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 : rouveyrollis@cerna.ensmp.fr

Nicolas Rouveyrollis15 September 2003

19/09/2003

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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