Catalogue of models for Electricity Markets

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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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Introduction

This document presents detailed simulations of some of the modelscurrently used for modelling electricity spot prices. In particular, we investigate the most relevant shapes and the impactof the parameters on the simulations. We have not attempted to be exhaustive, especially when the modelsare very flexible. We will focus on the simulation of forward/future price implied bythese models in forthcoming studies.

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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 AJD Model

Jump Diffusion Process with Erlang Distribution

Variation of Electricity Production

Stable/Instable Regime Model

Multifactor model based on Forward Curve

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Black & Scholes Model orGeometric Brownian Motion

0

( ) . ( ) ( ) ( )(0)

dS t S t dt S t dW tS S

α σ= +=

•Short description Classical model used for modelling financial assets whose log-returns follow a standard Brownian motion :

Its importance lies mainly in the fact that it is the simplest form of a diffusion that remains positive.

•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.

tdS dt dWS

α σ= +

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α = 0.1% σ = 0.001%

. α offsets σ

. Exponential shape

. Low volatility

α = 0.1% σ = 0.1%

. α offsets σ

. Exponential shape

Black & Scholes Model or Geometric Brownian Motion

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α = 0.1% σ = 1%

. High volatility

. Exponential shape is hidden

α = 1% σ = 1%

. High volatility

. α offsets σ

. Exponential shape

Black & Scholes Model or Geometric Brownian Motion

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Arithmetic Ornstein UhlenbeckProcess or Vasicek Model

•Short description Process used for modelling assets subject to supply / demand (for example, commodities).

In this equation the mean reversion component or drift term is governed by the difference between thecurrent price and the mean m as well as by the mean reversion rate a. If the spot price is below the mean,the mean reversion component will be positive, resulting in an upward influence on the spot price.Alternatively, if the spot price is above this level, the mean reversion component will be negative, thusexerting a downward influence on the spot price. Over time, this results in a price path that drifts towardsthe mean, at a speed determined by the mean reversion rate. Note that the spot price S can be negative.

•Reference Vasicek, O., "An equilibrium characterisation of the term structure", Journal of Financial Economics,

5:-88, 1977.

0

( ) .(0)

t t tdS S dt dWS S

α µ σ= − +

=

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Arithmetic Ornstein Uhlenbeck Process or Vasicek Model

α = 0.01 σ = 0.5 m = 10

. Slow reversion

. Short run variations far fromequilibrium level. Long term evolution close toequilibrium level

. Alpha is amplified

. Speed accelerated

. Concentration around equilibrium level

α = 0.9 σ = 0.5 m = 10

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Arithmetic Ornstein Uhlenbeck Process or Vasicek Model

. Sigma is increased

. Amplitudes of variations are greaterthan in the previous simulation. Concentration around equilibriumlevel

. Impact of varying alpha on short run dynamics of the process

α = 0.9 σ = 3 m = 10

α = 0.001, 0.1, 1 σ = 0.5 m = 10

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Geometric Ornstein UhlenbeckProcess or Mean RevertingProcess

•Short description

Process used for modelling assets showing a mean reversion logic whose log-returns follow anarithmetic Ornstein-Uhlenbeck process:

Its advantage is that prices are positive.

( ) tdS S dt dWS

α µ σ= − +

•Reference

Dixit, A.K. & R.S. Pindyck, "Investment under Uncertainty", Princeton UniversityPress, Princeton, N.J., 1994.

0

( ). . . .(0)

t t t t tdS S S dt S dWS S

α µ σ= − +=

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. High speed of reversion

. Low volatility

. Process is concentrated around theequilibrium level m with noisy variations

. α is decreased and σ is increased

. Slow reversion

. Process follows a stochastic trend

Geometric Ornstein Uhlenbeck Process

α = 1 σ = 0.00001 m = 1

α = 0.01 σ = 0.01 m = 1

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. Alpha is increased

. Reversion accelerated

. Concentration around equilibrium level

. Sigma is increased

. Compared to previous simulation, amplitudevaries, more spikes occur

Geometric Ornstein Uhlenbeck Process

α = 0.5 σ = 0.06 m = 1

α = 0.05 σ = 0.01 m = 1

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Spot Price based model of Lucia& Schwartz

•Short description Process of Onrstein Uhlenbeck type, contains an additive deterministic component to allow for seasonalbehaviour and trend. Under good regularity conditions on the deterministic function f, the dynamics ofthe model takes the following form :

This shows that when St deviates from the deterministic term f(t), it is pulled back at a rate proportionalto the difference f(t) - P. The only source of uncertainty is provided by the state variable X.

Process S can be view as the solution of the stochastic differential equation :

Consequently, it is an extension of the previous Vasicek Model with non constant equilibrium level

( ( )) ( ( ) ) Xt t td S f t K f t S dt dWσ− = − +

1 ( )( ( ) ) Xt t t

df tdS K f t S dt dWK dt

σ= + − +

1 ( )( ) ( )df tt f tK dt

µ = +

•Reference Lucia, J. J. & Schwartz, E. S., ""Electricity prices and power derivatives: Evidence from the Nordic powerexchange", Review of Derivatives Research 5(1), 5-50, 2002.

( )

.t t

Xt t X t

S f t Xd X K X d t d Wσ

= +

= − +

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Influence of volatility parameter sigma

σ = 0

σ = 1.5 σ = 0.95

σ = 0.4 σ = 0.2

σ = 0.1

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We use the Singular Spectrum Analysis 1decomposition method to compare these simulations. The idea is tocarryout a singular value decomposition of the `trajectory matrix' obtained from the original time series withsubsequent reconstruction of the series. In that way we reproduce the sinusoidal component and its variance. Wethen compare it to the total variance of the series simulated. In our example, we performed a SSA decomposition on50 components.

Variance of sinusoidal component retrieved % Total Variance (approximation)

0

20

40

60

80

100

120

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

va ria nc e

Variance of sinusoidal component retrieved % Total Variance (approximation)

0

20

40

60

80

100

120

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

va ria nc e

Influence of volatility parameter s

1 See : VAUTARD, R., YIOU, P. and GHIL, M. (1992) Singular-spectrum analysis: A toolkit for short, noisy chaotic signals,Physica D, 58, 95-126 andhttp://www.cf.ac.uk/maths/stats/changepoint/Methodology/Ch_p_method.html for computational algorithm

Simulation N° Variance of sinusoidalPrevious Page component retrieved

% Total Variance(approximation)

1 96% 2 76% 3 30% 4 20% 5 & 6 < 6%

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Influence of parameter K

σ = 0.1K = 0

σ = 0.5K = 0.1

σ = 0.1K = 0.1

σ = 0.1K = 1

σ = 0.5K = 0.5

σ = 0.5K = 1

σ = 0.5K = 1.5

σ = 0.1K = 0.5

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Parameters Influence

K . no real impact on sinusoidal component. can reduce a stochastic trend to an horizontal one

sigma . can produce a stochastic trend. can hide sinusoidal component. increase variations

Sinusoidal component . can reduce the noisy shape

Influence of parameters

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Log of Spot Price based model ofLucia & Schwartz

•Short description Similar to Vasicek Model , contains a multiplicative component allowing seasonal behaviour and trend.Under suitable conditions for f(t), spot process S can be viewed as the solution of a stochastic differentialequation :

On the other hand, let U = log(S) then U satisfies a stochastic differential equation :

Note that process U follows the dynamic defined by the Vasicek model with non constant equilibrium level.

²1 ( ) ( ) ( ) . .2

XXt t t X t t

df tdS f t Log S S dt S dWdt

σκ σκ = + + − +

( )( ( ) ( ))

( ) ( )

Xt t t X t

Xt X t

d Log S f t dX X dt dWLog S f t dt dW

κ σ

κ σ

− = = − +

= − − +

1 ( )( ) Xt t X t

df tdU f t U dt dWdt

κ σκ

= + − +

•Reference Lucia, J. J. & Schwartz, E. S., "Electricity prices and power derivatives: Evidence from the Nordic powerexchange", Review of Derivatives Research 5(1), 5-50, 2002.

( ) ( )

.t t

Xt t X t

L o g S f t Xd X K X d t d Wσ

= +

= − +

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σ = 0

In all cases, K = 0 & S(0) = 10

σ = 0.1

σ = 0.4

σ = 0.2

σ = 1

σ = 0.01

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σ = 0

In all cases, K = 0.5 & S(0) = 1

σ = 0.1

σ = 0.4

σ = 0.2

σ = 1

σ = 0.01

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σ = 0.1

σ = 0.4 σ = 1

σ = 0.01σ = 0

In all cases, K = 0.9 & S(0) = 1

σ = 0.2

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Parameters Influence

K . Can reduce the stochastic trend producedwith sigma. Can reduce amplitude of spikes

sigma . Can produce a stochastic trend and spikes. Gives random amplitude to sinusoidalcomponent

Sinusoidal component . Periodical max and min values caninfluence spikes and reduce the noisy shape

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Cox–Ingersoll–Ross Model

•Short description

Similar to Vasicek Model. It combines a mean reverting process with a square root process.The Cox-Ingersoll-Ross short term interest rate process remains positive, is mean-reverting and theabsolute variance of the interest rate increases with the interest rate itself.

•Reference

Cox, J.C., Ingersoll, Jr., J.E., Ross, S. A., "A theory of the term structure of interest rates", Econometrica53, 385-408., 1985.

( )t t t tS S dt S dWα µ σ= − +

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. Slow reversion and low volatility

. Low variations in both short and long run


α = 0.01 σ = 0.01 m = 30 α = 0.01 σ = 0.1 m = 30

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. Slow reversion and high volatility

. Reversion on long run

. High and low variations on short run depending on the price level

. Evolution of the variance is visible


α = 0.01 σ = 0.01 m = 30 α = 0.01 σ = 0.1 m = 30

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. Fast reversion

. Concentration close to equilibrium level m

. Variance seems to be constant over time except on last simulation where spikes occur


α = 0.1 σ = 0.01 m = 30 α = 0.01 σ = 0.5 m = 30

α = 0.01 σ = 0.1 m = 30 α = 0.01 σ = 1 m = 30

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•Short description Compared to previous 1-factor Model, this model :- allows a second source of uncertainty- reduces the correlation between spot and forward prices- gains in flexibility Notice that the two factors can be correlated but this correlation does not appear in the expression offorward prices.Factor can be interpreted as a stochastic trend driving the process as we will see in the next simulations.In some cases factor X can represent high frequency variations of spot price S. The factor f was set to zero.Simulations will show that we can give more or less importance to each factor.

Two Factor Model of Lucia &Schwartz

•Reference


( )t t tX

t t X t

t t

S f t Xd X K X d t d Wd d t d W ε

ε

ε

σ

ε µ σ

= + +

= − +

= +

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Two Factor Model of Lucia & Schwartz

. ε(0) = 1, µ = 0.001, σε = 0.05 and ρ = 0.7 (correlation)

. ε can be viewed as a stochastic trend

. The relative impact of X on the process is amplified via σX

σZ = 0.005 σZ = 0.05

In each case, the blue curve in the upper frame represents the spot price, the green one is thespot price less X. The middle one is the X process while the lower one is the e process

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. ε(0) = 1, µ = 0.001, σε = 0.05 , K=0.5 and ρ = 0.7(correlation)


. X offsets ε

. The relative impact of X on the process is amplified via σX

σZ = 0.5 σZ = 1


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. ε(0) = 1, µ = 0.1, σx = 0.2, K = 0.5 and ρ = 0.7 (correlation)

. ε offsets X

σZ = 1 σZ = 0.5


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. ε(0) = 1, µ = 0.1, σx = 0.2, K = 0.5 and ρ = 0.7(correlation)

. ε can be viewed as a stochastic trend (left side) and gradually becomes deterministic (right)

. Process S deviates from with respect to process X

σZ = 0.05 σZ = 0.005


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K= 0.2

. ε(0) = 0.01, µ = 0.01, σx = 0.2, σε = 0.095 and ρ = 0.7 (correlation)


K = 0.9


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. ε(0) = 0.01, µ = 0.01, σx = 0.2, σε = 0.095 and ρ = 0.7(correlation)

. X offsets ε

K = 0.02 K = 0.002


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•Short description Multiplicative version of previous model, the two factors can be correlated and this correlation appears inthe expression of forward prices.

The advantage is that the spot price S is positive. The factor f was set to zero.

Compared to previous model, spikes can occur via X. We will focus on this for the next simulations .

Two Factor Model of Lucia &Schwartz based on the log of spotprice

•Reference


( ) ( )t t tX

t t X t

t t

L o g S f t Xd X K X d t d Wd d t d W ε

ε

ε

σ

ε µ σ

= + +

= − +

= +

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Two Factor Model of Lucia & Schwartz based on the log of spot price

. Slow and fast reversion for X can induce spikes in the process (top row)

K = 0.001 σZ = 0.001 K = 0.01 σZ = 0.001 K = 0.1 σZ = 0.001

In each case, the upper frame represents the spot price, the middle one is the Xprocess while the lower one is the e process

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. Fast reversion for X can reduce the intensity of spikes (left column)

. Different values for σε can produce a stochastic trend (middle & right columns)

K = 0.5 σZ = 0.0001 K = 0.9 σZ = 0.0001 K = 0.9 σZ = 0.1


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Factor ε can add spikes to price process S for high volatility levels σε

K = 0.9 σZ = 0.5 K = 0.9 σZ = 0.9 K = 0.9 σZ = 1


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•Short description Model of Geometric Brownian type, incorporating a second factor with mean reverting dynamics.

Second factor is usually interpreted as a Stochastic Convenience Yield influencing the long termequilibrium.

Two Factor Model of GibsonSchwartz

•Reference

Gibson & Schwartz , "Stochastic Convenience Yield and the Pricing of Oil Contingent Claims",Journal of Finance, vol. 45, no 3, pp. 959-976, July 1990.

( )

( )

St t t S t t

Ct t C t

dS C S dt S dWdC K C dt dW

µ σ

α σ

= − +

= − +

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. Low convenience yield

. Small drift in the spot price process

. Convenience yield is a mean reverting process

. Low convenience yield

. Small drift in the spot price process

. Convenience yield is a medium mean reverting process

. Greater variation of spot price process (blue) comparedto same process without convenience yield (green)

Two Factor Model of Gibson Schwartz

In each case, the blue curve in the upper frame represents the spot price; the green is the spot priceless the convenience yield. The lower frame shows the convenience yield

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. Level of convenience yield is increased (C(0) and a)

. µ is amplified

. Spot price process (blue) is strongly reduced

. Volatility of convenience yield is amplified

. In some cases it can produce high variations

Two Factor Model of Gibson Schwartz

In each case, the blue curve in the upper frame represents the spot price; the green is the spot priceless the convenience yield. The lower frame shows the convenience yield

Catalogue of models for Electricity Markets

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