Dynamic models for electricity futures prices
Dr Paresh Date
Head of Financial Maths, OR and Statistics (FORS) group,
Department of Mathematics
Brunel University
February 2016
Dr Paresh Date
Dynamic models for electricity futures prices
Outline
Motivation: why model energy commodity prices?
Mathematical models for electricity spot price
Dr Paresh Date
Dynamic models for electricity futures prices
Outline
Motivation: why model energy commodity prices?
Mathematical models for electricity spot price
Description of models
Numerical experiments with market data
Summary
Dr Paresh Date
Dynamic models for electricity futures prices
Motivation: futures contracts
Futures contracts are standardized contracts to buy or sell a
commodity at a fixed date in the future; these are traded at
regulated exchanges.
Example: Buy today a 30th November futures contract for
500 MWh delivered over 1 month at 2MW per hour, at a
price fixed today.
Dr Paresh Date
Dynamic models for electricity futures prices
Motivation: futures contracts
Futures contracts are standardized contracts to buy or sell a
commodity at a fixed date in the future; these are traded at
regulated exchanges.
Example: Buy today a 30th November futures contract for
500 MWh delivered over 1 month at 2MW per hour, at a
price fixed today.
You open a margin account with a broker, where you keep a
proportion of the price of purchase, and keep posting profit or
loss w.r.t. the daily price.
Nearer 30th November, you would usually close the account
(by selling the same futures contract) and buy in the spot
market. > 90% positions are closed this way.
Dr Paresh Date
Dynamic models for electricity futures prices
Motivation: why do we need models?
At least two reasons:
We might need to forecast future prices, in order to manage
risk of a portfolio of futures contracts.
Futures are liquid; we need to price illiquid/ ‘over the counter’
contracts consistently with the liquid ones. This usually
involves calibrating a model to liquid contract prices, and then
using the same model to price the illiquid contract, e.g., an
Asian option.
Dr Paresh Date
Dynamic models for electricity futures prices
Modelling electricity prices
Aims
A new electricity futures pricing model.
Empirical study across three different models, including the
new one.
One-step ahead forecasting comparison.
Dr Paresh Date
Dynamic models for electricity futures prices
Modelling electricity prices
Aims
A new electricity futures pricing model.
Empirical study across three different models, including the
new one.
One-step ahead forecasting comparison.
Futures contracts are more liquid than spot itself.
Hence, they carry more information about the market
movement and market volatility.
We use the futures prices to infer a market-implied electricity
spot price; this is more useful in pricing illiquid/ negotiated
contracts.
Dr Paresh Date
Dynamic models for electricity futures prices
Electricity market
120
100
80
60
40
20
0
0
200
400
600
800
1000
1200
1400
1600
Figure: Electricity spot market prices
Dr Paresh Date
Dynamic models for electricity futures prices
Spot price modelling approaches
Classical modelling approach (two factor model with jumps)
Model a seasonality pattern (Fourier Series)
Model a long-term, continuous-time stochastic process (GBM)
Model another short-term, continuous-time stochastic process
(OU)
Model price spikes with a compound Poisson process
The resulting model is quite hard to calibrate (large number
of parameters, high nonlinearity).
Dr Paresh Date
Dynamic models for electricity futures prices
Spot price modelling approaches
Classical modelling approach (two factor model with jumps)
Model a seasonality pattern (Fourier Series)
Model a long-term, continuous-time stochastic process (GBM)
Model another short-term, continuous-time stochastic process
(OU)
Model price spikes with a compound Poisson process
The resulting model is quite hard to calibrate (large number
of parameters, high nonlinearity).
Our approach (random volatility model)
Model a seasonality pattern using a single sinusoid;
Single continuous-time stochastic process;
Model ‘fat tails’ of the spot price distribution using a random
volatility parameter.
Dr Paresh Date
Dynamic models for electricity futures prices
Two-factor model with jumps
Log spot price process
log St = xt + ζt + f (t),
(1)
dxt = (α − κxt )dt + σ1 dWt
+ dJt ,
(2)
dζt = µdt + σ2 dWt ,
f (t) = c1 + c2 sin(c3 t + c4 ),
(1)
(2)
ρdt =< dWt dWt
>, Jt =
Nt
X
qi , qi ∼ N (a, b 2 ),
i =1
Dr Paresh Date
Dynamic models for electricity futures prices
Two-factor model with jumps
Log spot price process
log St = xt + ζt + f (t),
(1)
dxt = (α − κxt )dt + σ1 dWt
+ dJt ,
(2)
dζt = µdt + σ2 dWt ,
f (t) = c1 + c2 sin(c3 t + c4 ),
(1)
(2)
ρdt =< dWt dWt
>, Jt =
Nt
X
qi , qi ∼ N (a, b 2 ),
i =1
k
and Nt is a Poisson process; P(Nt = k) = e k!(λt) .
Futures contract price for any maturity is given in terms of an
integral.
−λt
Dr Paresh Date
Dynamic models for electricity futures prices
Random volatility model (RVM)- Date, Islyaev 2015
Log spot price process (de-trended)
dxt = (α − κxt )dt + ζdWt ,
log ζ = N (µ, σ 2 ),
where α, κ, µ, σ are scalars and Wt is a Wiener process under
historical measure. Empirically, this was found to provide a good
explanation of futures price distribution and price volatility.
Dr Paresh Date
Dynamic models for electricity futures prices
Random volatility model (RVM)- Date, Islyaev 2015
Log spot price process (de-trended)
dxt = (α − κxt )dt + ζdWt ,
log ζ = N (µ, σ 2 ),
where α, κ, µ, σ are scalars and Wt is a Wiener process under
historical measure. Empirically, this was found to provide a good
explanation of futures price distribution and price volatility.
A closed-form, approximate futures pricing formula can be
obtained in terms of parameters.
Dr Paresh Date
Dynamic models for electricity futures prices
Interlude: particle filter
Particle filter (PF), also called a sequential Monte Carlo filter,
is a tool for recursive estimation of latent variables (spot price
here) from measured variables (vector of futures prices) which
themselves are functions of latent variables. The estimates
combine model predictions and noisy observations using Bayes
theorem.
Futures have more information about where the market thinks
the spot prices are heading; PF allows us to exploit this.
For our purpose, PF is a ‘black-box’ algorithm:
Dr Paresh Date
Dynamic models for electricity futures prices
Interlude: particle filter
Particle filter (PF), also called a sequential Monte Carlo filter,
is a tool for recursive estimation of latent variables (spot price
here) from measured variables (vector of futures prices) which
themselves are functions of latent variables. The estimates
combine model predictions and noisy observations using Bayes
theorem.
Futures have more information about where the market thinks
the spot prices are heading; PF allows us to exploit this.
For our purpose, PF is a ‘black-box’ algorithm:
PF takes past history of vector-valued daily futures prices, and
delivers a set of model parameters for the purpose of
forecasting. This is model calibration.
Once the model is calibrated, PF takes today’s futures prices
and delivers an ’arbitrage-free’ or internally consistent forecast
of tomorrow’s futures prices as well as the spot price.
Dr Paresh Date
Dynamic models for electricity futures prices
Numerical experiments: methodology and aims
Aims:
Calibration of both the models using Matlab Optimization
toolbox, using the PF.
Estimate the ‘market-implied’ spot price using the PF from
the prices of futures contracts.
Dr Paresh Date
Dynamic models for electricity futures prices
Numerical experiments: methodology and aims
Aims:
Calibration of both the models using Matlab Optimization
toolbox, using the PF.
Estimate the ‘market-implied’ spot price using the PF from
the prices of futures contracts.
Test out-of-sample one-step ahead futures price forecasts.
Compare models using statistical tests over in-sample data.
Dr Paresh Date
Dynamic models for electricity futures prices
Measures of comparison
Out-of-sample comparison for maturity T
N
MRAE
T
RMSE T
1 X |xiT − x̂iT |
=
,
N
xiT
i =1
v
u N
uX (x T − x̂ T )2
i
i
=t
N
i =1
Where:
N is a number of observations,
xiT is observed futures price at time ti for maturity T ,
Dr Paresh Date
Dynamic models for electricity futures prices
Measures of comparison
Out-of-sample comparison for maturity T
N
MRAE
T
RMSE T
1 X |xiT − x̂iT |
=
,
N
xiT
i =1
v
u N
uX (x T − x̂ T )2
i
i
=t
N
i =1
Where:
N is a number of observations,
xiT is observed futures price at time ti for maturity T ,
x̂i T is the expected price at time ti , derived using PF and past
futures prices.
Dr Paresh Date
Dynamic models for electricity futures prices
Model calibration
For both two factor and random volatility model: we use the
method of moments to find most of the parameters.
(minimize sum of squared errors between the first few sample
moments from data and theoretical moments from the model
parameters).
Some parameters which can be conveniently updated online,
are updated using the PF.
Dr Paresh Date
Dynamic models for electricity futures prices
Model calibration
For both two factor and random volatility model: we use the
method of moments to find most of the parameters.
(minimize sum of squared errors between the first few sample
moments from data and theoretical moments from the model
parameters).
Some parameters which can be conveniently updated online,
are updated using the PF.
We will calibrate and compare three models: two factor with
jumps, two factor without jumps and RVM.
Dr Paresh Date
Dynamic models for electricity futures prices
Numerical experiments: data
Nord Pool Data
Three data panels (300 data points in each panel)
Six futures contracts (22d, 44d, 66d, 88d, 110d, 132d)
Data range: 19/11/2007 to 17/12/2013.
Comparison using MRAE (% error), RMSE, in-sample and
out-of-sample, for futures contracts.
Dr Paresh Date
Dynamic models for electricity futures prices
Out-Of-Sample
Figure: Out-of-Sample error distribution between models across contracts
Out−Of−Sample Errors: Experiment 1
6
TFJ
TF
RVM
5
Error (%)
4
3
2
1
0
22
44
Dr Paresh Date
66
Maturity
88
110
132
Dynamic models for electricity futures prices
Out-Of-Sample
Figure: Out-of-Sample error distribution between models across contracts
Out−Of−Sample Errors: Experiment 1
6
TFJ
TF
RVM
5
Error (%)
4
3
2
1
0
22
44
66
Maturity
88
110
132
RVM is very competitive in terms of accuracy, while being
computationally far easier than the two factor models.
Dr Paresh Date
Dynamic models for electricity futures prices
Summary
Sophisticated financial models are necessary for accurate short
term forecasting and for pricing over-the-counter (i.e., illiquid)
financial derivatives in pool-based electricity markets.
Particle filtering is an extremely useful mathematical tool,
which allows us to use the past and present futures prices to
model the spot price evolution.
Reference: Suren Islyaev and Paresh Date, Electricity futures
price models: calibration and forecasting, European Journal of
Operational Research, vol. 247, pages 144-154, 2015.
◦ ◦ ◦
Dr Paresh Date
Dynamic models for electricity futures prices