain plot of  ,initial=0.3024,mean=0.2404,
8000
10000
x
x
0.3
0.7
0.2
0.6
p1

p0
0.25
0.5
0.15
0.4
4000
6000
iteration time
8000
8000
10000
0
2000
4000
6000
iteration time
8000
0
10000
0.2
0.4
x0
4000
6000
iteration time
8000
x
30
30
+
40
0.3
x
0.2
0.1
2000
4000
6000
iteration time
8000
10000
0
2000
4000
6000
iteration time
8000
x
20
10
10
0
2000
4000
6000
iteration time
8000
10000
0
0
2000
4000
6000
iteration time
8000
10000
std=0.0159,RMSE=0.0224
0.5
0.05
2000
chain plot of p-1, initial=0.2150,mean=0.3157,
0.25
0.15
0
chain plot of +, initial=18.7409,mean=13.1253,
std=1.5186,RMSE=3.4746
40
0
4000
6000
iteration time
0.1
10000
chain plot of -,initial=38.768,mean=27.9566,
std=0.16551,RMSE=2.6295
20
2000
chain plot of p1,initial=0.0560,mean=0.0856,
std=0.0103,RMSE=0.0177
0.8
2000
0
std=0.0173, RMSE=0.0173
0.35
-
4000
6000
iteration time
chain plot of p0, initial=0.73,mean=0.5988,
std=0.0103, RMSE=0.0177
0
1
x
0.2
-0.1
0.1
x
0
p-1

x

0.2
-0.2
chain plot of , initial=2.0525,mean=0.8452,
std=0.0556, RMSE=0.0716
0.2
10000
Figure 2 Chain plots of the model parameters for one simulation path.
34
10000
volatility
comparison of the true volatility and the estiamated volatility
4
true volatility
the estimated volatility
3
2
1
0
0
200
400
600
800
1000
time
Figure 3 Comparison of the true volatility variables and the mean of
the last 100 iterations of estimated volatility
comparison of the true positive jumps and the estimated jumps
50
true positive jumps
estimated positive jumps
-50
negative jump
postive jump
40
comparison of the true negative jumps and the estimated negative jumps
0
30
20
-100
-150
true negative jumps
estimated negative jumps
10
0
-200
0
200
400
600
time
800
1000
0
200
400
600
800
1000
time
Figure 4 Comparison of the true jumps and the mean of the last 100
iterations of estimated jumps
35
Level of the S&P 500 Index from 01/02/1990 to 12/31/2010
S&P 500 Index
800
750
700
650
600
1990
1995
2000
time
2005
2010
log change of S&P 500 Index
Empirical study
log change of the S&P 500 Index from 01/02/1990 to 12/31/2010
15
10
5
0
-5
-10
1990
1995
2000
time
2005
2010
Figure 5 Level and log changes of the S&P 500 index from January 2, 1990, to December 31, 2010.
mean
volatility
0.0236 1.1707
skewness
kurtosis
min
max
-0.1987
11.8688
-9.4695
10.9572
Table 3 This table summarizes statistics of the continuously compounded returns of the daily
return S&P 500 index return form 01/02/1990 to 12/31/2010
36
Initial Values
 μ(0)=0.05, θ(0)=1.8, κ(0)=0.05,ρ(0)=-0.4, and (σν) (0) =0.1
 (νt) (0) =VIX2/252, for t=1,2,…,T
 From Figure 5, we have
I (t 0)
 1, when ( y t 1  y t )  2

  1, when ( y t 1  y t )  2.5

0, otherwise

 The other initials are defined the same way as simulation
studies.
37
Empirical Studies
μ
κ
ρ
θ
συ
p1
p-1
p0
η+
η-
Initial
0.05
0.05
-0.4
1.8
0.1
0.0349
0.0221
0.953
3.000
3.731
Mean
0.0328
0.0734
-0.4022 1.6494
0.297
3.828*10-4 3.967*10-4 0.9992 1.791
3.573
Std
0.0114
0.006
0.0473
0.0727 5.912*10-4 7.989*10-4 0.001
0.0754
0.814
2.663
Table 4This table reports simulation results on the accuracy of MCMC estimators of
model parameters Θ={μ, κ, ρ, θ, συ, p1 , p-1 , p0, η+, η-} for the empirical studies.
38
chain plot of , mean=0.0734, standard error=0.006
0.03
p-1

0.08
0.06
0.02
0.04
0.01
0.02
0
0
1
2
3
iteration time
4
5
0
1
4
2
3
iteration time
4
15
10
5
0
5
4
0
1
x 10
x 10
chain plot of , mean=1.6494, standard error=0.0754
3.5
20
postive jump size +
0.1
0
chain plot of +,mean=1.791,std=0.814
chain plot of p-1, mean=3.9670*10-4, standard error=7.9670*10-4
0.04
0.12
2
3
iteration time
4
5
4
x 10
chain plot of -,mean=3.573, std=2.664
chain plot of p0, mean=0.9992, standard error=0.0010
80
1
negative jump size -
3
0.98

p0
2.5
2
0.96
1.5
0.94
1
0
1
2
3
iteration time
4
0
1
5
2
3
iteration time
4
60
40
20
5
0
4
x 10
0
1
4
x 10
2
3
iteration time
4
5
4
x 10
chain plot of , mean=0.0328,standard error= 0.0114
0.15
chain plot of , mean=-0.4022, standard error=0.0473
chain plot of p1, mean=3.8288*10-4,standard error=5.9115*10-4
0.05
0.1
0
-0.1
0.04
-0.2
0.05

0
0.02
-0.4
0.01
-0.05
-0.1
-0.3
p1

0.03
0
0
1
2
3
iteration time
4
5
4
x 10
chain plot of  , mean=0.3730,standard error=0.0120
-0.5
0
1
2
3
iteration time
4
5
4
x 10
-0.6
-0.7
0
1
2
3
iteration time
4
5
4
x 10
0.5

0.4
0.3
0.2
0.1
0
1
2
3
4
5
iteration time
x 10
Figure 6 Chain
plots of the model
parameters for the empirical studies. The mean and the RMSE of the model
parameters are the mean and RMSE of the last 20K iterations.
4
39
30
comparision of the implied volatility and the estimated volatility
implied volatility
the estimated volatility
volatility t
25
20
15
10
5
0
1990
1995
2000
time
2005
2010
Figure 7 Comparison of the true volatility variables and the
estimated volatility
the positive jumps
the estimated negative jumps
1.5
0
positive jumps
negative jumps
-0.5
-1
-1.5
-2
1
0.5
-2.5
-3
1990
1995
2000
time
2005
2010
0
1990
1995
2000
time
2005
2010
Figure 8 The estimated positive and negative jumps
40
Ability to Capture Many Small Jumps
ˆ   Ĵ ty1
y t 1  y t  

ˆ t

The model residuals ˆ
model is correctly specified.
y
t 1
follow N(0,1) if the
kernel density plots
0.7
kernel density estimates of the residuals
N(0,1)
0.6
0.5
0.4
0.3
0.2
0.1
0
-4
-3
-2
-1
0
1
2
3
4
5
Figure 9 Kernel density plot and QQ plot of the residuals
41
Conclusions
MCMC methods have very good performance
in estimating model parameters and high
dimensional latent variables
AJD model with double exponential jumps is
inadequate of modeling the many small jumps
in S&P 500 data
42
Thank you!
43
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