A Hidden Markov Stochastic Volatility Model for Energy … Framework Filtering and ... Scott (1987) and Stein&Stein(1991) Ball&Roma(1994) and Heston(1993) Fouque, Ppanicolaou,

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



Hong Miao


Empirical Findings





For Further Reading

Outline

1 MotivationPrevious WorkEmpirical Findings

2 Model and MethodologyModel FrameworkFiltering and Estimation

3 Future StudyOption PricingEmbed Other Characteristics of Energy Data



Hong Miao


Empirical Findings





For Further Reading

Outline






Hong Miao


Empirical Findings





For Further Reading

Outline






Hong Miao


Empirical Findings





For Further Reading

Outline






Hong Miao


Empirical Findings





For Further Reading

Volatility of Energy Data

Duffie, Gray & Hong(2004)Pindyck (2002) and Pindyck(2004)



Hong Miao


Empirical Findings





For Further Reading

Volatility of Energy Data

Duffie, Gray & Hong(2004)Pindyck (2002) and Pindyck(2004)



Hong Miao


Empirical Findings





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)



Hong Miao


Empirical Findings





For Further Reading

Volatility Models




Hong Miao


Empirical Findings





For Further Reading

Volatility Models




Hong Miao


Empirical Findings





For Further Reading

Volatility Models




Hong Miao


Empirical Findings





For Further Reading

Hidden Markov Models

Elliott, Fisher&Platen(1999)Lam&Li(1998)



Hong Miao


Empirical Findings





For Further Reading

Hidden Markov Models

Elliott, Fisher&Platen(1999)Lam&Li(1998)



Hong Miao


Empirical Findings





For Further Reading

Outline






Hong Miao


Empirical Findings





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



Hong Miao


Empirical Findings





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



Hong Miao


Empirical Findings





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

.



Hong Miao


Empirical Findings





For Further Reading


t

drt = µ(rt , t , . . . )dt + σ(rt , t , . . . )dwt ,

ddt

var(rt) = σ(rt , t , . . . )2 ≈dr2

tdt

.



Hong Miao


Empirical Findings





For Further Reading


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



Hong Miao


Empirical Findings





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



Hong Miao


Empirical Findings





For Further Reading

Outline






Hong Miao


Empirical Findings





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)〉.



Hong Miao


Empirical Findings





For Further Reading

Model Framework


Xk = AXk−1 + Mk . (2)

a = (a1, a2, . . . , aN)′

b = (b1, b2, . . . , bN)′

c = (c1, c2, . . . , cN)′.




Hong Miao


Empirical Findings





For Further Reading

Model Framework


Xk = AXk−1 + Mk . (2)

a = (a1, a2, . . . , aN)′

b = (b1, b2, . . . , bN)′

c = (c1, c2, . . . , cN)′.




Hong Miao


Empirical Findings





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



Hong Miao


Empirical Findings





For Further Reading

Outline






Hong Miao


Empirical Findings





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


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



Hong Miao


Empirical Findings





For Further Reading

Framework


|Fk = Λk ;


E[f (hk ) Xk |FY

k

]=


k]

E[Λk |FY

k

] . (3)



k

]=

∫ ∞

−∞f (z) qk (z)dz. (4)

qk (z) = A∫ ∞

−∞B(Yk , z, h, Yk−1)qk−1(h)dh. (5)




Hong Miao


Empirical Findings





For Further Reading

Framework


|Fk = Λk ;


E[f (hk ) Xk |FY

k

]=


k]

E[Λk |FY

k

] . (3)



k

]=

∫ ∞

−∞f (z) qk (z)dz. (4)

qk (z) = A∫ ∞

−∞B(Yk , z, h, Yk−1)qk−1(h)dh. (5)




Hong Miao


Empirical Findings





For Further Reading

Framework


|Fk = Λk ;


E[f (hk ) Xk |FY

k

]=


k]

E[Λk |FY

k

] . (3)



k

]=

∫ ∞

−∞f (z) qk (z)dz. (4)

qk (z) = A∫ ∞

−∞B(Yk , z, h, Yk−1)qk−1(h)dh. (5)




Hong Miao


Empirical Findings





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.



Hong Miao


Empirical Findings





For Further Reading

EM Algorithm




Hong Miao


Empirical Findings





For Further Reading

EM Algorithm




Hong Miao


Empirical Findings





For Further Reading

EM Algorithm




Hong Miao


Empirical Findings





For Further Reading

EM Algorithm




Hong Miao


Empirical Findings





For Further Reading

EM Algorithm




Hong Miao


Empirical Findings





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,



Hong Miao


Empirical Findings





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

].



Hong Miao


Empirical Findings





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

].



Hong Miao


Empirical Findings





For Further Reading

Outline






Hong Miao


Empirical Findings





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 ].



Hong Miao


Empirical Findings





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 ].



Hong Miao


Empirical Findings





For Further Reading

Outline






Hong Miao


Empirical Findings





For Further Reading

Extension of Our Model

Convenience yield;Seasonality;Correlation between price and stochastic volatility, etc.



Hong Miao


Empirical Findings





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.

Documents

A Hidden Markov Stochastic Volatility Model for Energy … Framework Filtering and ... Scott (1987) and Stein&Stein(1991) Ball&Roma(1994) and Heston(1993) Fouque, Ppanicolaou,