three electricity spot price models: evidence from pjm and alberta markets
DESCRIPTION
“Lunch at the Lab” Presentation Matt Lyle Department of mathematics&statistics University of Calgary, Alberta. Three electricity spot price models: Evidence from PJM and Alberta markets. Outline. Why these models? The random variable model The MR with Lapalcian motion and jumps model - PowerPoint PPT PresentationTRANSCRIPT
Three electricity spot price models: Evidence from PJM and Alberta markets
“Lunch at the Lab” Presentation
Matt Lyle
Department of mathematics&statistics
University of Calgary, Alberta
Outline
● Why these models?● The random variable model● The MR with Lapalcian motion and jumps model● The MR with many jumps model● Simulation results
Why these models?
● Standard models often overlook the unique characteristics seen in the electricity markets
● They capture more of the statistical characteristics of electricity price paths
● They are a result of the analysis found using the FFT method introduced last week
The random variable model
• Let us suppose the log spot price is as follows
( ) ( ) ( )
( ) :The deterministic component (established last week)
( ) :The stochastic component
P t D t S t
With
D t
S t
The random variable model
● Let
Where
is the Noise component
is the Jump component
( ) ( ) ( )S t N t J t
( )N t
( )J t
The random variable model
● We would like to be able to model directly, but we really have two components
● We use the recursive method suggested by Clewlow and Strickland to remove the jumps from the noise.
● This will allow us to establish the distribution for the noise…
( )S t
The random variable model
• The remaining noise after applying the recursive filter (anything greater then 3 std) for PJM
The random variable model
• The remaining noise after applying the recursive filter (anything greater then 3 std) for Alberta
The random variable model
• The density of the noise (the fit is logistic) PJM
The random variable model
• The density of the noise (the fit is logistic) Alberta
The random variable model
● We now have
( ) ( , )
Where is the logistic distribution
with parameters and
( )
With representing the Bernoulli process
and the exponential distribution
u u d d
N t Lg
Lg
J t B E B E
B
E
MR with Laplacian motion and jumps
• Since Brownian motion does not capture the statistical characteristics in energy markets perhaps an other type will…
MR with Laplacian motion and jumps
MR with Laplacian motion and jumps
t
t
We define the model in the following way
( )
: The log-price
: The speed of mean reversion
: The time dependent long term mean
: Volatility
L : Laplacian motion
Q: Expon
u u d dt t t t t t t t
t
dP P dt dL Q dJ Q dJ
P
ential distribution of jump applitudes
: Poisson processJ
The MR with many jumps model
● Similar to the standard MR jump diffusion models we simply increase the number of jumps.
● The idea comes from the Laplace distribution, which is similar to two exponential distributions spliced back-to-back
The MR with many jumps model
• So we get the following model
( )
Where the variables are the same as before except
we now have Brownian motion
u u d dt t t t t t tdP P dt dW Q dJ Q dJ
Results: random variable model
Results: random variable model
Results: random variable model
Sim Ab Real AB %errorSim PJM
Real PJM %error
Mean 55.64409 57.964664 4.00342465 44.5197 45.1509 1.39797
Std 59.60136 69.255757 13.9402014 32.3338 33.3573 3.06829
Skew 5.096807 5.2426489 2.78182826 2.7324 2.6196 -4.306
Kurtosis 51.50621 45.072841 -14.273285 18.2626 22.8215 19.9763
Results: MR with Laplacian motion
Results: MR with Laplacian motion
Results: MR with Laplacian motion
Sim Ab Real AB %errorSim PJM
Real PJM %error
Mean 58.01625 57.964664 -0.0890098 44.455 45.150 1.5406
Std 63.48928 69.255757 8.3263479 33.068 33.357 0.8648
Skew 4.664896 5.2426489 11.020242 3.0294 2.6196 -15.643
Kurtosis 43.93767 45.072841 2.5185096 24.410 22.821 -6.9622
Results: MR with many jumps
Results: MR with many jumps
