Page 1
August 3, 2015, Stochastic Processes and Applications, Ulaanbaatar, MONGOLIA Modeling Financial Time-Series with COGARCH processes - p. 1/51
Modeling Financial Time-Series withCOGARCH(p, q) processes
Erdenebaatar ChadraaMinnesota State University
Peter BrockwellColorado State University
Alex LindnerTechnische Universität München
Page 2
Outline
•Outline
Introduction
Continuous-time approaches
COGARCH(p, q) processes
Parameter Estimation
Real Data Analysis
Conclusions and Future Works
August 3, 2015, Stochastic Processes and Applications, Ulaanbaatar, MONGOLIA Modeling Financial Time-Series with COGARCH processes - p. 2/51
Outline
Page 3
Outline
•Outline
Introduction
Continuous-time approaches
COGARCH(p, q) processes
Parameter Estimation
Real Data Analysis
Conclusions and Future Works
August 3, 2015, Stochastic Processes and Applications, Ulaanbaatar, MONGOLIA Modeling Financial Time-Series with COGARCH processes - p. 2/51
Outline
• Introduction
◦ Stylized features of financial data
◦ GARCH models
◦ Lévy processes
Page 4
Outline
•Outline
Introduction
Continuous-time approaches
COGARCH(p, q) processes
Parameter Estimation
Real Data Analysis
Conclusions and Future Works
August 3, 2015, Stochastic Processes and Applications, Ulaanbaatar, MONGOLIA Modeling Financial Time-Series with COGARCH processes - p. 2/51
Outline
• Introduction
◦ Stylized features of financial data
◦ GARCH models
◦ Lévy processes
• Continuous-time approaches
◦ Nelson’s model
◦ COGARCH(1,1) model
Page 5
Outline
•Outline
Introduction
Continuous-time approaches
COGARCH(p, q) processes
Parameter Estimation
Real Data Analysis
Conclusions and Future Works
August 3, 2015, Stochastic Processes and Applications, Ulaanbaatar, MONGOLIA Modeling Financial Time-Series with COGARCH processes - p. 2/51
Outline
• Introduction
◦ Stylized features of financial data
◦ GARCH models
◦ Lévy processes
• Continuous-time approaches
◦ Nelson’s model
◦ COGARCH(1,1) model
• COGARCH(p, q) model◦ Construction◦ Properties
Page 6
Outline
•Outline
Introduction
Continuous-time approaches
COGARCH(p, q) processes
Parameter Estimation
Real Data Analysis
Conclusions and Future Works
August 3, 2015, Stochastic Processes and Applications, Ulaanbaatar, MONGOLIA Modeling Financial Time-Series with COGARCH processes - p. 2/51
Outline
• Introduction
◦ Stylized features of financial data
◦ GARCH models
◦ Lévy processes
• Continuous-time approaches
◦ Nelson’s model
◦ COGARCH(1,1) model
• COGARCH(p, q) model◦ Construction◦ Properties
• Parameter estimation
Page 7
Outline
•Outline
Introduction
Continuous-time approaches
COGARCH(p, q) processes
Parameter Estimation
Real Data Analysis
Conclusions and Future Works
August 3, 2015, Stochastic Processes and Applications, Ulaanbaatar, MONGOLIA Modeling Financial Time-Series with COGARCH processes - p. 2/51
Outline
• Introduction
◦ Stylized features of financial data
◦ GARCH models
◦ Lévy processes
• Continuous-time approaches
◦ Nelson’s model
◦ COGARCH(1,1) model
• COGARCH(p, q) model◦ Construction◦ Properties
• Parameter estimation
• Simulation and Application
Page 8
Outline
•Outline
Introduction
Continuous-time approaches
COGARCH(p, q) processes
Parameter Estimation
Real Data Analysis
Conclusions and Future Works
August 3, 2015, Stochastic Processes and Applications, Ulaanbaatar, MONGOLIA Modeling Financial Time-Series with COGARCH processes - p. 2/51
Outline
• Introduction
◦ Stylized features of financial data
◦ GARCH models
◦ Lévy processes
• Continuous-time approaches
◦ Nelson’s model
◦ COGARCH(1,1) model
• COGARCH(p, q) model◦ Construction◦ Properties
• Parameter estimation
• Simulation and Application
• Conclusions and Future work
Page 9
August 3, 2015, Stochastic Processes and Applications, Ulaanbaatar, MONGOLIA Modeling Financial Time-Series with COGARCH processes - p. 3/51
Introduction
Page 10
Outline
Introduction
•Stylized Features
•Dow Jones Industrial
Average•GARCH model
•Example
•Lévy processes
•Lévy processes
•Example: Brownian Motion
•Example: Compound
Poisson
Continuous-time approaches
COGARCH(p, q) processes
Parameter Estimation
Real Data Analysis
Conclusions and Future Works
August 3, 2015, Stochastic Processes and Applications, Ulaanbaatar, MONGOLIA Modeling Financial Time-Series with COGARCH processes - p. 4/51
Stylized Features
Page 11
Outline
Introduction
•Stylized Features
•Dow Jones Industrial
Average•GARCH model
•Example
•Lévy processes
•Lévy processes
•Example: Brownian Motion
•Example: Compound
Poisson
Continuous-time approaches
COGARCH(p, q) processes
Parameter Estimation
Real Data Analysis
Conclusions and Future Works
August 3, 2015, Stochastic Processes and Applications, Ulaanbaatar, MONGOLIA Modeling Financial Time-Series with COGARCH processes - p. 4/51
Stylized Features
• Tail heaviness
Page 12
Outline
Introduction
•Stylized Features
•Dow Jones Industrial
Average•GARCH model
•Example
•Lévy processes
•Lévy processes
•Example: Brownian Motion
•Example: Compound
Poisson
Continuous-time approaches
COGARCH(p, q) processes
Parameter Estimation
Real Data Analysis
Conclusions and Future Works
August 3, 2015, Stochastic Processes and Applications, Ulaanbaatar, MONGOLIA Modeling Financial Time-Series with COGARCH processes - p. 4/51
Stylized Features
• Tail heaviness
• Volatility clustering
Page 13
Outline
Introduction
•Stylized Features
•Dow Jones Industrial
Average•GARCH model
•Example
•Lévy processes
•Lévy processes
•Example: Brownian Motion
•Example: Compound
Poisson
Continuous-time approaches
COGARCH(p, q) processes
Parameter Estimation
Real Data Analysis
Conclusions and Future Works
August 3, 2015, Stochastic Processes and Applications, Ulaanbaatar, MONGOLIA Modeling Financial Time-Series with COGARCH processes - p. 4/51
Stylized Features
• Tail heaviness
• Volatility clustering
• Dependence without correlation
Jul 2004 Jul 2005 Jul 20067000
8000
9000
10000
11000
12000
DJIA recorded from 02/13/2003 to 05/12/2006.
0 1 2 3 4 5 6
x 104
−20
−10
0
10
20
0 50 100 150 200 250 300
0
0.05
0.1
0.15
0.2
0.25
DJIA log-returns and the ACF of the squared log returns.
Page 14
Outline
Introduction
•Stylized Features
•Dow Jones Industrial
Average•GARCH model
•Example
•Lévy processes
•Lévy processes
•Example: Brownian Motion
•Example: Compound
Poisson
Continuous-time approaches
COGARCH(p, q) processes
Parameter Estimation
Real Data Analysis
Conclusions and Future Works
August 3, 2015, Stochastic Processes and Applications, Ulaanbaatar, MONGOLIA Modeling Financial Time-Series with COGARCH processes - p. 5/51
Dow Jones Industrial Average
Jul 2004 Jul 2005 Jul 20067000
8000
9000
10000
11000
12000
DJIA recorded from 02/13/2003 to 05/12/2006.
0 2000 4000 6000 8000 100000
200
400
600
800
1000
0 2000 4000 6000 8000 100000
50
100
150
200
DJIA squared log-returns, and the estimated volatilities.
Page 15
Outline
Introduction
•Stylized Features
•Dow Jones Industrial
Average•GARCH model
•Example
•Lévy processes
•Lévy processes
•Example: Brownian Motion
•Example: Compound
Poisson
Continuous-time approaches
COGARCH(p, q) processes
Parameter Estimation
Real Data Analysis
Conclusions and Future Works
August 3, 2015, Stochastic Processes and Applications, Ulaanbaatar, MONGOLIA Modeling Financial Time-Series with COGARCH processes - p. 6/51
GARCH model
Xn = lnPn − lnPn−1 where Pn are asset price, n = 1, 2, . . .
GARCH(p, q) model:
Xn =√vn εn,
vn = α0 +
p∑
i=1
αiX2n−i +
q∑
j=1
βjvn−j ,
where {εn} i.i.d.∼ N(0, 1), α0 > 0, αi ≥ 0, i = 1, . . . , p
Page 16
Outline
Introduction
•Stylized Features
•Dow Jones Industrial
Average•GARCH model
•Example
•Lévy processes
•Lévy processes
•Example: Brownian Motion
•Example: Compound
Poisson
Continuous-time approaches
COGARCH(p, q) processes
Parameter Estimation
Real Data Analysis
Conclusions and Future Works
August 3, 2015, Stochastic Processes and Applications, Ulaanbaatar, MONGOLIA Modeling Financial Time-Series with COGARCH processes - p. 7/51
Example
ARCH(1) process:
Xn =√vn εn,
vn = α0 + α1X2n−1.
Page 17
Outline
Introduction
•Stylized Features
•Dow Jones Industrial
Average•GARCH model
•Example
•Lévy processes
•Lévy processes
•Example: Brownian Motion
•Example: Compound
Poisson
Continuous-time approaches
COGARCH(p, q) processes
Parameter Estimation
Real Data Analysis
Conclusions and Future Works
August 3, 2015, Stochastic Processes and Applications, Ulaanbaatar, MONGOLIA Modeling Financial Time-Series with COGARCH processes - p. 7/51
Example
ARCH(1) process:
Xn =√vn εn,
vn = α0 + α1X2n−1.
• Xn with |α1| < 1 is strictly stationary white noise,
Page 18
Outline
Introduction
•Stylized Features
•Dow Jones Industrial
Average•GARCH model
•Example
•Lévy processes
•Lévy processes
•Example: Brownian Motion
•Example: Compound
Poisson
Continuous-time approaches
COGARCH(p, q) processes
Parameter Estimation
Real Data Analysis
Conclusions and Future Works
August 3, 2015, Stochastic Processes and Applications, Ulaanbaatar, MONGOLIA Modeling Financial Time-Series with COGARCH processes - p. 7/51
Example
ARCH(1) process:
Xn =√vn εn,
vn = α0 + α1X2n−1.
• Xn with |α1| < 1 is strictly stationary white noise,
• Xn is not i.i.d.,
Page 19
Outline
Introduction
•Stylized Features
•Dow Jones Industrial
Average•GARCH model
•Example
•Lévy processes
•Lévy processes
•Example: Brownian Motion
•Example: Compound
Poisson
Continuous-time approaches
COGARCH(p, q) processes
Parameter Estimation
Real Data Analysis
Conclusions and Future Works
August 3, 2015, Stochastic Processes and Applications, Ulaanbaatar, MONGOLIA Modeling Financial Time-Series with COGARCH processes - p. 7/51
Example
ARCH(1) process:
Xn =√vn εn,
vn = α0 + α1X2n−1.
• Xn with |α1| < 1 is strictly stationary white noise,
• Xn is not i.i.d.,
• Xn is “heavy-tailed”,
Page 20
Outline
Introduction
•Stylized Features
•Dow Jones Industrial
Average•GARCH model
•Example
•Lévy processes
•Lévy processes
•Example: Brownian Motion
•Example: Compound
Poisson
Continuous-time approaches
COGARCH(p, q) processes
Parameter Estimation
Real Data Analysis
Conclusions and Future Works
August 3, 2015, Stochastic Processes and Applications, Ulaanbaatar, MONGOLIA Modeling Financial Time-Series with COGARCH processes - p. 7/51
Example
ARCH(1) process:
Xn =√vn εn,
vn = α0 + α1X2n−1.
• Xn with |α1| < 1 is strictly stationary white noise,
• Xn is not i.i.d.,
• Xn is “heavy-tailed”,
• X2n has the ACF of an AR(1).
Page 21
Outline
Introduction
•Stylized Features
•Dow Jones Industrial
Average•GARCH model
•Example
•Lévy processes
•Lévy processes
•Example: Brownian Motion
•Example: Compound
Poisson
Continuous-time approaches
COGARCH(p, q) processes
Parameter Estimation
Real Data Analysis
Conclusions and Future Works
August 3, 2015, Stochastic Processes and Applications, Ulaanbaatar, MONGOLIA Modeling Financial Time-Series with COGARCH processes - p. 8/51
Lévy processes
An adapted process L := {Lt, t ≥ 0} with L0 = 0 a.s. is
a real valued Lévy process if
Page 22
Outline
Introduction
•Stylized Features
•Dow Jones Industrial
Average•GARCH model
•Example
•Lévy processes
•Lévy processes
•Example: Brownian Motion
•Example: Compound
Poisson
Continuous-time approaches
COGARCH(p, q) processes
Parameter Estimation
Real Data Analysis
Conclusions and Future Works
August 3, 2015, Stochastic Processes and Applications, Ulaanbaatar, MONGOLIA Modeling Financial Time-Series with COGARCH processes - p. 8/51
Lévy processes
An adapted process L := {Lt, t ≥ 0} with L0 = 0 a.s. is
a real valued Lévy process if
• L has independent increments,
Page 23
Outline
Introduction
•Stylized Features
•Dow Jones Industrial
Average•GARCH model
•Example
•Lévy processes
•Lévy processes
•Example: Brownian Motion
•Example: Compound
Poisson
Continuous-time approaches
COGARCH(p, q) processes
Parameter Estimation
Real Data Analysis
Conclusions and Future Works
August 3, 2015, Stochastic Processes and Applications, Ulaanbaatar, MONGOLIA Modeling Financial Time-Series with COGARCH processes - p. 8/51
Lévy processes
An adapted process L := {Lt, t ≥ 0} with L0 = 0 a.s. is
a real valued Lévy process if
• L has independent increments,
• L has stationary increments,
Page 24
Outline
Introduction
•Stylized Features
•Dow Jones Industrial
Average•GARCH model
•Example
•Lévy processes
•Lévy processes
•Example: Brownian Motion
•Example: Compound
Poisson
Continuous-time approaches
COGARCH(p, q) processes
Parameter Estimation
Real Data Analysis
Conclusions and Future Works
August 3, 2015, Stochastic Processes and Applications, Ulaanbaatar, MONGOLIA Modeling Financial Time-Series with COGARCH processes - p. 8/51
Lévy processes
An adapted process L := {Lt, t ≥ 0} with L0 = 0 a.s. is
a real valued Lévy process if
• L has independent increments,
• L has stationary increments,
• L is continuous in probability.
Page 25
Outline
Introduction
•Stylized Features
•Dow Jones Industrial
Average•GARCH model
•Example
•Lévy processes
•Lévy processes
•Example: Brownian Motion
•Example: Compound
Poisson
Continuous-time approaches
COGARCH(p, q) processes
Parameter Estimation
Real Data Analysis
Conclusions and Future Works
August 3, 2015, Stochastic Processes and Applications, Ulaanbaatar, MONGOLIA Modeling Financial Time-Series with COGARCH processes - p. 9/51
Lévy processes
Characteristic function of Lt:
ϕt(θ) = E(exp(iθLt)) = exp(tξ(θ)), θ ∈ R,
where ξ(θ) satisfies the Lévy - Khinchin formula,
ξ(θ) = iγLθ − τ2Lθ2
2+
∫
R
(eiθx − 1− iθx1|x|≤1
)dνL(x).
ξ(θ) is called a characteristic exponent or Lévy symbol.
(γL, τ2L, νL) is called the characteristic triplet of L.
νL on R is called the Lévy measure.
Page 26
Outline
Introduction
•Stylized Features
•Dow Jones Industrial
Average•GARCH model
•Example
•Lévy processes
•Lévy processes
•Example: Brownian Motion
•Example: Compound
Poisson
Continuous-time approaches
COGARCH(p, q) processes
Parameter Estimation
Real Data Analysis
Conclusions and Future Works
August 3, 2015, Stochastic Processes and Applications, Ulaanbaatar, MONGOLIA Modeling Financial Time-Series with COGARCH processes - p. 10/51
Example: Brownian Motion
The Lévy symbol: ξ(θ) = iγBθ − τ2Bθ2
2 .
The characteristic triplet: (γB , τ2B , 0).
0 0.2 0.4 0.6 0.8 1−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
A sample path of a standard Brownian motion(γB = 0, τ2B = 1).
Page 27
Outline
Introduction
•Stylized Features
•Dow Jones Industrial
Average•GARCH model
•Example
•Lévy processes
•Lévy processes
•Example: Brownian Motion
•Example: Compound
Poisson
Continuous-time approaches
COGARCH(p, q) processes
Parameter Estimation
Real Data Analysis
Conclusions and Future Works
August 3, 2015, Stochastic Processes and Applications, Ulaanbaatar, MONGOLIA Modeling Financial Time-Series with COGARCH processes - p. 11/51
Example: Compound Poisson
Lt :=Nt∑i=1
Yi, where {Yn} i.i.d.∼ FY , indep. of N = (Nt)t≥0.
The Lévy symbol: ξ(θ) =∫R(eiθx − 1)λFY (dx).
The characteristic triplet: (γL, 0, λFY ).
0 0.2 0.4 0.6 0.8 1−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
A sample path of a compound Poisson process with jumprate λ = 25 and standard normal jumps.
Page 28
August 3, 2015, Stochastic Processes and Applications, Ulaanbaatar, MONGOLIA Modeling Financial Time-Series with COGARCH processes - p. 12/51
Continuous-time approaches
Page 29
Outline
Introduction
Continuous-time approaches
•Nelson’s diffusion limit
•COGARCH(1,1) model
•COGARCH(1,1) model
•COGARCH(1,1) model
COGARCH(p, q) processes
Parameter Estimation
Real Data Analysis
Conclusions and Future Works
August 3, 2015, Stochastic Processes and Applications, Ulaanbaatar, MONGOLIA Modeling Financial Time-Series with COGARCH processes - p. 13/51
Nelson’s diffusion limit
Gt = lnPt where Pt, t ≥ 0 are asset price.
GARCH(1, 1) diffusion limit satisfies
dGt = σt dW(1)t ,
dσ2t = θ(γ − σ2
t )dt+ ρσ2t dW
(2)t , t ≥ 0,
where (W(1)t )t≥0 and (W
(2)t )t≥0 are independent Brownian
motions.
Page 30
Outline
Introduction
Continuous-time approaches
•Nelson’s diffusion limit
•COGARCH(1,1) model
•COGARCH(1,1) model
•COGARCH(1,1) model
COGARCH(p, q) processes
Parameter Estimation
Real Data Analysis
Conclusions and Future Works
August 3, 2015, Stochastic Processes and Applications, Ulaanbaatar, MONGOLIA Modeling Financial Time-Series with COGARCH processes - p. 13/51
Nelson’s diffusion limit
Gt = lnPt where Pt, t ≥ 0 are asset price.
GARCH(1, 1) diffusion limit satisfies
dGt = σt dW(1)t ,
dσ2t = θ(γ − σ2
t )dt+ ρσ2t dW
(2)t , t ≥ 0,
where (W(1)t )t≥0 and (W
(2)t )t≥0 are independent Brownian
motions.
• GARCH process is driven by a single noise sequence.
Page 31
Outline
Introduction
Continuous-time approaches
•Nelson’s diffusion limit
•COGARCH(1,1) model
•COGARCH(1,1) model
•COGARCH(1,1) model
COGARCH(p, q) processes
Parameter Estimation
Real Data Analysis
Conclusions and Future Works
August 3, 2015, Stochastic Processes and Applications, Ulaanbaatar, MONGOLIA Modeling Financial Time-Series with COGARCH processes - p. 13/51
Nelson’s diffusion limit
Gt = lnPt where Pt, t ≥ 0 are asset price.
GARCH(1, 1) diffusion limit satisfies
dGt = σt dW(1)t ,
dσ2t = θ(γ − σ2
t )dt+ ρσ2t dW
(2)t , t ≥ 0,
where (W(1)t )t≥0 and (W
(2)t )t≥0 are independent Brownian
motions.
• GARCH process is driven by a single noise sequence.
• The diffusion limit is driven by two independent Brownian
motions.
Page 32
Outline
Introduction
Continuous-time approaches
•Nelson’s diffusion limit
•COGARCH(1,1) model
•COGARCH(1,1) model
•COGARCH(1,1) model
COGARCH(p, q) processes
Parameter Estimation
Real Data Analysis
Conclusions and Future Works
August 3, 2015, Stochastic Processes and Applications, Ulaanbaatar, MONGOLIA Modeling Financial Time-Series with COGARCH processes - p. 14/51
COGARCH(1,1) model
Due to Klüppelberg et. al. (2005). Start with GARCH(1,1):
ξn = σnεn, σ2n = α0 + α1ξ
2n−1 + β1σ
2n−1, n ∈ N0,
From the recursion,
σ2n = α0
n−1∑
i=0
n−1∏
j=i+1
(β1 + α1ε2j ) + σ2
0
n−1∏
j=0
(β1 + α1ε2j )
=(σ20+α0
∫ n
0
exp[−
⌊s⌋∑
j=0
log(β1+α1ε2j )]ds)exp
[ n−1∑
j=0
log(β1+α1ε2j )].
Denote Xn := −n−1∑j=0
log(β1 + α1ε2j ).
Page 33
Outline
Introduction
Continuous-time approaches
•Nelson’s diffusion limit
•COGARCH(1,1) model
•COGARCH(1,1) model
•COGARCH(1,1) model
COGARCH(p, q) processes
Parameter Estimation
Real Data Analysis
Conclusions and Future Works
August 3, 2015, Stochastic Processes and Applications, Ulaanbaatar, MONGOLIA Modeling Financial Time-Series with COGARCH processes - p. 15/51
COGARCH(1,1) model
dGt = σt dLt, t > 0, G0 = 0,
σ2t =
(σ20 + α0
∫ t
0
eXs ds
)e−Xt− , t ≥ 0,
whereXt := −
∑
0<s≤t
log(β1 + α1(∆Ls)
2).
The volatility process σ2t satisfies
σ2t = α0t−log β1
∫ t
0
σ2sds+(α1/β1)
∑
0<s<t
σ2s(∆Ls)
2+σ20 , t ≥ 0.
Page 34
Outline
Introduction
Continuous-time approaches
•Nelson’s diffusion limit
•COGARCH(1,1) model
•COGARCH(1,1) model
•COGARCH(1,1) model
COGARCH(p, q) processes
Parameter Estimation
Real Data Analysis
Conclusions and Future Works
August 3, 2015, Stochastic Processes and Applications, Ulaanbaatar, MONGOLIA Modeling Financial Time-Series with COGARCH processes - p. 16/51
COGARCH(1,1) model
Page 35
Outline
Introduction
Continuous-time approaches
•Nelson’s diffusion limit
•COGARCH(1,1) model
•COGARCH(1,1) model
•COGARCH(1,1) model
COGARCH(p, q) processes
Parameter Estimation
Real Data Analysis
Conclusions and Future Works
August 3, 2015, Stochastic Processes and Applications, Ulaanbaatar, MONGOLIA Modeling Financial Time-Series with COGARCH processes - p. 16/51
COGARCH(1,1) model
• Stationary and uncorrelated increments: (Gt+r −Gt)t≥0,r > 0
Page 36
Outline
Introduction
Continuous-time approaches
•Nelson’s diffusion limit
•COGARCH(1,1) model
•COGARCH(1,1) model
•COGARCH(1,1) model
COGARCH(p, q) processes
Parameter Estimation
Real Data Analysis
Conclusions and Future Works
August 3, 2015, Stochastic Processes and Applications, Ulaanbaatar, MONGOLIA Modeling Financial Time-Series with COGARCH processes - p. 16/51
COGARCH(1,1) model
• Stationary and uncorrelated increments: (Gt+r −Gt)t≥0,r > 0
• Heavy tails and volatility clusters at high levels
Page 37
Outline
Introduction
Continuous-time approaches
•Nelson’s diffusion limit
•COGARCH(1,1) model
•COGARCH(1,1) model
•COGARCH(1,1) model
COGARCH(p, q) processes
Parameter Estimation
Real Data Analysis
Conclusions and Future Works
August 3, 2015, Stochastic Processes and Applications, Ulaanbaatar, MONGOLIA Modeling Financial Time-Series with COGARCH processes - p. 16/51
COGARCH(1,1) model
• Stationary and uncorrelated increments: (Gt+r −Gt)t≥0,r > 0
• Heavy tails and volatility clusters at high levels
• Volatility process has an ACF of an CAR(1)
Page 38
Outline
Introduction
Continuous-time approaches
•Nelson’s diffusion limit
•COGARCH(1,1) model
•COGARCH(1,1) model
•COGARCH(1,1) model
COGARCH(p, q) processes
Parameter Estimation
Real Data Analysis
Conclusions and Future Works
August 3, 2015, Stochastic Processes and Applications, Ulaanbaatar, MONGOLIA Modeling Financial Time-Series with COGARCH processes - p. 16/51
COGARCH(1,1) model
• Stationary and uncorrelated increments: (Gt+r −Gt)t≥0,r > 0
• Heavy tails and volatility clusters at high levels
• Volatility process has an ACF of an CAR(1)
• Squared increment process has an ACF of an ARMA(1,1)
Page 39
August 3, 2015, Stochastic Processes and Applications, Ulaanbaatar, MONGOLIA Modeling Financial Time-Series with COGARCH processes - p. 17/51
COGARCH(p, q) processes
Page 40
Outline
Introduction
Continuous-time approaches
COGARCH(p, q) processes
•Construction
•Construction cont.d
•Construction cont.d
•COGARCH(p, q)
equations•Non-negativity of the
volatility•Stationarity property
•Second-order property
•Squared increment
process•Squared increment
process•ACF of the squared
increments•Example: COGARCH(1,3)
•Example: COGARCH(1,3)
•Mixing: COGARCH(2,2)
Parameter Estimation
Real Data Analysis
Conclusions and Future Works
August 3, 2015, Stochastic Processes and Applications, Ulaanbaatar, MONGOLIA Modeling Financial Time-Series with COGARCH processes - p. 18/51
Construction
Start with a GARCH(p, q) process (ξn)n∈N0defined by
ξn = σnεn,
σ2n = α0 +
p∑
i=1
αiξ2n−i +
q∑
j=1
βjσ2n−j , n ≥ max{p, q}.
The volatility process can be viewed as a “self-exciting”
ARMA(q, p− 1) process driven by the noise sequence
(σ2n−1ε
2n−1)n∈N.
Page 41
Outline
Introduction
Continuous-time approaches
COGARCH(p, q) processes
•Construction
•Construction cont.d
•Construction cont.d
•COGARCH(p, q)
equations•Non-negativity of the
volatility•Stationarity property
•Second-order property
•Squared increment
process•Squared increment
process•ACF of the squared
increments•Example: COGARCH(1,3)
•Example: COGARCH(1,3)
•Mixing: COGARCH(2,2)
Parameter Estimation
Real Data Analysis
Conclusions and Future Works
August 3, 2015, Stochastic Processes and Applications, Ulaanbaatar, MONGOLIA Modeling Financial Time-Series with COGARCH processes - p. 19/51
Construction cont.d
CARMA(q, p−1) process (ψt)t≥0 with driving Lévy process L:
ψt = c+ a′ζt,
dζt = Bζtdt+ e dLt,
where a′ = [α1, . . . , αq], αj := 0 for j > p, e = [0, · · · , 0, 1]′
and
B =
0 1 0 · · · 0
0 0 1 · · · 0...
......
. . . 0
0 0 0 · · · 1
−βq −βq−1 −βq−2 · · · −β1
.
Need to replace (Lt) by a continuous-time analog of the
driving process (σ2n−1ε
2n−1)n∈N.
Page 42
Outline
Introduction
Continuous-time approaches
COGARCH(p, q) processes
•Construction
•Construction cont.d
•Construction cont.d
•COGARCH(p, q)
equations•Non-negativity of the
volatility•Stationarity property
•Second-order property
•Squared increment
process•Squared increment
process•ACF of the squared
increments•Example: COGARCH(1,3)
•Example: COGARCH(1,3)
•Mixing: COGARCH(2,2)
Parameter Estimation
Real Data Analysis
Conclusions and Future Works
August 3, 2015, Stochastic Processes and Applications, Ulaanbaatar, MONGOLIA Modeling Financial Time-Series with COGARCH processes - p. 20/51
Construction cont.d
In discrete-time,
R(d)n :=
n−1∑
i=0
ξ2i =
n−1∑
i=0
σ2i ε
2i .
Its continuous-time analog:
Rt =∑
0<s≤t
σ2s−(∆Ls)
2 =
∫ t
0
σ2s−d[L,L]
(d)s ,
i.e.dRt = σ2
t− d[L,L](d)t ,
where [L,L](d)t is the discrete part of the quadratic
covariation of L.
Page 43
Outline
Introduction
Continuous-time approaches
COGARCH(p, q) processes
•Construction
•Construction cont.d
•Construction cont.d
•COGARCH(p, q)
equations•Non-negativity of the
volatility•Stationarity property
•Second-order property
•Squared increment
process•Squared increment
process•ACF of the squared
increments•Example: COGARCH(1,3)
•Example: COGARCH(1,3)
•Mixing: COGARCH(2,2)
Parameter Estimation
Real Data Analysis
Conclusions and Future Works
August 3, 2015, Stochastic Processes and Applications, Ulaanbaatar, MONGOLIA Modeling Financial Time-Series with COGARCH processes - p. 21/51
COGARCH(p, q) equations
COGARCH(p, q) (q ≥ p ≥ 1) process with parameters B, a,α0 and driving Lévy process L is defined as follows:
dGt =√Vt dLt, t > 0, G0 = 0,
Vt = α0 + a′Yt−, t > 0, V0 = α0 + a′Y0,
where the state process Y = (Yt)t≥0 is the unique càdlàgsolution of the stochastic differential equation
dYt = BYt− dt+ e(α0 + a′Yt−) d[L,L](d)t , t > 0,
with initial value Y0, independent of the driving Lévy process(Lt)t≥0.
• When p = q = 1 it is reduced to the COGARCH(1, 1) model.
Page 44
Outline
Introduction
Continuous-time approaches
COGARCH(p, q) processes
•Construction
•Construction cont.d
•Construction cont.d
•COGARCH(p, q)
equations•Non-negativity of the
volatility•Stationarity property
•Second-order property
•Squared increment
process•Squared increment
process•ACF of the squared
increments•Example: COGARCH(1,3)
•Example: COGARCH(1,3)
•Mixing: COGARCH(2,2)
Parameter Estimation
Real Data Analysis
Conclusions and Future Works
August 3, 2015, Stochastic Processes and Applications, Ulaanbaatar, MONGOLIA Modeling Financial Time-Series with COGARCH processes - p. 22/51
Non-negativity of the volatility
Page 45
Outline
Introduction
Continuous-time approaches
COGARCH(p, q) processes
•Construction
•Construction cont.d
•Construction cont.d
•COGARCH(p, q)
equations•Non-negativity of the
volatility•Stationarity property
•Second-order property
•Squared increment
process•Squared increment
process•ACF of the squared
increments•Example: COGARCH(1,3)
•Example: COGARCH(1,3)
•Mixing: COGARCH(2,2)
Parameter Estimation
Real Data Analysis
Conclusions and Future Works
August 3, 2015, Stochastic Processes and Applications, Ulaanbaatar, MONGOLIA Modeling Financial Time-Series with COGARCH processes - p. 22/51
Non-negativity of the volatility
Vt > 0 a.s. ∀ t ≥ 0
for any driving Lévy process if
a′eBte ≥ 0, ∀ t ≥ 0.
Page 46
Outline
Introduction
Continuous-time approaches
COGARCH(p, q) processes
•Construction
•Construction cont.d
•Construction cont.d
•COGARCH(p, q)
equations•Non-negativity of the
volatility•Stationarity property
•Second-order property
•Squared increment
process•Squared increment
process•ACF of the squared
increments•Example: COGARCH(1,3)
•Example: COGARCH(1,3)
•Mixing: COGARCH(2,2)
Parameter Estimation
Real Data Analysis
Conclusions and Future Works
August 3, 2015, Stochastic Processes and Applications, Ulaanbaatar, MONGOLIA Modeling Financial Time-Series with COGARCH processes - p. 22/51
Non-negativity of the volatility
Vt > 0 a.s. ∀ t ≥ 0
for any driving Lévy process if
a′eBte ≥ 0, ∀ t ≥ 0.
• In the COGARCH(2, 2) case, the necessary and sufficientcondition is
α2 ≥ 0 and α1 ≥ −α2λ(B).
Page 47
Outline
Introduction
Continuous-time approaches
COGARCH(p, q) processes
•Construction
•Construction cont.d
•Construction cont.d
•COGARCH(p, q)
equations•Non-negativity of the
volatility•Stationarity property
•Second-order property
•Squared increment
process•Squared increment
process•ACF of the squared
increments•Example: COGARCH(1,3)
•Example: COGARCH(1,3)
•Mixing: COGARCH(2,2)
Parameter Estimation
Real Data Analysis
Conclusions and Future Works
August 3, 2015, Stochastic Processes and Applications, Ulaanbaatar, MONGOLIA Modeling Financial Time-Series with COGARCH processes - p. 23/51
Stationarity property
Page 48
Outline
Introduction
Continuous-time approaches
COGARCH(p, q) processes
•Construction
•Construction cont.d
•Construction cont.d
•COGARCH(p, q)
equations•Non-negativity of the
volatility•Stationarity property
•Second-order property
•Squared increment
process•Squared increment
process•ACF of the squared
increments•Example: COGARCH(1,3)
•Example: COGARCH(1,3)
•Mixing: COGARCH(2,2)
Parameter Estimation
Real Data Analysis
Conclusions and Future Works
August 3, 2015, Stochastic Processes and Applications, Ulaanbaatar, MONGOLIA Modeling Financial Time-Series with COGARCH processes - p. 23/51
Stationarity property
Ytd−→ Y∞, as t→ ∞,
where Y∞ is a finite r.v., if∫
R
log(1 + ‖S−1ea′S‖r y2) dνL(y) < −λ = −λ(B),
for some r ∈ [1,∞] and some matrix S such that S−1BS is
diagonal.
Page 49
Outline
Introduction
Continuous-time approaches
COGARCH(p, q) processes
•Construction
•Construction cont.d
•Construction cont.d
•COGARCH(p, q)
equations•Non-negativity of the
volatility•Stationarity property
•Second-order property
•Squared increment
process•Squared increment
process•ACF of the squared
increments•Example: COGARCH(1,3)
•Example: COGARCH(1,3)
•Mixing: COGARCH(2,2)
Parameter Estimation
Real Data Analysis
Conclusions and Future Works
August 3, 2015, Stochastic Processes and Applications, Ulaanbaatar, MONGOLIA Modeling Financial Time-Series with COGARCH processes - p. 23/51
Stationarity property
Ytd−→ Y∞, as t→ ∞,
where Y∞ is a finite r.v., if∫
R
log(1 + ‖S−1ea′S‖r y2) dνL(y) < −λ = −λ(B),
for some r ∈ [1,∞] and some matrix S such that S−1BS is
diagonal.
• In the COGARCH(1,1) case, this is a necessary andsufficient condition for existence of a stationary solution.
Page 50
Outline
Introduction
Continuous-time approaches
COGARCH(p, q) processes
•Construction
•Construction cont.d
•Construction cont.d
•COGARCH(p, q)
equations•Non-negativity of the
volatility•Stationarity property
•Second-order property
•Squared increment
process•Squared increment
process•ACF of the squared
increments•Example: COGARCH(1,3)
•Example: COGARCH(1,3)
•Mixing: COGARCH(2,2)
Parameter Estimation
Real Data Analysis
Conclusions and Future Works
August 3, 2015, Stochastic Processes and Applications, Ulaanbaatar, MONGOLIA Modeling Financial Time-Series with COGARCH processes - p. 23/51
Stationarity property
Ytd−→ Y∞, as t→ ∞,
where Y∞ is a finite r.v., if∫
R
log(1 + ‖S−1ea′S‖r y2) dνL(y) < −λ = −λ(B),
for some r ∈ [1,∞] and some matrix S such that S−1BS is
diagonal.
• In the COGARCH(1,1) case, this is a necessary andsufficient condition for existence of a stationary solution.
• If Y0d= Y∞ then (Yt)t≥0 and (Vt)t≥0 are strictly stationary.
Page 51
Outline
Introduction
Continuous-time approaches
COGARCH(p, q) processes
•Construction
•Construction cont.d
•Construction cont.d
•COGARCH(p, q)
equations•Non-negativity of the
volatility•Stationarity property
•Second-order property
•Squared increment
process•Squared increment
process•ACF of the squared
increments•Example: COGARCH(1,3)
•Example: COGARCH(1,3)
•Mixing: COGARCH(2,2)
Parameter Estimation
Real Data Analysis
Conclusions and Future Works
August 3, 2015, Stochastic Processes and Applications, Ulaanbaatar, MONGOLIA Modeling Financial Time-Series with COGARCH processes - p. 24/51
Second-order property
cov(Y∞) exists if EL41 <∞ and
‖S−1ea′S‖2r ρ < 2(−λ− ‖S−1ea′S‖rµ),
where µ := EL21, ρ := EL4
1.
If (Vt)t≥0 is the stationary volatility process, then
cov(Vt+h, Vt) =α20β
2q
(βq − µα1)2(1−m)cov(ψt+h, ψt), t, h ≥ 0,
where m := var (ψt).
Page 52
Outline
Introduction
Continuous-time approaches
COGARCH(p, q) processes
•Construction
•Construction cont.d
•Construction cont.d
•COGARCH(p, q)
equations•Non-negativity of the
volatility•Stationarity property
•Second-order property
•Squared increment
process•Squared increment
process•ACF of the squared
increments•Example: COGARCH(1,3)
•Example: COGARCH(1,3)
•Mixing: COGARCH(2,2)
Parameter Estimation
Real Data Analysis
Conclusions and Future Works
August 3, 2015, Stochastic Processes and Applications, Ulaanbaatar, MONGOLIA Modeling Financial Time-Series with COGARCH processes - p. 24/51
Second-order property
cov(Y∞) exists if EL41 <∞ and
‖S−1ea′S‖2r ρ < 2(−λ− ‖S−1ea′S‖rµ),
where µ := EL21, ρ := EL4
1.
If (Vt)t≥0 is the stationary volatility process, then
cov(Vt+h, Vt) =α20β
2q
(βq − µα1)2(1−m)cov(ψt+h, ψt), t, h ≥ 0,
where m := var (ψt).
• Volatility process has an ACF of a CARMA(q, p− 1).
(Volatility process of a GARCH(p, q) has an ACF of an
ARMA(q, p− 1)).
Page 53
Outline
Introduction
Continuous-time approaches
COGARCH(p, q) processes
•Construction
•Construction cont.d
•Construction cont.d
•COGARCH(p, q)
equations•Non-negativity of the
volatility•Stationarity property
•Second-order property
•Squared increment
process•Squared increment
process•ACF of the squared
increments•Example: COGARCH(1,3)
•Example: COGARCH(1,3)
•Mixing: COGARCH(2,2)
Parameter Estimation
Real Data Analysis
Conclusions and Future Works
August 3, 2015, Stochastic Processes and Applications, Ulaanbaatar, MONGOLIA Modeling Financial Time-Series with COGARCH processes - p. 25/51
Squared increment process
G(r)t := Gt+r −Gt =
∫
(t,t+r]
√Vs dLs, t ≥ 0.
For any t ≥ 0 and h ≥ r > 0,
E(G(r)t ) = 0,
E((G(r)t )2) =
α0βqr
βq − µα1EL2
1,
cov(G(r)t , G
(r)t+h) = 0.
Page 54
Outline
Introduction
Continuous-time approaches
COGARCH(p, q) processes
•Construction
•Construction cont.d
•Construction cont.d
•COGARCH(p, q)
equations•Non-negativity of the
volatility•Stationarity property
•Second-order property
•Squared increment
process•Squared increment
process•ACF of the squared
increments•Example: COGARCH(1,3)
•Example: COGARCH(1,3)
•Mixing: COGARCH(2,2)
Parameter Estimation
Real Data Analysis
Conclusions and Future Works
August 3, 2015, Stochastic Processes and Applications, Ulaanbaatar, MONGOLIA Modeling Financial Time-Series with COGARCH processes - p. 26/51
Squared increment process
cov((G(r)t )2, (G
(r)t+h)
2) = EL21 a
′eBhB−1(I − e−Br) cov(Yr, G2r), h ≥ r
var((G(r)t )2) = 6EL2
1 a′Kr + 2(rEL2
1EV∞)2 + rEL41EV
2∞,
whereB := B + µea′,
cov(Yr, G2r) = [(I − eBr)cov(Y∞)− B−1(eBr − I)cov(Y∞)B′]e
and
Kr :=[(rI−B−1(eBr−I))cov(Y∞)−B−1(B−1(eBr−I)−rI)cov(Y∞)B
Page 55
Outline
Introduction
Continuous-time approaches
COGARCH(p, q) processes
•Construction
•Construction cont.d
•Construction cont.d
•COGARCH(p, q)
equations•Non-negativity of the
volatility•Stationarity property
•Second-order property
•Squared increment
process•Squared increment
process•ACF of the squared
increments•Example: COGARCH(1,3)
•Example: COGARCH(1,3)
•Mixing: COGARCH(2,2)
Parameter Estimation
Real Data Analysis
Conclusions and Future Works
August 3, 2015, Stochastic Processes and Applications, Ulaanbaatar, MONGOLIA Modeling Financial Time-Series with COGARCH processes - p. 27/51
ACF of the squared increments
The ACF of the squared increment process:
ρ(h) = c1eλ1h + · · ·+ cqe
λqh, h ≥ r,
where λ1, . . . , λq are the eigenvalues of B. The ACF can alsobe written as
ρ(h) =a′Ph a+ a′Qh
a′P0 a+ a′Q0 +R0, h ≥ r,
where Ph,Qh, P0,Q0 and R0 depend only on β1, . . . , βq.
Page 56
Outline
Introduction
Continuous-time approaches
COGARCH(p, q) processes
•Construction
•Construction cont.d
•Construction cont.d
•COGARCH(p, q)
equations•Non-negativity of the
volatility•Stationarity property
•Second-order property
•Squared increment
process•Squared increment
process•ACF of the squared
increments•Example: COGARCH(1,3)
•Example: COGARCH(1,3)
•Mixing: COGARCH(2,2)
Parameter Estimation
Real Data Analysis
Conclusions and Future Works
August 3, 2015, Stochastic Processes and Applications, Ulaanbaatar, MONGOLIA Modeling Financial Time-Series with COGARCH processes - p. 27/51
ACF of the squared increments
The ACF of the squared increment process:
ρ(h) = c1eλ1h + · · ·+ cqe
λqh, h ≥ r,
where λ1, . . . , λq are the eigenvalues of B. The ACF can alsobe written as
ρ(h) =a′Ph a+ a′Qh
a′P0 a+ a′Q0 +R0, h ≥ r,
where Ph,Qh, P0,Q0 and R0 depend only on β1, . . . , βq.
• Squared increment process has an ACF of an ARMA(q, q).
(Square of a GARCH(p, q) process has an ACF of an
ARMA(q, q).
Page 57
Outline
Introduction
Continuous-time approaches
COGARCH(p, q) processes
•Construction
•Construction cont.d
•Construction cont.d
•COGARCH(p, q)
equations•Non-negativity of the
volatility•Stationarity property
•Second-order property
•Squared increment
process•Squared increment
process•ACF of the squared
increments•Example: COGARCH(1,3)
•Example: COGARCH(1,3)
•Mixing: COGARCH(2,2)
Parameter Estimation
Real Data Analysis
Conclusions and Future Works
August 3, 2015, Stochastic Processes and Applications, Ulaanbaatar, MONGOLIA Modeling Financial Time-Series with COGARCH processes - p. 28/51
Example: COGARCH(1,3)
0 1000 2000 3000 4000 5000 6000 7000 8000−100
0
100
200
Gt
0 1000 2000 3000 4000 5000 6000 7000 8000
−5
0
5
G(1
)t
0 1000 2000 3000 4000 5000 6000 7000 8000
2
4
6
8V
t
The simulated compound-Poisson driven COGARCH(1,3)process with jump-rate 2, normally distributed jumps with
mean zero and variance 0.74 and coefficients α0 = α1 = 1,β1 = 1.2, β2 = .48 + π2 and β3 = .064 + .4π2.
Page 58
Outline
Introduction
Continuous-time approaches
COGARCH(p, q) processes
•Construction
•Construction cont.d
•Construction cont.d
•COGARCH(p, q)
equations•Non-negativity of the
volatility•Stationarity property
•Second-order property
•Squared increment
process•Squared increment
process•ACF of the squared
increments•Example: COGARCH(1,3)
•Example: COGARCH(1,3)
•Mixing: COGARCH(2,2)
Parameter Estimation
Real Data Analysis
Conclusions and Future Works
August 3, 2015, Stochastic Processes and Applications, Ulaanbaatar, MONGOLIA Modeling Financial Time-Series with COGARCH processes - p. 29/51
Example: COGARCH(1,3)
0 10 20 30 40
0
0.2
0.4
0.6
0.8
1
0 10 20 30 40
0
0.015
0.03
0.045
0.06
0.075
The sample ACFs of the volatilities (Vt) (left) and of thesquared COGARCH increments ((Gt+1 −Gt)
2) (right) of arealisation of length 1,000,000 of the COGARCH(1,3)
process.
Page 59
Outline
Introduction
Continuous-time approaches
COGARCH(p, q) processes
•Construction
•Construction cont.d
•Construction cont.d
•COGARCH(p, q)
equations•Non-negativity of the
volatility•Stationarity property
•Second-order property
•Squared increment
process•Squared increment
process•ACF of the squared
increments•Example: COGARCH(1,3)
•Example: COGARCH(1,3)
•Mixing: COGARCH(2,2)
Parameter Estimation
Real Data Analysis
Conclusions and Future Works
August 3, 2015, Stochastic Processes and Applications, Ulaanbaatar, MONGOLIA Modeling Financial Time-Series with COGARCH processes - p. 30/51
Mixing: COGARCH(2,2)
Page 60
Outline
Introduction
Continuous-time approaches
COGARCH(p, q) processes
•Construction
•Construction cont.d
•Construction cont.d
•COGARCH(p, q)
equations•Non-negativity of the
volatility•Stationarity property
•Second-order property
•Squared increment
process•Squared increment
process•ACF of the squared
increments•Example: COGARCH(1,3)
•Example: COGARCH(1,3)
•Mixing: COGARCH(2,2)
Parameter Estimation
Real Data Analysis
Conclusions and Future Works
August 3, 2015, Stochastic Processes and Applications, Ulaanbaatar, MONGOLIA Modeling Financial Time-Series with COGARCH processes - p. 30/51
Mixing: COGARCH(2,2)
• For a compound-Poisson driven COGARCH(2,2) process,
(Yt)t≥0 is strongly mixing with geometric rate.
Page 61
Outline
Introduction
Continuous-time approaches
COGARCH(p, q) processes
•Construction
•Construction cont.d
•Construction cont.d
•COGARCH(p, q)
equations•Non-negativity of the
volatility•Stationarity property
•Second-order property
•Squared increment
process•Squared increment
process•ACF of the squared
increments•Example: COGARCH(1,3)
•Example: COGARCH(1,3)
•Mixing: COGARCH(2,2)
Parameter Estimation
Real Data Analysis
Conclusions and Future Works
August 3, 2015, Stochastic Processes and Applications, Ulaanbaatar, MONGOLIA Modeling Financial Time-Series with COGARCH processes - p. 30/51
Mixing: COGARCH(2,2)
• For a compound-Poisson driven COGARCH(2,2) process,
(Yt)t≥0 is strongly mixing with geometric rate.
• The volatility process (Vt)t≥0 and the squared increment
process (G(r)rn )n∈N also inherits the strong mixing property
as well as the rate.
Page 62
August 3, 2015, Stochastic Processes and Applications, Ulaanbaatar, MONGOLIA Modeling Financial Time-Series with COGARCH processes - p. 31/51
Parameter Estimation
Page 63
Outline
Introduction
Continuous-time approaches
COGARCH(p, q) processes
Parameter Estimation
•Least squares method
•Strong consistency and
asymptotic normality•Estimating the parameters
•Example: COGARCH(2,2)
•Example: COGARCH(2,2)
•Simulation result
•Simulation result
•Simulation result
•Volatility Estimation
•Volatility Estimation
Real Data Analysis
Conclusions and Future Works
August 3, 2015, Stochastic Processes and Applications, Ulaanbaatar, MONGOLIA Modeling Financial Time-Series with COGARCH processes - p. 32/51
Least squares method
• The noise sequence in the squared increment process((G
(r)t )2)t≥0 is not an i.i.d. or a martingale difference.
Page 64
Outline
Introduction
Continuous-time approaches
COGARCH(p, q) processes
Parameter Estimation
•Least squares method
•Strong consistency and
asymptotic normality•Estimating the parameters
•Example: COGARCH(2,2)
•Example: COGARCH(2,2)
•Simulation result
•Simulation result
•Simulation result
•Volatility Estimation
•Volatility Estimation
Real Data Analysis
Conclusions and Future Works
August 3, 2015, Stochastic Processes and Applications, Ulaanbaatar, MONGOLIA Modeling Financial Time-Series with COGARCH processes - p. 32/51
Least squares method
• The noise sequence in the squared increment process((G
(r)t )2)t≥0 is not an i.i.d. or a martingale difference.
• Under strong mixing and moment conditions, the least
squares estimators (LSE) of ARMA representations in
which the noise is the linear innovation process are
strongly consistent and asymptotically normal.
(Francq and Zakoïan (1998)).
Page 65
Outline
Introduction
Continuous-time approaches
COGARCH(p, q) processes
Parameter Estimation
•Least squares method
•Strong consistency and
asymptotic normality•Estimating the parameters
•Example: COGARCH(2,2)
•Example: COGARCH(2,2)
•Simulation result
•Simulation result
•Simulation result
•Volatility Estimation
•Volatility Estimation
Real Data Analysis
Conclusions and Future Works
August 3, 2015, Stochastic Processes and Applications, Ulaanbaatar, MONGOLIA Modeling Financial Time-Series with COGARCH processes - p. 33/51
Strong consistency and asymptotic normality
Francq and Zakoïan (1998):
• For a strictly stationary ergodic process (Xt)t∈Z satisfying
the weak ARMA representation the LSE are strongly
consistent.
Page 66
Outline
Introduction
Continuous-time approaches
COGARCH(p, q) processes
Parameter Estimation
•Least squares method
•Strong consistency and
asymptotic normality•Estimating the parameters
•Example: COGARCH(2,2)
•Example: COGARCH(2,2)
•Simulation result
•Simulation result
•Simulation result
•Volatility Estimation
•Volatility Estimation
Real Data Analysis
Conclusions and Future Works
August 3, 2015, Stochastic Processes and Applications, Ulaanbaatar, MONGOLIA Modeling Financial Time-Series with COGARCH processes - p. 33/51
Strong consistency and asymptotic normality
Francq and Zakoïan (1998):
• For a strictly stationary ergodic process (Xt)t∈Z satisfying
the weak ARMA representation the LSE are strongly
consistent.
• If in addition, (Xt)t∈Z satisfies E|Xt|4+2ν <∞ and strongly
mixing with the mixing rate such that∑∞
k=0 αν/(2+ν)k <∞
for some ν > 0 then the LSE are asymptotically normal.
Page 67
Outline
Introduction
Continuous-time approaches
COGARCH(p, q) processes
Parameter Estimation
•Least squares method
•Strong consistency and
asymptotic normality•Estimating the parameters
•Example: COGARCH(2,2)
•Example: COGARCH(2,2)
•Simulation result
•Simulation result
•Simulation result
•Volatility Estimation
•Volatility Estimation
Real Data Analysis
Conclusions and Future Works
August 3, 2015, Stochastic Processes and Applications, Ulaanbaatar, MONGOLIA Modeling Financial Time-Series with COGARCH processes - p. 34/51
Estimating the parameters
Page 68
Outline
Introduction
Continuous-time approaches
COGARCH(p, q) processes
Parameter Estimation
•Least squares method
•Strong consistency and
asymptotic normality•Estimating the parameters
•Example: COGARCH(2,2)
•Example: COGARCH(2,2)
•Simulation result
•Simulation result
•Simulation result
•Volatility Estimation
•Volatility Estimation
Real Data Analysis
Conclusions and Future Works
August 3, 2015, Stochastic Processes and Applications, Ulaanbaatar, MONGOLIA Modeling Financial Time-Series with COGARCH processes - p. 34/51
Estimating the parameters
• Estimate β by fitting an ARMA(2,2) to the observed
squared increments and minimizing the least-squares sum.
Page 69
Outline
Introduction
Continuous-time approaches
COGARCH(p, q) processes
Parameter Estimation
•Least squares method
•Strong consistency and
asymptotic normality•Estimating the parameters
•Example: COGARCH(2,2)
•Example: COGARCH(2,2)
•Simulation result
•Simulation result
•Simulation result
•Volatility Estimation
•Volatility Estimation
Real Data Analysis
Conclusions and Future Works
August 3, 2015, Stochastic Processes and Applications, Ulaanbaatar, MONGOLIA Modeling Financial Time-Series with COGARCH processes - p. 34/51
Estimating the parameters
• Estimate β by fitting an ARMA(2,2) to the observed
squared increments and minimizing the least-squares sum.
• Estimate a by matching the ACF of the fitted ARMA(2,2)
model and
ρ(h) =a′Ph a+ a′Qh
a′P0 a+ a′Q0 +R0, h ≥ r.
Page 70
Outline
Introduction
Continuous-time approaches
COGARCH(p, q) processes
Parameter Estimation
•Least squares method
•Strong consistency and
asymptotic normality•Estimating the parameters
•Example: COGARCH(2,2)
•Example: COGARCH(2,2)
•Simulation result
•Simulation result
•Simulation result
•Volatility Estimation
•Volatility Estimation
Real Data Analysis
Conclusions and Future Works
August 3, 2015, Stochastic Processes and Applications, Ulaanbaatar, MONGOLIA Modeling Financial Time-Series with COGARCH processes - p. 34/51
Estimating the parameters
• Estimate β by fitting an ARMA(2,2) to the observed
squared increments and minimizing the least-squares sum.
• Estimate a by matching the ACF of the fitted ARMA(2,2)
model and
ρ(h) =a′Ph a+ a′Qh
a′P0 a+ a′Q0 +R0, h ≥ r.
• Finally, estimate α0 using
E((G(r)t )2) =
α0βqr
βq − µα1EL2
1.
(Assume µ = EL21 = 1).
Page 71
Outline
Introduction
Continuous-time approaches
COGARCH(p, q) processes
Parameter Estimation
•Least squares method
•Strong consistency and
asymptotic normality•Estimating the parameters
•Example: COGARCH(2,2)
•Example: COGARCH(2,2)
•Simulation result
•Simulation result
•Simulation result
•Volatility Estimation
•Volatility Estimation
Real Data Analysis
Conclusions and Future Works
August 3, 2015, Stochastic Processes and Applications, Ulaanbaatar, MONGOLIA Modeling Financial Time-Series with COGARCH processes - p. 35/51
Example: COGARCH(2,2)
α0 = 1, α1 = 0.1, α2 = 0, β1 = 1, β2 = 0.2 and compoundPoisson driving process with std normal jumps and EL2
1 = 1.
ACF of the squared increment:
ρ(h) = 0.1040e−0.1127h − 0.0811e−0.8873h, h = 1, 2, . . .
ARMA(2,2) parameters:
φ1 = 1.3052, φ2 = −0.3679, θ1 = −1.2642, θ2 = 0.3669.
Hence,
β1 = log ξ1ξ2 = 1 and β2 = log ξ1 log ξ2 = 0.1,
where ξ−11 = 0.8934 and ξ−1
2 = 0.4118 are the autoregressiveroots of the ARMA(2,2).
Page 72
Outline
Introduction
Continuous-time approaches
COGARCH(p, q) processes
Parameter Estimation
•Least squares method
•Strong consistency and
asymptotic normality•Estimating the parameters
•Example: COGARCH(2,2)
•Example: COGARCH(2,2)
•Simulation result
•Simulation result
•Simulation result
•Volatility Estimation
•Volatility Estimation
Real Data Analysis
Conclusions and Future Works
August 3, 2015, Stochastic Processes and Applications, Ulaanbaatar, MONGOLIA Modeling Financial Time-Series with COGARCH processes - p. 36/51
Example: COGARCH(2,2)
ACF as a function of a:
ρ(h) =17.2007α2
1 − 0.2185α22 + 3.8771α1 − 0.4370α2
14.6900α21 + 0.2853α2
2 + 2.3673α1 + 6.5707α2 + 5e−0.1127h
+−2.3295α2
1 + 1.8340α22 − 4.1338α1 + 3.6680α2
14.6900α21 + 0.2853α2
2 + 2.3673α1 + 6.5707α2 + 5e−0.8873h
Matching the two ACFs yields
−150.7545α21 + 2.3868α2
2 − 34.9245α1 + 10.7736α2 + 5 = 0,
−14.0288α21 + 22.8957α2
2 − 48.5971α1 + 51.7914α2 + 5 = 0,
giving a = (0.1, 0). Further, find
β1 = β1 + α2 = 1, β2 = β2 + α1 = 0.2 and α0 = 1.
Page 73
Outline
Introduction
Continuous-time approaches
COGARCH(p, q) processes
Parameter Estimation
•Least squares method
•Strong consistency and
asymptotic normality•Estimating the parameters
•Example: COGARCH(2,2)
•Example: COGARCH(2,2)
•Simulation result
•Simulation result
•Simulation result
•Volatility Estimation
•Volatility Estimation
Real Data Analysis
Conclusions and Future Works
August 3, 2015, Stochastic Processes and Applications, Ulaanbaatar, MONGOLIA Modeling Financial Time-Series with COGARCH processes - p. 37/51
Simulation result
α0 = 1, α1 = 0.1, α2 = 0, β1 = 1, β2 = 0.2 and compoundPoisson driving process with std normal jumps and EL2
1 = 1.
1, 000, 000 realizations of G(1)i , i = 0, . . . , 999, 999 were
simulated. The estimation was repeated 2000 times.
α0 α1 β1 β2
Mean 1.0683 0.0961 1.0123 0.1986(0.0100) (0.0010) (0.0054) (0.0009)
Bias 0.0683 -0.0039 0.0123 -0.0014(0.0100) (0.0013) (0.0054) (0.0009)
MSE 0.1099 0.0018 0.0300 0.0009(0.0059) (0.00001) (0.0014) (0.0001)
MAE 0.2509 0.0337 0.1383 0.0242(0.0069) (0.0008) (0.0033) (0.0006)
Page 74
Outline
Introduction
Continuous-time approaches
COGARCH(p, q) processes
Parameter Estimation
•Least squares method
•Strong consistency and
asymptotic normality•Estimating the parameters
•Example: COGARCH(2,2)
•Example: COGARCH(2,2)
•Simulation result
•Simulation result
•Simulation result
•Volatility Estimation
•Volatility Estimation
Real Data Analysis
Conclusions and Future Works
August 3, 2015, Stochastic Processes and Applications, Ulaanbaatar, MONGOLIA Modeling Financial Time-Series with COGARCH processes - p. 38/51
Simulation result
1 1.2 1.4 1.60
50
100
150
φ1
−0.6 −0.4 −0.2 00
50
100
150
φ2
−1.6 −1.4 −1.2 −10
50
100
150
θ1
0 0.2 0.4 0.60
50
100
150
θ2
Least-squares estimators of the ARMA(2,2) parameters. Thetrue values are φ1 = 1.3052, φ2 = −0.3679, θ1 = −1.2642 and
θ2 = 0.3669.
Page 75
Outline
Introduction
Continuous-time approaches
COGARCH(p, q) processes
Parameter Estimation
•Least squares method
•Strong consistency and
asymptotic normality•Estimating the parameters
•Example: COGARCH(2,2)
•Example: COGARCH(2,2)
•Simulation result
•Simulation result
•Simulation result
•Volatility Estimation
•Volatility Estimation
Real Data Analysis
Conclusions and Future Works
August 3, 2015, Stochastic Processes and Applications, Ulaanbaatar, MONGOLIA Modeling Financial Time-Series with COGARCH processes - p. 39/51
Simulation result
0 0.5 1 1.5 20
50
100
150
α0
0 0.1 0.20
50
100
150
α1
0.5 1 1.50
50
100
150
β1
0.1 0.15 0.2 0.25 0.30
50
100
150
β2
Least-squares estimators of the COGARCH parameters. Thetrue values are α0 = 1, α1 = 0.1, β1 = 1 and β2 = 0.2.
Page 76
Outline
Introduction
Continuous-time approaches
COGARCH(p, q) processes
Parameter Estimation
•Least squares method
•Strong consistency and
asymptotic normality•Estimating the parameters
•Example: COGARCH(2,2)
•Example: COGARCH(2,2)
•Simulation result
•Simulation result
•Simulation result
•Volatility Estimation
•Volatility Estimation
Real Data Analysis
Conclusions and Future Works
August 3, 2015, Stochastic Processes and Applications, Ulaanbaatar, MONGOLIA Modeling Financial Time-Series with COGARCH processes - p. 40/51
Volatility Estimation
Let h be a positive integer and G0, G 1
h, . . . , G1, G1+ 1
h, . . . are
the observed log returns. Then
Yt+1 = eBYt +h∑
i=1
(I +1
hB)h−ie
(G
( 1
h)
t+ i−1
h
)2
andVt+1 = α0 + a′Yt+1, t = 0, 1, . . .
where Y0 is an initial starting value.
Page 77
Outline
Introduction
Continuous-time approaches
COGARCH(p, q) processes
Parameter Estimation
•Least squares method
•Strong consistency and
asymptotic normality•Estimating the parameters
•Example: COGARCH(2,2)
•Example: COGARCH(2,2)
•Simulation result
•Simulation result
•Simulation result
•Volatility Estimation
•Volatility Estimation
Real Data Analysis
Conclusions and Future Works
August 3, 2015, Stochastic Processes and Applications, Ulaanbaatar, MONGOLIA Modeling Financial Time-Series with COGARCH processes - p. 41/51
Volatility Estimation
0 100 200 300 400 5000
20
40
60
(G(1
)t
)20 100 200 300 400 500
0
2
4
6
Vt
(whe
nh
=10
)
0 100 200 300 400 5000
2
4
6
Vt
(whe
nh
=1)
Sample path of the squared increment process (top graph),theoretical (blue line) and the estimated volatility (red line)
based on the observations G 1
h, G 2
h, . . . , G500 and the
estimated coefficients. The middle graph is for h = 10 andthe bottom graph is for h = 1.
Page 78
August 3, 2015, Stochastic Processes and Applications, Ulaanbaatar, MONGOLIA Modeling Financial Time-Series with COGARCH processes - p. 42/51
Real Data Analysis
Page 79
Outline
Introduction
Continuous-time approaches
COGARCH(p, q) processes
Parameter Estimation
Real Data Analysis
•Dow Jones 5-minute data
•Dow Jones 5-minute data
•Estimated Coefficients
•Volatility Estimation
•Goodness of fit
•Goodness of fit
Conclusions and Future Works
August 3, 2015, Stochastic Processes and Applications, Ulaanbaatar, MONGOLIA Modeling Financial Time-Series with COGARCH processes - p. 43/51
Dow Jones 5-minute data
Jul 2004 Jul 2005 Jul 20067000
8000
9000
10000
11000
12000
Dow Jones Industrial Average recorded from February12th, 2003 to May 12th, 2006. (813 trading days with 785-minute observations per day, resulting in total of 63414
5-minute observations).
Page 80
Outline
Introduction
Continuous-time approaches
COGARCH(p, q) processes
Parameter Estimation
Real Data Analysis
•Dow Jones 5-minute data
•Dow Jones 5-minute data
•Estimated Coefficients
•Volatility Estimation
•Goodness of fit
•Goodness of fit
Conclusions and Future Works
August 3, 2015, Stochastic Processes and Applications, Ulaanbaatar, MONGOLIA Modeling Financial Time-Series with COGARCH processes - p. 44/51
Dow Jones 5-minute data
0 1 2 3 4 5 6
x 104
−20
−10
0
10
20
0 50 100 150 200 250 300
0
0.05
0.1
0.15
0.2
0.25
Dow Jones log returns (the top graph) and the ACF of thesquared log returns.
Page 81
Outline
Introduction
Continuous-time approaches
COGARCH(p, q) processes
Parameter Estimation
Real Data Analysis
•Dow Jones 5-minute data
•Dow Jones 5-minute data
•Estimated Coefficients
•Volatility Estimation
•Goodness of fit
•Goodness of fit
Conclusions and Future Works
August 3, 2015, Stochastic Processes and Applications, Ulaanbaatar, MONGOLIA Modeling Financial Time-Series with COGARCH processes - p. 45/51
Estimated Coefficients
COGARCH(2, 2) model was fitted to the 30-minute log
returns G(1)0 , G
(1)1 , . . . , G
(1)10568.
The driving compound Poisson process has jump-rate c = 2
and normally distributed jumps with mean zero and variance
0.7071.
α0 = 0.9760, α1 = 0.0117, α2 = 0.1860,
β1 = 0.6088, β2 = 0.0122.
Page 82
Outline
Introduction
Continuous-time approaches
COGARCH(p, q) processes
Parameter Estimation
Real Data Analysis
•Dow Jones 5-minute data
•Dow Jones 5-minute data
•Estimated Coefficients
•Volatility Estimation
•Goodness of fit
•Goodness of fit
Conclusions and Future Works
August 3, 2015, Stochastic Processes and Applications, Ulaanbaatar, MONGOLIA Modeling Financial Time-Series with COGARCH processes - p. 46/51
Volatility Estimation
0 2000 4000 6000 8000 100000
200
400
600
800
1000
0 2000 4000 6000 8000 100000
50
100
150
200
Dow Jones squared 30-minute log returns (top graph), theestimated volatilities based on the estimated coefficients and
5-minute log returns (middle graph). The unit of time is 30minutes and there are h = 6 observations per unit interval.
Page 83
Outline
Introduction
Continuous-time approaches
COGARCH(p, q) processes
Parameter Estimation
Real Data Analysis
•Dow Jones 5-minute data
•Dow Jones 5-minute data
•Estimated Coefficients
•Volatility Estimation
•Goodness of fit
•Goodness of fit
Conclusions and Future Works
August 3, 2015, Stochastic Processes and Applications, Ulaanbaatar, MONGOLIA Modeling Financial Time-Series with COGARCH processes - p. 47/51
Goodness of fit
Estimated residuals:
G(1)t−1/
√Vt, t = 1, 2, . . . , 10569.
Sample mean: −0.0232,
Sample standard deviation: 0.9325,
Ljung-Box test statistic: QLB = 222.0025 with lag 189,
McLeod-Li test statistic: QML = 214.0934 with lag 189.
The critical value at 0.05 level was 222.0757.
Page 84
Outline
Introduction
Continuous-time approaches
COGARCH(p, q) processes
Parameter Estimation
Real Data Analysis
•Dow Jones 5-minute data
•Dow Jones 5-minute data
•Estimated Coefficients
•Volatility Estimation
•Goodness of fit
•Goodness of fit
Conclusions and Future Works
August 3, 2015, Stochastic Processes and Applications, Ulaanbaatar, MONGOLIA Modeling Financial Time-Series with COGARCH processes - p. 48/51
Goodness of fit
0 50 100 150 200 250−0.1
0
0.1
0.2
0 50 100 150 200 250−0.1
0
0.1
0.2
0 50 100 150 200 250−0.1
0
0.1
0.2
ACF of the squared 30-minute log returns (top graph), theresiduals (middle graph) and the squared residuals (bottom
graph).
Page 85
August 3, 2015, Stochastic Processes and Applications, Ulaanbaatar, MONGOLIA Modeling Financial Time-Series with COGARCH processes - p. 49/51
Conclusions and Future Works
Page 86
Outline
Introduction
Continuous-time approaches
COGARCH(p, q) processes
Parameter Estimation
Real Data Analysis
Conclusions and Future Works
•Conclusions
•Future Works
August 3, 2015, Stochastic Processes and Applications, Ulaanbaatar, MONGOLIA Modeling Financial Time-Series with COGARCH processes - p. 50/51
Conclusions
• A family of continuous time GARCH processes,generalizing the COGARCH(1, 1) process was introducedand studied.
• A least-squares method to estimate the parameters of aCOGARCH(2,2) process was proposed making use of theACF structure of the squared increment process.
• When the driving Lévy process is compound Poisson, thenthe state process and the squared increments are stronglymixing with exponential rate, ensuring strong consistencyand asymptotic normality of the LSE.
• The COGARCH(2, 2) model with compound poissondriving process was applied to a real data.
Page 87
Outline
Introduction
Continuous-time approaches
COGARCH(p, q) processes
Parameter Estimation
Real Data Analysis
Conclusions and Future Works
•Conclusions
•Future Works
August 3, 2015, Stochastic Processes and Applications, Ulaanbaatar, MONGOLIA Modeling Financial Time-Series with COGARCH processes - p. 51/51
Future Works
• Investigate the degree to which the stationarity conditioncan be relaxed when q > 1.
• Investigate the strong mixing property of COGARCH(p, q)processes with q > 2 and with general drivingLévy process.
• Investigate the connections between higher orderCOGARCH and GARCH processes.
• Comparisons of COGARCH models fitted to observationsof the same process made at different frequencies.
Thank you!