A Hidden Markov Stochastic Volatility Model for Energy … Framework Filtering and ... Scott (1987)...
Transcript of A Hidden Markov Stochastic Volatility Model for Energy … Framework Filtering and ... Scott (1987)...
Energy PriceModeling
Robert J.Elliott, Tao Lin,
Hong Miao
MotivationPrevious Work
Empirical Findings
Model andMethodologyModel Framework
Filtering andEstimation
Future StudyOption Pricing
Embed OtherCharacteristics ofEnergy Data
For Further Reading
A Hidden Markov Stochastic Volatility Modelfor Energy Prices
Robert J. Elliott1 Tao Lin2 Hong Miao1
1Haskayne School of BusinessUniversity of Calgary
2Department of Finance and Management ScienceNorges Handelshøyskole
FIBE 2007 på NHH
Energy PriceModeling
Robert J.Elliott, Tao Lin,
Hong Miao
MotivationPrevious Work
Empirical Findings
Model andMethodologyModel Framework
Filtering andEstimation
Future StudyOption Pricing
Embed OtherCharacteristics ofEnergy Data
For Further Reading
Outline
1 MotivationPrevious WorkEmpirical Findings
2 Model and MethodologyModel FrameworkFiltering and Estimation
3 Future StudyOption PricingEmbed Other Characteristics of Energy Data
Energy PriceModeling
Robert J.Elliott, Tao Lin,
Hong Miao
MotivationPrevious Work
Empirical Findings
Model andMethodologyModel Framework
Filtering andEstimation
Future StudyOption Pricing
Embed OtherCharacteristics ofEnergy Data
For Further Reading
Outline
1 MotivationPrevious WorkEmpirical Findings
2 Model and MethodologyModel FrameworkFiltering and Estimation
3 Future StudyOption PricingEmbed Other Characteristics of Energy Data
Energy PriceModeling
Robert J.Elliott, Tao Lin,
Hong Miao
MotivationPrevious Work
Empirical Findings
Model andMethodologyModel Framework
Filtering andEstimation
Future StudyOption Pricing
Embed OtherCharacteristics ofEnergy Data
For Further Reading
Outline
1 MotivationPrevious WorkEmpirical Findings
2 Model and MethodologyModel FrameworkFiltering and Estimation
3 Future StudyOption PricingEmbed Other Characteristics of Energy Data
Energy PriceModeling
Robert J.Elliott, Tao Lin,
Hong Miao
MotivationPrevious Work
Empirical Findings
Model andMethodologyModel Framework
Filtering andEstimation
Future StudyOption Pricing
Embed OtherCharacteristics ofEnergy Data
For Further Reading
Outline
1 MotivationPrevious WorkEmpirical Findings
2 Model and MethodologyModel FrameworkFiltering and Estimation
3 Future StudyOption PricingEmbed Other Characteristics of Energy Data
Energy PriceModeling
Robert J.Elliott, Tao Lin,
Hong Miao
MotivationPrevious Work
Empirical Findings
Model andMethodologyModel Framework
Filtering andEstimation
Future StudyOption Pricing
Embed OtherCharacteristics ofEnergy Data
For Further Reading
Volatility of Energy Data
Duffie, Gray & Hong(2004)Pindyck (2002) and Pindyck(2004)
Energy PriceModeling
Robert J.Elliott, Tao Lin,
Hong Miao
MotivationPrevious Work
Empirical Findings
Model andMethodologyModel Framework
Filtering andEstimation
Future StudyOption Pricing
Embed OtherCharacteristics ofEnergy Data
For Further Reading
Volatility of Energy Data
Duffie, Gray & Hong(2004)Pindyck (2002) and Pindyck(2004)
Energy PriceModeling
Robert J.Elliott, Tao Lin,
Hong Miao
MotivationPrevious Work
Empirical Findings
Model andMethodologyModel Framework
Filtering andEstimation
Future StudyOption Pricing
Embed OtherCharacteristics ofEnergy Data
For Further Reading
Volatility Models
Hull and White(1987)Scott (1987) and Stein&Stein(1991)Ball&Roma(1994) and Heston(1993)Fouque, Ppanicolaou, Sircar &Solna(2003)
Energy PriceModeling
Robert J.Elliott, Tao Lin,
Hong Miao
MotivationPrevious Work
Empirical Findings
Model andMethodologyModel Framework
Filtering andEstimation
Future StudyOption Pricing
Embed OtherCharacteristics ofEnergy Data
For Further Reading
Volatility Models
Hull and White(1987)Scott (1987) and Stein&Stein(1991)Ball&Roma(1994) and Heston(1993)Fouque, Ppanicolaou, Sircar &Solna(2003)
Energy PriceModeling
Robert J.Elliott, Tao Lin,
Hong Miao
MotivationPrevious Work
Empirical Findings
Model andMethodologyModel Framework
Filtering andEstimation
Future StudyOption Pricing
Embed OtherCharacteristics ofEnergy Data
For Further Reading
Volatility Models
Hull and White(1987)Scott (1987) and Stein&Stein(1991)Ball&Roma(1994) and Heston(1993)Fouque, Ppanicolaou, Sircar &Solna(2003)
Energy PriceModeling
Robert J.Elliott, Tao Lin,
Hong Miao
MotivationPrevious Work
Empirical Findings
Model andMethodologyModel Framework
Filtering andEstimation
Future StudyOption Pricing
Embed OtherCharacteristics ofEnergy Data
For Further Reading
Volatility Models
Hull and White(1987)Scott (1987) and Stein&Stein(1991)Ball&Roma(1994) and Heston(1993)Fouque, Ppanicolaou, Sircar &Solna(2003)
Energy PriceModeling
Robert J.Elliott, Tao Lin,
Hong Miao
MotivationPrevious Work
Empirical Findings
Model andMethodologyModel Framework
Filtering andEstimation
Future StudyOption Pricing
Embed OtherCharacteristics ofEnergy Data
For Further Reading
Hidden Markov Models
Elliott, Fisher&Platen(1999)Lam&Li(1998)
Energy PriceModeling
Robert J.Elliott, Tao Lin,
Hong Miao
MotivationPrevious Work
Empirical Findings
Model andMethodologyModel Framework
Filtering andEstimation
Future StudyOption Pricing
Embed OtherCharacteristics ofEnergy Data
For Further Reading
Hidden Markov Models
Elliott, Fisher&Platen(1999)Lam&Li(1998)
Energy PriceModeling
Robert J.Elliott, Tao Lin,
Hong Miao
MotivationPrevious Work
Empirical Findings
Model andMethodologyModel Framework
Filtering andEstimation
Future StudyOption Pricing
Embed OtherCharacteristics ofEnergy Data
For Further Reading
Outline
1 MotivationPrevious WorkEmpirical Findings
2 Model and MethodologyModel FrameworkFiltering and Estimation
3 Future StudyOption PricingEmbed Other Characteristics of Energy Data
Energy PriceModeling
Robert J.Elliott, Tao Lin,
Hong Miao
MotivationPrevious Work
Empirical Findings
Model andMethodologyModel Framework
Filtering andEstimation
Future StudyOption Pricing
Embed OtherCharacteristics ofEnergy Data
For Further Reading
Characteristics of Energy DataStatistic Facts
Tabelle: Descriptive Statistics of Energy Commodities
Products Mean Std. Dev. Skewness KurtosisWTI 0.000273 0.0211187 -0.227757 6.00646Brent 0.000265 0.024241 -1.541471 34.04660Alberta 0.000402 0.453598 0.002155 4.889515Nord Pool 0.000501 0.099274 1.713652 29.31280
Jarque-Bera Test reject the normality assumption
Energy PriceModeling
Robert J.Elliott, Tao Lin,
Hong Miao
MotivationPrevious Work
Empirical Findings
Model andMethodologyModel Framework
Filtering andEstimation
Future StudyOption Pricing
Embed OtherCharacteristics ofEnergy Data
For Further Reading
Characteristics of Energy DataStatistic Facts
Tabelle: Descriptive Statistics of Energy Commodities
Products Mean Std. Dev. Skewness KurtosisWTI 0.000273 0.0211187 -0.227757 6.00646Brent 0.000265 0.024241 -1.541471 34.04660Alberta 0.000402 0.453598 0.002155 4.889515Nord Pool 0.000501 0.099274 1.713652 29.31280
Jarque-Bera Test reject the normality assumption
Energy PriceModeling
Robert J.Elliott, Tao Lin,
Hong Miao
MotivationPrevious Work
Empirical Findings
Model andMethodologyModel Framework
Filtering andEstimation
Future StudyOption Pricing
Embed OtherCharacteristics ofEnergy Data
For Further Reading
Characteristics of Energy DataPlots of Volatility – dr 2
t
drt = µ(rt , t , . . . )dt + σ(rt , t , . . . )dwt ,
ddt
var(rt) = σ(rt , t , . . . )2 ≈dr2
tdt
.
Energy PriceModeling
Robert J.Elliott, Tao Lin,
Hong Miao
MotivationPrevious Work
Empirical Findings
Model andMethodologyModel Framework
Filtering andEstimation
Future StudyOption Pricing
Embed OtherCharacteristics ofEnergy Data
For Further Reading
Characteristics of Energy DataPlots of Volatility – dr 2
t
drt = µ(rt , t , . . . )dt + σ(rt , t , . . . )dwt ,
ddt
var(rt) = σ(rt , t , . . . )2 ≈dr2
tdt
.
Energy PriceModeling
Robert J.Elliott, Tao Lin,
Hong Miao
MotivationPrevious Work
Empirical Findings
Model andMethodologyModel Framework
Filtering andEstimation
Future StudyOption Pricing
Embed OtherCharacteristics ofEnergy Data
For Further Reading
Characteristics of Energy DataPlots of Volatility – dr 2
t
.00
.04
.08
.12
.16
.20
1990 1992 1994 1996 1998 2000 2002 2004
Daily Volatility of Brent Crude
(a) Brent
.000
.004
.008
.012
.016
.020
500 1000 1500 2000 2500 3000 3500
Daily Volatility of W TI Crude
(b) WTI
0
1
2
3
4
5
2002 2003 2004 2005
Daily Volatility of Alberta Electricity
(c) Alberta
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
500 1000 1500 2000
Daily Volatility of Nord Pool Spot
(d) NordPool
Abbildung: Daily Volatility of Energy Products
Energy PriceModeling
Robert J.Elliott, Tao Lin,
Hong Miao
MotivationPrevious Work
Empirical Findings
Model andMethodologyModel Framework
Filtering andEstimation
Future StudyOption Pricing
Embed OtherCharacteristics ofEnergy Data
For Further Reading
Characteristics of Energy DataPlots of Volatility – Moving Average
.000
.001
.002
.003
.004
.005
.006
500 1000 1500 2000 2500 3000 3500
60-Day-W indow Volatility of Brent Crude
(a) Brent
.0000
.0002
.0004
.0006
.0008
.0010
.0012
.0014
500 1000 1500 2000 2500 3000 3500
60-Day-W indow Volatility of W TI Crude
(b) WTI
.0
.1
.2
.3
.4
.5
.6
.7
250 500 750 1000 1250
60-Day-W indow Volatility of Alberta Electricity
(c) Alberta
.00
.01
.02
.03
.04
.05
.06
.07
.08
.09
500 1000 1500 2000
60-Day-W indow Volatility of Nord Pool Spot
(d) NordPool
Abbildung: Daily Volatility of Energy Products
Energy PriceModeling
Robert J.Elliott, Tao Lin,
Hong Miao
MotivationPrevious Work
Empirical Findings
Model andMethodologyModel Framework
Filtering andEstimation
Future StudyOption Pricing
Embed OtherCharacteristics ofEnergy Data
For Further Reading
Outline
1 MotivationPrevious WorkEmpirical Findings
2 Model and MethodologyModel FrameworkFiltering and Estimation
3 Future StudyOption PricingEmbed Other Characteristics of Energy Data
Energy PriceModeling
Robert J.Elliott, Tao Lin,
Hong Miao
MotivationPrevious Work
Empirical Findings
Model andMethodologyModel Framework
Filtering andEstimation
Future StudyOption Pricing
Embed OtherCharacteristics ofEnergy Data
For Further Reading
Model Framework
dYt = γ(L− Yt)dt + σtdWt (1)dlogσt = (a(t)− b(t)logσt)dt + c(t)dBt
Xk = AXk−1 + Mk . (2)
a = (a1, a2, . . . , aN)′
b = (b1, b2, . . . , bN)′
c = (c1, c2, . . . , cN)′.
a(t) = 〈a, X (t)〉b(t) = 〈b, X (t)〉c(t) = 〈c, X (t)〉.
Energy PriceModeling
Robert J.Elliott, Tao Lin,
Hong Miao
MotivationPrevious Work
Empirical Findings
Model andMethodologyModel Framework
Filtering andEstimation
Future StudyOption Pricing
Embed OtherCharacteristics ofEnergy Data
For Further Reading
Model Framework
dYt = γ(L− Yt)dt + σtdWt (1)dlogσt = (a(t)− b(t)logσt)dt + c(t)dBt
Xk = AXk−1 + Mk . (2)
a = (a1, a2, . . . , aN)′
b = (b1, b2, . . . , bN)′
c = (c1, c2, . . . , cN)′.
a(t) = 〈a, X (t)〉b(t) = 〈b, X (t)〉c(t) = 〈c, X (t)〉.
Energy PriceModeling
Robert J.Elliott, Tao Lin,
Hong Miao
MotivationPrevious Work
Empirical Findings
Model andMethodologyModel Framework
Filtering andEstimation
Future StudyOption Pricing
Embed OtherCharacteristics ofEnergy Data
For Further Reading
Model Framework
dYt = γ(L− Yt)dt + σtdWt (1)dlogσt = (a(t)− b(t)logσt)dt + c(t)dBt
Xk = AXk−1 + Mk . (2)
a = (a1, a2, . . . , aN)′
b = (b1, b2, . . . , bN)′
c = (c1, c2, . . . , cN)′.
a(t) = 〈a, X (t)〉b(t) = 〈b, X (t)〉c(t) = 〈c, X (t)〉.
Energy PriceModeling
Robert J.Elliott, Tao Lin,
Hong Miao
MotivationPrevious Work
Empirical Findings
Model andMethodologyModel Framework
Filtering andEstimation
Future StudyOption Pricing
Embed OtherCharacteristics ofEnergy Data
For Further Reading
Simulation
Abbildung: Simulated Sample Path
0 50 100 150 200−5
0
5Volatility Driven Process
0 50 100 150 2000
10
20Price Process
0 50 100 150 200
1
2
State Process
Energy PriceModeling
Robert J.Elliott, Tao Lin,
Hong Miao
MotivationPrevious Work
Empirical Findings
Model andMethodologyModel Framework
Filtering andEstimation
Future StudyOption Pricing
Embed OtherCharacteristics ofEnergy Data
For Further Reading
Outline
1 MotivationPrevious WorkEmpirical Findings
2 Model and MethodologyModel FrameworkFiltering and Estimation
3 Future StudyOption PricingEmbed Other Characteristics of Energy Data
Energy PriceModeling
Robert J.Elliott, Tao Lin,
Hong Miao
MotivationPrevious Work
Empirical Findings
Model andMethodologyModel Framework
Filtering andEstimation
Future StudyOption Pricing
Embed OtherCharacteristics ofEnergy Data
For Further Reading
Framework
Changing measuredPdP
|Fk = Λk ;
Calculating conditional expectation
E[f (hk ) Xk |FY
k
]=
E[Λk f (hk ) Xk |FY
k]
E[Λk |FY
k
] . (3)
Deriving recursive estimator
E[Λk f (hk ) Xk |FY
k
]=
∫ ∞
−∞f (z) qk (z)dz. (4)
qk (z) = A∫ ∞
−∞B(Yk , z, h, Yk−1)qk−1(h)dh. (5)
Re-estimating model parameters
Energy PriceModeling
Robert J.Elliott, Tao Lin,
Hong Miao
MotivationPrevious Work
Empirical Findings
Model andMethodologyModel Framework
Filtering andEstimation
Future StudyOption Pricing
Embed OtherCharacteristics ofEnergy Data
For Further Reading
Framework
Changing measuredPdP
|Fk = Λk ;
Calculating conditional expectation
E[f (hk ) Xk |FY
k
]=
E[Λk f (hk ) Xk |FY
k]
E[Λk |FY
k
] . (3)
Deriving recursive estimator
E[Λk f (hk ) Xk |FY
k
]=
∫ ∞
−∞f (z) qk (z)dz. (4)
qk (z) = A∫ ∞
−∞B(Yk , z, h, Yk−1)qk−1(h)dh. (5)
Re-estimating model parameters
Energy PriceModeling
Robert J.Elliott, Tao Lin,
Hong Miao
MotivationPrevious Work
Empirical Findings
Model andMethodologyModel Framework
Filtering andEstimation
Future StudyOption Pricing
Embed OtherCharacteristics ofEnergy Data
For Further Reading
Framework
Changing measuredPdP
|Fk = Λk ;
Calculating conditional expectation
E[f (hk ) Xk |FY
k
]=
E[Λk f (hk ) Xk |FY
k]
E[Λk |FY
k
] . (3)
Deriving recursive estimator
E[Λk f (hk ) Xk |FY
k
]=
∫ ∞
−∞f (z) qk (z)dz. (4)
qk (z) = A∫ ∞
−∞B(Yk , z, h, Yk−1)qk−1(h)dh. (5)
Re-estimating model parameters
Energy PriceModeling
Robert J.Elliott, Tao Lin,
Hong Miao
MotivationPrevious Work
Empirical Findings
Model andMethodologyModel Framework
Filtering andEstimation
Future StudyOption Pricing
Embed OtherCharacteristics ofEnergy Data
For Further Reading
Framework
Changing measuredPdP
|Fk = Λk ;
Calculating conditional expectation
E[f (hk ) Xk |FY
k
]=
E[Λk f (hk ) Xk |FY
k]
E[Λk |FY
k
] . (3)
Deriving recursive estimator
E[Λk f (hk ) Xk |FY
k
]=
∫ ∞
−∞f (z) qk (z)dz. (4)
qk (z) = A∫ ∞
−∞B(Yk , z, h, Yk−1)qk−1(h)dh. (5)
Re-estimating model parameters
Energy PriceModeling
Robert J.Elliott, Tao Lin,
Hong Miao
MotivationPrevious Work
Empirical Findings
Model andMethodologyModel Framework
Filtering andEstimation
Future StudyOption Pricing
Embed OtherCharacteristics ofEnergy Data
For Further Reading
EM Algorithm
The basic idea of the EM algorithm is:Separating observations into several groups;Starting with appropriate initial values, which satisfyconstraints for the parameters;Calculating estimates with the first group of data;Re-estimating the parameters iteratively until somestopping criterion is satisfied;Repeating the process with new group of data;Converging to true value, proved by Elliott et al. 1997.
Energy PriceModeling
Robert J.Elliott, Tao Lin,
Hong Miao
MotivationPrevious Work
Empirical Findings
Model andMethodologyModel Framework
Filtering andEstimation
Future StudyOption Pricing
Embed OtherCharacteristics ofEnergy Data
For Further Reading
EM Algorithm
The basic idea of the EM algorithm is:Separating observations into several groups;Starting with appropriate initial values, which satisfyconstraints for the parameters;Calculating estimates with the first group of data;Re-estimating the parameters iteratively until somestopping criterion is satisfied;Repeating the process with new group of data;Converging to true value, proved by Elliott et al. 1997.
Energy PriceModeling
Robert J.Elliott, Tao Lin,
Hong Miao
MotivationPrevious Work
Empirical Findings
Model andMethodologyModel Framework
Filtering andEstimation
Future StudyOption Pricing
Embed OtherCharacteristics ofEnergy Data
For Further Reading
EM Algorithm
The basic idea of the EM algorithm is:Separating observations into several groups;Starting with appropriate initial values, which satisfyconstraints for the parameters;Calculating estimates with the first group of data;Re-estimating the parameters iteratively until somestopping criterion is satisfied;Repeating the process with new group of data;Converging to true value, proved by Elliott et al. 1997.
Energy PriceModeling
Robert J.Elliott, Tao Lin,
Hong Miao
MotivationPrevious Work
Empirical Findings
Model andMethodologyModel Framework
Filtering andEstimation
Future StudyOption Pricing
Embed OtherCharacteristics ofEnergy Data
For Further Reading
EM Algorithm
The basic idea of the EM algorithm is:Separating observations into several groups;Starting with appropriate initial values, which satisfyconstraints for the parameters;Calculating estimates with the first group of data;Re-estimating the parameters iteratively until somestopping criterion is satisfied;Repeating the process with new group of data;Converging to true value, proved by Elliott et al. 1997.
Energy PriceModeling
Robert J.Elliott, Tao Lin,
Hong Miao
MotivationPrevious Work
Empirical Findings
Model andMethodologyModel Framework
Filtering andEstimation
Future StudyOption Pricing
Embed OtherCharacteristics ofEnergy Data
For Further Reading
EM Algorithm
The basic idea of the EM algorithm is:Separating observations into several groups;Starting with appropriate initial values, which satisfyconstraints for the parameters;Calculating estimates with the first group of data;Re-estimating the parameters iteratively until somestopping criterion is satisfied;Repeating the process with new group of data;Converging to true value, proved by Elliott et al. 1997.
Energy PriceModeling
Robert J.Elliott, Tao Lin,
Hong Miao
MotivationPrevious Work
Empirical Findings
Model andMethodologyModel Framework
Filtering andEstimation
Future StudyOption Pricing
Embed OtherCharacteristics ofEnergy Data
For Further Reading
EM Algorithm
The basic idea of the EM algorithm is:Separating observations into several groups;Starting with appropriate initial values, which satisfyconstraints for the parameters;Calculating estimates with the first group of data;Re-estimating the parameters iteratively until somestopping criterion is satisfied;Repeating the process with new group of data;Converging to true value, proved by Elliott et al. 1997.
Energy PriceModeling
Robert J.Elliott, Tao Lin,
Hong Miao
MotivationPrevious Work
Empirical Findings
Model andMethodologyModel Framework
Filtering andEstimation
Future StudyOption Pricing
Embed OtherCharacteristics ofEnergy Data
For Further Reading
Define:
ρk =φ
(Yk−γL+(γ−1)Yk−1
ehk
)ehk φ(Yk )
, k = 0, 1, 2, ..,
ρ′
k =φ
(hk−〈α,Xk−1〉−〈β,Xk−1〉hk−1
〈θ,Xk−1〉
)〈θ, Xk−1〉φ (hk )
, k = 1, 2, ..,
and
λ0 = 1,
λk = ρkρ′
k =φ
(Yk−γL+(γ−1)Yk−1
ehk
)ehk φ(Yk )
φ(
hk−〈α,Xk−1〉−〈β,Xk−1〉hk−1〈θ,Xk−1〉
)〈θ, Xk−1〉φ (hk )
for k > 1,
Energy PriceModeling
Robert J.Elliott, Tao Lin,
Hong Miao
MotivationPrevious Work
Empirical Findings
Model andMethodologyModel Framework
Filtering andEstimation
Future StudyOption Pricing
Embed OtherCharacteristics ofEnergy Data
For Further Reading
dPdP
|Fk = Λk =∏k
t=1 λt To determine a new set
θ :={
pji , µ, α, β, θ 1 ≤ i , j ≤ N}
.which maximizes theexpected log-likelihood:
Q(θ, θ
)= Eθ
[log
dPθ
dPθ
| FYt
].
Energy PriceModeling
Robert J.Elliott, Tao Lin,
Hong Miao
MotivationPrevious Work
Empirical Findings
Model andMethodologyModel Framework
Filtering andEstimation
Future StudyOption Pricing
Embed OtherCharacteristics ofEnergy Data
For Further Reading
dPdP
|Fk = Λk =∏k
t=1 λt To determine a new set
θ :={
pji , µ, α, β, θ 1 ≤ i , j ≤ N}
.which maximizes theexpected log-likelihood:
Q(θ, θ
)= Eθ
[log
dPθ
dPθ
| FYt
].
Energy PriceModeling
Robert J.Elliott, Tao Lin,
Hong Miao
MotivationPrevious Work
Empirical Findings
Model andMethodologyModel Framework
Filtering andEstimation
Future StudyOption Pricing
Embed OtherCharacteristics ofEnergy Data
For Further Reading
Outline
1 MotivationPrevious WorkEmpirical Findings
2 Model and MethodologyModel FrameworkFiltering and Estimation
3 Future StudyOption PricingEmbed Other Characteristics of Energy Data
Energy PriceModeling
Robert J.Elliott, Tao Lin,
Hong Miao
MotivationPrevious Work
Empirical Findings
Model andMethodologyModel Framework
Filtering andEstimation
Future StudyOption Pricing
Embed OtherCharacteristics ofEnergy Data
For Further Reading
Option Pricing
Monte Carlo Simulation
P = E [C(YT , hT , XT )|Gt ]
= E [C(YT , hT , XT )|Yt ∨Ht ∨ Xt ]
= E [E [C(YT , hT , XT )|Yt ∨HT ]|Ht ∨ Xt ]
= E [E [E [C(YT , hT , XT )|Yt ∨HT ∨ XT ]|Ht ∨ XT ]|Xt ].
Energy PriceModeling
Robert J.Elliott, Tao Lin,
Hong Miao
MotivationPrevious Work
Empirical Findings
Model andMethodologyModel Framework
Filtering andEstimation
Future StudyOption Pricing
Embed OtherCharacteristics ofEnergy Data
For Further Reading
Option Pricing
Monte Carlo Simulation
P = E [C(YT , hT , XT )|Gt ]
= E [C(YT , hT , XT )|Yt ∨Ht ∨ Xt ]
= E [E [C(YT , hT , XT )|Yt ∨HT ]|Ht ∨ Xt ]
= E [E [E [C(YT , hT , XT )|Yt ∨HT ∨ XT ]|Ht ∨ XT ]|Xt ].
Energy PriceModeling
Robert J.Elliott, Tao Lin,
Hong Miao
MotivationPrevious Work
Empirical Findings
Model andMethodologyModel Framework
Filtering andEstimation
Future StudyOption Pricing
Embed OtherCharacteristics ofEnergy Data
For Further Reading
Outline
1 MotivationPrevious WorkEmpirical Findings
2 Model and MethodologyModel FrameworkFiltering and Estimation
3 Future StudyOption PricingEmbed Other Characteristics of Energy Data
Energy PriceModeling
Robert J.Elliott, Tao Lin,
Hong Miao
MotivationPrevious Work
Empirical Findings
Model andMethodologyModel Framework
Filtering andEstimation
Future StudyOption Pricing
Embed OtherCharacteristics ofEnergy Data
For Further Reading
Extension of Our Model
Convenience yield;Seasonality;Correlation between price and stochastic volatility, etc.
Energy PriceModeling
Robert J.Elliott, Tao Lin,
Hong Miao
MotivationPrevious Work
Empirical Findings
Model andMethodologyModel Framework
Filtering andEstimation
Future StudyOption Pricing
Embed OtherCharacteristics ofEnergy Data
For Further Reading
For Further Reading I
R. J. Elliott, L. Aggoun, and J. B. Moore.Hidden Markov Models: Estimation and Control, 1997.
D. Duffie, S. Gray, and P. Hoang.Volatility in Energy Prices.Barclays, 2004.