Catalogue of Models for Electricity Prices Part 2
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Transcript of Catalogue of Models for Electricity Prices Part 2
19/09/2003
CERNA, Centre d’économie industrielle
Ecole Nationale Supérieure des Mines de Paris - 60, bld St Michel - 75272 Paris cedex 06 - France
Téléphone : (33) 01 40 51 91 26 - Télécopie : (33) 01 44 07 10 46 - E-mail : [email protected]
Nicolas Rouveyrollis15 September 2003
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Table of Contents
PART 1
Black & Scholes Model or Geometric Brownian Motion 5
Arithmetic Ornstein Uhlenbeck Process or Vasicek Model 8
Geometric Ornstein Uhlenbeck Process or Mean Reverting Process 11
Spot Price based model of Lucia & Schwartz 14
Log of Spot Price based model of Lucia & Schwartz 19
Cox–Ingersoll–Ross Model 24
Two Factors Model of Lucia & Schwartz 28
Two Factors Model of Lucia & Schwartz based on the log of spot price 35
Two Factors Model of Gibson-Schwartz 39
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Table of Contents
PART 2
Simple 1-Factor Affine Jump Diffusion Model 45
Jump Diffusion Process with Erlang Distribution 52
Variation of Electricity Production 55
Stable/Instable Regime Model 59
Multifactor Model based on Forward Curve 62
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Simple 1 Factor Affine JumpDiffusion Model
•Short description Ornstein-Uhlenbeck process with a jump component Compared to previous 1-factor models, a jump component is added allowing arrivals of episodic strongvariation in the price level.
Amplitude of jumps follows a Gaussian distribution.
•Reference
Lucia, J. J. & Schwartz, E. S., "Electricity prices and power derivatives: Evidence from the Nordicpower exchange", Review of Derivatives Research 5(1), 5-50, 2002.
( ) ( ). ( , ) ( )
( ) or ( )
t t
t t X X J
G S f t XdX K X dt dW J dNG x x Log x
σ µ σ λ= +
=− + +=
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. Lambda controls the occurrence of jumps
. Right: few jumps
. Left: frequent jumps
X(0)=0, K=1, σX=0.8, µ = 4, σJ=0.9, λ=0.001 X(0)=0, K=1, σX=0.8, µ = 4, σJ=0.9, λ=0.2
Simple 1 Factor Affine Jump Diffusion Model
In each case G(x)=x, the upper frame represents the spot price, while the lower one is theinfinitesimal process for jumps
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Simple 1 Factor Affine Jump Diffusion Model
. Parameter K controls price level persistence and the influence of jumps
. Left : small value for K, after occurrence of jump, level of price is changed
. Right : value of K close to 1, after a jump, price process comes back to its previous level,occurrence of spikes
X(0)=0, K=0.5, σX=0.8, µ = 4, σJ=0.9, λ=0.2 X(0)=0, K=1, σX=0.8, µ = 4, σJ=0.9, λ=0.2
In each case G(x)=x, the upper frame represents the spot price, while the lower one is theinfinitesimal process for jumps
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. Setting µ close to value 0 can produce up and down jumps. The random amplitude of jumps is controlled by sJ
. Left : mixture of positive and negative jumps with small amplitudes, no real impact on the process
. Right: mixture of positive and negative jumps with large amplitudes
X(0)=0, K=1, σX=0.8, µ = 0, σJ=0.01, λ=0.2 X(0)=0, K=1, σX=0.8, µ = 0.1, σJ=10, λ=0.2
Simple 1 Factor Affine Jump Diffusion Model
In each case G(x)=x, the upper frame represents the spot price, while the lower one is theinfinitesimal process for jumps
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Simple 1 Factor Affine Jump Diffusion Model
. Lambda controls the occurrence of jumps
. Left : few jumps
. Right : frequent jumps, note that only few jumps are visible
X(0)=1, K=1, σX=0.8, µ = 4, σJ=0.9, λ=0.001 X(0)=1, K=1, σX=0.8, µ = 4, σJ=0.9, λ=0.2
In each case G(x)=Log(x), the upper frame represents the spot price, while the lower one isthe infinitesimal process for jumps
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Simple 1 Factor Affine Jump Diffusion Model
. Parameter K controls price level persistence and the influence of jumps
. Left: small value for K, change in price level after the occurrence of jumps
. Right: value of K close to 1, price process comes back to its previous level after a jump,occurrence of spikes
X(0)=1, K=0.02, σX=0.8, µ = 1, σJ=0.0001, λ=0.2 X(0)=1, K=1, σX=0.8, µ = 1, σJ=0.0001, λ=0.2
In each case G(x)=Log(x), the upper frame represents the spot price, while the lower one isthe infinitesimal process for jumps
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Simple 1 Factor Affine Jump Diffusion Model
. Setting µ close to value 0 can produce upward and downward jumps.
. The random amplitude of jumps is controlled by sJ
. Left : mixture of positive and negative jumps with small amplitudes, no real impact on the process.
. Right : mixture of positive and negative jumps with strong amplitudes .
X(0)=1, K=1, σX=0.8, µ = 0, σJ=0.0001, λ=0.2 X(0)=1, K=1, σX=0.8, µ = 0, σJ=0.01, λ=0.2
In each case G(x)=Log(x), the upper frame represents the spot price, while the lower one isthe infinitesimal process for jumps
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Jump Diffusion Process withErlang Distribution
•Short description Ornstein-Uhlenbeck process with a jump component
Amplitude of jumps follows an Erlang(n) distribution, particular case n =1 gives an exponential(β)distribution.
In the following simulations, we will focus of the impact of parameters n and λ on the spot price process
•Reference
For an application of Erlang distribution see :Dickson, D.C.M. and Hipp, C. (1998). Ruin probabilities for Erlang(2) risk processes.
Insurance: Mathematics and Economics 22, 251-262.
( ) ( , ) ( )~ ( , )t t S SdS K S dt dW J n dN
J Erlang nϕ σ β λ
β= − + +
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. Parameter n increases magnitude of spikes
. Left : spikes are not visible
S(0)=2, K=0.8, σ=1, ϕ = 2, λ=0.2, P=0.01 n=1 S(0)=2, K=0.8, σ=1, ϕ = 2, λ=0.2, P=0.01 n=10
Jump Diffusion Model with Erlang distribution
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. Parameter n increases magnitude of spikes
. Left : high value for n
. Right : high value for n, λ is amplified and reduces the magnitude of spikes
S(0)=2, K=0.8, σ=1, ϕ = 2, λ=0.2, P=0.01 n=100 S(0)=2, K=0.8, σ=1, ϕ = 2, λ=2, P=0.01 n=100
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Jump Diffusion Model with Erlang distribution
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Markov Chain Model :Variation of electricity production
•Short description This model allows for variations in the spot price due to the percentage of online generators For the simulations, we choose : f(t) = exp(a * [cos(phi) - cos(2pi*t / 365 + phi) ]) ) g(t) = exp(b * [cos(psi) - cos(2pi*t + psi) ] ) And we will focus on effects of functions f and g on the simulation.
•Reference
Elliott, R. J. , G. Sick and M. Stein, Pricing Electricity Calls, Haskayne School of Business, Universityof Calgary, University of Oregon, March 9, 2003 .
( ) ( ) exp( ) ,, : determ in ist functions
( ): 2 -state M arkov C hain
t t t
t t t
t
S f t g t X a Zf gdX X dt dWZ
α µ
= < >
= − − +
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a=0 b=0 a = 0.1 b=0 a =1 b=0
. Function f influences amplitude of jumps produced by the Markov chain,especiallywhen function f reaches its max or min
Markov Chain Model :Variation of electricity production
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a = 0.1 b = 0 a = 0.1 b = 0.1 a = 0.1 b = 3
. Function g influences amplitude of jumps produced by the Markov chain, in particularwhen function g reaches its max or min
. Function g can hide jumps
Markov Chain Model :Variation of electricity production
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σ = 0, a=0.1, b=0.1 σ = 0.1 a=0.1, b=0.1 σ = 0.5 a=0.1, b=0.1
. Parameter σ disturbs the periodicities
Markov Chain Model :Variation of electricity production
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Markov model : Stable/InstableRegime Model
•Short description Electricity spot price switches between stable and unstable regimes.In the stable regime it follows a mean reversion dynamics :
whereas in the unstable regime it follows a log-normal distribution :
In the next simulation we will focus on the two parameters of unstable regime.
•Reference
de Jong, C., Huisman, R. Option Formulas for Mean-Reverting Power Prices with SpikesEnergy Global, Rotterdam School of Management at Erasmus University , 2002
1 1( ) ( ) ( ( ))t t t tLn S Ln S Ln Sα µ ε− −= + − +
2 2,( )t tLn S µ ε= +
1~ (0, )Nε σ
2 2~ (0, )Nε σ
1 1
2 2,
[1 ( )]exp(log( ) ( log( )) )( ) exp( )
( ) 2-state Markov Chain
t t t t
t
S c t S Sc t
c t
α µ εµ ε
− −= − + − +
+ +
=
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. Different values for µ2 show different level of spikes
µ2 = 100 µ2=200
In each case the upper frame represents the stable regime, the medium one is the instableregime and the lower one is the spot process
Markov model : Stable/Instable Regime Model
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. σ2 controls the magnitude of jumps : high values intensify randomness while low values accentuatesmall variations of magnitude
σ2 = 0.9 σ2 = 0.001
In each case the upper frame represents the stable regime, the medium one is the instableregime and the lower one is the spot process
Markov model : Stable/Instable Regime Model
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Multifactor model based onForward Curve
•Short description Model with spot price based on forward curve. This allows multiple sources of uncertainty. In the following simulations we chose n = 2
•Reference
Cortazar G. and Schwartz E “The Valuation of Commodity Contingent Claims”, The Journal of Derivatives, Vol 1, No 4, pp 27-39, 1994
1 0 0
( , )
(0, ).exp 0.5 ( , )² ( , )
t
t tni
i i ui
S F t t
F t u t du u t dWσ σ=
=
= − +
∑ ∫ ∫
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•Contango on Forward Curve
•Contango on Forward Curve•Backwardation + Contango onForward Curve
•Backwardation + Contango onForward Curve
Multifactor model based on Forward Curve
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